I am simulating water level at every 1 minute interval from an event of rainfall. I am using Jacobian method of 5 row and 5 column

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Zulhash Uddin
Zulhash Uddin on 1 Mar 2011
I am simulating a biorention swale wherein rainfall runoff is coming into catch basin and catch basin is connected to a pipe (upper pipe) which is perforated and water is flowing into a sand filled trench through the perforation. At the bottom of the trench there is another perforated pipe (lower pipe) which is draining water from the trench to city sewer through the lower pipe. In my problem I have 5 mass balance equation which I put as F1 matrix and I developed Jacobian matrix by differentiating those 5 equation. I want to run the model for 12 hrs period even though the rainfall period is 4 hrs.I wrote the problem is as follows:
clc
clear all
format long
% Solve nonlinear system F(x)=0 using NewtonÌs method
% input data
%Rainfall data
i_n=[0.00000099
0.00000101
0.00000102
0.00000103
0.00000105
0.00000106
0.00000107
0.00000108
0.00000110
0.00000111
0.00000111
0.00000112
0.00000114
0.00000115
0.00000117
0.00000118
0.00000119
0.00000121
0.00000122
0.00000124
0.00000125
0.00000127
0.00000129
0.00000131
0.00000133
0.00000135
0.00000137
0.00000139
0.00000141
0.00000143
0.00000145
0.00000148
0.00000151
0.00000154
0.00000157
0.00000159
0.00000162
0.00000165
0.00000168
0.00000171
0.00000174
0.00000179
0.00000184
0.00000189
0.00000193
0.00000198
0.00000203
0.00000208
0.00000213
0.00000217
0.00000222
0.00000232
0.00000242
0.00000251
0.00000261
0.00000271
0.00000281
0.00000290
0.00000300
0.00000310
0.00000320
0.00000357
0.00000394
0.00000431
0.00000468
0.00000505
0.00000542
0.00000579
0.00000616
0.00000652
0.00000689
0.00000992
0.00001294
0.00001596
0.00001898
0.00002200
0.00002502
0.00002805
0.00003107
0.00003409
0.00003711
0.00003429
0.00003147
0.00002864
0.00002582
0.00002300
0.00002018
0.00001736
0.00001453
0.00001171
0.00000889
0.00000850
0.00000811
0.00000772
0.00000733
0.00000693
0.00000654
0.00000615
0.00000576
0.00000537
0.00000498
0.00000484
0.00000470
0.00000457
0.00000443
0.00000429
0.00000415
0.00000401
0.00000387
0.00000374
0.00000360
0.00000286
0.00000282
0.00000277
0.00000273
0.00000268
0.00000263
0.00000259
0.00000254
0.00000250
0.00000245
0.00000241
0.00000237
0.00000234
0.00000231
0.00000228
0.00000225
0.00000222
0.00000218
0.00000215
0.00000212
0.00000209
0.00000206
0.00000204
0.00000202
0.00000199
0.00000197
0.00000194
0.00000192
0.00000190
0.00000187
0.00000185
0.00000183
0.00000181
0.00000180
0.00000178
0.00000176
0.00000175
0.00000173
0.00000171
0.00000170
0.00000167
0.00000166
0.00000164
0.00000163
0.00000161
0.00000160
0.00000158
0.00000157
0.00000155
0.00000154
0.00000153
0.00000151
0.00000150
0.00000149
0.00000148
0.00000146
0.00000145
0.00000144
0.00000143
0.00000142
0.00000140
0.00000139
0.00000138
0.00000137
0.00000136
0.00000135
0.00000134
0.00000133
0.00000133
0.00000132
0.00000131
0.00000130
0.00000129
0.00000128
0.00000127
0.00000126
0.00000125
0.00000125
0.00000124
0.00000123
0.00000122
0.00000121
0.00000121
0.00000120
0.00000119
0.00000118
0.00000118
0.00000117
0.00000116
0.00000116
0.00000115
0.00000114
0.00000114
0.00000113
0.00000112
0.00000112
0.00000111
0.00000111
0.00000110
0.00000109
0.00000109
0.00000108
0.00000108
0.00000107
0.00000106
0.00000106
0.00000105
0.00000105
0.00000104
0.00000104
0.00000103
0.0000010400
0.0000010300
0.0000010200
0.0000010100
0.0000010000
0.0000009900
0.0000009800
0.0000009700
0.0000009600
0.0000009500
0.0000009400
0.0000009300
0.0000009200
0.0000009100
0.0000009000
0
0
0
0
0];
A_catmt=385;
%disp(i_n1)
QET_n=0;
QET_n1=0;
g=9.1;
W=3.55;
V=.68;
K_ns=.0000018;
K_uns=0.0084;
K=.1;
r=0.8;
s=1;
rs=r*s;
I_n=1;
I_n1=2;
d_p1=0.15;
d_p2=0.2;
%Hdes=dead storage height of catc basin
Hdes_n1=0.8;
Hdes_n=0.8;
L1=35;
L2=17.33;
L3=16;
fi=.98;
I_a=0;
%area of bioretention cell calculation
A_bc=L1*.30;
A_cb=0.61^2;
%cross sectional area
A_p1=pi*d_p1^2/4;
A_p2=pi*d_p2^2/4;
A_mh=pi*.8^2/4;
A_mhexit=pi*.2^2/4;
H_bc=0.8;
D_p1=.15;
D_p2=0.2;
D_mhexit=0.2;
C_d=0.65;
C=0.65;
i_n1=0;
%time step
delta_t=60;
% heights at t=0, boundary situation
