{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":43621,"title":"Get derivarive of polynomial given as vector array.","description":"Get derivarive of polynomial given as vector array.\r\n\r\nExample  \r\n\r\np=[ 1     2     0     5     0     3 ];\r\n\r\nresult=[ 5 8 0 10 0 ];","description_html":"\u003cp\u003eGet derivarive of polynomial given as vector array.\u003c/p\u003e\u003cp\u003eExample\u003c/p\u003e\u003cp\u003ep=[ 1     2     0     5     0     3 ];\u003c/p\u003e\u003cp\u003eresult=[ 5 8 0 10 0 ];\u003c/p\u003e","function_template":"function y = PolyPol(x)\r\n  y = x;\r\nend","test_suite":"%%\r\np = [ 1     2     0     5     0     3 ];\r\ny_correct = [ 5 8 0 10 0 ];\r\nassert(isequal(PolyPol(p),y_correct))\r\n%%\r\np = [ 3     2     5     1     0     2];\r\ny_correct = [ 15     8    15     2     0 ];\r\nassert(isequal(PolyPol(p),y_correct))\r\n%%\r\np = [ 15     8    15     2     0 ];\r\ny_correct = [  60    24    30     2 ];\r\nassert(isequal(PolyPol(p),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":90467,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":83,"test_suite_updated_at":"2016-10-25T09:14:14.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2016-10-25T09:10:39.000Z","updated_at":"2026-03-22T02:28:51.000Z","published_at":"2016-10-25T09:14:14.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGet derivarive of polynomial given as vector array.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ep=[ 1 2 0 5 0 3 ];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eresult=[ 5 8 0 10 0 ];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1370,"title":"Derivative of polynomial","description":"Compute the derivative of a given polynomial. The input is an \u003chttp://www.mathworks.com/help/matlab/math/representing-polynomials.html array of coefficients of polynomials\u003e.\r\n\r\nExample:\r\n\r\n Input is  [2 0 5]\r\n Output is [4 0]","description_html":"\u003cp\u003eCompute the derivative of a given polynomial. The input is an \u003ca href = \"http://www.mathworks.com/help/matlab/math/representing-polynomials.html\"\u003earray of coefficients of polynomials\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e Input is  [2 0 5]\r\n Output is [4 0]\u003c/pre\u003e","function_template":"function y = your_fcn_name(x)\r\n  y = x;\r\nend","test_suite":"%%\r\n%p = 4x^4 + 3x^3 + x + 19\r\nx=[4 3 0 1 19]\r\ny_correct = [16 9 0 1]\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n%%\r\n%p = 5x^6 + 3x^4 + 8x^2 + 1\r\nx=[5 0 3 0 8 0 1]\r\ny_correct = [30     0    12     0    16     0]\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n%%\r\nx=[[3 zeros(1,399) 15 zeros(1,100)]]\r\ny_correct = [1500\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t1500\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0]\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n%%\r\nx=[1 -2 3 -4 -3 2 -1]\r\ny_correct = [6   -10    12   -12    -6     2]\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":5217,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":273,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-03-21T10:14:39.000Z","updated_at":"2026-03-02T14:25:37.000Z","published_at":"2013-03-21T10:14:39.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCompute the derivative of a given polynomial. The input is an\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/help/matlab/math/representing-polynomials.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003earray of coefficients of polynomials\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input is  [2 0 5]\\n Output is [4 0]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1912,"title":"Local Minima","description":"Given a vector of data x, find the values of local minimum that is smaller than its neighbor elements. \r\nFor example, if\r\n\r\n x = [ 1 2 5 8 7 5 9 10]\r\n\r\na local minimum equal to 5 can be found. Return a vector of local minima. ","description_html":"\u003cp\u003eGiven a vector of data x, find the values of local minimum that is smaller than its neighbor elements. \r\nFor example, if\u003c/p\u003e\u003cpre\u003e x = [ 1 2 5 8 7 5 9 10]\u003c/pre\u003e\u003cp\u003ea local minimum equal to 5 can be found. Return a vector of local minima.\u003c/p\u003e","function_template":"function y = loc_min(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [ 1 2 5 8 7 5 9 10];\r\ny_correct = 5;\r\nassert(isequal(loc_min(x),y_correct))\r\n\r\n%%\r\nx = [.7 -3 -15 8 -6 7];\r\ny_correct = [-15 -6];\r\nassert(isequal(loc_min(x),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":2,"created_by":18066,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":69,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-10-03T21:12:50.000Z","updated_at":"2026-03-05T10:48:03.000Z","published_at":"2013-10-03T21:12:50.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a vector of data x, find the values of local minimum that is smaller than its neighbor elements. For example, if\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [ 1 2 5 8 7 5 9 10]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ea local minimum equal to 5 can be found. Return a vector of local minima.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":561,"title":"Find the jerk","description":"No, it's not the author of this problem...\r\n\r\nJerk is the rate of change in acceleration over time of an object.  So, if given the position of an object over time in the form of a 1-by-N vector, return the indices i where there is nonzero jerk.  \r\n\r\nSuper rad bonus hint: The signal you need to find the jerk of will be given by the variable sig, created with the commands \r\n\r\n  h = 0.065; % stepsize\r\n  t = -10:h:10;\r\n  sigCoefs = 2*rand(1,3)-1;\r\n  sig = polyval(sigCoefs,t);\r\n  breakPoint = randi(length(sig)-2)+1;\r\n  sig(breakPoint) = (1.01)*sig(breakPoint); % this creates a nonzero jerk\r\n \r\nCheck the signal visually with\r\n\r\n  plot(t,sig,'k.-')\r\n\r\nNow, using just sig, determine breakPoint.\r\n ","description_html":"\u003cp\u003eNo, it's not the author of this problem...\u003c/p\u003e\u003cp\u003eJerk is the rate of change in acceleration over time of an object.  So, if given the position of an object over time in the form of a 1-by-N vector, return the indices i where there is nonzero jerk.\u003c/p\u003e\u003cp\u003eSuper rad bonus hint: The signal you need to find the jerk of will be given by the variable sig, created with the commands\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eh = 0.065; % stepsize\r\nt = -10:h:10;\r\nsigCoefs = 2*rand(1,3)-1;\r\nsig = polyval(sigCoefs,t);\r\nbreakPoint = randi(length(sig)-2)+1;\r\nsig(breakPoint) = (1.01)*sig(breakPoint); % this creates a nonzero jerk\r\n\u003c/pre\u003e\u003cp\u003eCheck the signal visually with\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eplot(t,sig,'k.-')\r\n\u003c/pre\u003e\u003cp\u003eNow, using just sig, determine breakPoint.\u003c/p\u003e","function_template":"function idx = findAJerk(sig)\r\n  idx = find(sig\u003e0);\r\nend","test_suite":"%% \r\n\r\n  h = 0.065; % stepsize\r\n  t = -10:h:10;\r\n\r\nfor tr = 1:1000\r\n  sigCoefs = 2*rand(1,3)-1;\r\n  sig = polyval(sigCoefs,t);\r\n  breakPoint = randi(length(sig)-2)+1;\r\n  sig(breakPoint) = (1.01)*sig(breakPoint);\r\n  assert(any(abs(findAJerk(sig) - breakPoint)\u003c=6)) % extra wide window out of kindness\r\nend\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":4,"created_by":2688,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":73,"test_suite_updated_at":"2012-04-07T16:14:29.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-04-07T03:32:53.000Z","updated_at":"2026-01-31T12:36:27.000Z","published_at":"2012-04-07T03:37:20.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNo, it's not the author of this problem...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eJerk is the rate of change in acceleration over time of an object. So, if given the position of an object over time in the form of a 1-by-N vector, return the indices i where there is nonzero jerk.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuper rad bonus hint: The signal you need to find the jerk of will be given by the variable sig, created with the commands\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[h = 0.065; % stepsize\\nt = -10:h:10;\\nsigCoefs = 2*rand(1,3)-1;\\nsig = polyval(sigCoefs,t);\\nbreakPoint = randi(length(sig)-2)+1;\\nsig(breakPoint) = (1.01)*sig(breakPoint); % this creates a nonzero jerk]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCheck the signal visually with\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[plot(t,sig,'k.-')]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNow, using just sig, determine breakPoint.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46918,"title":"Numerical differentiation with high precision","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 447.9px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 223.95px; transform-origin: 407px 223.95px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 98.2px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 49.1px; text-align: left; transform-origin: 384px 49.1px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 264.658px 8px; transform-origin: 264.658px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA basic approach for numerical differentiation is by calculating the difference quotient \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(x) = (f(x+h) - f(x-h)) / (2h)\" style=\"width: 178px; height: 35px;\" width=\"178\" height=\"35\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2167px 8px; transform-origin: 27.2167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, see for example \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/2892\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 2892\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 233.242px 8px; transform-origin: 233.