using lsqnonlin with multiple functions

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joshua payne
joshua payne on 26 Apr 2023
Answered: Alex Sha on 4 Aug 2023
im trying to optimize parameters,x, using 3 functions, fun1, fun2, fun3.
the optimized parameters must be the best fit for all 3 functions, not individually.
% ogden
clc
clear all
D=readmatrix('Treloar_data.xlsx');
stretch=D(1:end,1); %this is lambda
lambda=stretch;
stress=D(1:end,2); %this is stress
T_0=stress;
lambda_EBT=D(1:17,5);
T_0_EBT=D(1:17,6);
lambda_shear=D(1:14,3);
T_0_shear=D(1:14,4);
fun1=@(x)((x(1)*(lambda.^(x(4)-1) -lambda.^(-.5*x(4)-1))+x(2)*(lambda.^(x(5)-1) -lambda.^(-.5*x(5)-1))+x(3)*(lambda.^(x(6)-1) -lambda.^(-.5*x(6)-1)))-T_0);
fun2=@(x)((x(1)*(lambda_EBT.^(x(4)-1) -lambda_EBT.^((-7/2)*x(4)-(5/2)))+x(2)*(lambda_EBT.^(x(5)-1) -lambda_EBT.^((-7/2)*x(5)-(5/2)))+x(3)*(lambda_EBT.^(x(6)-1) -lambda_EBT.^((-7/2)*x(6)-(5/2))))-T_0_EBT);
fun3=@(x)((x(1)*(lambda_shear.^(x(4)-1) -lambda_shear.^((-5/2)*x(4)-(3/2)))+x(2)*(lambda_shear.^(x(5)-1) -lambda_shear.^((-5/2)*x(5)-(3/2)))+x(3)*(lambda_shear.^(x(6)-1) -lambda_shear.^((-5/2)*x(6)-(3/2))))-T_0_shear);
fun=[fun1; fun2; fun3]
% x0=[1, 1, 1, 1, 1, 1];%initial geuss is sensitive
x0=[.5649 ,(3.856*10^(-3)) 0 ,1.297, 4.342, 15.13];
x=lsqnonlin(fun,x0);
c(1)=x(1) %mu1
c(2)=x(2) %mu2
c(3)=x(3) %mu3
c(4)=x(4) %alpha1
c(5)=x(5) %alpha2
c(6)=x(6) %alpha3
T_0_ogden=(c(1)*(lambda.^(c(4)-1) -lambda.^(-.5*c(4)-1))+c(2)*(lambda.^(c(5)-1) -lambda.^(-.5*c(5)-1))+c(3)*(lambda.^(c(6)-1) -lambda.^(-.5*c(6)-1)));
T_0_ogden_EBT=(x(1)*(lambda_EBT.^(x(4)-1) -lambda_EBT.^((-7/2)*x(4)-(5/2)))+x(2)*(lambda_EBT.^(x(5)-1) -lambda_EBT.^((-7/2)*x(5)-(5/2)))+x(3)*(lambda_EBT.^(x(6)-1) -lambda_EBT.^((-7/2)*x(6)-(5/2))));
T_0_ogden_shear=(x(1)*(lambda_shear.^(x(4)-1) -lambda_shear.^((-5/2)*x(4)-(3/2)))+x(2)*(lambda_shear.^(x(5)-1) -lambda_shear.^((-5/2)*x(5)-(3/2)))+x(3)*(lambda_shear.^(x(6)-1) -lambda_shear.^((-5/2)*x(6)-(3/2))));
figure
plot(lambda,T_0,'r')
hold on
plot(lambda,T_0_ogden,'b')
plot(lambda_EBT,T_0_ogden_EBT,'g')
plot(lambda_shear,T_0_ogden_shear,'c')

Answers (2)

Torsten
Torsten on 26 Apr 2023
Moved: Torsten on 26 Apr 2023
So you want
sum(((x(1)*(lambda.^(x(4)-1) -lambda.^(-.5*x(4)-1))+x(2)*(lambda.^(x(5)-1) -lambda.^(-.5*x(5)-1))+x(3)*(lambda.^(x(6)-1) -lambda.^(-.5*x(6)-1)))-T_0).^2) +
sum((((x(1)*(lambda_EBT.^(x(4)-1) -lambda_EBT.^((-7/2)*x(4)-(5/2)))+x(2)*(lambda_EBT.^(x(5)-1) -lambda_EBT.^((-7/2)*x(5)-(5/2)))+x(3)*(lambda_EBT.^(x(6)-1) -lambda_EBT.^((-7/2)*x(6)-(5/2))))-T_0_EBT).^2) +
sum(((x(1)*(lambda_shear.^(x(4)-1) -lambda_shear.^((-5/2)*x(4)-(3/2)))+x(2)*(lambda_shear.^(x(5)-1) -lambda_shear.^((-5/2)*x(5)-(3/2)))+x(3)*(lambda_shear.^(x(6)-1) -lambda_shear.^((-5/2)*x(6)-(3/2))))-T_0_shear).^2)
to be minimzed with respect to x ?
Then define
fun = @(x)[fun1(x);fun2(x);fun3(x)];
  2 Comments
joshua payne
joshua payne on 26 Apr 2023
im confused why the code has sum's around each function and then youre still calling fun1, fun2, fun3
Torsten
Torsten on 26 Apr 2023
Edited: Torsten on 26 Apr 2023
The long expression with the sums over the arrays squared would not be part of the code.
I asked the question if this sum expression is the function you want to minimize with respect to x.
If this is the case, use
fun = @(x)[fun1(x);fun2(x);fun3(x)];

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Alex Sha
Alex Sha on 4 Aug 2023
@joshua payne refer to the results below
Sum Squared Error (SSE): 0.377485784540046
Root of Mean Square Error (RMSE): 0.0821024821858161
Correlation Coef. (R): 0.996098098188982
R-Square: 0.992211421215707
Parameter Best Estimate
--------- -------------
c1 0.0323483986102822
c2 -1.23856186333989
c3 1.19422161081346E-14
c4 3.36896942510881
c5 -0.571016034599837
c6 17.1390641835716
T_0_ogde:
T_0_ogde
T_0_ogden
T_0_ogden_shear

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