H1_n = 0;
H2_n = 0;
H3_n = 0;
H4_n = 0;
H5_n = 0;
% constant value of H(primes)
% substitue height variables into x variables in order to eliminate square root expression in orifice equation
X1_n = sqrt(H1_n);
X2_n = sqrt(H2_n);
X3_n = H3_n;
X4_n = sqrt(H4_n);
X5_n = sqrt(H5_n);
% set initial iteration for the next time step, variables X_n1
X1_n1_old = [1
1
1
1
1];
X1_n1 = X1_n1_old(1,1);
X2_n1 = X1_n1_old(2,1);
X3_n1 = X1_n1_old(3,1);
X4_n1 = X1_n1_old(4,1);
X5_n1 = X1_n1_old(5,1);
fprintf('step X1_n1 X2_n1 X3_n1 X4_n1 X5_n1 X1_n X2_n X3_n X4_n X5_n \n')
% set stopping conditions and maximum iteration runs
error = 1*10^(-5);
iter=0;
itermax=5;
aa1=0.5*delta_t*A_p1*sqrt(2*g)*(C_d/D_p1)*sqrt(C_d/1.656)
aa2=A_cb
aa3=A_cb*Hdes_n1
aa4=0.5*delta_t*fi*A_catmt
bb1=.6*0.5*delta_t*2756
bb2=.6*0.5*delta_t*245.47
bb3=.6*0.5*delta_t*6.75
bb4=0.1177*L1
cc1=0.2*L2*W
cc2=delta_t*(L2+W)*K_ns*r*s
cc3=0.5*delta_t*QET_n
cc4=0.5*delta_t*0.0113*C*sqrt(2*g)
dd1=0.1177*L3
dd2=0.5*delta_t*A_p2*sqrt(2*g)*(C_d/D_p2)*sqrt(C_d/1.656)
ee1=0.5*delta_t*A_mhexit*sqrt(2*g)*(C_d/D_mhexit)*sqrt(C_d/1.656)
ee2=A_mh
%X1_n=0.13670; X2_n=-1.15261; X3_n=-0.00013; X4_n=-0.02598; X5_n=-0.07425;
% X1_n1=0.11496; X2_n1=-0.13479; X3_n1=-0.00022; X4_n1=0.00640; X5_n1=0.02197;
%FF=aa1*(X1_n1^3)+aa2*(X1_n1^2)+aa3-aa2*X1_n^2+aa1*X1_n^3-aa3-aa4*i_n(2+1,1)-aa4*i_n(2,1)+aa4*I_a
F1=[aa1*X1_n1^3+aa2*X1_n1^2+aa3+aa1*X1_n^3-aa2*X1_n^2-aa3-aa4*i_n(1+1,1)-aa4*i_n(1,1)
aa1*X1_n1^3-bb1*X2_n1^6+bb2*X2_n1^4-bb3*X2_n1^2-bb4*X2_n1^2+aa1*X1_n^3-bb1*X2_n^6+bb2*X2_n^4-bb3*X2_n^2+bb4*X2_n^2
bb1*X2_n1^6-bb2*X2_n1^4+bb3*X2_n1^2-cc1*X3_n1-cc2*X3_n1+bb1*X2_n^6-bb2*X2_n^4+bb3*X2_n^2-cc2*X3_n+cc1*X3_n-cc3*QET_n1-cc3*QET_n-cc4*X3_n1^1.5-cc4*X3_n^1.5
dd1*X4_n1^2+dd2*X4_n1^4-cc4*X3_n1^1.5+dd2*X4_n^4-cc4*X3_n^1.5-dd1*X4_n^2
ee1*X5_n1^3+ee2*X5_n1^2-dd2*X4_n1^3-ee2*X5_n^2+ee2*X5_n^3-dd2*X4_n^3];
for kk=1:241
iteration = 1;
max_iteration_runs =15;
while(iteration<=max_iteration_runs)
iteration = iteration+1;
J1=[3*aa1*X1_n1^2+2*aa2*X1_n1 0 0 0 0
3*aa1*X1_n1^2 -6*bb1*X2_n1^5+4*bb2*X2_n1^3-2*bb3*X2_n1-2*bb4*X2_n1 0 0 0
0 6*bb1*X2_n1^5-4*bb2*X2_n1^3+2*bb3*X2_n1 -cc1-cc2-cc3-1.5*cc4*X3_n1^0.5 0 0
0 0 1.5*cc4*X3_n1^0.5 2*dd1*X4_n1+3*dd2*X4_n1^2 0
0 0 0 -3*dd2*X4_n1^2 3*ee1*X5_n1^2+2*ee2*X5_n1];
X1_n1_new = X1_n1_old-inv(J1)*F1;
X1_n1_old = X1_n1_new;
X1_n1 = X1_n1_old(1,1);
X2_n1 = X1_n1_old(2,1);
X3_n1 = X1_n1_old(3,1);
X4_n1 = X1_n1_old(4,1);
X5_n1 = X1_n1_old(5,1);
F1=[aa1*X1_n1^3+aa2*X1_n1^2+aa3+aa1*X1_n^3-aa2*X1_n^2-aa3-aa4*i_n(kk+1,1)-aa4*i_n(kk,1)
aa1*X1_n1^3-bb1*X2_n1^6+bb2*X2_n1^4-bb3*X2_n1^2-bb4*X2_n1^2+aa1*X1_n^3-bb1*X2_n^6+bb2*X2_n^4-bb3*X2_n^2+bb4*X2_n^2
bb1*X2_n1^6-bb2*X2_n1^4+bb3*X2_n1^2-cc1*X3_n1-cc2*X3_n1+bb1*X2_n^6-bb2*X2_n^4+bb3*X2_n^2-cc2*X3_n+cc1*X3_n-cc3*QET_n1-cc3*QET_n-cc4*X3_n1^1.5-cc4*X3_n^1.5
dd1*X4_n1^2+dd2*X4_n1^4-cc4*X3_n1^1.5+dd2*X4_n^4-cc4*X3_n^1.5-dd1*X4_n^2
ee1*X5_n1^3+ee2*X5_n1^2-dd2*X4_n1^3-ee2*X5_n^2+ee2*X5_n^3-dd2*X4_n^3];
end
fprintf('%8.5f %8.5f %8.5f %8.5f %8.5f %8.5f %8.5f %8.5f %8.5f %8.5f %8.5f\n',kk,X1_n1,X2_n1,X3_n1,X4_n1,X5_n1,X1_n,X2_n,X3_n,X4_n,X5_n)
X1_n = X1_n1;
X2_n = X2_n1;
X3_n = X3_n1;
X4_n = X4_n1;
X5_n = X5_n1;
end
Please help me if any one can to run the model simultaneously for 12 hrs or more. I sincerely appreciate your help. Anticipated thanks.
Zulhash

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Answers (1)

Walter Roberson
Walter Roberson on 1 Mar 2011
If your rainfall data is per minute, then your line
for kk=1:241
would be stepping minute by minute over 240 minutes = 4 hours.
In two places in the body of the "for kk" loop, you reference
i_n(kk+1,1)
Replace those references by initializing
i_n_kk = 0;
if kk+1 <= length(i_n); i_n_kk = i_n(kk); end
then use i_n_kk instead of i_n(kk+1,1) . This will use 0 for all later rain data after the end of your i_n matrix.
  2 Comments
Zulhash Uddin
Zulhash Uddin on 2 Mar 2011
Dear Walter Roberson,
I have used code like you. My current program and output is like below. But still it has some error.Its not giving correct value for 0 rainfall data. If you kindly check my program and output below I belive you could give me the right direction. Thanks for kind help.