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Unfortunately, this approach leads to the problem that for small step sizes \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eh\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.5583px 8px; transform-origin: 43.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e a subtractive cancellation error occurs, while for large step sizes the truncation error of the numerical scheme dominates. The same problem occurs also for similar numerical schemes of higher order.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 238.7px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 119.35px; text-align: left; transform-origin: 384px 119.35px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline;width: 273px;height: 233px\" 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\" data-image-state=\"image-loaded\" width=\"273\" height=\"233\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.7667px 8px; transform-origin: 17.7667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTask:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 355.55px 8px; transform-origin: 355.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e Can you find a numerical approach that eliminates the subtractive cancellation error and enables accuracies up to machine accuracy for small step sizes? Inputs to your function are a function handle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 8px; transform-origin: 11.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003efun\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.9px 8px; transform-origin: 108.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to a scalar function, as well as the point \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 8px; transform-origin: 7.7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ex0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 173.108px 8px; transform-origin: 173.108px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, at which the numerical derivative should be calculated.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.9417px 8px; transform-origin: 15.9417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHint:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 121.733px 8px; transform-origin: 121.733px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e If you do not have any idea, check out \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://blogs.mathworks.com/cleve/\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCleve's Corner\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 178.15px 8px; transform-origin: 178.15px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Back in the year 2013, you will find an interesting article.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function Df = numerical_diff(fun,x0)\r\n  % this basic approach will not be good enough.. :-(\r\n  % h  = 1e-5;\r\n  % Df = (fun(x0+h) - fun(x0-h)) / (2*h);\r\nend","test_suite":"%%\r\nfun = @sin;\r\nfor k = 1:5\r\n  x = randn;\r\n  assert(abs(numerical_diff(fun,x) - cos(x)) \u003c 1e-12)\r\nend\r\n%%\r\nfun = @(x)x^(9/2);\r\nfor k = 1:5\r\n  x = abs(randn);\r\n  assert(abs(numerical_diff(fun,x) - (9/2)*x^(7/2)) \u003c 1e-12);\r\nend\r\n%%\r\nfun = @(x)exp(x)/(cos(x)^3+sin(x)^3);\r\nx  = [ 0 pi/4 pi/2 pi ];\r\nDf = [ 1 exp(pi/4)*sqrt(2) exp(pi/2) -exp(pi)];\r\nfor k = 1:4\r\n  assert(abs(numerical_diff(fun,x(k)) - Df(k)) \u003c 1e-12);\r\nend\r\n%%\r\nfun = @(x)exp(x)/sqrt(cos(x)^3+sin(x)^3);\r\nx  = [ -pi/8 0 pi/8 pi/4 pi/2 ];\r\nDf = [ 0.042678767134941 1 2.15903261751524 exp(pi/4)*2^(1/4) exp(pi/2) -exp(pi)];\r\nfor k = 1:5\r\n  assert(abs(numerical_diff(fun,x(k)) - Df(k)) \u003c 1e-12)\r\nend\r\n%%\r\nstr = fileread('numerical_diff.m'); % sorry, no regexp hacks :-)\r\nassert(isempty(regexp(str,'regexp')));","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":11486,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-10-18T17:51:29.000Z","updated_at":"2020-10-18T17:56:54.000Z","published_at":"2020-10-18T17:56:16.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA basic approach for numerical differentiation is by calculating the difference quotient \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(x) = (f(x+h) - f(x-h)) / (2h)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef^\\\\prime(x) \\\\approx \\\\frac{f(x+h) - f(x-h)}{2h}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, see for example \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/2892\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 2892\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Unfortunately, this approach leads to the problem that for small step sizes \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e a subtractive cancellation error occurs, while for large step sizes the truncation error of the numerical scheme dominates. The same problem occurs also for similar numerical schemes of higher order.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"233\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"273\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTask:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Can you find a numerical approach that eliminates the subtractive cancellation error and enables accuracies up to machine accuracy for small step sizes? Inputs to your function are a function handle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003efun\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e to a scalar function, as well as the point \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, at which the numerical derivative should be calculated.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHint:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e If you do not have any idea, check out \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://blogs.mathworks.com/cleve/\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCleve's Corner\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Back in the year 2013, you will find an interesting 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\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52562,"title":"Easy Sequences 6: Coefficient sums of derivatives","description":"Consider the polynomial function  and its first-order derivative . The sums of the coefficients of P and P', are  and , respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows:  etc.  The total sum of this sequence converge to .\r\nFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, . Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 191px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 98px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eConsider the polynomial function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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YCAwEBgIDFSJgf8D3Giv49mFrO8AAAAASUVORK5CYII=\" width=\"163\" height=\"20\" style=\"width: 163px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e and its first-order derivative \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"129.5\" height=\"35\" style=\"width: 129.5px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e. The sums of the coefficients of P and P', are \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"120\" height=\"18\" style=\"width: 120px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANgAAAAkCAYAAADrcPSPAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAA2KADAAQAAAABAAAAJAAAAABhLo42AAAH2ElEQVR4Ae2ae8zXUxzHu0gPKZVoaV1VzIiW3Ld02TSXdFEkZc1iY1pDxuSuy4bNH5qhzB9szLoTi0h/5ClKEuZaDQm1LpRu4vXO97TzfPvev79vz/fpOZ/t/XzPOZ/POedzOfff06CBI+cB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAeeBKA80jGL6eJ3JTwBTwRaQhC5AKKiP5pQ/7GFZkoZqWaYR/d8EqsDMBLo0QaYf6Al6gPXgY7AS7AZlppYo1z2FgmuR3ZtCvrZE88akHYpfD7qBf8D7YCnIHc8ONPIC2Af+BeogCV2JkOTDsAmejC4zaWLdCL4GsuNVEEfnIbAGBNm9lfKL4hqoZf6zIboH2aOyjrWsb5Lu88bkLjrRInIQLAXLgWzfA8aCTNSeWs8BNWw7N+kEW+KrZ7eh9COgrKRddzj4Ath6x00wrfxa0ew6/vSf8PuDMtIJKLUN+HUOy39bRiN8OuWNya20p4klDLDavpq0/KLd7GarPHFyNpJPgDFgPzBOTjLBenvyH/J9PAA6ajYDZaWrUGwekOMWAGN71ATTpFwG5PBpQEeKxuAML29OAGrrU1BGGodS0u8TMBEo9vKBH1okJDcdlJnyxmQgximeslWnOD/NokC8A6CXn5kmvw5hNSQkmWCvIyfFtHoUTYPpQAN7YUEd6dxtbI+aYJd4cpND9BhvtSPftAqRq83iFXS+GBwfocSp8DSg5JM+EXJlYOWNyVuenbI1aCz3sPiabJmpmppmkMVNsK7IKgBapVuAoul2OpBuSR9e0uozyGtffURNsCnwdVdrBIJIq+l3wPhRC0OZqApllgDtvFF0G0zZsDFKqCS8PDGRH8xCsinCns3w5A9dDVr75cIGg18uTf5uhHU00jFR53m9Mj0NeoJjmVpi3KNAZ/UgUhA+txhhcpbIUU3qwj4A/BrT6wiPPydGrgzsPDEZjQEaxyKdkMLI8HR/HeUXqvQEO4UOdI43pPbPBfcADa75oC04FulOjHojxjANYkM/mkQd+up42M/TV3f0slOemNgbgk4mYfSlxTjLSh9KVnqCnUSrH4ANQCu2n3Qs0gX6dD+jnuQ7e3bq9e2rOmjzEHTWqq5j0fI6qH+Qyp29Qn9M9IpuaLtJBHx3WGUdrfSh5HH+gpx5ncuv9dpoxvdscAcYC8xk7kBal0ddkHXZry90Goaa38Cer6NGm+PhXPQ/mMEGHSv7ZqjnrzKSAt0X81JUTOxNIGqC2bwjJlhSBasR1I4kxD1yBLXZi0KteKYNfXXGrQSV5ZEjzpZJCMhuHZWbxAmXkN8GnfYD2aC7WhZ6j0r2GMiaHpSl84A6UTGxfw/Uzh1Gw2EYO37xC1V6B/O3b/KfkZBT9LLY3SvUXe01L32sf/Qi9SDQHUx2a6DWNdIg03jZCj7KqPx06r2Zsa5dbZWdyZiOi8nftNvSa7thRB/2HDKPIofFbebhwoISO2l3GFgDpEgnkIR0rFoUIVjl8VrzVfDDSDuoOb6GyRRRruDMAC3ADWA1SEsLqXBp2ko++dHk3/WVpcma4+ECKun5OgvpWFeJo12Wvu06SWLyGxU0CUUn//8J/KvXQ0OqU4OO5gRTx+vAN0B3M51/k5B01OSJIzktSk4DvDboMTodCu4DWVdv6R5lG+xYynMs1etwf6+H2bE9lV8gSUzsyWJ2siDLbN4RP3Ec7QkmBbWCa4JtVyYB6Xg5HphHEn8V7UrXgL1ggp9p5Y8w3uIVldTjzkNAR6OncnTyJHW75Kivqmtz1B9CXY2VnUD3qLpMSWPys2WkPYms4kNJmxf1g7S/Xo18NTlzketWg5M+85LXltqsBJX1keM6jNNdq66+GNqx0dFS8c97Z55DGzrG50XWR5Y0MbEfL+aicxjJJjM3xoQJxZVXcoKt9BR6IK7ThPwyTjANAD1oaECG7byt4GlnKjvpeKiFQoNoWE5ltfuZwZjnOyiDHmljorvVX56+W/jqCuInlWmxkC27gH4HrkHa9pNQUONJ6vllLqegD1DAZvmZJc2ntf1i7JgPFoNbQNDvRboPvQJ2g7KTVn2NE+ma55FEdk4HWe+hqm9olUkk/GaJiV4R3wYjgRaZnkA/sdh0PhlzN55HWhOyBiWdYG2tWlVW2k42JSOFdL9aASaB74EhKTnDy4j3u2GU/JvEdmOCgrAINAMK0EzgJwXkMqBv1qOOv80i8yO8xt/hm3dB0AuicDQpT0wmo+hAoFhNAbrr2zTVy+zgO81mJE03RnAisLfzZ8hrBfaTDLHlNMB0DDwTKEj6VxTx7wWVpCKPiF1RVKuWsesP0rIniHpQuBkY2bivFp+0u2NQv0WWaWDtA7JlVJEdFdR2JWLSF930gCYfaGNo6uF+r2wP3ytAatJ2bs6g/sGiGbsuoMWXKdORyC+vsjmgN6g0FTHB2qPkTyDMlvXw1K9Nae8XClDZaRwKKpYaRM3LrmyAfpWKyVDa3gjki20elFbZYBBKRaygXejtHNAO7AQbwA9Aq38RNJ5GXwS6bLYpooN63GYnbO8IdoHV9dgPMl2ntgtBN2UgjelqcECZMCpigoX1VVT5iTSsy6YM1QulI+cB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAeeB/wA2WWDXBrcNEgAAAABJRU5ErkJggg==\" width=\"108\" height=\"18\" style=\"width: 