This stems purely from some play on my part. Suppose I asked you to work with the sequence formed as 2*n*F_n + 1, where F_n is the n'th Fibonacci number? Part of me would not be surprised to find there is nothing simple we could do. But, then it costs nothing to try, to see where MATLAB can take me in an explorative sense.
n = sym(0:100).';
Fn = fibonacci(n);
Sn = 2*n.*Fn + 1;
Sn(1:10) % A few elements
ans =
For kicks, I tried asking ChatGPT. Giving it nothing more than the first 20 members of thse sequence as integers, it decided this is a Perrin sequence, and gave me a recurrence relation, but one that is in fact incorrect. Good effort from the Ai, but a fail in the end.
Is there anything I can do? Try null! (Look carefully at the array generated by Toeplitz. It is at least a pretty way to generate the matrix I needed.)
X = toeplitz(Sn,[1,zeros(1,4)]);
rank(X(5:end,:))
ans = 5
Hmm. So there is no linear combination of those columns that yields all zeros, since the resulting matrix was full rank.
X = toeplitz(Sn,[1,zeros(1,5)]);
rank(X(6:end,:))
ans = 5
But if I take it one step further, we see the above matrix is now rank deficient. What does that tell me? It says there is some simple linear combination of the columns of X(6:end,:) that always yields zero. The previous test tells me there is no shorter constant coefficient recurrence releation, using fewer terms.
null(X(6:end,:))
ans =
Let me explain what those coefficients tell me. In fact, they yield a very nice recurrence relation for the sequence S_n, not unlike the original Fibonacci sequence it was based upon.
where the first 5 members of that sequence are given as [1 3 5 13 25]. So a 6 term linear constant coefficient recurrence relation. If it reminds you of the generating relation for the Fibonacci sequence, that is good, because it should. (Remember I started the sequence at n==0, IF you decide to test it out.) We can test it out, like this:
SfunM = memoize(@(N) Sfun(N));
SfunM(25)
ans = 3751251
2*25*fibonacci(sym(25)) + 1
ans =
3751251
And indeed, it works as expected.
function Sn = Sfun(n)
switch n
case 0
Sn = 1;
case 1
Sn = 3;
case 2
Sn = 5;
case 3
Sn = 13;
case 4
Sn = 25;
otherwise
Sn = Sfun(n-5) + Sfun(n-4) - 3*Sfun(n-3) - Sfun(n-2) +3*Sfun(n-1);
end
end
A beauty of this, is I started from nothing but a sequence of integers, derived from an expression where I had no rational expectation of finding a formula, and out drops something pretty. I might call this explorational mathematics.
The next step of course is to go in the other direction. That is, given the derived recurrence relation, if I substitute the formula for S_n in terms of the Fibonacci numbers, can I prove it is valid in general? (Yes.) After all, without some proof, it may fail for n larger than 100. (I'm not sure how much I can cram into a single discussion, so I'll stop at this point for now. If I see interest in the ideas here, I can proceed further. For example, what was I doing with that sequence in the first place? And of course, can I prove the relation is valid? Can I do so using MATLAB?)
(I'll be honest, starting from scratch, I'm not sure it would have been obvious to find that relation, so null was hugely useful here.)
It's frustrating when a long function or script runs and prints unexpected outputs to the command window. The line producing those outputs can be difficult to find.
Starting in R2024b, use dbstop to find the line with unsuppressed outputs!
Run this line of code before running the script or function. Execution will pause when the line is hit and the file will open to that line. Outputs that are intentionaly displayed by functions such as disp() or fprintf() will be ignored.
You can download these toolkits from the provided links.
The reason for writing this article is that many people have started using the chord diagram plotting toolkit that I developed. However, some users are unsure about customizing certain styles. As the developer, I have a good understanding of the implementation principles of the toolkit and can apply it flexibly. This has sparked the idea of challenging myself to create various styles of chord diagrams. Currently, the existing code is quite lengthy. In the future, I may integrate some of this code into the toolkit, enabling users to achieve the effects of many lines of code with just a few lines.
Without further ado, let's see the extent to which this MATLAB toolkit can currently perform.
Hi everyone, I wrote several fancy functions that may help your coding experience, since they are in very early developing stage, I will be thankful if anyone could try them and give some feedbacks. Currently I have following:
fstr: a Python f-string like expression
printf: an easy to use fprintf function, accepts multiple arguments with seperator, end string control.
I will bring more functions or packages like logger when I am available.
If you have a folder with an enormous number of files and want to use the uigetfile function to select specific files, you may have noticed a significant delay in displaying the file list.
Thanks to the assistance from MathWorks support, an interesting behavior was observed.
For example, if a folder such as Z:\Folder1\Folder2\data contains approximately 2 million files, and you attempt to use uigetfile to access files with a specific extension (e.g., *.ext), the following behavior occurs:
Method 1: This takes minutes to show me the list of all files
Method 3: This method also takes minutes to display the file list. What is intertesting is that this method is the same as Method 2, except that a file seperator "\" is added at the end of the folder string.