Zulhash
clc
clear all
format long
% Solve nonlinear system F(x)=0 using NewtonÌs method
% input data
%Rainfall data
i_n=[0.00000099
0.00000101
0.00000102
0.00000103
0.00000105
0.00000106
0.00000107
0.00000108
0.00000110
0.00000111
0.00000111
0.00000112
0.00000114
0.00000115
0.00000117
0.00000118
0.00000119
0.00000121
0.00000122
0.00000124
0.00000125
0.00000127
0.00000129
0.00000131
0.00000133
0.00000135
0.00000137
0.00000139
0.00000141
0.00000143
0.00000145
0.00000148
0.00000151
0.00000154
0.00000157
0.00000159
0.00000162
0.00000165
0.00000168
0.00000171
0.00000174
0.00000179
0.00000184
0.00000189
0.00000193
0.00000198
0.00000203
0.00000208
0.00000213
0.00000217
0.00000222
0.00000232
0.00000242
0.00000251
0.00000261
0.00000271
0.00000281
0.00000290
0.00000300
0.00000310
0.00000320
0.00000357
0.00000394
0.00000431
0.00000468
0.00000505
0.00000542
0.00000579
0.00000616
0.00000652
0.00000689
0.00000992
0.00001294
0.00001596
0.00001898
0.00002200
0.00002502
0.00002805
0.00003107
0.00003409
0.00003711
0.00003429
0.00003147
0.00002864
0.00002582
0.00002300
0.00002018
0.00001736
0.00001453
0.00001171
0.00000889
0.00000850
0.00000811
0.00000772
0.00000733
0.00000693
0.00000654
0.00000615
0.00000576
0.00000537
0.00000498
0.00000484
0.00000470
0.00000457
0.00000443
0.00000429
0.00000415
0.00000401
0.00000387
0.00000374
0.00000360
0.00000286
0.00000282
0.00000277
0.00000273
0.00000268
0.00000263
0.00000259
0.00000254
0.00000250
0.00000245
0.00000241
0.00000237
0.00000234
0.00000231
0.00000228
0.00000225
0.00000222
0.00000218
0.00000215
0.00000212
0.00000209
0.00000206
0.00000204
0.00000202
0.00000199
0.00000197
0.00000194
0.00000192
0.00000190
0.00000187
0.00000185
0.00000183
0.00000181
0.00000180
0.00000178
0.00000176
0.00000175
0.00000173
0.00000171
0.00000170
0.00000167
0.00000166
0.00000164
0.00000163
0.00000161
0.00000160
0.00000158
0.00000157
0.00000155
0.00000154
0.00000153
0.00000151
0.00000150
0.00000149
0.00000148
0.00000146
0.00000145
0.00000144
0.00000143
0.00000142
0.00000140
0.00000139
0.00000138
0.00000137
0.00000136
0.00000135
0.00000134
0.00000133
0.00000133
0.00000132
0.00000131
0.00000130
0.00000129
0.00000128
0.00000127
0.00000126
0.00000125
0.00000125
0.00000124
0.00000123
0.00000122
0.00000121
0.00000121
0.00000120
0.00000119
0.00000118
0.00000118
0.00000117
0.00000116
0.00000116
0.00000115
0.00000114
0.00000114
0.00000113
0.00000112
0.00000112
0.00000111
0.00000111
0.00000110
0.00000109
0.00000109
0.00000108
0.00000108
0.00000107
0.00000106
0.00000106
0.00000105
0.00000105
0.00000104
0.00000104
0.00000103
0
0
0
0
0];
A_catmt=385;
%disp(i_n1)
QET_n=0;
QET_n1=0;
g=9.1;
W=3.55;
V=.68;
K_ns=.0000018;
K_uns=0.0084;
K=.1;
r=0.8;
s=1;
rs=r*s;
I_n=1;
I_n1=2;
d_p1=0.15;
d_p2=0.2;
%Hdes=dead storage height of catc basin
Hdes_n1=0.8;
Hdes_n=0.8;
L1=35;
L2=17.33;
L3=16;
fi=.98;
I_a=0;
%area of bioretention cell calculation
A_bc=L1*.30;
A_cb=0.61^2;
%cross sectional area
A_p1=pi*d_p1^2/4;
A_p2=pi*d_p2^2/4;
A_mh=pi*.8^2/4;
A_mhexit=pi*.2^2/4;
H_bc=0.8;
D_p1=.15;
D_p2=0.2;
D_mhexit=0.2;
C_d=0.65;
C=0.65;
i_n1=0;
%time step
delta_t=60;
% heights at t=0, boundary situation
H1_n = 0;
H2_n = 0;
H3_n = 0;
H4_n = 0;
H5_n = 0;
% constant value of H(primes)
% substitue height variables into x variables in order to eliminate square root expression in orifice equation
X1_n = sqrt(H1_n);
X2_n = sqrt(H2_n);
X3_n = H3_n;
X4_n = sqrt(H4_n);
X5_n = sqrt(H5_n);
% set initial iteration for the next time step, variables X_n1
X1_n1_old = [1
1
1
1
1];
X1_n1 = X1_n1_old(1,1);
X2_n1 = X1_n1_old(2,1);
X3_n1 = X1_n1_old(3,1);
X4_n1 = X1_n1_old(4,1);
X5_n1 = X1_n1_old(5,1);
fprintf('step X1_n1 X2_n1 X3_n1 X4_n1 X5_n1 X1_n X2_n X3_n X4_n X5_n \n')
% set stopping conditions and maximum iteration runs
error = 1*10^(-5);
iter=0;
itermax=5;
aa1=0.5*delta_t*A_p1*sqrt(2*g)*(C_d/D_p1)*sqrt(C_d/1.656);
aa2=A_cb;
aa3=A_cb*Hdes_n1;
aa4=0.5*delta_t*fi*A_catmt;
bb1=.6*0.5*delta_t*2756;
bb2=.6*0.5*delta_t*245.47;
%%
bb3=.6*0.5*delta_t*6.75;
bb4=0.1177*L1;
cc1=0.2*L2*W;
cc2=delta_t*(L2+W)*K_ns*r*s;
cc3=0.5*delta_t*QET_n;
cc4=0.5*delta_t*0.0113*C*sqrt(2*g);
dd1=0.1177*L3;
dd2=0.5*delta_t*A_p2*sqrt(2*g)*(C_d/D_p2)*sqrt(C_d/1.656);
ee1=0.5*delta_t*A_mhexit*sqrt(2*g)*(C_d/D_mhexit)*sqrt(C_d/1.656);
ee2=A_mh;
%X1_n=0.13670; X2_n=-1.15261; X3_n=-0.00013; X4_n=-0.02598; X5_n=-0.07425;
% X1_n1=0.11496; X2_n1=-0.13479; X3_n1=-0.00022; X4_n1=0.00640; X5_n1=0.02197;
%FF=aa1*(X1_n1^3)+aa2*(X1_n1^2)+aa3-aa2*X1_n^2+aa1*X1_n^3-aa3-aa4*i_n(2+1,1)-aa4*i_n(2,1)+aa4*I_a
F1=[aa1*X1_n1^3+aa2*X1_n1^2+aa3+aa1*X1_n^3-aa2*X1_n^2-aa3-aa4*i_n(1+1,1)-aa4*i_n(1,1)
aa1*X1_n1^3-bb1*X2_n1^6+bb2*X2_n1^4-bb3*X2_n1^2-bb4*X2_n1^2+aa1*X1_n^3-bb1*X2_n^6+bb2*X2_n^4-bb3*X2_n^2+bb4*X2_n^2
bb1*X2_n1^6-bb2*X2_n1^4+bb3*X2_n1^2-cc1*X3_n1-cc2*X3_n1+bb1*X2_n^6-bb2*X2_n^4+bb3*X2_n^2-cc2*X3_n+cc1*X3_n-cc3*QET_n1-cc3*QET_n-cc4*X3_n1^1.5-cc4*X3_n^1.5
dd1*X4_n1^2+dd2*X4_n1^4-cc4*X3_n1^1.5+dd2*X4_n^4-cc4*X3_n^1.5-dd1*X4_n^2
ee1*X5_n1^3+ee2*X5_n1^2-dd2*X4_n1^3-ee2*X5_n^2+ee2*X5_n^3-dd2*X4_n^3];
%i_n(kk+1,1)
%Replace those references by initializing
%i_n_kk = 0;
%if kk+1 <= length(i_n); i_n_kk = i_n(kk); end
%then use i_n_kk instead of i_n(kk+1,1) . This will use 0 for all later rain data after the end of your i_n matrix.