108px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e, respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALQAAAAkCAYAAAAgqxBxAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAtKADAAQAAAABAAAAJAAAAACH07FpAAAG+ElEQVR4Ae2ZaahWRRjHXdLUyrTFMEOvuVTiErSY2XrNkjBLKKPMun4w+6QRFC2SSWX2oSRSCipQyRYTbwUFaQgqtGcrRiUp2mKLCq65tPz+OoPDcM5533vf98w973Ue+L8z8zzPzDzznzlz5szbpk2UyEBkIDIQGYgMRAYiA5GByEBkIDIQGYgMRAYiA62BgfYFGMQpxDAPDAGrWyCefvQ5FkwC48DJYDfYDkpJWxzGANWdCPoC1dsKiiLl8tuBgK8C40EDGAyOAX+AAyBKmQy8id9/4IMy/avlpgmcCfYD9e9jAbqsB74H9i9NPU36UrDNlDWWbqAIUg6/wwjUjsXnQQ/n8CIMpBZimEKQlsDQC/oFp28bg5824tMxgcju6L4y9d8n1cMhORasAWrnE3AiaEkph98BBLgH+GN3yzux17fkQGqhbxG5C1jiQi7o0abfb0lHgU5Ai+9q8DWwMSnVccSXVShk00Lo5xkHUt4LZF/m2UIWy+FXRyYd8/4BT4CeQG8ljUll9+31GeUoKQzobPYx2AD+Bpr8kAv6Pfr7GZwGfOmMYj1QTMJc4Mr5FKxtsWtw8q8YHy2UOkcfKlsuvyMISGOZkRLYFGOXj8aiN1OUBAYeRSeCLgU6f4qwUAv6ePraB24BaTIJg120az2n+Y5tqmezxbscnzlWGTAtl9/Hiek70C4lNu3gPwLLhT6ao3gMjKR8EOiVJgm9oIfS5wqgV2uaDMZgJ9F91XZAv82xnZ3SwCDHR+MLKU3hVw/nzSWCW4rdcjG2hO9RZz6BEf8EtOtpcUhCL+jDvWb/9sdsJ3GJ43qmo9cbRjtYkkgvu21DV4EhJA9+XyZwOw49qIWWtFdNXkHPo+Ge4DZwIK9OqtBundOGrr2s9LIZ0h1AE50k0utmwEpvm8k5zYPfOhPzD6Trco6/ppq/iWg10dO8qIu4Qz9jYt1C2smJV69nu1ttcPRJ2Y2O7/VJDlXW5cGv7tq18WjMd1c53ppu7gyi19lzOfBf00Vb0F2I0cY0gbwr0ynYBa0/IrLkG4zWV7cFeUpe/N5L0BqD7tztETHPcVTcdogjhxbwQiBiGkxKUliZSWSngiUGbqB7nYL/YDqmQ1ldnVnJ+gC1Ps1N8+JXR8OHgK5VJ4MiHxEJ77C4pFvdUDIrQakJs/5Jqf6cuNIY7iGtB9rtfjW6oia6Y1a8n4IG4MvvjkJ/xGSJ7rOtuPWsrlppHvxq7ueDrkDHrLWgJiRpQeuLvNKvchEh0bXWbPA92AGuAb7oy1zSDVj7HvJrpAwoem2/BTaBccDdjSkeEndhKt4sce2/ZTlWYMuL31nENB7cB96oIL7gVZN24dOJotL7xr9oYxm4ATSCpoqu9vo1tVIF/noAV4Ee4BKwASSJFv1mY9ARShvCv6bsJjrKHQSW3z7k9aBUW/Lg93aCXAjmgAeqHXCtt6eve93HZkELw8L66YoolOhooMWsh3BQGZ26H3vDUvyHordjWp/iUw11tflVezorP1eN4I7WNuyNQqi/vl2e9eX+DtBx6ALX4OUbKF9odA+T2sU6zej8ZLrjM8s3Bi6Xy+8o4tIH4GKgN0ySdEf5WJIh6o4wUIpwLTodgW4Exx2pVnFOk/Y60Fn5CpAml2HQRJ9lHM4htQu60ej8RGdx6zPAM/alPBkM8/R5FUvxq34vArvA2yDpuwr1oWs7jetVFYw0Z27aU3c06G8b8VId664FHT29iqXqJlQJrypFuEi2i+Nz8jrvViptaeBFoHZ187IgAYvQfQF0DFoNXHmWgurKNtw1kL/Y6GV/ybOdR1lnbtl0xr4D5C2l+NXxaBtQTHrAFyRAc7AVyKceWGnO3OiNqHbEw7m2IZOOIBUvsidtFll1TRMtn2QR3o7w9gMN0GJCFUJ+2mnPtpuVTvL61E7xrmljHekAYx9IqhsdtSXy/d1uhrHZvlQ3b8niV/FuATaeUul6fLUZSJozN3rD2gdafc1WQ448Sd7GoM2ik2MrVddxbdnsJrrXIFamhKFrIztIpY+k+JWrHomj216p/Hb8Oyc0ruvG14CIVxsbTaqyXstdgC96te8Etk/5amHkKVn8rqBjG0s56f1eoM2Zmw9Nn9qJL/faG0PZLnjF5ktWXd+30GXtJBOBSL+1YJH2Jp7rwFQwDqicJbqfHgQ+AiFvc7JiqsTW1LnRg66F2yulU32j1IOkB71U3ZQmi6l+nrD+BF2LGV6TouqD9z7wYJNqFde5Nc1N7izrSZ0LdgPtgLUuQxjAZrAcJB1laml8rW1ugnB/J71sBP4XcZDOc+jkF9p8CujDstaltc1NkPnQEeOkID2F6aQuTDdBemltcxOEtNhJZCAyEBmIDEQGIgORgchAZKBIDPwPOJEp8UY8O6EAAAAASUVORK5CYII=\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e etc.  The total sum of this sequence converge to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"18\" height=\"18\" style=\"width: 18px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"98.5\" height=\"19\" style=\"width: 98.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e. Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function totSum = tot_dCoefSum(coef)\r\n  y = x;\r\nend","test_suite":"%%\r\ncs = [5 6 -7 -8];\r\nts = '88';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [3 15 -2 1];\r\nts = '120';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [-7 22 43 6 -75 3 1 0 -80 10 5];\r\nts = '-42698751';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = 1:25;\r\nts = '1836856501837772435875025';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = repmat([2,-1],1,15);\r\nts = '47298214022376392514505945712317';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [ones(1,20) zeros(1,10)];\r\nts = '24893912605687593731774059567276';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = repmat([-2,-25,1],1,10);\r\nts = '-68761759219969440143678420163128';\r\nassert(isequal(tot_dCoefSum(cs),ts))","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":"2021-08-17T17:53:33.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-16T19:00:56.000Z","updated_at":"2025-11-30T19:39:34.000Z","published_at":"2021-08-17T12:43:32.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider the polynomial function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eP\\\\left(x\\\\right)=5x^3+6x^2-7x-8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its first-order derivative \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{dP}{dx}=15x^2+12x-7\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The sums of the coefficients of P and P', are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e5 + 6 - 7 - 8 = -4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e15+12-7= 20\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e-4,\\\\ 20,\\\\ 42, ...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc.  The total sum of this sequence converge to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e88\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e[5\\\\ 6\\\\ -7\\\\ -8]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":42588,"title":"Derivative function","description":"Given a function handle f, generate a function that evaluates the derivative of f\r\n\r\n\r\nExamples:\r\n    \r\n   f = @sin;\r\n   df = Derivative(f);\r\n   df([0 pi])\r\n   ans =\r\n        1    -1\r\n\r\nDerivative of sin is cos, cos([0 pi]) = [1 -1]\r\n\r\nHint(Added 2015/9/17):\r\n\u003chttps://en.wikipedia.org/wiki/Numerical_differentiation\u003e","description_html":"\u003cp\u003eGiven a function handle f, generate a function that evaluates the derivative of f\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003cpre\u003e   f = @sin;\r\n   df = Derivative(f);\r\n   df([0 pi])\r\n   ans =\r\n        1    -1\u003c/pre\u003e\u003cp\u003eDerivative of sin is cos, cos([0 pi]) = [1 -1]\u003c/p\u003e\u003cp\u003eHint(Added 2015/9/17): \u003ca href = \"https://en.wikipedia.org/wiki/Numerical_differentiation\"\u003ehttps://en.wikipedia.org/wiki/Numerical_differentiation\u003c/a\u003e\u003c/p\u003e","function_template":"function df = Derivative(f)\r\n  df = f;\r\nend","test_suite":"%%\r\nf = @sin;\r\ndf = Derivative(f);\r\nx = 10*rand(10000,1);\r\ndy = cos(x);\r\nassert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\n%% \r\nf = @log;\r\ndf = Derivative(f);\r\nx = exp(500*rand(10000,1));\r\ndy = 1./x;\r\nassert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\n%%\r\nf = @exp;\r\ndf = Derivative(f);\r\nx = 50*(rand(10000,1)-0.5);\r\ndy = exp(x);\r\nassert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\n%%\r\nt = 10*rand;\r\nf = @(x)1./(x+t);\r\ndf = Derivative(f);\r\nx = 100*(rand(10000,1)-0.5); x(x==-t) = 1;\r\ndy = -1./(x+t).^2;\r\nassert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\n%%\r\nt = 10*rand-5;\r\nf = @(x)atan(t*x);\r\ndf = Derivative(f);\r\nx = 100*(rand(10000,1)-0.5);\r\ndy = t./(t^2*x.^2 + 1);\r\nassert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\n%%\r\nfor t = 2:6\r\n    f = @(x)t.^x./(sin(x).^t+cos(x).^t);\r\n    df = Derivative(f);\r\n    x = 3+rand(10000,1);\r\n    dy = (t.^x.*log(t))./(cos(x).^t+sin(x).^t)-...\r\n        (t.^x.*(t.*cos(x).*sin(x).^(t-1)-...\r\n        t.*cos(x).^(t-1).*sin(x)))./(cos(x).^t+sin(x).^t).^2;\r\n    assert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\nend","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":1434,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":11,"test_suite_updated_at":"2015-09-08T18:13:33.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-09-07T05:46:00.000Z","updated_at":"2015-09-17T09:10:17.000Z","published_at":"2015-09-07T05:47:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a function handle f, generate a function that evaluates the derivative of f\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   f = @sin;\\n   df = Derivative(f);\\n   df([0 pi])\\n   ans =\\n        1    -1]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDerivative of sin is cos, cos([0 pi]) = [1 -1]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHint(Added 2015/9/17):\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Numerical_differentiation\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;https://en.wikipedia.org/wiki/Numerical_differentiation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46053,"title":"Construct finite difference approximations of derivatives","description":"In solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is\r\n\r\n f' = (f_{j+1} - f_j)/h\r\n\r\nwhere j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to \u003chttp://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf use Taylor series\u003e. In the example above, one can write\r\n\r\n f_{j+1} = f_j + h f' + h^2 f''/2 + h^3 f'''/6 + ...\r\n\r\nThen solving for f' and neglecting terms of order h^2 and higher gives\r\n\r\n f' = (f_{j+1} - f_j)/h - h f''/2\r\n\r\nBecause the exponent on h in the last term is 1, the method is called a first order method.