This post is more of a "tips and tricks" guide than a question.
If you have a folder with an enormous number of files and want to use the uigetfile function to select specific files, you may have noticed a significant delay in displaying the file list.
Thanks to the assistance from MathWorks support, an interesting behavior was observed.
For example, if a folder such as Z:\Folder1\Folder2\data contains approximately 2 million files, and you attempt to use uigetfile to access files with a specific extension (e.g., *.ext), the following behavior occurs:
Method 1: This takes minutes to show me the list of all files
Method 3: This method also takes minutes to display the file list. What is intertesting is that this method is the same as Method 2, except that a file seperator "\" is added at the end of the folder string.
Hi!My name is Mike McLernon, and I’m a product marketing engineer with MathWorks.In my role, I look at the trends ongoing in the wireless industry, across lots of different standards (think 5G, WLAN, SatCom, Bluetooth, etc.), and I seek to shape and guide the software that MathWorks builds to respond to these trends.That’s all about communicating within the Mathworks organization, but every so often it’s worth communicating these trends to our audience in the world at large.Many of the people reading this are engineers (or engineers at heart), and we all want to know what’s happening in the geek world around us.I think that now is one of these times to communicate an important milestone.So, without further ado . . .
Bluetooth 6.0 is here!Announced in September by the Bluetooth Special Interest Group (SIG), it’s making more advances in its quest to create a “world without wires”.A few of the salient features in Bluetooth 6.0 are:
Channel sounding (for accurate distance measurements)
Decision-based advertising filtering (for more efficient channel scanning)
Monitoring advertisers (for improved energy efficiency when devices come into and go out of range)
Frame space updates (for both higher throughput and better coexistence management)
Bluetooth 6.0 includes other features as well, but the SIG has put special promotional emphasison channel sounding (CS), which once upon a time was called High Accuracy Distance Measurement (HADM).The SIG has said that CS is a step towards true distance awareness, and 10 cm ranging accuracy.I think their emphasis is in exactly the right place, so let’s learn more about this technology and how it is used.
CS can be used for the following use cases:
Keyless vehicle entry, performed by communication between a key fob or phone and the car’s anchor points
Smart locks, to permit access only when an authorized device is within a designated proximity to the locks
Geofencing, to limit access to designated areas
Warehouse management, to monitor inventory and manage logistics
Asset tracking for virtually any object of interest
In the past, Bluetooth devices would use a received signal strength indicator (RSSI) measurement to infer the distance between two of them.They would assume a free space path loss on the link, and use a straightforward equation to estimate distance:
where
received power in dBm,
transmitted power in dBm,
propagation loss in dB,
distance between the two devices, in m,
Bluetooth signal wavelength, in m,
Bluetooth signal frequency, in Hz,
speed of light (3 x 1e8 m/s).
So in this method, .But if the signal suffers more loss from multipath or shadowing, then the distance would be overestimated.Something better needed to be found.
Bluetooth 6.0 introduces not one, but two ways to accurately measure distance:
Round-trip time (RTT) measurement
Phase-based ranging (PBR) measurement
The RTT measurement method uses the fact that the Bluetooth signal time of flight (TOF) between two devices is half the RTT.It can then accurately compute the distance das
, where c is again the speed of light.This method requires accurate measurements of the time of departure (TOD) of the outbound signal from device 1 (the Initiator), time of arrival (TOA) of the outbound signal to device 2 (the Reflector), TOD of the return signal from device 2, and TOA of the return signal to device 1.The diagram below shows the signal paths and the times.
The PBR method uses two Bluetooth signals of different frequencies to measure distance.These signals are simply tones – sine waves.Without going through the derivation, PBR calculates distance das
, where
distance between the two devices, in m,
speed of light (3 x 1e8 m/s),
phase measured at the Initiator after the tone at completes a round trip, in radians,
phase measured at the Initiator after the tone at completes a round trip, in radians,
frequency of the first tone, in Hz,
frequency of the second tone, in Hz.
The mod() operation is needed to eliminate ambiguities in the distance calculation and the final division by two is to convert a round trip distance to a one-way distance.Because a given phase difference value can hold true for an infinite number of distance values, the mod() operation chooses the smallest distance that satisfies the equation.Also, these tones can be as close as 1 MHz apart.In that case, the maximum resolvable distance measurement is about 150 m.The plot below shows that ambiguity and repetition in distance measurement.
Bluetooth 6.0 outlines RTT and PBR distance measurement methods, but CS does not mandate a specific algorithm for calculating distance estimates. This flexibility allows device manufacturers to tailor solutions to various use cases, balancing computational complexity with required accuracy and adapting to different radio environments. Examples include simple phase difference calculations and FFT-based methods.
