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Atom on 10 Jun 2024 at 12:54
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Commented: Star Strider on 10 Jun 2024 at 16:42
Accepted Answer: Star Strider
Open in MATLAB Online
I have two equations F1(u,v)=0,F2(u,v)=0 with 4 parameters a,b,c,d, 3 of which (a,c,d) are given. For the range b=0.1:0.1:9, I have to collect all positive real solutions of the system F1(u,v)=0,F2(u,v)=0. How to then plot each solutions obtained for b=0.1:0.1:9
How to do that. Note that the solutions (u,v) should be positive.
a = 1.5; c = 0.4; d = 0.9;
b=0.1:0.1:9;
syms u v
F1= u*(1 - u) - b*u*v/(c*u + 1);
F2= 0.5*b*u*v/(c*u + 1) + u*v/(u + 1) - d*v^2;
E=[F1==0,F2==0];
solve(E,u,v)
plot(u,v)
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Accepted Answer
Star Strider on 10 Jun 2024 at 13:21
Open in MATLAB Online
a = 1.5; c = 0.4; d = 0.9;
b=0.1:0.1:9;
syms u v
F1 = @(u,v,b) u*(1 - u) - b*u*v/(c*u + 1);
F2 = @(u,v,b) 0.5*b*u*v/(c*u + 1) + u*v/(u + 1) - d*v^2;
% E=[F1==0,F2==0];
for k = 1:numel(b)
[uv(:,k),fv(k)] = fmincon(@(p)norm([F1(p(1),p(2),b(k)); F2(p(1),p(2),b(k))]), rand(2,1), [],[],[],[], zeros(2,1));
end
Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. 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format shortG
uv
uv = 2x90
1.9593e-06 0.91147 0.86145 0.8088 0.75463 0.70008 0.6463 0.59434 0.54505 0.49901 0.45659 0.4179 0.38287 0.35132 0.323 0.29762 0.27487 0.25447 0.23614 0.21965 0.20479 0.19135 0.17917 0.16811 0.99999 0.14884 0.14042 0.1327 0.1256 0.11906 0.0012415 0.60404 0.62098 0.63263 0.63888 0.63985 0.63591 0.62762 0.61571 0.60098 0.58423 0.56617 0.54742 0.52845 0.50964 0.49125 0.47345 0.45634 0.44001 0.42445 0.40969 0.3957 0.38246 0.36993 5.7414e-06 0.34686 0.33624 0.32619 0.31666 0.30763
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
fv
fv = 1x90
1.0e+00 * 2.399e-06 2.6284e-09 1.6137e-09 2.9782e-09 5.5332e-09 5.0974e-09 1.6383e-09 1.7269e-09 5.4861e-09 3.6968e-09 2.5587e-09 2.2643e-09 1.7091e-09 2.8115e-09 2.2959e-09 4.0353e-09 5.9438e-09 6.0736e-09 3.0128e-09 3.4053e-09 5.7217e-09 8.3134e-10 3.362e-09 5.8099e-09 7.9969e-06 4.2445e-09 2.3542e-09 6.0049e-09 3.598e-09 1.9788e-09
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
figure
plot(b, uv(1,:), 'DisplayName','u(b)')
hold on
plot(b,uv(2,:), 'DisplayName','v(b)')
hold off
grid
xlabel('b')
ylabel('Parameter Value')
legend('Location','best')
If you want to elimiinate the ‘spike’ discontinuities, use the ismissing function to detect the ‘outlier’ values (that appear to be either 0 or 1 in the ‘uv’ matrix, other functions may also be useful), set them to NaN, and then use the fillmissing function to interpolate them. I leave that to you.
.
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Atom on 10 Jun 2024 at 15:42
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https://matlabcentral.mathworks.com/matlabcentral/answers/2127071-f1-u-v-0-f2-u-v-0-with-4-parameters-a-b-c-d-3-of-which-a-c-d-are-given#comment_3183331
Thank you. I would like to eliminate the 'spike' discontinuities. Could you please provide more detailed suggestions? I would appreciate it if you could provide the necessary modifications to the code.
Star Strider on 10 Jun 2024 at 16:31
Direct link to this comment
https://matlabcentral.mathworks.com/matlabcentral/answers/2127071-f1-u-v-0-f2-u-v-0-with-4-parameters-a-b-c-d-3-of-which-a-c-d-are-given#comment_3183346
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Open in MATLAB Online
As always, my pleasure!
Sure!
This generally works, although not as well as I would like it to. The reason is that the outliers change from run to run, and the isoutlier calls do no always work as I would like them to. (The ismissing approach failed because the outlier spikes are not always single isolated values.)