\r\n\r\nWrite a function that takes the order |n| of the derivative and a vector |terms| indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example, |n = 1| and |terms = [1 0]|, and \r\n\r\n coeffs   = [1 -1]\r\n errOrder = 1\r\n errCoeff = -0.5; \r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 435.55px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 217.775px; transform-origin: 407px 217.775px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.183px 7.91667px; transform-origin: 380.183px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37.5833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.7917px; text-align: left; transform-origin: 384px 18.7917px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.5917px 7.91667px; transform-origin: 13.5917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKkAAABLCAYAAAALWaemAAAEa0lEQVR4nO2d3ZHiMBCEOwdnQAJOYCNwBGRABtQlsO/3RAzUReAcuBCIgRS4B9GnwYvBsq0f2/1VqbZg+RHtsTQzHsmAEEIIIYQQQggRiQrA1+OvGI50S0AFoAVwBXAEcAdwytqjZSDdElEBuMEJXQGo4cS+w40O4jXSLSEtnLDN4/EOXvxdrk4tAOmWiC84oW+5O7IwpFtCzpAfNQbplogdvA+1z9yXJSHdEnKAF1s+1HCkW2QauFTJEcAFXuyjaRL+J9ItIXt4USn0Bc9ii59ItwwwOuVoIIYh3RJi/Solnocj3RLCFMoNy77efJyhhbAW3RbBFd6vyk0Fl28c05f7xBaajC9Jt1VTwR+k74HvaeAO6NDXD+3HAT5aHnPgp46ih8D+lqDbJmjgxW4+vLb7nvOH14WUrFVwaRsGI6WPTjF1Ex1sGmVuv4oGN/Qg2veUbqQxdRMdWL1zjfDZazbSmLqJDjcML474ghtBTnAH6dMIsmYjjambMNjC3KHFEQwYWNz7jrUaaWzdRmHTIhc4x7cvebt7dGQJzvEe4cURNKIhv2+tRhpbt2BodDwD2MG+M4JXIdoYnZmZb4QbBAOGbrrGBhJDWl8aZglGOqdus8C8Xf14TCPsS/zy9SGjRy7Y1xDhGDDUneeZPrKNWh1e/K/7frIEI51Tt8lw1OyOin25PxbAliwwYV9vCCsrC/Gr1jjdp9AtCF72GnrG0KhLHEWPcNMUReIoF3K9OtSvWoOR5tAt+INDhugzyhHXYn9LAyc4/ewQQv2qMUbKqzIl6JhLt8GcEF50EMXnmIEK3o/aw7kvF4RPPaF+VYiRMijtVrrn1DOXbm+p4R17W+FiHf6+DjYoO+1EIxh74JnnCxnhdhO+rxRy6PaWFv5s4dl8Nc+/q15hdDsWFmNMbbHW1nAajpJCWTHRdLMV16mCIOv/TGlzikEdGoyf6rZIEt3oj6ascKnhR+wpbc6TqobzyVs4TZY8backiW6quBZFM6biWoikWN9Q26aIIrGFEil3oUgd3f8C8EctW5u0BDpXxXXq6P7vTN+nNq79/nyI+mHFderEfInRvSgQu42fktaiSGzFtXKCokhYca1tpUWxMGgquVBE+Kr/Tc52nOq1Q1p57OE3CNusS8ZqFU31ZcMKtU0cpz2eR0wWlegqU9mEbMqwaOyKzwr+bmYqKCmbMZsyLBZO7bxbGe8HqVrJstncJmFc7nHGBs7KlcDsi2Y8USwcRbvLibl0R7elEVmxm9Yy4K3wnJa6YYNpKVEOvBp4fzzmEo1vPBuwCtVFNpgfPcMbKEdNG/WrMEhkwS7p4a0SbUmijfp1tVBkwVannfFzSm/x7AoIkRy7xPxV+skasBBZ4BLzV0UldtmN/FGRBRsUvRopcy2cFOI/n24yS390Cdu/i5VibzL7iu5Uf4IWI4rEvCvNs/7oDs5QV1/CJ8ri024y1h/lrYuESAp3V+7zN2s4w2yhSjYhhBBCCCGEEEIIIYQQQsTkHzIr3fuGMmo5AAAAAElFTkSuQmCC\" alt=\"f' = (f_{j+1} - f_j})/h\" style=\"width: 84.5px; height: 37.5px;\" width=\"84.5\" height=\"37.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 365.642px 7.91667px; transform-origin: 365.642px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003euse Taylor series\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 116.3px 7.91667px; transform-origin: 116.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In the example above, one can write\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.0833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.0417px; text-align: left; transform-origin: 384px 18.0417px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.5917px 7.91667px; transform-origin: 13.5917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f_{j+1} = f_j + hf' + h^2 f''/2 + h^3 f'''/6 + ...\" style=\"width: 226.5px; height: 36px;\" width=\"226.5\" height=\"36\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.7333px 7.91667px; transform-origin: 51.7333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThen solving for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACMAAAAkCAYAAAAD3IPhAAABNklEQVRYhe2XXbGEMAxGj4c6wAAGVsEqwEEdrINrAQ1IwAMWqgEL3IcmU3Zm2W2ZBnjgPPET2kyS5gtwc3MDQAM8Mu0aKydaIAAj0AMz4L/YLsBk4chDFh/k3sv9DLgP9n/yvq/tiJNNZ1LYn7LZuPFNkPdtbWdevEflF5qiXPsiZlm8y7TXFFaPSicLL+SfjAmjqAziSMi0d2L/qah34Yl18iKlKKye+S+bdcSTVA3dtCelaFg9/1U7Jo1OT9FCXsc1Retlq7EditbLVmM7jIaUoqoFuQdt94tcn4oKXdWesZcRwxGgFI1K9RGgFFXdEnE0Q1W3RBzNUBnIFUdTJi7SX3QEMBmOStFh6vAj7Yj1sf7dUHE8XKXXI4IjHelTektLVOZATE84yxGlIabJc4GecnMp/gGjCl04ApTc8QAAAABJRU5ErkJggg==\" alt=\"f'\" style=\"width: 17.5px; height: 18px;\" width=\"17.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 94.9083px 7.91667px; transform-origin: 94.9083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and neglecting terms of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"h^2\" style=\"width: 15.5px; height: 19.5px;\" width=\"15.5\" height=\"19.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 53.3px 7.91667px; transform-origin: 53.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and higher gives\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 38.5833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.2917px; text-align: left; transform-origin: 384px 19.2917px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.5917px 7.91667px; transform-origin: 13.5917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f' = (f_{j+1} - f_j)/h - hf''/2\" style=\"width: 130px; height: 38.5px;\" width=\"130\" height=\"38.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.3px 7.91667px; transform-origin: 81.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBecause the exponent on \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eh\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 191.35px 7.91667px; transform-origin: 191.35px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the last term is 1, the method is called a first order method.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.717px 7.91667px; transform-origin: 110.717px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes the order\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91.7917px 7.91667px; transform-origin: 91.7917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the derivative and a vector\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eterms\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 130.3px 7.91667px; transform-origin: 130.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003en = 1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.91667px; transform-origin: 13.6167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.05px 7.91667px; transform-origin: 50.05px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eterms = [1 0]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 61.3px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 30.65px; transform-origin: 404px 30.65px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 69.3px 7.91667px; transform-origin: 69.3px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e coeffs   = [1 -1]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 50.05px 7.91667px; transform-origin: 50.05px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e errOrder = 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 65.45px 7.91667px; transform-origin: 65.45px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e errCoeff = -0.5;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [coeff,errOrder,errCoeff] = FDderiv(n,terms)\r\n%  n        = order of the derivative sought\r\n%  terms    = grid indices of terms to include\r\n%  coeff    = coefficients of the terms in the formula \r\n%  errOrder = exponent of h in the first non-zero higher-order term\r\n%  errCoeff = coefficient of the first non-zero higher-order term\r\n\r\n  coeff    = ...\r\n  errOrder = ...\r\n  errCoeff = ...\r\nend","test_suite":"%%\r\n%  First-order forward difference for the first derivative\r\nn = 1;\r\nterms = [1 0];\r\ncoeff_correct = [1 -1];\r\nerrOrder_correct = 1;\r\nerrCoeff_correct = -1/2;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  First-order backward difference for the first derivative\r\nn = 1;\r\nterms = [0 -1];\r\ncoeff_correct = [1 -1];\r\nerrOrder_correct = 1;\r\nerrCoeff_correct = 1/2;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order centered difference for the first derivative\r\nn = 1;\r\nterms = [1 -1];\r\ncoeff_correct = [1/2 -1/2];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = -1/6;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order forward difference for the first derivative\r\nn = 1;\r\nterms = [2 1 0];\r\ncoeff_correct = [-1/2 2 -3/2];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = 1/3;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order backward difference for the first derivative\r\nn = 1;\r\nterms = [0 -1 -2];\r\ncoeff_correct = [3/2 -2 1/2];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = 1/3;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Fourth-order centered difference for the first derivative\r\nn = 1;\r\nterms = [2 1 0 -1 -2];\r\ncoeff_correct = [-1 8 0 -8 1]/12\r\nerrOrder_correct = 4;\r\nerrCoeff_correct = 1/30;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order centered difference for the second derivative\r\nn = 2;\r\nterms = [1 0 -1];\r\ncoeff_correct = [1 -2 1];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = -1/12;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Fourth-order centered difference for the second derivative\r\nn = 2;\r\nterms = [2 1 0 -1 -2];\r\ncoeff_correct = [-1 16 -30 16 -1]/12;\r\nerrOrder_correct = 4;\r\nerrCoeff_correct = 1/90;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order centered difference for the third derivative\r\nn = 3;\r\nterms = [2 1 0 -1 -2];\r\ncoeff_correct = [1/2 -1 0 1 -1/2];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = -1/4;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order centered difference for the fourth derivative\r\nn = 4;\r\nterms = [2 1 0 -1 -2];\r\ncoeff_correct = [1 -4 6 -4 1];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = -1/6;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-22T03:37:57.000Z","updated_at":"2020-11-15T13:41:30.000Z","published_at":"2020-07-22T05:20:57.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f' = (f_{j+1} - f_j})/h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime=\\\\frac{f_{j+1}-f_j}{h}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf\\\"\u003e\u003cw:r\u003e\u003cw:t\u003euse Taylor series\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. In the example above, one can write\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f_{j+1} = f_j + hf' + h^2 f''/2 + h^3 f'''/6 + ...