i_n_kk = 0;
for kk=1:226
if kk+1<=length(i_n);
i_n_kk = i_n(kk);
iteration = 1;
max_iteration_runs =15;
while(iteration<=max_iteration_runs)
iteration = iteration+1;
J1=[3*aa1*X1_n1^2+2*aa2*X1_n1 0 0 0 0
3*aa1*X1_n1^2 -6*bb1*X2_n1^5+4*bb2*X2_n1^3-2*bb3*X2_n1-2*bb4*X2_n1 0 0 0
0 6*bb1*X2_n1^5-4*bb2*X2_n1^3+2*bb3*X2_n1 -cc1-cc2-cc3-1.5*cc4*X3_n1^0.5 0 0
0 0 1.5*cc4*X3_n1^0.5 2*dd1*X4_n1+3*dd2*X4_n1^2 0
0 0 0 -3*dd2*X4_n1^2 3*ee1*X5_n1^2+2*ee2*X5_n1];
X1_n1_new = X1_n1_old-inv(J1)*F1;
X1_n1_old = X1_n1_new;
X1_n1 = X1_n1_old(1,1);
X2_n1 = X1_n1_old(2,1);
X3_n1 = X1_n1_old(3,1);
X4_n1 = X1_n1_old(4,1);
X5_n1 = X1_n1_old(5,1);
F1=[aa1*X1_n1^3+aa2*X1_n1^2+aa3+aa1*X1_n^3-aa2*X1_n^2-aa3-aa4*i_n_kk-aa4*i_n(kk,1)
aa1*X1_n1^3-bb1*X2_n1^6+bb2*X2_n1^4-bb3*X2_n1^2-bb4*X2_n1^2+aa1*X1_n^3-bb1*X2_n^6+bb2*X2_n^4-bb3*X2_n^2+bb4*X2_n^2
bb1*X2_n1^6-bb2*X2_n1^4+bb3*X2_n1^2-cc1*X3_n1-cc2*X3_n1+bb1*X2_n^6-bb2*X2_n^4+bb3*X2_n^2-cc2*X3_n+cc1*X3_n-cc3*QET_n1-cc3*QET_n-cc4*X3_n1^1.5-cc4*X3_n^1.5
dd1*X4_n1^2+dd2*X4_n1^4-cc4*X3_n1^1.5+dd2*X4_n^4-cc4*X3_n^1.5-dd1*X4_n^2
ee1*X5_n1^3+ee2*X5_n1^2-dd2*X4_n1^3-ee2*X5_n^2+ee2*X5_n^3-dd2*X4_n^3];
end
fprintf('%8.5f %8.5f %8.5f %8.5f %8.5f %8.5f %8.5f %8.5f %8.5f %8.5f %8.5f\n',kk,X1_n1,X2_n1,X3_n1,X4_n1,X5_n1,X1_n,X2_n,X3_n,X4_n,X5_n)
X1_n = X1_n1;
X2_n = X2_n1;
X3_n = X3_n1;
X4_n = X4_n1;
X5_n = X5_n1;
end
end
Result:
step X1_n1 X2_n1 X3_n1 X4_n1 X5_n1 X1_n X2_n X3_n X4_n X5_n
1.00000 0.13619 0.01114 0.00122 0.00463 0.00009 0.00000 0.00000 0.00000 0.00000 0.00000
2.00000 0.11499 0.00909 0.00323 0.01144 0.00478 0.13619 0.01114 0.00122 0.00463 0.00009
3.00000 0.12725 0.00992 0.00497 0.01990 0.01210 0.11499 0.00909 0.00323 0.01144 0.00478
4.00000 0.12201 0.00991 0.00683 0.02914 0.02227 0.12725 0.00992 0.00497 0.01990 0.01210
5.00000 0.12591 0.00976 0.00863 0.03896 0.03444 0.12201 0.00991 0.00683 0.02914 0.02227
6.00000 0.12456 0.01019 0.01044 0.04915 0.04770 0.12591 0.00976 0.00863 0.03896 0.03444
7.00000 0.12584 0.00978 0.01222 0.05952 0.06138 0.12456 0.01019 0.01044 0.04915 0.04770
8.00000 0.12578 0.01030 0.01397 0.06987 0.07507 0.12584 0.00978 0.01222 0.05952 0.06138
9.00000 0.12698 0.00995 0.01570 0.08004 0.08850 0.12578 0.01030 0.01397 0.06987 0.07507
10.00000 0.12694 0.01041 0.01741 0.08985 0.10146 0.12698 0.00995 0.01570 0.08004 0.08850
11.00000 0.12696 0.00998 0.01907 0.09916 0.11379 0.12694 0.01041 0.01741 0.08985 0.10146
12.00000 0.12753 0.01045 0.02069 0.10782 0.12533 0.12696 0.00998 0.01907 0.09916 0.11379
13.00000 0.12838 0.01018 0.02230 0.11578 0.13597 0.12753 0.01045 0.02069 0.10782 0.12533
14.00000 0.12851 0.01054 0.02387 0.12301 0.14567 0.12838 0.01018 0.02230 0.11578 0.13597
15.00000 0.12956 0.01034 0.02541 0.12952 0.15443 0.12851 0.01054 0.02387 0.12301 0.14567
16.00000 0.12957 0.01065 0.02693 0.13535 0.16228 0.12956 0.01034 0.02541 0.12952 0.15443
17.00000 0.13012 0.01043 0.02840 0.14057 0.16930 0.12957 0.01065 0.02693 0.13535 0.16228
18.00000 0.13094 0.01080 0.02985 0.14523 0.17555 0.13012 0.01043 0.02840 0.14057 0.16930
19.00000 0.13106 0.01057 0.03127 0.14942 0.18115 0.13094 0.01080 0.02985 0.14523 0.17555
20.00000 0.13208 0.01092 0.03266 0.15320 0.18618 0.13106 0.01057 0.03127 0.14942 0.18115
21.00000 0.13208 0.01072 0.03402 0.15664 0.19073 0.13208 0.01092 0.03266 0.15320 0.18618
22.00000 0.13315 0.01103 0.03535 0.15980 0.19487 0.13208 0.01072 0.03402 0.15664 0.19073
23.00000 0.13365 0.01093 0.03667 0.16271 0.19868 0.13315 0.01103 0.03535 0.15980 0.19487
24.00000 0.13444 0.01118 0.03795 0.16543 0.20220 0.13365 0.01093 0.03667 0.16271 0.19868
25.00000 0.13506 0.01111 0.03923 0.16799 0.20550 0.13444 0.01118 0.03795 0.16543 0.20220
26.00000 0.13576 0.01133 0.04048 0.17040 0.20860 0.13506 0.01111 0.03923 0.16799 0.20550
27.00000 0.13641 0.01129 0.04171 0.17269 0.21153 0.13576 0.01133 0.04048 0.17040 0.20860
28.00000 0.13707 0.01149 0.04292 0.17487 0.21431 0.13641 0.01129 0.04171 0.17269 0.21153
29.00000 0.13772 0.01147 0.04410 0.17695 0.21697 0.13707 0.01149 0.04292 0.17487 0.21431
30.00000 0.13837 0.01164 0.04527 0.17895 0.21951 0.13772 0.01147 0.04410 0.17695 0.21697
31.00000 0.13901 0.01163 0.04642 0.18087 0.22194 0.13837 0.01164 0.04527 0.17895 0.21951
32.00000 0.14013 0.01186 0.04756 0.18272 0.22428 0.13901 0.01163 0.04642 0.18087 0.22194
33.00000 0.14097 0.01189 0.04871 0.18451 0.22654 0.14013 0.01186 0.04756 0.18272 0.22428
34.00000 0.14194 0.01208 0.04984 0.18626 0.22874 0.14097 0.01189 0.04871 0.18451 0.22654
35.00000 0.14282 0.01213 0.05098 0.18797 0.23089 0.14194 0.01208 0.04984 0.18626 0.22874
36.00000 0.14327 0.01225 0.05210 0.18964 0.23298 0.14282 0.01213 0.05098 0.18797 0.23089
37.00000 0.14442 0.01234 0.05321 0.19127 0.23503 0.14327 0.01225 0.05210 0.18964 0.23298
38.00000 0.14516 0.01249 0.05431 0.19286 0.23703 0.14442 0.01234 0.05321 0.19127 0.23503