Try this —
a = 1.5; c = 0.4; d = 0.9;
b=0.1:0.1:9;
syms u v
F1 = @(u,v,b) u*(1 - u) - b*u*v/(c*u + 1);
F2 = @(u,v,b) 0.5*b*u*v/(c*u + 1) + u*v/(u + 1) - d*v^2;
% E=[F1==0,F2==0];
for k = 1:numel(b)
[uv(:,k),fv(k)] = fmincon(@(p)norm([F1(p(1),p(2),b(k)); F2(p(1),p(2),b(k))]), rand(2,1), [],[],[],[], zeros(2,1));
end
Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. 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Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. 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Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.Local minimum possible. Constraints satisfied.fmincon stopped because the size of the current step is less thanthe value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
format shortG
uv
uv = 2x90
9.8128e-06 1.9595e-06 0.86145 0.8088 0.75463 0.70008 0.6463 0.59434 0.54505 0.49901 0.45659 0.4179 0.38287 0.35132 1 0.29762 9.8747e-06 0.25447 0.23614 0.21965 0.20479 0.19135 1 0.16811 0.15804 0.14884 0.99999 0.1327 0.1256 0.11906 0.0027794 0.0012416 0.62098 0.63263 0.63888 0.63985 0.63591 0.62762 0.61571 0.60098 0.58423 0.56617 0.54742 0.52845 1.5443e-06 0.49125 0.0027873 0.45634 0.44001 0.42445 0.40969 0.3957 9.6446e-07 0.36993 0.35808 0.34686 4.3672e-06 0.32619 0.31666 0.30763
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
fv
fv = 1x90
1.0e+00 * 1.2007e-05 2.399e-06 5.3921e-09 5.1788e-09 3.3437e-09 4.1734e-09 4.9515e-09 3.9618e-10 2.296e-09 4.9309e-09 4.2973e-09 2.1425e-09 7.2797e-09 5.6279e-09 1.5995e-06 5.0248e-09 1.2032e-05 6.9628e-09 1.9764e-09 3.4825e-09 1.538e-09 3.1273e-09 1.2745e-06 5.1597e-09 7.7891e-10 4.6436e-09 6.3948e-06 1.1631e-09 4.1181e-09 5.8897e-10
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
figure
plot(b, uv(1,:), 'DisplayName','u(b)')
hold on
plot(b,uv(2,:), 'DisplayName','v(b)')
hold off
grid
xlabel('b')
ylabel('Parameter Value')
legend('Location','best')
Lvu = isoutlier(uv(1,:), 'movmedian', 7);
Nru = nnz(Lvu)
Nru =
13
uv(1,Lvu) = NaN;
Lvv = isoutlier(uv(2,:), 'movmedian', 7);
Nrv = nnz(Lvv)
Nrv =
13
uv(2,Lvv) = NaN;
uv(1,:) = fillmissing(uv(1,:), 'linear');
uv(2,:) = fillmissing(uv(2,:), 'linear');
uv = [uv(1,:); uv(2,:)]
uv = 2x90
9.8128e-06 0.43073 0.86145 0.8088 0.75463 0.70008 0.6463 0.59434 0.54505 0.49901 0.45659 0.4179 0.38287 0.35132 0.32447 0.29762 0.27604 0.25447 0.23614 0.21965 0.20479 0.19135 0.17973 0.16811 0.15804 0.14884 0.14077 0.1327 0.1256 0.11906 0.0027794 0.31188 0.62098 0.63263 0.63888 0.63985 0.63591 0.62762 0.61571 0.60098 0.58423 0.56617 0.54742 0.52845 0.50985 0.49125 0.4738 0.45634 0.44001 0.42445 0.40969 0.3957 0.38282 0.36993 0.35808 0.34686 0.33653 0.32619 0.31666 0.30763
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
save('uvb.mat', 'uv','b')
whos('-file', 'uvb')
Name Size Bytes Class Attributes b 1x90 720 double uv 2x90 1440 double
which('uvb.mat')
/users/mss.system.z28zz/uvb.mat
figure
plot(b, uv(1,:), 'DisplayName','u(b)')
hold on
plot(b,uv(2,:), 'DisplayName','v(b)')
hold off
grid
xlabel('b')
ylabel('Parameter Value')
legend('Location','best')
It may be necesssary to run it several times to get good data. Those values.will have been saved in ‘uvb.mat’, so you can use them in other applications.
.
Atom on 10 Jun 2024 at 16:38
Direct link to this comment
https://matlabcentral.mathworks.com/matlabcentral/answers/2127071-f1-u-v-0-f2-u-v-0-with-4-parameters-a-b-c-d-3-of-which-a-c-d-are-given#comment_3183351
This is amazing... Thank you greatly.
Star Strider on 10 Jun 2024 at 16:42
Direct link to this comment
https://matlabcentral.mathworks.com/matlabcentral/answers/2127071-f1-u-v-0-f2-u-v-0-with-4-parameters-a-b-c-d-3-of-which-a-c-d-are-given#comment_3183356
As always, my pleasure!
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