\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e f_{j+1} = f_j + h f\\\\prime +  \\\\frac{h^2}{2}f\\\\prime\\\\prime +  \\\\frac{h^3}{6}f\\\\prime\\\\prime\\\\prime + \\\\ldots\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThen solving for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$f\\\\prime$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and neglecting terms of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h^2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$h^2$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and higher gives\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f' = (f_{j+1} - f_j)/h - hf''/2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime = \\\\frac{f_{j+1} - f_j}{h} - \\\\frac{h}{2}f\\\\prime\\\\prime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBecause the exponent on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in the last term is 1, the method is called a first order method.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes the order\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of the derivative and a vector\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eterms\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eterms = [1 0]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ coeffs   = [1 -1]\\n errOrder = 1\\n errCoeff = -0.5;]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":43621,"title":"Get derivarive of polynomial given as vector array.","description":"Get derivarive of polynomial given as vector array.\r\n\r\nExample  \r\n\r\np=[ 1     2     0     5     0     3 ];\r\n\r\nresult=[ 5 8 0 10 0 ];","description_html":"\u003cp\u003eGet derivarive of polynomial given as vector array.\u003c/p\u003e\u003cp\u003eExample\u003c/p\u003e\u003cp\u003ep=[ 1     2     0     5     0     3 ];\u003c/p\u003e\u003cp\u003eresult=[ 5 8 0 10 0 ];\u003c/p\u003e","function_template":"function y = PolyPol(x)\r\n  y = x;\r\nend","test_suite":"%%\r\np = [ 1     2     0     5     0     3 ];\r\ny_correct = [ 5 8 0 10 0 ];\r\nassert(isequal(PolyPol(p),y_correct))\r\n%%\r\np = [ 3     2     5     1     0     2];\r\ny_correct = [ 15     8    15     2     0 ];\r\nassert(isequal(PolyPol(p),y_correct))\r\n%%\r\np = [ 15     8    15     2     0 ];\r\ny_correct = [  60    24    30     2 ];\r\nassert(isequal(PolyPol(p),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":90467,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":83,"test_suite_updated_at":"2016-10-25T09:14:14.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2016-10-25T09:10:39.000Z","updated_at":"2026-03-22T02:28:51.000Z","published_at":"2016-10-25T09:14:14.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGet derivarive of polynomial given as vector array.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ep=[ 1 2 0 5 0 3 ];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eresult=[ 5 8 0 10 0 ];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1370,"title":"Derivative of polynomial","description":"Compute the derivative of a given polynomial. The input is an \u003chttp://www.mathworks.com/help/matlab/math/representing-polynomials.html array of coefficients of polynomials\u003e.\r\n\r\nExample:\r\n\r\n Input is  [2 0 5]\r\n Output is [4 0]","description_html":"\u003cp\u003eCompute the derivative of a given polynomial. The input is an \u003ca href = \"http://www.mathworks.com/help/matlab/math/representing-polynomials.html\"\u003earray of coefficients of polynomials\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e Input is  [2 0 5]\r\n Output is [4 0]\u003c/pre\u003e","function_template":"function y = your_fcn_name(x)\r\n  y = x;\r\nend","test_suite":"%%\r\n%p = 4x^4 + 3x^3 + x + 19\r\nx=[4 3 0 1 19]\r\ny_correct = [16 9 0 1]\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n%%\r\n%p = 5x^6 + 3x^4 + 8x^2 + 1\r\nx=[5 0 3 0 8 0 1]\r\ny_correct = [30     0    12     0    16     0]\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n%%\r\nx=[[3 zeros(1,399) 15 zeros(1,100)]]\r\ny_correct = [1500\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t1500\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0\t0]\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n%%\r\nx=[1 -2 3 -4 -3 2 -1]\r\ny_correct = [6   -10    12   -12    -6     2]\r\nassert(isequal(your_fcn_name(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":5217,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":273,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-03-21T10:14:39.000Z","updated_at":"2026-03-02T14:25:37.000Z","published_at":"2013-03-21T10:14:39.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCompute the derivative of a given polynomial. The input is an\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/help/matlab/math/representing-polynomials.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003earray of coefficients of polynomials\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input is  [2 0 5]\\n Output is [4 0]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1912,"title":"Local Minima","description":"Given a vector of data x, find the values of local minimum that is smaller than its neighbor elements. \r\nFor example, if\r\n\r\n x = [ 1 2 5 8 7 5 9 10]\r\n\r\na local minimum equal to 5 can be found. Return a vector of local minima. ","description_html":"\u003cp\u003eGiven a vector of data x, find the values of local minimum that is smaller than its neighbor elements. \r\nFor example, if\u003c/p\u003e\u003cpre\u003e x = [ 1 2 5 8 7 5 9 10]\u003c/pre\u003e\u003cp\u003ea local minimum equal to 5 can be found. Return a vector of local minima.\u003c/p\u003e","function_template":"function y = loc_min(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [ 1 2 5 8 7 5 9 10];\r\ny_correct = 5;\r\nassert(isequal(loc_min(x),y_correct))\r\n\r\n%%\r\nx = [.7 -3 -15 8 -6 7];\r\ny_correct = [-15 -6];\r\nassert(isequal(loc_min(x),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":2,"created_by":18066,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":69,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-10-03T21:12:50.000Z","updated_at":"2026-03-05T10:48:03.000Z","published_at":"2013-10-03T21:12:50.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a vector of data x, find the values of local minimum that is smaller than its neighbor elements. For example, if\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [ 1 2 5 8 7 5 9 10]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ea local minimum equal to 5 can be found. Return a vector of local minima.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":561,"title":"Find the jerk","description":"No, it's not the author of this problem...\r\n\r\nJerk is the rate of change in acceleration over time of an object.  So, if given the position of an object over time in the form of a 1-by-N vector, return the indices i where there is nonzero jerk.  \r\n\r\nSuper rad bonus hint: The signal you need to find the jerk of will be given by the variable sig, created with the commands \r\n\r\n  h = 0.065; % stepsize\r\n  t = -10:h:10;\r\n  sigCoefs = 2*rand(1,3)-1;\r\n  sig = polyval(sigCoefs,t);\r\n  breakPoint = randi(length(sig)-2)+1;\r\n  sig(breakPoint) = (1.01)*sig(breakPoint); % this creates a nonzero jerk\r\n \r\nCheck the signal visually with\r\n\r\n  plot(t,sig,'k.-')\r\n\r\nNow, using just sig, determine breakPoint.\r\n ","description_html":"\u003cp\u003eNo, it's not the author of this problem...\u003c/p\u003e\u003cp\u003eJerk is the rate of change in acceleration over time of an object.  So, if given the position of an object over time in the form of a 1-by-N vector, return the indices i where there is nonzero jerk.\u003c/p\u003e\u003cp\u003eSuper rad bonus hint: The signal you need to find the jerk of will be given by the variable sig, created with the commands\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eh = 0.065; % stepsize\r\nt = -10:h:10;\r\nsigCoefs = 2*rand(1,3)-1;\r\nsig = polyval(sigCoefs,t);\r\nbreakPoint = randi(length(sig)-2)+1;\r\nsig(breakPoint) = (1.01)*sig(breakPoint); % this creates a nonzero jerk\r\n\u003c/pre\u003e\u003cp\u003eCheck the signal visually with\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eplot(t,sig,'k.-')\r\n\u003c/pre\u003e\u003cp\u003eNow, using just sig, determine breakPoint.\u003c/p\u003e","function_template":"function idx = findAJerk(sig)\r\n  idx = find(sig\u003e0);\r\nend","test_suite":"%% \r\n\r\n  h = 0.065; % stepsize\r\n  t = -10:h:10;\r\n\r\nfor tr = 1:1000\r\n  sigCoefs = 2*rand(1,3)-1;\r\n  sig = polyval(sigCoefs,t);\r\n  breakPoint = randi(length(sig)-2)+1;\r\n  sig(breakPoint) = (1.01)*sig(breakPoint);\r\n  assert(any(abs(findAJerk(sig) - breakPoint)\u003c=6)) % extra wide window out of kindness\r\nend\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":4,"created_by":2688,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":73,"test_suite_updated_at":"2012-04-07T16:14:29.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-04-07T03:32:53.000Z","updated_at":"2026-01-31T12:36:27.000Z","published_at":"2012-04-07T03:37:20.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNo, it's not the author of this problem...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eJerk is the rate of change in acceleration over time of an object. So, if given the position of an object over time in the form of a 1-by-N vector, return the indices i where there is nonzero jerk.