Although Bluetooth 6.0 is now out, it is far from a finished version.The SIG is actively working through the ratification process for two major extensions:
High Data Throughput, up to 8 Mbps
5 and 6 GHz operation
See the last few minutes of this videofrom the SIG to learn more about these exciting future developments.And if you want to see more Bluetooth blogs, give a review of this one!Finally, if you have specific Bluetooth questions, give me a shout and let’s start a discussion!
isinf(A) - check if the array elements are infinity
isnan(A)
Rounding Functions
ceil(x) - rounds each element of x to nearest integer >= to element
floor(x) - rounds each element of x to nearest integer <= to element
fix(x) - rounds each element of x to nearest integer towards 0
round(x) - rounds each element of x to nearest integer. if round(x, N), rounds N digits to the right of the decimal point.
rem(dividend, divisor) - produces a result that is either 0 or has the same sign as the dividen.
mod(dividend, divisor) - produces a result that is either 0 or same result as divisor.
Ex: 12/2, 12 is dividend, 2 is divisor
sum(inputArray) - sums all entires in array
Complex Number Functions
abs(z) - absolute value, is magnitude of complex number (phasor form r*exp(j0)
angle(z) - phase angle, corresponds to 0 in r*exp(j0)
complex(a,b) - creates complex number z = a + jb
conj(z) - given complex conjugate a - jb
real(z) - extracts real part from z
imag(z) - extracts imaginary part from z
unwrap(z) - removes the modulus 2pi from an array of phase angles.
Statistics Functions
mean(xAr) - arithmetic mean calculated.
std(xAr) - calculated standard deviation
median(xAr) - calculated the median of a list of numbers
mode(xAr) - calculates the mode, value that appears most often
max(xAr)
min(xAr)
If using &&, this means that if first false, don't bother evaluating second
Random Number Functions
rand(numRand, 1) - creates column array
rand(1, numRand) - creates row array, both with numRand elements, between 0 and 1
randi(maxRandVal, numRan, 1) - creates a column array, with numRand elements, between 1 and maxRandValue.
randn(numRand, 1) - creates a column array with normally distributed values.
Ex: 10 * rand(1, 3) + 6
"10*rand(1, 3)" produces a row array with 3 random numbers between 0 and 10. Adding 6 to each element results in random values between 6 and 16.
randi(20, 1, 5)
Generates 5 (last argument) random integers between 1 (second argument) and 20 (first argument). The arguments 1 and 5 specify a 1 × 5 row array is returned.
Discrete Integer Mathematics
primes(maxVal) - returns prime numbers less than or equal to maxVal
isprime(inputNums) - returns a logical array, indicating whether each element is a prime number
factor(intVal) - returns the prime factors of a number
gcd(aVals, bVals) - largest integer that divides both a and b without a remainder
lcm(aVals, bVals) - smallest positive integer that is divisible by both a and b
factorial(intVals) - returns the factorial
perms(intVals) - returns all the permutations of the elements int he array intVals in a 2D array pMat.
outRot90 = rot90(A) - Rotates array by 90 degrees counter clockwise around element at index (1,1).
outTril = tril(A) - Returns the lower triangular part of an array.
outTriU = triu(A) - Returns the upper triangular part of an array.
arrayOut = repmat(subarrayIn, mRow, nCol), creates a large array by replicating a smaller array, with mRow x nCol tiling of copies of subarrayIn
reshapeOut - reshape(arrayIn, numRow, numCol) - returns array with modifid dimensions. Product must equal to arrayIn of numRow and numCol.
outLin = find(inputAr) - locates all nonzero elements of inputAr and returns linear indices of these elements in outLin.
[sortOut, sortIndices] = sort(inArray) - sorts elements in ascending order, results result in sortOut. specify 'descend' if you want descending order. sortIndices hold the sorted indices of the array elements, which are row indices of the elements of sortOut in the original array
[sortOut, sortIndices] = sortrows(inArray, colRef) - sorts array based on values in column colRef while keeping the rows together. Bases on first column by default.
isequal(inArray1, inarray2, ..., inArrayN)
isequaln(inArray1, inarray2, ..., inarrayn)
arrayChar = ischar(inArray) - ischar tests if the input is a character array.
arrayLogical = islogical(inArray) - islogical tests for logical array.
arrayNumeric = isnumeric(inArray) - isnumeric tests for numeric array.
arrayInteger = isinteger(inArray) - isinteger tests whether the input is integer type (Ex: uint8, uint16, ...)
arrayFloat = isfloat(inArray) - isfloat tests for floating-point array.
arrayReal= isreal(inArray) - isreal tests for real array.
objIsa = isa(obj,ClassName) - isa determines whether input obj is an object of specified class ClassName.
arrayScalar = isscalar(inArray) - isscalar tests for scalar type.
arrayVector = isvector(inArray) - isvector tests for a vector (a 1D row or column array).
arrayColumn = iscolumn(inArray) - iscolumn tests for column 1D arrays.
arrayMatrix = ismatrix(inArray) - ismatrix returns true for a scalar or array up to 2D, but false for an array of more than 2 dimensions.
arrayEmpty = isempty(inArray) - isempty tests whether inArray is empty.
primeArray = isprime(inArray) - isprime returns a logical array primeArray, of the same size as inArray. The value at primeArray(index) is true when inArray(index) is a prime number. Otherwise, the values are false.