39.00000 0.14610 0.01256 0.05541 0.19442 0.23898 0.14516 0.01249 0.05431 0.19286 0.23703
40.00000 0.14692 0.01272 0.05651 0.19595 0.24089 0.14610 0.01256 0.05541 0.19442 0.23898
41.00000 0.14779 0.01279 0.05761 0.19745 0.24277 0.14692 0.01272 0.05651 0.19595 0.24089
42.00000 0.14949 0.01305 0.05872 0.19894 0.24462 0.14779 0.01279 0.05761 0.19745 0.24277
43.00000 0.15068 0.01317 0.05987 0.20042 0.24645 0.14949 0.01305 0.05872 0.19894 0.24462
44.00000 0.15212 0.01339 0.06104 0.20191 0.24829 0.15068 0.01317 0.05987 0.20042 0.24645
45.00000 0.15297 0.01348 0.06223 0.20342 0.25015 0.15212 0.01339 0.06104 0.20191 0.24829
46.00000 0.15454 0.01371 0.06343 0.20492 0.25200 0.15297 0.01348 0.06223 0.20342 0.25015
47.00000 0.15567 0.01384 0.06466 0.20643 0.25387 0.15454 0.01371 0.06343 0.20492 0.25200
48.00000 0.15702 0.01404 0.06590 0.20795 0.25574 0.15567 0.01384 0.06466 0.20643 0.25387
49.00000 0.15820 0.01419 0.06717 0.20947 0.25762 0.15702 0.01404 0.06590 0.20795 0.25574
50.00000 0.15906 0.01432 0.06843 0.21099 0.25950 0.15820 0.01419 0.06717 0.20947 0.25762
51.00000 0.16048 0.01450 0.06971 0.21251 0.26138 0.15906 0.01432 0.06843 0.21099 0.25950
52.00000 0.16341 0.01491 0.07107 0.21404 0.26326 0.16048 0.01450 0.06971 0.21251 0.26138
53.00000 0.16535 0.01517 0.07254 0.21564 0.26519 0.16341 0.01491 0.07107 0.21404 0.26326
54.00000 0.16741 0.01546 0.07408 0.21731 0.26720 0.16535 0.01517 0.07254 0.21564 0.26519
55.00000 0.16967 0.01578 0.07570 0.21905 0.26930 0.16741 0.01546 0.07408 0.21731 0.26720
56.00000 0.17173 0.01607 0.07740 0.22085 0.27149 0.16967 0.01578 0.07570 0.21905 0.26930
57.00000 0.17382 0.01637 0.07918 0.22273 0.27376 0.17173 0.01607 0.07740 0.22085 0.27149
58.00000 0.17549 0.01661 0.08101 0.22465 0.27610 0.17382 0.01637 0.07918 0.22273 0.27376
59.00000 0.17768 0.01692 0.08289 0.22661 0.27850 0.17549 0.01661 0.08101 0.22465 0.27610
60.00000 0.17948 0.01718 0.08484 0.22861 0.28094 0.17768 0.01692 0.08289 0.22661 0.27850
61.00000 0.18145 0.01746 0.08683 0.23064 0.28343 0.17948 0.01718 0.08484 0.22861 0.28094
62.00000 0.19104 0.01885 0.08926 0.23282 0.28602 0.18145 0.01746 0.08683 0.23064 0.28343
63.00000 0.19534 0.01956 0.09224 0.23535 0.28891 0.19104 0.01885 0.08926 0.23282 0.28602
64.00000 0.20223 0.02056 0.09565 0.23829 0.29227 0.19534 0.01956 0.09224 0.23535 0.28891
65.00000 0.20697 0.02136 0.09950 0.24162 0.29612 0.20223 0.02056 0.09565 0.23829 0.29227
66.00000 0.21260 0.02219 0.10373 0.24529 0.30042 0.20697 0.02136 0.09950 0.24162 0.29612
67.00000 0.21724 0.02300 0.10832 0.24925 0.30511 0.21260 0.02219 0.10373 0.24529 0.30042
68.00000 0.22215 0.02374 0.11323 0.25343 0.31013 0.21724 0.02300 0.10832 0.24925 0.30511
69.00000 0.22655 0.02452 0.11843 0.25779 0.31540 0.22215 0.02374 0.11323 0.25343 0.31013
70.00000 0.23079 0.02518 0.12388 0.26228 0.32087 0.22655 0.02452 0.11843 0.25779 0.31540
71.00000 0.23507 0.02595 0.12955 0.26686 0.32647 0.23079 0.02518 0.12388 0.26228 0.32087
72.00000 0.28085 0.03392 0.13935 0.27270 0.33288 0.23507 0.02595 0.12955 0.26686 0.32647
73.00000 0.29293 0.03667 0.15424 0.28149 0.34185 0.28085 0.03392 0.13935 0.27270 0.33288
74.00000 0.31981 0.04168 0.17281 0.29315 0.35438 0.29293 0.03667 0.15424 0.28149 0.34185
75.00000 0.33252 0.04490 0.19525 0.30690 0.37004 0.31981 0.04168 0.17281 0.29315 0.35438
76.00000 0.35182 0.04873 0.22064 0.32200 0.38787 0.33252 0.04490 0.19525 0.30690 0.37004
77.00000 0.36387 0.05204 0.24883 0.33775 0.40696 0.35182 0.04873 0.22064 0.32200 0.38787
78.00000 0.37925 0.05530 0.27915 0.35366 0.42657 0.36387 0.05204 0.24883 0.33775 0.40696
79.00000 0.39037 0.05858 0.31130 0.36939 0.44617 0.37925 0.05530 0.27915 0.35366 0.42657
80.00000 0.40326 0.06154 0.34476 0.38473 0.46542 0.39037 0.05858 0.31130 0.36939 0.44617
81.00000 0.41356 0.06478 0.37923 0.39953 0.48409 0.40326 0.06154 0.34476 0.38473 0.46542
82.00000 0.38502 0.05727 0.40562 0.41170 0.50077 0.41356 0.06478 0.37923 0.39953 0.48409
83.00000 0.38880 0.05779 0.42257 0.41953 0.51317 0.38502 0.05727 0.40562 0.41170 0.50077
84.00000 0.36337 0.05224 0.43315 0.42422 0.52096 0.38880 0.05779 0.42257 0.41953 0.51317
85.00000 0.36163 0.05089 0.43678 0.42642 0.52527 0.36337 0.05224 0.43315 0.42422 0.52096
86.00000 0.33772 0.04652 0.43560 0.42648 0.52670 0.36163 0.05089 0.43678 0.42642 0.52527
87.00000 0.33071 0.04376 0.42937 0.42480 0.52574 0.33772 0.04652 0.43560 0.42648 0.52670
88.00000 0.30643 0.03999 0.41950 0.42154 0.52273 0.33071 0.04376 0.42937 0.42480 0.52574
89.00000 0.29345 0.03597 0.40593 0.41691 0.51790 0.30643 0.03999 0.41950 0.42154 0.52273
90.00000 0.26606 0.03229 0.38957 0.41099 0.51143 0.29345 0.03597 0.40593 0.41691 0.51790
91.00000 0.24447 0.02676 0.37046 0.40389 0.50343 0.26606 0.03229 0.38957 0.41099 0.51143
92.00000 0.25423 0.02990 0.35256 0.39641 0.49446 0.24447 0.02676 0.37046 0.40389 0.50343