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuper rad bonus hint: The signal you need to find the jerk of will be given by the variable sig, created with the commands\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[h = 0.065; % stepsize\\nt = -10:h:10;\\nsigCoefs = 2*rand(1,3)-1;\\nsig = polyval(sigCoefs,t);\\nbreakPoint = randi(length(sig)-2)+1;\\nsig(breakPoint) = (1.01)*sig(breakPoint); % this creates a nonzero jerk]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCheck the signal visually with\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[plot(t,sig,'k.-')]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNow, using just sig, determine breakPoint.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46918,"title":"Numerical differentiation with high precision","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 447.9px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 223.95px; transform-origin: 407px 223.95px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 98.2px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 49.1px; text-align: left; transform-origin: 384px 49.1px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 264.658px 8px; transform-origin: 264.658px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA basic approach for numerical differentiation is by calculating the difference quotient \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(x) = (f(x+h) - f(x-h)) / (2h)\" style=\"width: 178px; height: 35px;\" width=\"178\" height=\"35\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2167px 8px; transform-origin: 27.2167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, see for example \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/2892\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 2892\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 233.242px 8px; transform-origin: 233.242px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Unfortunately, this approach leads to the problem that for small step sizes \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eh\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.5583px 8px; transform-origin: 43.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e a subtractive cancellation error occurs, while for large step sizes the truncation error of the numerical scheme dominates. The same problem occurs also for similar numerical schemes of higher order.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 238.7px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 119.35px; text-align: left; transform-origin: 384px 119.35px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline;width: 273px;height: 233px\" 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\" data-image-state=\"image-loaded\" width=\"273\" height=\"233\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.7667px 8px; transform-origin: 17.7667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTask:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 355.55px 8px; transform-origin: 355.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e Can you find a numerical approach that eliminates the subtractive cancellation error and enables accuracies up to machine accuracy for small step sizes? Inputs to your function are a function handle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 8px; transform-origin: 11.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003efun\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.9px 8px; transform-origin: 108.9px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to a scalar function, as well as the point \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 8px; transform-origin: 7.7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ex0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 173.108px 8px; transform-origin: 173.108px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, at which the numerical derivative should be calculated.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.9417px 8px; transform-origin: 15.9417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eHint:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 121.733px 8px; transform-origin: 121.733px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e If you do not have any idea, check out \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://blogs.mathworks.com/cleve/\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eCleve's Corner\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 178.15px 8px; transform-origin: 178.15px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Back in the year 2013, you will find an interesting article.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function Df = numerical_diff(fun,x0)\r\n  % this basic approach will not be good enough.. :-(\r\n  % h  = 1e-5;\r\n  % Df = (fun(x0+h) - fun(x0-h)) / (2*h);\r\nend","test_suite":"%%\r\nfun = @sin;\r\nfor k = 1:5\r\n  x = randn;\r\n  assert(abs(numerical_diff(fun,x) - cos(x)) \u003c 1e-12)\r\nend\r\n%%\r\nfun = @(x)x^(9/2);\r\nfor k = 1:5\r\n  x = abs(randn);\r\n  assert(abs(numerical_diff(fun,x) - (9/2)*x^(7/2)) \u003c 1e-12);\r\nend\r\n%%\r\nfun = @(x)exp(x)/(cos(x)^3+sin(x)^3);\r\nx  = [ 0 pi/4 pi/2 pi ];\r\nDf = [ 1 exp(pi/4)*sqrt(2) exp(pi/2) -exp(pi)];\r\nfor k = 1:4\r\n  assert(abs(numerical_diff(fun,x(k)) - Df(k)) \u003c 1e-12);\r\nend\r\n%%\r\nfun = @(x)exp(x)/sqrt(cos(x)^3+sin(x)^3);\r\nx  = [ -pi/8 0 pi/8 pi/4 pi/2 ];\r\nDf = [ 0.042678767134941 1 2.15903261751524 exp(pi/4)*2^(1/4) exp(pi/2) -exp(pi)];\r\nfor k = 1:5\r\n  assert(abs(numerical_diff(fun,x(k)) - Df(k)) \u003c 1e-12)\r\nend\r\n%%\r\nstr = fileread('numerical_diff.m'); % sorry, no regexp hacks :-)\r\nassert(isempty(regexp(str,'regexp')));","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":11486,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-10-18T17:51:29.000Z","updated_at":"2020-10-18T17:56:54.000Z","published_at":"2020-10-18T17:56:16.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA basic approach for numerical differentiation is by calculating the difference quotient \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(x) = (f(x+h) - f(x-h)) / (2h)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef^\\\\prime(x) \\\\approx \\\\frac{f(x+h) - f(x-h)}{2h}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, see for example \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/2892\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 2892\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Unfortunately, this approach leads to the problem that for small step sizes \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e a subtractive cancellation error occurs, while for large step sizes the truncation error of the numerical scheme dominates. The same problem occurs also for similar numerical schemes of higher order.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"233\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"273\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTask:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Can you find a numerical approach that eliminates the subtractive cancellation error and enables accuracies up to machine accuracy for small step sizes? Inputs to your function are a function handle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003efun\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e to a scalar function, as well as the point \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, at which the numerical derivative should be calculated.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eHint:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e If you do not have any idea, check out \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://blogs.mathworks.com/cleve/\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCleve's Corner\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Back in the year 2013, you will find an interesting 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\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":52562,"title":"Easy Sequences 6: Coefficient sums of derivatives","description":"Consider the polynomial function  and its first-order derivative . The sums of the coefficients of P and P', are  and , respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows:  etc.  The total sum of this sequence converge to .\r\nFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, . Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 191px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 98px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eConsider the polynomial function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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YCAwEBgIDFSJgf8D3Giv49mFrO8AAAAASUVORK5CYII=\" width=\"163\" height=\"20\" style=\"width: 163px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e and its first-order derivative \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"129.5\" height=\"35\" style=\"width: 129.5px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e. The sums of the coefficients of P and P', are \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"120\" height=\"18\" style=\"width: 