finiteArray = isfinite(inArray) - isfinite returns a logical array finiteArray, of the same size as inArray. The value at finiteArray(index) is true when inArray(index) is finite. Otherwise, the values are false.
infiniteArray = isinf(inArray) - isinf returns a logical array infiniteArray, of the same size as inArray. The value at infiniteArray(index) is true when inArray(index) is infinite. Otherwise, the values are false.
nanArray = isnan(inArray) - isnan returns a logical array nanArray, of the same size as inArray. The value at nanArray(index) is true when inArray(index) is NaN. Otherwise, the values are false.
allNonzero = all(inArray) - all identifies whether all array elements are non-zero (true). Instead of testing elements along the columns, all(inArray, 2) tests along the rows. all(inArray,1) is equivalent to all(inArray).
anyNonzero = any(inArray) - any identifies whether any array elements are non-zero (true), and false otherwise. Instead of testing elements along the columns, any(inArray, 2) tests along the rows. any(inArray,1) is equivalent to any(inArray).
logicArray = ismember(inArraySet,areElementsMember) - ismember returns a logical array logicArray the same size as inArraySet. The values at logicArray(i) are true where the elements of the first array inArraySet are found in the second array areElementsMember. Otherwise, the values are false. Similar values found by ismember can be extracted with inArraySet(logicArray).
any(x) - Returns true if x is nonzero; otherwise, returns false.
isnan(x) - Returns true if x is NaN (Not-a-Number); otherwise, returns false.
isfinite(x) - Returns true if x is finite; otherwise, returns false. Ex: isfinite(Inf) is false, and isfinite(10) is true.
isinf(x) - Returns true if x is +Inf or -Inf; otherwise, returns false.
fctName = @(arglist) expression - anonymous function
nargin - keyword returns the number of input arguments passed to the function.
Looping
while condition
% code
end
for index = startVal:endVal
% code
end
continue: Skips the rest of the current loop iteration and begins the next iteration.
break: Exits a loop before it has finished all iterations.
switch expression
case value1
% code
case value2
% code
otherwise
% code
end
Comprehensive Overview (may repeat)
Built in functions/constants
abs(x) - absolute value
pi - 3.1415...
inf - ∞
eps - floating point accuracy 1e6 106
sum(x) - sums elements in x
cumsum(x) - Cumulative sum
prod - Product of array elements cumprod(x) cumulative product
diff - Difference of elements round/ceil/fix/floor Standard functions..
*Standard functions: sqrt, log, exp, max, min, Bessel *Factorial(x) is only precise for x < 21
Variable Generation
j:k - row vector
j:i:k - row vector incrementing by i
linspace(a,b,n) - n points linearly spaced and including a and b
NaN(a,b) - axb matrix of NaN values
ones(a,b) - axb matrix with all 1 values
zeros(a,b) - axb matrix with all 0 values
meshgrid(x,y) - 2d grid of x and y vectors
global x
Ch. 11 - Custom Functions
function [ outputArgs ] = MainFunctionName (inputArgs)
% statements go here
end
function [ outputArgs ] = LocalFunctionName (inputArgs)
% statements go here
end
You are allowed to have nested functions in MATLAB
Anonymous Function:
fctName = @(argList) expression
Ex: RaisedCos = @(angle) (cosd(angle))^2;
global variables - can be accessed from anywhere in the file
Persistent variables
persistent variable - only known to function where it was declared, maintains value between calls to function.
Recursion - base case, decreasing case, ending case
nargin - evaluates to the number of arguments the function was called with
Ch. 12 - Plotting
plot(xArray, yArray)
refer to help plot for more extensive documentation, chapter 12 only briefly covers plotting
plot - Line plot
yyaxis - Enables plotting with y-axes on both left and right sides
loglog - Line plot with logarithmic x and y axes
semilogy - Line plot with linear x and logarithmic y axes
semilogx - Line plot with logarithmic x and linear y axes
stairs - Stairstep graph
axis - sets the aspect ratio of x and y axes, ticks etc.
grid - adds a grid to the plot
gtext - allows positioning of text with the mouse
text - allows placing text at specified coordinates of the plot
xlabel labels the x-axis
ylabel labels the y-axis
title sets the graph title
figure(n) makes figure number n the current figure
hold on allows multiple plots to be superimposed on the same axes
hold off releases hold on current plot and allows a new graph to be drawn
close(n) closes figure number n
subplot(a, b, c) creates an a x b matrix of plots with c the current figure
orient specifies the orientation of a figure
Animated plot example:
for j = 1:360
pause(0.02)
plot3(x(j),y(j),z(j),'O')
axis([minx maxx miny maxy minz maxz]);
hold on;
end
Ch. 13 - Strings
stringArray = string(inputArray) - converts the input array inputArray to a string array
number = strLength(stringIn) - returns the number of characters in the input string
stringArray = strings(n, m) - returns an n-by-m array of strings with no characters,
doing strings(sz) returns an array of strings with no characters, where sz defines the size.