93.00000 0.24035 0.02629 0.33674 0.38953 0.48560 0.25423 0.02990 0.35256 0.39641 0.49446
94.00000 0.24373 0.02790 0.32188 0.38306 0.47735 0.24035 0.02629 0.33674 0.38953 0.48560
95.00000 0.23408 0.02535 0.30842 0.37691 0.46956 0.24373 0.02790 0.32188 0.38306 0.47735
96.00000 0.23355 0.02607 0.29571 0.37106 0.46214 0.23408 0.02535 0.30842 0.37691 0.46956
97.00000 0.22622 0.02411 0.28395 0.36543 0.45505 0.23355 0.02607 0.29571 0.37106 0.46214
98.00000 0.22354 0.02434 0.27277 0.36002 0.44824 0.22622 0.02411 0.28395 0.36543 0.45505
99.00000 0.21713 0.02268 0.26225 0.35477 0.44165 0.22354 0.02434 0.27277 0.36002 0.44824
100.00000 0.21303 0.02260 0.25218 0.34966 0.43526 0.21713 0.02268 0.26225 0.35477 0.44165
101.00000 0.20685 0.02107 0.24257 0.34465 0.42901 0.21303 0.02260 0.25218 0.34966 0.43526
102.00000 0.20779 0.02170 0.23363 0.33982 0.42293 0.20685 0.02107 0.24257 0.34465 0.42901
103.00000 0.20375 0.02064 0.22547 0.33526 0.41713 0.20779 0.02170 0.23363 0.33982 0.42293
104.00000 0.20330 0.02096 0.21786 0.33095 0.41164 0.20375 0.02064 0.22547 0.33526 0.41713
105.00000 0.20008 0.02011 0.21083 0.32686 0.40645 0.20330 0.02096 0.21786 0.33095 0.41164
106.00000 0.19867 0.02021 0.20423 0.32297 0.40152 0.20008 0.02011 0.21083 0.32686 0.40645
107.00000 0.19595 0.01950 0.19807 0.31925 0.39682 0.19867 0.02021 0.20423 0.32297 0.40152
108.00000 0.19404 0.01949 0.19224 0.31569 0.39232 0.19595 0.01950 0.19807 0.31925 0.39682
109.00000 0.19149 0.01884 0.18673 0.31226 0.38799 0.19404 0.01949 0.19224 0.31569 0.39232
110.00000 0.18956 0.01880 0.18151 0.30895 0.38383 0.19149 0.01884 0.18673 0.31226 0.38799
111.00000 0.18685 0.01816 0.17655 0.30575 0.37981 0.18956 0.01880 0.18151 0.30895 0.38383
112.00000 0.16495 0.01536 0.17095 0.30240 0.37577 0.18685 0.01816 0.17655 0.30575 0.37981
113.00000 0.17798 0.01671 0.16540 0.29882 0.37151 0.16495 0.01536 0.17095 0.30240 0.37577
114.00000 0.16840 0.01591 0.16051 0.29535 0.36716 0.17798 0.01671 0.16540 0.29882 0.37151
115.00000 0.17315 0.01598 0.15582 0.29208 0.36299 0.16840 0.01591 0.16051 0.29535 0.36716
116.00000 0.16852 0.01593 0.15155 0.28898 0.35905 0.17315 0.01598 0.15582 0.29208 0.36299
117.00000 0.16970 0.01549 0.14750 0.28604 0.35531 0.16852 0.01593 0.15155 0.28898 0.35905
118.00000 0.16757 0.01577 0.14375 0.28324 0.35176 0.16970 0.01549 0.14750 0.28604 0.35531
119.00000 0.16713 0.01516 0.14022 0.28057 0.34838 0.16757 0.01577 0.14375 0.28324 0.35176
120.00000 0.16598 0.01552 0.13690 0.27803 0.34516 0.16713 0.01516 0.14022 0.28057 0.34838
121.00000 0.16488 0.01488 0.13378 0.27560 0.34208 0.16598 0.01552 0.13690 0.27803 0.34516
122.00000 0.16409 0.01523 0.13083 0.27327 0.33914 0.16488 0.01488 0.13378 0.27560 0.34208
123.00000 0.16310 0.01466 0.12805 0.27104 0.33632 0.16409 0.01523 0.13083 0.27327 0.33914
124.00000 0.16259 0.01499 0.12542 0.26891 0.33362 0.16310 0.01466 0.12805 0.27104 0.33632
125.00000 0.16178 0.01450 0.12296 0.26688 0.33105 0.16259 0.01499 0.12542 0.26891 0.33362
126.00000 0.16114 0.01477 0.12063 0.26493 0.32858 0.16178 0.01450 0.12296 0.26688 0.33105
127.00000 0.16039 0.01433 0.11843 0.26308 0.32623 0.16114 0.01477 0.12063 0.26493 0.32858
128.00000 0.15969 0.01455 0.11634 0.26129 0.32398 0.16039 0.01433 0.11843 0.26308 0.32623
129.00000 0.15856 0.01411 0.11435 0.25958 0.32181 0.15969 0.01455 0.11634 0.26129 0.32398
130.00000 0.15807 0.01431 0.11244 0.25793 0.31973 0.15856 0.01411 0.11435 0.25958 0.32181
131.00000 0.15718 0.01394 0.11063 0.25633 0.31772 0.15807 0.01431 0.11244 0.25793 0.31973
132.00000 0.15652 0.01409 0.10889 0.25479 0.31578 0.15718 0.01394 0.11063 0.25633 0.31772
133.00000 0.15571 0.01376 0.10723 0.25330 0.31391 0.15652 0.01409 0.10889 0.25479 0.31578
134.00000 0.15539 0.01392 0.10565 0.25187 0.31210 0.15571 0.01376 0.10723 0.25330 0.31391
135.00000 0.15477 0.01364 0.10415 0.25049 0.31036 0.15539 0.01392 0.10565 0.25187 0.31210
136.00000 0.15391 0.01371 0.10271 0.24916 0.30869 0.15477 0.01364 0.10415 0.25049 0.31036
137.00000 0.15359 0.01349 0.10133 0.24788 0.30707 0.15391 0.01371 0.10271 0.24916 0.30869
138.00000 0.15253 0.01352 0.10000 0.24664 0.30551 0.15359 0.01349 0.10133 0.24788 0.30707
139.00000 0.15232 0.01333 0.09872 0.24543 0.30399 0.15253 0.01352 0.10000 0.24664 0.30551
140.00000 0.15160 0.01339 0.09749 0.24426 0.30253 0.15232 0.01333 0.09872 0.24543 0.30399
141.00000 0.15075 0.01313 0.09631 0.24313 0.30110 0.15160 0.01339 0.09749 0.24426 0.30253
142.00000 0.15039 0.01322 0.09516 0.24203 0.29972 0.15075 0.01313 0.09631 0.24313 0.30110
143.00000 0.14974 0.01301 0.09406 0.24095 0.29837 0.15039 0.01322 0.09516 0.24203 0.29972
144.00000 0.14925 0.01306 0.09300 0.23991 0.29707 0.14974 0.01301 0.09406 0.24095 0.29837
145.00000 0.14910 0.01293 0.09199 0.23891 0.29580 0.14925 0.01306 0.09300 0.23991 0.29707
146.00000 0.14831 0.01293 0.09101 0.23793 0.29457 0.14910 0.01293 0.09199 0.23891 0.29580