120px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANgAAAAkCAYAAADrcPSPAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAA2KADAAQAAAABAAAAJAAAAABhLo42AAAH2ElEQVR4Ae2ae8zXUxzHu0gPKZVoaV1VzIiW3Ld02TSXdFEkZc1iY1pDxuSuy4bNH5qhzB9szLoTi0h/5ClKEuZaDQm1LpRu4vXO97TzfPvev79vz/fpOZ/t/XzPOZ/POedzOfff06CBI+cB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAeeBKA80jGL6eJ3JTwBTwRaQhC5AKKiP5pQ/7GFZkoZqWaYR/d8EqsDMBLo0QaYf6Al6gPXgY7AS7AZlppYo1z2FgmuR3ZtCvrZE88akHYpfD7qBf8D7YCnIHc8ONPIC2Af+BeogCV2JkOTDsAmejC4zaWLdCL4GsuNVEEfnIbAGBNm9lfKL4hqoZf6zIboH2aOyjrWsb5Lu88bkLjrRInIQLAXLgWzfA8aCTNSeWs8BNWw7N+kEW+KrZ7eh9COgrKRddzj4Ath6x00wrfxa0ew6/vSf8PuDMtIJKLUN+HUOy39bRiN8OuWNya20p4klDLDavpq0/KLd7GarPHFyNpJPgDFgPzBOTjLBenvyH/J9PAA6ajYDZaWrUGwekOMWAGN71ATTpFwG5PBpQEeKxuAML29OAGrrU1BGGodS0u8TMBEo9vKBH1okJDcdlJnyxmQgximeslWnOD/NokC8A6CXn5kmvw5hNSQkmWCvIyfFtHoUTYPpQAN7YUEd6dxtbI+aYJd4cpND9BhvtSPftAqRq83iFXS+GBwfocSp8DSg5JM+EXJlYOWNyVuenbI1aCz3sPiabJmpmppmkMVNsK7IKgBapVuAoul2OpBuSR9e0uozyGtffURNsCnwdVdrBIJIq+l3wPhRC0OZqApllgDtvFF0G0zZsDFKqCS8PDGRH8xCsinCns3w5A9dDVr75cIGg18uTf5uhHU00jFR53m9Mj0NeoJjmVpi3KNAZ/UgUhA+txhhcpbIUU3qwj4A/BrT6wiPPydGrgzsPDEZjQEaxyKdkMLI8HR/HeUXqvQEO4UOdI43pPbPBfcADa75oC04FulOjHojxjANYkM/mkQd+up42M/TV3f0slOemNgbgk4mYfSlxTjLSh9KVnqCnUSrH4ANQCu2n3Qs0gX6dD+jnuQ7e3bq9e2rOmjzEHTWqq5j0fI6qH+Qyp29Qn9M9IpuaLtJBHx3WGUdrfSh5HH+gpx5ncuv9dpoxvdscAcYC8xk7kBal0ddkHXZry90Goaa38Cer6NGm+PhXPQ/mMEGHSv7ZqjnrzKSAt0X81JUTOxNIGqC2bwjJlhSBasR1I4kxD1yBLXZi0KteKYNfXXGrQSV5ZEjzpZJCMhuHZWbxAmXkN8GnfYD2aC7WhZ6j0r2GMiaHpSl84A6UTGxfw/Uzh1Gw2EYO37xC1V6B/O3b/KfkZBT9LLY3SvUXe01L32sf/Qi9SDQHUx2a6DWNdIg03jZCj7KqPx06r2Zsa5dbZWdyZiOi8nftNvSa7thRB/2HDKPIofFbebhwoISO2l3GFgDpEgnkIR0rFoUIVjl8VrzVfDDSDuoOb6GyRRRruDMAC3ADWA1SEsLqXBp2ko++dHk3/WVpcma4+ECKun5OgvpWFeJo12Wvu06SWLyGxU0CUUn//8J/KvXQ0OqU4OO5gRTx+vAN0B3M51/k5B01OSJIzktSk4DvDboMTodCu4DWVdv6R5lG+xYynMs1etwf6+H2bE9lV8gSUzsyWJ2siDLbN4RP3Ec7QkmBbWCa4JtVyYB6Xg5HphHEn8V7UrXgL1ggp9p5Y8w3uIVldTjzkNAR6OncnTyJHW75Kivqmtz1B9CXY2VnUD3qLpMSWPys2WkPYms4kNJmxf1g7S/Xo18NTlzketWg5M+85LXltqsBJX1keM6jNNdq66+GNqx0dFS8c97Z55DGzrG50XWR5Y0MbEfL+aicxjJJjM3xoQJxZVXcoKt9BR6IK7ThPwyTjANAD1oaECG7byt4GlnKjvpeKiFQoNoWE5ltfuZwZjnOyiDHmljorvVX56+W/jqCuInlWmxkC27gH4HrkHa9pNQUONJ6vllLqegD1DAZvmZJc2ntf1i7JgPFoNbQNDvRboPvQJ2g7KTVn2NE+ma55FEdk4HWe+hqm9olUkk/GaJiV4R3wYjgRaZnkA/sdh0PhlzN55HWhOyBiWdYG2tWlVW2k42JSOFdL9aASaB74EhKTnDy4j3u2GU/JvEdmOCgrAINAMK0EzgJwXkMqBv1qOOv80i8yO8xt/hm3dB0AuicDQpT0wmo+hAoFhNAbrr2zTVy+zgO81mJE03RnAisLfzZ8hrBfaTDLHlNMB0DDwTKEj6VxTx7wWVpCKPiF1RVKuWsesP0rIniHpQuBkY2bivFp+0u2NQv0WWaWDtA7JlVJEdFdR2JWLSF930gCYfaGNo6uF+r2wP3ytAatJ2bs6g/sGiGbsuoMWXKdORyC+vsjmgN6g0FTHB2qPkTyDMlvXw1K9Nae8XClDZaRwKKpYaRM3LrmyAfpWKyVDa3gjki20elFbZYBBKRaygXejtHNAO7AQbwA9Aq38RNJ5GXwS6bLYpooN63GYnbO8IdoHV9dgPMl2ntgtBN2UgjelqcECZMCpigoX1VVT5iTSsy6YM1QulI+cB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAecB5wHnAeeB/wA2WWDXBrcNEgAAAABJRU5ErkJggg==\" width=\"108\" height=\"18\" style=\"width: 108px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e, respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e etc.  The total sum of this sequence converge to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"18\" height=\"18\" style=\"width: 18px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"98.5\" height=\"19\" style=\"width: 98.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e. Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function totSum = tot_dCoefSum(coef)\r\n  y = x;\r\nend","test_suite":"%%\r\ncs = [5 6 -7 -8];\r\nts = '88';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [3 15 -2 1];\r\nts = '120';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [-7 22 43 6 -75 3 1 0 -80 10 5];\r\nts = '-42698751';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = 1:25;\r\nts = '1836856501837772435875025';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = repmat([2,-1],1,15);\r\nts = '47298214022376392514505945712317';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = [ones(1,20) zeros(1,10)];\r\nts = '24893912605687593731774059567276';\r\nassert(isequal(tot_dCoefSum(cs),ts))\r\n%%\r\ncs = repmat([-2,-25,1],1,10);\r\nts = '-68761759219969440143678420163128';\r\nassert(isequal(tot_dCoefSum(cs),ts))","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":16,"test_suite_updated_at":"2021-08-17T17:53:33.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-08-16T19:00:56.000Z","updated_at":"2025-11-30T19:39:34.000Z","published_at":"2021-08-17T12:43:32.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider the polynomial function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eP\\\\left(x\\\\right)=5x^3+6x^2-7x-8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its first-order derivative \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\frac{dP}{dx}=15x^2+12x-7\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The sums of the coefficients of P and P', are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e5 + 6 - 7 - 8 = -4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e15+12-7= 20\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, respectively. If we keep summing up coefficients for all higher derivatives the sums sequence will be as follows: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e-4,\\\\ 20,\\\\ 42, ...\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc.  The total sum of this sequence converge to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e88\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this exercise, you are given an array corresponding to the coefficients of a polynomial function. In the example above, the coefficient array is therefore, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e[5\\\\ 6\\\\ -7\\\\ -8]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Your task is to find the total of the sum of the coefficients of the given polynomial function plus the sum of the coefficients of its first derivative plus the sum of cefficients of all its higher degree derivatives.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":42588,"title":"Derivative function","description":"Given a function handle f, generate a function that evaluates the derivative of f\r\n\r\n\r\nExamples:\r\n    \r\n   f = @sin;\r\n   df = Derivative(f);\r\n   df([0 pi])\r\n   ans =\r\n        1    -1\r\n\r\nDerivative of sin is cos, cos([0 pi]) = [1 -1]\r\n\r\nHint(Added 2015/9/17):\r\n\u003chttps://en.wikipedia.org/wiki/Numerical_differentiation\u003e","description_html":"\u003cp\u003eGiven a function handle f, generate a function that evaluates the derivative of f\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003cpre\u003e   f = @sin;\r\n   df = Derivative(f);\r\n   df([0 pi])\r\n   ans =\r\n        1    -1\u003c/pre\u003e\u003cp\u003eDerivative of sin is cos, cos([0 pi]) = [1 -1]\u003c/p\u003e\u003cp\u003eHint(Added 2015/9/17): \u003ca href = \"https://en.wikipedia.org/wiki/Numerical_differentiation\"\u003ehttps://en.wikipedia.org/wiki/Numerical_differentiation\u003c/a\u003e\u003c/p\u003e","function_template":"function df = Derivative(f)\r\n  df = f;\r\nend","test_suite":"%%\r\nf = @sin;\r\ndf = Derivative(f);\r\nx = 10*rand(10000,1);\r\ndy = cos(x);\r\nassert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\n%% \r\nf = @log;\r\ndf = Derivative(f);\r\nx = exp(500*rand(10000,1));\r\ndy = 1./x;\r\nassert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\n%%\r\nf = @exp;\r\ndf = Derivative(f);\r\nx = 50*(rand(10000,1)-0.5);\r\ndy = exp(x);\r\nassert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\n%%\r\nt = 10*rand;\r\nf = @(x)1./(x+t);\r\ndf = Derivative(f);\r\nx = 100*(rand(10000,1)-0.5); x(x==-t) = 1;\r\ndy = -1./(x+t).^2;\r\nassert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\n%%\r\nt = 10*rand-5;\r\nf = @(x)atan(t*x);\r\ndf = Derivative(f);\r\nx = 100*(rand(10000,1)-0.5);\r\ndy = t./(t^2*x.^2 + 1);\r\nassert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\n%%\r\nfor t = 2:6\r\n    f = @(x)t.^x./(sin(x).^t+cos(x).^t);\r\n    df = Derivative(f);\r\n    x = 3+rand(10000,1);\r\n    dy = (t.^x.*log(t))./(cos(x).^t+sin(x).^t)-...\r\n        (t.^x.*(t.*cos(x).*sin(x).^(t-1)-...\r\n        t.*cos(x).^(t-1).*sin(x)))./(cos(x).^t+sin(x).^t).^2;\r\n    assert(max(abs((dy-df(x))./dy))\u003c1e-13)\r\nend","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":1434,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":11,"test_suite_updated_at":"2015-09-08T18:13:33.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-09-07T05:46:00.000Z","updated_at":"2015-09-17T09:10:17.000Z","published_at":"2015-09-07T05:47:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a function handle f, generate a function that evaluates the derivative of f\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[   f = @sin;\\n   df = Derivative(f);\\n   df([0 pi])\\n   ans =\\n        1    -1]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDerivative of sin is cos, cos([0 pi]) = [1 -1]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHint(Added 2015/9/17):\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Numerical_differentiation\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;https://en.wikipedia.org/wiki/Numerical_differentiation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026gt;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46053,"title":"Construct finite difference approximations of derivatives","description":"In solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is\r\n\r\n f' = (f_{j+1} - f_j)/h\r\n\r\nwhere j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to \u003chttp://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf use Taylor series\u003e. In the example above, one can write\r\n\r\n f_{j+1} = f_j + h f' + h^2 f''/2 + h^3 f'''/6 + ...\r\n\r\nThen solving for f' and neglecting terms of order h^2 and higher gives\r\n\r\n f' = (f_{j+1} - f_j)/h - h f''/2\r\n\r\nBecause the exponent on h in the last term is 1, the method is called a first order method.\r\n\r\nWrite a function that takes the order |n| of the derivative and a vector |terms| indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example, |n = 1| and |terms = [1 0]|, and \r\n\r\n coeffs   = [1 -1]\r\n errOrder = 1\r\n errCoeff = -0.5; \r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 435.55px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 217.775px; transform-origin: 407px 217.775px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.183px 7.91667px; transform-origin: 380.183px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37.5833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.7917px; text-align: left; transform-origin: 384px 18.7917px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.5917px 7.91667px; transform-origin: 13.5917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f' = (f_{j+1} - f_j})/h\" style=\"width: 84.5px; height: 37.5px;\" width=\"84.5\" height=\"37.