charVec1 = char(stringScalar) char(stringScalar) converts the string scalar stringScalar into a character vector charVec1.
charVec2 = char(numericArray) char(numericArray) converts the numeric array numericArray into a character vector charVec2 corresponding to the Unicode transformation format 16 (UTF-16) code.
stringOut = string1 + string2 - combines the text in both strings
stringOut = join(textSrray) - consecutive elements of array joined, placing a space character between them.
stringOut = blanks(n) - creates a string of n blank characters
stringOut = strcar(string1, string2) - horizontally concatenates strings in arrays.
sprintf(formatSpec, number) - for printing out strings
strExp = sprintf("%0.6e", pi)
stringArrayOur = compose(formatSpec, arrayIn) - formats data in arrayIn.
lower(string) - converts to lowercase
upper(string) - converts to uppercase
num2str(inputArray, precision) - returns a character vector with the maximum number of digits specified by precision
mat2str(inputMat, precision), converts matrix into a character vector.
numberOut = sscanf(inputText, format) - convert inputText according to format specifier
strncmpi(str1, str2, n) - case-insensitive comparison of first n characters.
isstring(string) - logical output
isStringScalar(string) - logical output
ischar(inputVar) - logical output
iscellstr(inputVar) - logical output.
isstrprop(stringIn, categoryString) - returns a logical array of the same size as stringIn, with true at indices where elements of the stringIn belong to specified category:
iskeyword(stringIn) - returns true if string is a keyword in the matlab language
isletter(charVecIn)
isspace(charVecIn)
ischar(charVecIn)
contains(string1, testPattern) - boolean outputs if string contains a specific pattern
startsWith(string1, testPattern) - also logical output
endsWith(string1, testPattern) - also logical output
strfind(stringIn, pattern) - returns INDEX of each occurence in array
number = count(stringIn, patternSeek) - returns the number of occurences of string scalar in the string scalar stringIn.
strip(strArray) - removes all consecutive whitespace characters from begining and end of each string in Array, with side argument 'left', 'right', 'both'.
pad(stringIn) - pad with whitespace characters, can also specify where.
replace(stringIn, oldStr, newStr) - replaces all occurrences of oldStr in string array stringIn with newStr.
replaceBetween(strIn, startStr, endStr, newStr)
strrep(origStr, oldSubsr, newSubstr) - searches original string for substric, and if found, replace with new substric.
insertAfter(stringIn, startStr, newStr) - insert a new string afte the substring specified by startStr.
insertBefore(stringIn, endPos, newStr)
extractAfter(stringIn, startStr)
extractBefore(stringIn, startPos)
split(stringIn, delimiter) - divides string at whitespace characters.
Time to time I need to filll an existing array with NaNs using logical indexing. A trick I discover is using arithmetics rather than filling. It is quite faster in some circumtances
A=rand(10000);
b=A>0.5;
tic; A(b) = NaN; toc
Elapsed time is 0.737291 seconds.
tic; A = A + 0./~b; toc;
Elapsed time is 0.027666 seconds.
If you know trick for other value filling feel free to post.
In the spirit of warming up for this year's minihack contest, I'm uploading a walkthrough for how to design an airship using pure Matlab script. This is commented and uncondensed; half of the challenge for the minihacks is how minimize characters. But, maybe it will give people some ideas.
The actual airship design is from one of my favorite original NES games that I played when I was a kid - Little Nemo: The Dream Master. The design comes from the intro of the game when Nemo sees the Slumberland airship leave for Slumberland:
(Snip from a frame of the opening scene in Capcom's game Little Nemo: The Dream Master, showing the Slumberland airship).
I spent hours playing this game with my two sisters, when we were little. It's fun and tough, but the graphics sparked the imagination. On to the code walkthrough, beginning with the color palette: these four colors are the only colors used for the airship:
c1=cat(3,1,.7,.4); % Cream color
c2=cat(3,.7,.1,.3); % Magenta
c3=cat(3,0.7,.5,.1); % Gold
c4=cat(3,.5,.3,0); % bronze
We start with the airship carriage body. We want something rectangular but smoothed on the corners. To do this we are going to start with the separate derivatives of the x and y components, which can be expressed using separate blocks of only three levels: [1, 0, -1]. You could integrate to create a rectangle, but if we smooth the derivatives prior to integrating we will get rounded edges. This is done in the following code:
% Binary components for x & y vectors
z=zeros(1,30);
o=ones(1,100);
% X and y vectors
x=[z,o,z,-o];
y=[1+z,1-o,z-1,1-o];
% Smoother function (fourier / circular)
s=@(x)ifft(fft(x).*conj(fft(hann(45)'/22,260)));
% Integrator function with replication and smoothing to form mesh matrices
u=@(x)repmat(cumsum(s(x)),[30,1]);
% Construct x and y components of carriage with offsets
x3=u(x)-49.35;
y3=u(y)+6.35;
y3 = y3*1.25; % Make it a little fatter
% Add a z-component to make the full set of matrices for creating a 3D
% surface:
z3=linspace(0,1,30)'.*ones(1,260)*30;
z3(14,:)=z3(15,:); % These two lines are for adding platforms
z3(2,:)=z3(3,:); % to the carriage, later.