147.00000 0.14788 0.01278 0.09007 0.23699 0.29339 0.14831 0.01293 0.09101 0.23793 0.29457
148.00000 0.14769 0.01284 0.08916 0.23607 0.29224 0.14788 0.01278 0.09007 0.23699 0.29339
149.00000 0.14691 0.01266 0.08829 0.23519 0.29112 0.14769 0.01284 0.08916 0.23607 0.29224
150.00000 0.14646 0.01268 0.08743 0.23432 0.29004 0.14691 0.01266 0.08829 0.23519 0.29112
151.00000 0.14627 0.01258 0.08661 0.23348 0.28898 0.14646 0.01268 0.08743 0.23432 0.29004
152.00000 0.14502 0.01249 0.08581 0.23265 0.28795 0.14627 0.01258 0.08661 0.23348 0.28898
153.00000 0.14527 0.01245 0.08502 0.23184 0.28694 0.14502 0.01249 0.08581 0.23265 0.28795
154.00000 0.14421 0.01239 0.08426 0.23105 0.28595 0.14527 0.01245 0.08502 0.23184 0.28694
155.00000 0.14435 0.01233 0.08352 0.23028 0.28498 0.14421 0.01239 0.08426 0.23105 0.28595
156.00000 0.14335 0.01227 0.08280 0.22952 0.28404 0.14435 0.01233 0.08352 0.23028 0.28498
157.00000 0.14345 0.01221 0.08210 0.22879 0.28311 0.14335 0.01227 0.08280 0.22952 0.28404
158.00000 0.14245 0.01216 0.08142 0.22806 0.28221 0.14345 0.01221 0.08210 0.22879 0.28311
159.00000 0.14254 0.01209 0.08076 0.22736 0.28132 0.14245 0.01216 0.08142 0.22806 0.28221
160.00000 0.14154 0.01204 0.08011 0.22666 0.28046 0.14254 0.01209 0.08076 0.22736 0.28132
161.00000 0.14163 0.01198 0.07948 0.22598 0.27961 0.14154 0.01204 0.08011 0.22666 0.28046
162.00000 0.14110 0.01198 0.07887 0.22531 0.27877 0.14163 0.01198 0.07948 0.22598 0.27961
163.00000 0.14043 0.01183 0.07827 0.22466 0.27796 0.14110 0.01198 0.07887 0.22531 0.27877
164.00000 0.14032 0.01188 0.07768 0.22402 0.27716 0.14043 0.01183 0.07827 0.22466 0.27796
165.00000 0.13990 0.01177 0.07711 0.22339 0.27638 0.14032 0.01188 0.07768 0.22402 0.27716
166.00000 0.13964 0.01179 0.07656 0.22278 0.27561 0.13990 0.01177 0.07711 0.22339 0.27638
167.00000 0.13880 0.01163 0.07601 0.22218 0.27486 0.13964 0.01179 0.07656 0.22278 0.27561
168.00000 0.13877 0.01167 0.07547 0.22159 0.27412 0.13880 0.01163 0.07601 0.22218 0.27486
169.00000 0.13829 0.01157 0.07495 0.22100 0.27339 0.13877 0.01167 0.07547 0.22159 0.27412
170.00000 0.13806 0.01158 0.07444 0.22043 0.27268 0.13829 0.01157 0.07495 0.22100 0.27339
171.00000 0.13769 0.01150 0.07395 0.21987 0.27198 0.13806 0.01158 0.07444 0.22043 0.27268
172.00000 0.13688 0.01143 0.07345 0.21932 0.27129 0.13769 0.01150 0.07395 0.21987 0.27198
173.00000 0.13682 0.01139 0.07296 0.21877 0.27061 0.13688 0.01143 0.07345 0.21932 0.27129
174.00000 0.13635 0.01136 0.07249 0.21823 0.26993 0.13682 0.01139 0.07296 0.21877 0.27061
175.00000 0.13609 0.01130 0.07202 0.21769 0.26927 0.13635 0.01136 0.07249 0.21823 0.26993
176.00000 0.13572 0.01128 0.07157 0.21717 0.26862 0.13609 0.01130 0.07202 0.21769 0.26927
177.00000 0.13541 0.01121 0.07112 0.21666 0.26797 0.13572 0.01128 0.07157 0.21717 0.26862
178.00000 0.13506 0.01120 0.07068 0.21615 0.26734 0.13541 0.01121 0.07112 0.21666 0.26797
179.00000 0.13473 0.01113 0.07026 0.21565 0.26672 0.13506 0.01120 0.07068 0.21615 0.26734
180.00000 0.13491 0.01117 0.06985 0.21516 0.26611 0.13473 0.01113 0.07026 0.21565 0.26672
181.00000 0.13429 0.01108 0.06945 0.21469 0.26552 0.13491 0.01117 0.06985 0.21516 0.26611
182.00000 0.13410 0.01107 0.06906 0.21423 0.26494 0.13429 0.01108 0.06945 0.21469 0.26552
183.00000 0.13367 0.01101 0.06867 0.21378 0.26438 0.13410 0.01107 0.06906 0.21423 0.26494
184.00000 0.13337 0.01098 0.06829 0.21333 0.26382 0.13367 0.01101 0.06867 0.21378 0.26438
185.00000 0.13300 0.01092 0.06791 0.21288 0.26327 0.13337 0.01098 0.06829 0.21333 0.26382
186.00000 0.13267 0.01089 0.06754 0.21244 0.26272 0.13300 0.01092 0.06791 0.21288 0.26327
187.00000 0.13231 0.01084 0.06717 0.21201 0.26218 0.13267 0.01089 0.06754 0.21244 0.26272
188.00000 0.13196 0.01080 0.06680 0.21157 0.26164 0.13231 0.01084 0.06717 0.21201 0.26218
189.00000 0.13215 0.01082 0.06646 0.21114 0.26111 0.13196 0.01080 0.06680 0.21157 0.26164
190.00000 0.13151 0.01075 0.06612 0.21073 0.26058 0.13215 0.01082 0.06646 0.21114 0.26111
191.00000 0.13130 0.01071 0.06577 0.21032 0.26007 0.13151 0.01075 0.06612 0.21073 0.26058
192.00000 0.13087 0.01067 0.06543 0.20991 0.25957 0.13130 0.01071 0.06577 0.21032 0.26007
193.00000 0.13055 0.01062 0.06510 0.20951 0.25907 0.13087 0.01067 0.06543 0.20991 0.25957
194.00000 0.13072 0.01065 0.06477 0.20911 0.25857 0.13055 0.01062 0.06510 0.20951 0.25907
195.00000 0.13008 0.01057 0.06446 0.20872 0.25808 0.13072 0.01065 0.06477 0.20911 0.25857
196.00000 0.12986 0.01054 0.06414 0.20833 0.25760 0.13008 0.01057 0.06446 0.20872 0.25808
197.00000 0.12942 0.01049 0.06382 0.20795 0.25712 0.12986 0.01054 0.06414 0.20833 0.25760
198.00000 0.12965 0.01052 0.06352 0.20756 0.25665 0.12942 0.01049 0.06382 0.20795 0.25712
199.00000 0.12897 0.01044 0.06322 0.20719 0.25619 0.12965 0.01052 0.06352 0.20756 0.25665