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 365.642px 7.91667px; transform-origin: 365.642px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003euse Taylor series\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 116.3px 7.91667px; transform-origin: 116.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In the example above, one can write\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 36.0833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.0417px; text-align: left; transform-origin: 384px 18.0417px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.5917px 7.91667px; transform-origin: 13.5917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f_{j+1} = f_j + hf' + h^2 f''/2 + h^3 f'''/6 + ...\" style=\"width: 226.5px; height: 36px;\" width=\"226.5\" height=\"36\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.7333px 7.91667px; transform-origin: 51.7333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThen solving for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACMAAAAkCAYAAAAD3IPhAAABNklEQVRYhe2XXbGEMAxGj4c6wAAGVsEqwEEdrINrAQ1IwAMWqgEL3IcmU3Zm2W2ZBnjgPPET2kyS5gtwc3MDQAM8Mu0aKydaIAAj0AMz4L/YLsBk4chDFh/k3sv9DLgP9n/yvq/tiJNNZ1LYn7LZuPFNkPdtbWdevEflF5qiXPsiZlm8y7TXFFaPSicLL+SfjAmjqAziSMi0d2L/qah34Yl18iKlKKye+S+bdcSTVA3dtCelaFg9/1U7Jo1OT9FCXsc1Retlq7EditbLVmM7jIaUoqoFuQdt94tcn4oKXdWesZcRwxGgFI1K9RGgFFXdEnE0Q1W3RBzNUBnIFUdTJi7SX3QEMBmOStFh6vAj7Yj1sf7dUHE8XKXXI4IjHelTektLVOZATE84yxGlIabJc4GecnMp/gGjCl04ApTc8QAAAABJRU5ErkJggg==\" alt=\"f'\" style=\"width: 17.5px; height: 18px;\" width=\"17.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 94.9083px 7.91667px; transform-origin: 94.9083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and neglecting terms of order \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"h^2\" style=\"width: 15.5px; height: 19.5px;\" width=\"15.5\" height=\"19.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 53.3px 7.91667px; transform-origin: 53.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and higher gives\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 38.5833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.2917px; text-align: left; transform-origin: 384px 19.2917px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.5917px 7.91667px; transform-origin: 13.5917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e       \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f' = (f_{j+1} - f_j)/h - hf''/2\" style=\"width: 130px; height: 38.5px;\" width=\"130\" height=\"38.5\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.3px 7.91667px; transform-origin: 81.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBecause the exponent on \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eh\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 191.35px 7.91667px; transform-origin: 191.35px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in the last term is 1, the method is called a first order method.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.717px 7.91667px; transform-origin: 110.717px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes the order\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91.7917px 7.91667px; transform-origin: 91.7917px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of the derivative and a vector\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eterms\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 130.3px 7.91667px; transform-origin: 130.3px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003en = 1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.91667px; transform-origin: 13.6167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.05px 7.91667px; transform-origin: 50.05px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eterms = [1 0]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 61.3px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 30.65px; transform-origin: 404px 30.65px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 69.3px 7.91667px; transform-origin: 69.3px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e coeffs   = [1 -1]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 50.05px 7.91667px; transform-origin: 50.05px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e errOrder = 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 65.45px 7.91667px; transform-origin: 65.45px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e errCoeff = -0.5;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [coeff,errOrder,errCoeff] = FDderiv(n,terms)\r\n%  n        = order of the derivative sought\r\n%  terms    = grid indices of terms to include\r\n%  coeff    = coefficients of the terms in the formula \r\n%  errOrder = exponent of h in the first non-zero higher-order term\r\n%  errCoeff = coefficient of the first non-zero higher-order term\r\n\r\n  coeff    = ...\r\n  errOrder = ...\r\n  errCoeff = ...\r\nend","test_suite":"%%\r\n%  First-order forward difference for the first derivative\r\nn = 1;\r\nterms = [1 0];\r\ncoeff_correct = [1 -1];\r\nerrOrder_correct = 1;\r\nerrCoeff_correct = -1/2;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  First-order backward difference for the first derivative\r\nn = 1;\r\nterms = [0 -1];\r\ncoeff_correct = [1 -1];\r\nerrOrder_correct = 1;\r\nerrCoeff_correct = 1/2;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order centered difference for the first derivative\r\nn = 1;\r\nterms = [1 -1];\r\ncoeff_correct = [1/2 -1/2];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = -1/6;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order forward difference for the first derivative\r\nn = 1;\r\nterms = [2 1 0];\r\ncoeff_correct = [-1/2 2 -3/2];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = 1/3;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order backward difference for the first derivative\r\nn = 1;\r\nterms = [0 -1 -2];\r\ncoeff_correct = [3/2 -2 1/2];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = 1/3;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Fourth-order centered difference for the first derivative\r\nn = 1;\r\nterms = [2 1 0 -1 -2];\r\ncoeff_correct = [-1 8 0 -8 1]/12\r\nerrOrder_correct = 4;\r\nerrCoeff_correct = 1/30;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order centered difference for the second derivative\r\nn = 2;\r\nterms = [1 0 -1];\r\ncoeff_correct = [1 -2 1];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = -1/12;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Fourth-order centered difference for the second derivative\r\nn = 2;\r\nterms = [2 1 0 -1 -2];\r\ncoeff_correct = [-1 16 -30 16 -1]/12;\r\nerrOrder_correct = 4;\r\nerrCoeff_correct = 1/90;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order centered difference for the third derivative\r\nn = 3;\r\nterms = [2 1 0 -1 -2];\r\ncoeff_correct = [1/2 -1 0 1 -1/2];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = -1/4;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)\r\n\r\n%%\r\n%  Second-order centered difference for the fourth derivative\r\nn = 4;\r\nterms = [2 1 0 -1 -2];\r\ncoeff_correct = [1 -4 6 -4 1];\r\nerrOrder_correct = 2;\r\nerrCoeff_correct = -1/6;\r\n[coeff,errOrder,errCoeff] = FDderiv(n,terms);\r\nassert(all(abs(coeff-coeff_correct) \u003c 1e-6))\r\nassert(isequal(errOrder,errOrder_correct))\r\nassert(abs((errCoeff-errCoeff_correct)/errCoeff_correct) \u003c 1e-4)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-22T03:37:57.000Z","updated_at":"2020-11-15T13:41:30.000Z","published_at":"2020-07-22T05:20:57.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f' = (f_{j+1} - f_j})/h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime=\\\\frac{f_{j+1}-f_j}{h}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf\\\"\u003e\u003cw:r\u003e\u003cw:t\u003euse Taylor series\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. In the example above, one can write\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f_{j+1} = f_j + hf' + h^2 f''/2 + h^3 f'''/6 + ...\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e f_{j+1} = f_j + h f\\\\prime +  \\\\frac{h^2}{2}f\\\\prime\\\\prime +  \\\\frac{h^3}{6}f\\\\prime\\\\prime\\\\prime + \\\\ldots\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThen solving for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$f\\\\prime$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and neglecting terms of order \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h^2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$h^2$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and higher gives\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e       \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f' = (f_{j+1} - f_j)/h - hf''/2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime = \\\\frac{f_{j+1} - f_j}{h} - \\\\frac{h}{2}f\\\\prime\\\\prime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBecause the exponent on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"h\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in the last term is 1, the method is called a first order method.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes the order\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of the derivative and a vector\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eterms\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eterms = [1 0]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ coeffs   = [1 -1]\\n errOrder = 1\\n errCoeff = 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