Plotting x, y, and the top row of the smoothed, integrated, and replicated matrices x3 and y3 gives the following:
We now have the x and y components for a 3D mesh of the carriage, let's make it more interesting by adding a color scheme including doors, and texture for the trim around the doors. Let's also add platforms beneath the doors for passengers to walk on.
Starting with the color values, let's make doors by convolving points in a color-matrix by a door shaped function.
m=0*z3; % Image matrix that will be overlayed on carriage surface
m(7,10:12:end)=1; % Door locations (lower deck)
m(21,10:12:end)=1; % Door locations (upper deck)
drs = ones(9, 5); % Door shape
m=1-conv2(m,ones(9,5),'same'); % Applying
To add the trim, we will convolve matrix "m" (the color matrix) with a square function, and look for values that lie between the extrema. We will use this to create a displacement function that bumps out the -x, and -y values of the carriage surface in intermediary polar coordinate format.
rm=conv2(m,ones(5)/25,'same'); % Smoothing the door function
rm(~m)=0; % Preserving only the region around the doors
rds=0*m; % Radial displacement function
rds(rm<1&rm>0)=1; % Preserving region around doors
rds(m==0)=0;
rds(13:14,:)=6; % Adding walkways
rds(1:2,:)=6;
% Apply radial displacement function
[th,rd]=cart2pol(x3,y3);
[x3T,y3T]=pol2cart(th,(rd+rds)*.89);
If we plot the color function (m) and radial displacement function (rds) we get the following:
In the upper plot you can see the doors, and in the bottom map you can see the walk way and door trim.
Next, we are going to add some flags draped from the bottom and top of the carriage. We are going to recycle the values in "z3" to do this, by multiplying that matrix with the absolute value of a sine-wave, squished a bit with the soft-clip erf() function.
We add a keel to the airship carriage using a canonical sphere turned on its side, again using the soft-clip erf() function to make it roughly rectangular in x and y, and multiplying with a vector that is half nan's to make the top half transparent.
At this point, since we are beginning the plotting of the ship, we also need to create our hgtransform objects. These allow us to move all of the components of the airship in unison, and also link objects with pivot points to the airship, such as the propeller.
% Now we need some flags extending around the top and bottom of the
% carriage. We can do this my multiplying the height function (z3) with the
% absolute value of a sine-wave, rounded with a compression function
% The carriage is done. Now we can make the blimp above it.
We haven't adjusted the shading of the image yet, but you can see the design features that have been created:
Next, we start working on the blimp. This is going to use a few more vertices & faces. We are going to use a tapered cylinder for this part, and will start by making the overlaid image, which will have 2 colors plus radial rings, circles, and squiggles for ornamentation.
M=525; % Blimp (matrix dimensions)
N=700;
% Assign the blimp the cream and magenta colors
t=122; % Transition point
b=ones(M,N,3); % Blimp color map template
bc=b.*c1; % Blimp color map
bc(:,t+1:end-t,:)=b(:,t+1:end-t,:).*c2;
% Add axial rings around blimp
l=[.17,.3,.31,.49];
l=round([l,1-fliplr(l)]*N); % Mirroring
lnw=ones(1,N); % Mask
lnw(l)=0;
lnw=rescale(conv(lnw,hann(7)','same'));
bc=bc.*lnw;
% Now add squiggles. We're going to do this by making an even function in
% the x-dimension (N, 725) added with a sinusoidal oscillation in the
The figures below show the color scheme and mask used to apply the squiggles and circles generated in the code above:
Finally, for the colormap we are going to smooth the binary mask to avoid hard transitions, and use it to to add a "puffy" texture to the blimp shape. This will be done by diffusing the mask iteratively in a loop with a non-linear min() operator.
Notice that the parent of the blimp surface plot is the same as the carriage (e.g. hgtransform object "P"). Plotting at this point using flat shading and adding some lighting gives the image below:
Next, we need to add a propeller so it can move. This will include the creation of a shaft using the cylinder() function. The rest of the pieces (the propeller blades, collars and shaft tip) all use the same canonical sphere with distortions applied using various math functions. Note that when the propeller is made it is linked to hgtransform object "S" rather than "P." This will allow the propeller to rotate, but still be joined to the airship.
% Next, the propeller. First, we start with the shaft. This is a simple
% cylinder. We add an offset variable and a scale variable to move our
% propeller components around, as well.
shx = -70; % This is our x-shifter for components
scl = 3; % Component size scaler
[x,y,z]=cylinder(1, 20); % Canonical cylinder for prop shaft.
Now for some final details including the ropes to the blimp, a flag hung on one of the ropes, and railings around the walkways so that passengers don't plummet to their doom. This will make use of the ad-hoc "ropeG" function, which takes a 3D vector of points and makes a conforming cylinder around it, so that you get lighting functions etc. that don't work on simple lines. This function is added to the script at the end to do this:
% Rope function for making a 3D curve have thickness, like a rope.