200.00000 0.12876 0.01041 0.06292 0.20682 0.25573 0.12897 0.01044 0.06322 0.20719 0.25619
201.00000 0.12887 0.01042 0.06263 0.20646 0.25527 0.12876 0.01041 0.06292 0.20682 0.25573
202.00000 0.12825 0.01035 0.06235 0.20610 0.25483 0.12887 0.01042 0.06263 0.20646 0.25527
203.00000 0.12801 0.01032 0.06206 0.20574 0.25438 0.12825 0.01035 0.06235 0.20610 0.25483
204.00000 0.12813 0.01033 0.06178 0.20539 0.25395 0.12801 0.01032 0.06206 0.20574 0.25438
205.00000 0.12749 0.01026 0.06151 0.20505 0.25352 0.12813 0.01033 0.06178 0.20539 0.25395
206.00000 0.12725 0.01022 0.06123 0.20470 0.25309 0.12749 0.01026 0.06151 0.20505 0.25352
207.00000 0.12738 0.01024 0.06096 0.20436 0.25267 0.12725 0.01022 0.06123 0.20470 0.25309
208.00000 0.12674 0.01016 0.06070 0.20403 0.25225 0.12738 0.01024 0.06096 0.20436 0.25267
209.00000 0.12707 0.01020 0.06045 0.20370 0.25184 0.12674 0.01016 0.06070 0.20403 0.25225
210.00000 0.12632 0.01012 0.06019 0.20338 0.25144 0.12707 0.01020 0.06045 0.20370 0.25184
211.00000 0.12612 0.01008 0.05993 0.20305 0.25104 0.12632 0.01012 0.06019 0.20338 0.25144
212.00000 0.12622 0.01010 0.05969 0.20273 0.25064 0.12612 0.01008 0.05993 0.20305 0.25104
213.00000 0.12558 0.01002 0.05944 0.20241 0.25024 0.12622 0.01010 0.05969 0.20273 0.25064
214.00000 0.12591 0.01006 0.05920 0.20210 0.24986 0.12558 0.01002 0.05944 0.20241 0.25024
215.00000 0.12515 0.00997 0.05896 0.20179 0.24947 0.12591 0.01006 0.05920 0.20210 0.24986
216.00000 0.12495 0.00994 0.05872 0.20149 0.24909 0.12515 0.00997 0.05896 0.20179 0.24947
217.00000 0.12505 0.00996 0.05848 0.20118 0.24871 0.12495 0.00994 0.05872 0.20149 0.24909
218.00000 0.12440 0.00988 0.05825 0.20087 0.24833 0.12505 0.00996 0.05848 0.20118 0.24871
219.00000 0.12474 0.00992 0.05802 0.20057 0.24796 0.12440 0.00988 0.05825 0.20087 0.24833
220.00000 0.12397 0.00983 0.05779 0.20028 0.24759 0.12474 0.00992 0.05802 0.20057 0.24796
221.00000 0.12436 0.00987 0.05757 0.19998 0.24722 0.12397 0.00983 0.05779 0.20028 0.24759
222.00000 0.12356 0.00979 0.05735 0.19969 0.24686 0.12436 0.00987 0.05757 0.19998 0.24722
223.00000 -0.12426 -0.00823 0.05535 0.19977 0.24671 0.12356 0.00979 0.05735 0.19969 0.24686
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 4.987888e-023.
> In Mar1 at 388
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 1.093629e-022.
> In Mar1 at 388
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 4.435033e-019.
> In Mar1 at 388
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 6.377615e-019.
> In Mar1 at 388
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 2.854821e-029.
> In Mar1 at 388
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 2.651359e-076.
> In Mar1 at 388
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 7.026973e-118.
> In Mar1 at 388
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 5.895068e-306.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
224.00000 NaN NaN NaN NaN NaN -0.12426 -0.00823 0.05535 0.19977 0.24671
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
225.00000 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
Warning: Matrix is singular, close to singular or badly scaled.
Results may be inaccurate. RCOND = NaN.
> In Mar1 at 388
226.00000 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
>>
Walter Roberson
Walter Roberson on 2 Mar 2011
You can get better accuracy in your solutions by replacing
inv(J1)*F
with
J1\F
However, even with that change, you will start getting warnings of singularity at exactly the same place.
Is your data really only available to 3 decimal places, or is that an artifact of the way you displayed it after you read it in from some file? You may wish to switch to "format long g" instead of "format long" and re-examine the rainfall data you obtained.
Unfortunately, with those equations, I have no "feel" for what is being calculated, and so am not able to advise much on what is going on.
In your first line of your construction of F1, you have replaced one occurrence of i_n(kk,1) with i_n_kk, but you missed the second replacement on that line.
Also, your logic for i_n_kk is off a bit. You should move
i_n_kk = 0
down one line to inside the "for kk" loop. You would then have the "if" statement and the assignment to i_n_kk for the situation where kk is in range. Your "end" statement for that "if" should go after that assignment rather than being where it is at the second last line of your code.
It appears that your code is falling over the _second_ time i_n(kk) is 0, which suggests that the first time it is 0, the values you are calculating go wacko, and then are just too far out of range to calculate by the second time around. I suggest you examine more carefully your mathematics for the case where i_n first becomes 0.

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