% Inputs:
% - xyz (3D curve vector, M points in 3 x M format)
% - N (Number of radial points in cylinder function around the curve
% - W (Width of the rope)
%
% Outputs:
% - xf, yf, zf (Matrices that can be used with surf())
function [xf, yf, zf] = RopeG(xyz, N, W)
% Canonical cylinder with N points in circumference
[xt,yt,zt] = cylinder(1, N);
% Extract just the first ring and make (W) wide
xyzt = [xt(1, :); yt(1, :); zt(1, :)]*W;
% Get local orientation vector between adjacent points in rope
dxyz = xyz(:, 2:end) - xyz(:, 1:end-1);
dxyz(:, end+1) = dxyz(:, end);
vcs = dxyz./vecnorm(dxyz);
% We need to orient circle so that its plane normal is parallel to
Using this function we can define the ropes and balconies. Note that the balconies simply recycle one of the rows of the original carriage surface, defining the outer rim of the walkway, but bumping up in the z-dimension.
And, very last, we are going to add a flag attached to the outer cable. Let's make it flap in the wind. To make it we will recycle the z3 matrix again, but taper it based on its x-value. Then we will sinusoidally oscillate the flag in the y dimension as a function of x, constraining the y-position to be zero where it meets the cable. Lastly, we will displace it quadratically in the x-dimension as a function of z so that it lines up nicely with the rope. The phase of the sine-function is modified in the animation loop to give it a flapping motion.
Putting all the pieces together reveals the full airship:
A note about lighting: lighting and material properties really change the feel of the image you create. The above picture is rendered in a cartoony style by setting the specular exponent to a very low value (1), and adding lots of diffuse and ambient reflectivity as well. A light below the airship was also added, albeit with lower strength. Settings used for this plot were:
What about all the rest of the stuff (clouds, moon, atmospheric haze etc.) These were all (mostly) recycled bits from previous minihack entries. The clouds were made using power-law noise as explained in Adam Danz' blog post. The moon was borrowed from moonrun, but with an increased number of points. Atmospheric haze was recycled from Matlon5. The rest is just camera angles, hgtransform matrix updates, and updating alpha-maps or vertex coordinates.
Finally, the use of hann() adds the signal processing toolbox as a dependency. To avoid this use the following anonymous function:
Create a struct arrays where each struct has field names "a," "b," and "c," which store different types of data. What efficient methods do you have to assign values from individual variables "a," "b," and "c" to each struct element? Here are five methods I've provided, listed in order of decreasing efficiency. What do you think?
Create an array of 10,000 structures, each containing each of the elements corresponding to the a,b,c variables.
Suppose you need to do a computation many times. We are going to assume that this computation cannot be vectorized. The simplest case is to use a for loop:
number_of_elements = 1e6;
test_fcn = @(x) sqrt(x) / x;
tic
for i = 1:number_of_elements
x(i) = test_fcn(i);
end
t_forward = toc;
disp(t_forward + " seconds")
0.10925 seconds
Preallocation:
This can easily be sped up by preallocating the variable that houses results:
tic
x = zeros(number_of_elements, 1);
for i = 1:number_of_elements
x(i) = test_fcn(i);
end
t_forward_prealloc = toc;
disp(t_forward_prealloc + " seconds")
0.035106 seconds
In this example, preallocation speeds up the loop by a factor of about three to four (running in R2024a). Comment below if you get dramatically different results.
Is there a way to skip the explicit preallocation and still be fast? Indeed, there is.
clear x
tic
for i = number_of_elements:-1:1
x(i) = test_fcn(i);
end
t_backward = toc;
disp(t_backward + " seconds")
0.032392 seconds
By running the loop backwards, the preallocation is implicitly performed during the first iteration and the loop runs in about the same time (within statistical noise):
Do you get similar results when running this code? Let us know your thoughts in the comments below.
Beneficial side effect:
Have you ever had to use a for loop to delete elements from a vector? If so, keeping track of index offsets can be tricky, as deleting any element shifts all those that come after. By running the for loop in reverse, you don't need to worry about index offsets while deleting elements.
The idea is a simple one, embodied in these 5 steps.
1. Take any 4 digit integer, reduce to its decimal digits.
2. Sort the digits in decreasing order.
3. Flip the sequence of those digits, then recompose the two sets of sorted digits into 4 digit numbers. If there were any 0 digits, they will become leading zeros on the smaller number. In this case, a leading zero is acceptable to consider a number as a 4 digit integer.
4. Subtract the two numbers, smaller from the larger. The result will always have no more than 4 decimal digits. If it is less than 1000, then presume there are leading zero digits.
5. If necessary, repeat the above operation, until the result converges to a stable result, or until you see a cycle.
Since this process is deterministic, and must always result in a new 4 digit integer, it must either terminate at either an absorbing state, or in a cycle.
For example, consider the number 6174.
7641 - 1467
ans = 6174
We get 6174 directly back. That seems rather surprising to me. But even more interesting is you will find all 4 digit numbers (excluding the pure rep-digit nmbers) will always terminate at 6174, after at most a few steps. For example, if we start with 1234
4321 - 1234
ans = 3087
8730 - 0378
ans = 8352
8532 - 2358
ans = 6174
and we see that after 3 iterations of this process, we end at 6174. Similarly, if we start with 9998, it too maps to 6174 after 5 iterations.
9998 ==> 999 ==> 8991 ==> 8082 ==> 8532 ==> 6174
Why should that happen? That is, why should 6174 always drop out in the end? Clearly, since this is a deterministic proces which always produces another 4 digit integer (Assuming we treat integers with a leading zero as 4 digit integers), we must either end in some cycle, or we must end at some absorbing state. But for all (non-pure rep-digit) starting points to end at the same place, it seems just a bit surprising.
I always like to start a problem by working on a simpler problem, and see if it gives me some intuition about the process. I'll do the same thing here, but with a pair of two digit numbers. There are 100 possible two digit numbers, since we must treat all one digit numbers as having a "tens" digit of 0.
N = (0:99)';
Next, form the Kaprekar mapping for 2 digit numbers. This is easier than you may think, since we can do it in a very few lines of code on all possible inputs.
Do you see what happens? All of the rep-digit numbers, like 11, 44, 55, etc., all map directly to 0, and they stay there, since 0 also maps into 0. We can see that in the star on the lower right.
G2cycles = cyclebasis(G2)
G2cycles = 2x1 cell array
{1x1 cell}
{1x5 cell}
G2cycles{1}
ans = 1x1 cell array
{'00'}
All other numbers eventually end up in the cycle:
G2cycles{2}
ans = 1x5 cell array
{'09'} {'45'} {'27'} {'63'} {'81'}
That is
81 ==> 63 ==> 27 ==> 45 ==> 09 ==> and back to 81
looping forever.
Another way of trying to visualize what happens with 2 digit numbers is to use symbolics. Thus, if we assume any 2 digit number can be written as 10*T+U, where I'll assume T>=U, since we always sort the digits first
syms T U
(10*T + U) - (10*U+T)
ans =
So after one iteration for 2 digit numbers, the result maps ALWAYS to a new 2 digit number that is divisible by 9. And there are only 10 such 2 digit numbers that are divisible by 9. So the 2-digit case must resolve itself rather quickly.
What happens when we move to 3 digit numbers? Note that for any 3 digit number abc (without loss of generality, assume a >= b >= c) it almost looks like it reduces to the 2 digit probem, aince we have abc - cba. The middle digit will always cancel itself in the subtraction operation. Does that mean we should expect a cycle at the end, as happens with 2 digit numbers? A simple modification to our previous code will tell us the answer.
So the 3 digit number 100 required 6 iterations to eventually reach 495.
shortestpath(G3,101,496) - 1
ans = 1x7
100 99 891 792 693 594 495
<mw-icon class=""></mw-icon>
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I think I've rather exhausted the 3 digit case. It is time now to move to the 4 digit problem, but we've already done all the hard work. The same scheme will apply to compute a graph. And the graph theory tools do all the hard work for us.
And here we see the behavior, with one stable final point, 6174 as the only non-zero ending state. There are no circular cycles as we had for the 2-digit case.
How many iterations were necessary at most before termination?
D4 = distances(G4,6175);
D4(isinf(D4)) = -inf;
plot(D4)
The plot tells the story here. The maximum number of iterations before termination is 7 for the 4 digit case.
find(D4 == 7,1,'last') - 1
ans = 9985
shortestpath(G4,9986,6175) - 1
ans = 1x8
9985 4086 8172 7443 3996 6264 4176 6174
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
Can you go further? Are there 5 or 6 digit Kaprekar constants? Sadly, I have read that for more than 4 digits, things break down a bit, there is no 5 digit (or higher) Kaprekar constant.
We can verify that fact, at least for 5 digit numbers.
The result here are 4 disjoint cycles. Of course the rep-digit cycle must always be on its own, but the other three cycles are also fully disjoint, and are of respective length 2, 4, and 4.
Studying the attached document Duffing Equation from the University of Colorado, I noticed that there is an analysis of The Non-Chaotic Duffing Equation and all the graphs were created with Matlab. And since the code is not given I took the initiative to try to create the same graphs with the following code.
Plotting the Potential Energy and Identifying Extrema
An attractor is called strange if it has a fractal structure, that is if it has non-integer Hausdorff dimension. This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist. If a strange attractor is chaotic, exhibiting sensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.
The term strange attractor was coined by David Ruelle and Floris Takens to describe the attractor resulting from a series of bifurcations of a system describing fluid flow. Strange attractors are often differentiable in a few directions, but some are like a Cantor dust, and therefore not differentiable. Strange attractors may also be found in the presence of noise, where they may be shown to support invariant random probability measures of Sinai–Ruelle–Bowen type.
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