Solving a nonlinear equation with numerical integration

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Michael Bond 2017-11-30

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https://ww2.mathworks.cn/matlabcentral/answers/370203-solving-a-nonlinear-equation-with-numerical-integration

评论： Michael Bond 2017-12-1

Hello everybody,

I am pretty new in Matlab and struggling at the moment. I want to calculate two unknown parameters (x1 and x2) from two equations with integrals inside.

The Values P1, P2, S1, S2, C1, C2 are known and have the datatype double.

B(lambda) and e(lambda) are datasets (x,y) stored in arrays that describe the filter curves (also double).

Many thanks for your help.

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Answer 1

Torsten 2017-11-30

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https://ww2.mathworks.cn/matlabcentral/answers/370203-solving-a-nonlinear-equation-with-numerical-integration#answer_293953

编辑：Torsten 2017-11-30

在 MATLAB Online 中打开

First method:

1. Fit functions B1(lambda), B2(lambda), e1(lambda) and e2(lambda) to your data in the different ranges (500 to 550 and 600 to 650).

2. Call fsolve as

fun = @(x)[P1-S1*integral(@(lambda)B1(lambda).*e1(lambda).*(1-exp(-x(1)./lambda.^1.39)).*c1./(lambda.^5.*(exp(c2./(lambda*x(2)))-1)),500,550),P2-S2*integral(@(lambda)B2(lambda).*e2(lambda).*(1-exp(-x(1)./lambda.^1.39)).*c1./(lambda.^5.*(exp(c2./(lambda*x(2)))-1)),600,650)];
x0 = [1, 1];
xsol = fsolve(fun,x0)

Second method:

function main
S1 = ...;
S2 = ...;
P1 = ...;
P2 = ...;
l = ...;
e = ...;
B = ...;
x0 = [1, 1];
xsol = fsolve(@(x)fun(x,S1,S2,P1,P2,l,e,B),x0);
function res = fun(x,S1,S2,P1,P2,l,e,B)
interpe = @(lambda)interp1(l,e,lambda);
interpB = @(lambda)interp1(l,B,lambda);
funI = @(lambda)interpe(lambda).*interpB(lambda).*(1-exp(-x(1)./lambda.^1.39)).*c1./(lambda.^5.*(exp(c2./(lambda*x(2)))-1));
res = [P1-S1*integral(funI,500,550),P2-S2*integral(funI,600,650)];

Best wishes

Torsten.

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Michael Bond 2017-12-1

编辑：Michael Bond 2017-12-1

在 MATLAB Online 中打开

Hello Torsten, Update: I've managed to get a licence for the optimization toolbox, so now, I can use fsolve and I think its the right solver for my problem.

Here the important part of my code:

P1=100;
P2=150; 
S1=1;
S2=2;
fun = @(x)[P1-S1*integral(@(B1_x)B1(B1_x).*E(E_x).*(1-exp(-x(1)./B1_x.^1.39)).*c1./(B1_x.^5.*(exp(c2./(B1_x*x(2)))-1)),500,550),P2-S2*integral(@(B2_x)B2(B2_x).*E(E_x).*(1-exp(-x(1)./B2_x.^1.39)).*c1./(B2_x.^5.*(exp(c2./(B2_x*x(2)))-1)),600,650)];
x0 = [1, 1];
xsol = fsolve(fun,x0);

B1, B2 and E are 1751x2 double arrays, B1_x, B2_x and E_x are 1751x1 double arrays (x-Values), B1_y, B2_y, E_y have the same size.

After trying to compile I'll get another Errors:

Subscript indices must either be real positive integers or logicals.
Error in solve_equation>@(B1_x)B1(B1_x).*E(E_x).*(1-exp(-x(1)./B1_x.^1.39)).*c1./(B1_x.^5.*(exp(c2./(B1_x*x(2)))-1))
Error in integralCalc/iterateScalarValued (line 314)
                fx = FUN(t);
Error in integralCalc/vadapt (line 132)
            [q,errbnd] = iterateScalarValued(u,tinterval,pathlen);
Error in integralCalc (line 75)
        [q,errbnd] = vadapt(@AtoBInvTransform,interval);
Error in integral (line 88)
Q = integralCalc(fun,a,b,opstruct);
Error in
solve_equation>@(x)[P1-S1*integral(@(B1_x)B1(B1_x).*E(E_x).*(1-exp(-x(1)./B1_x.^1.39)).*c1./(B1_x.^5.*(exp(c2./(B1_x*x(2)))-1)),400,750),P2-S2*integral(@(B2_x)B2(B2_x).*E(E_x).*(1-exp(-x(1)./B2_x.^1.39)).*c1./(B2_x.^5.*(exp(c2./(B2_x*x(2)))-1)),400,750)]
Error in fsolve (line 230)
            fuser = feval(funfcn{3},x,varargin{:});
Error in solve_equation (line 84)
xsol = fsolve(fun,x0);
Caused by:
    Failure in initial objective function evaluation. FSOLVE cannot continue.

Michael Bond 2017-12-1

在 MATLAB Online 中打开

There is still no error and no answer. I'll try to explain my variables clearlier:

E, B1, B2 are 1751x2 double arrays (x,y Data); with x =400:0.2:750. I have extracted the x and y values to new one dimensional arrays (1751x1 double), they are named as B1_x, B1_y, B2_x, B2_y, E_x, E_y. In principle it doesn't matter which x-Variable will be assigned to "e" because they have all the same values. Also the integration limits can be the same (400-750).

 function main
  S1 = 1.2;
  S2 = 1.7;
  P1 = 100;
  P2 = 150;
  l = B1_x;
  e = E_y;
  B1 = B1_y;
  B2 = B2_y;
x0 = [1, 1];
xsol = fsolve(@(x)fun(x,S1,S2,P1,P2,l,e,B1,B2),x0)
end 
function res = fun(x,S1,S2,P1,P2,l,e,B1,B2);
interpe = @(lambda)interp1(l,e,lambda);
interpB1 = @(lambda)interp1(l,B1,lambda);
interpB2 = @(lambda)interp1(l,B2,lambda);
funI = @(lambda)interpe(lambda).*interpB1(lambda).*(1-exp(-x(1)./lambda.^1.39)).*c1./(lambda.^5.*(exp(c2./(lambda*x(2)))-1));
funII = @(lambda)interpe(lambda).*interpB2(lambda).*(1-exp(-x(1)./lambda.^1.39)).*c1./(lambda.^5.*(exp(c2./(lambda*x(2)))-1));
res = [P1-S1*integral(funI,400,750),P2-S2*integral(funII,400,750)];
end

Torsten 2017-12-1

编辑：Torsten 2017-12-1

在 MATLAB Online 中打开

Remove the two "end" statements in your code.

Remove the semicolon after

function res = fun(x,S1,S2,P1,P2,l,e,B1,B2);

Best wishes

Torsten.

Michael Bond 2017-12-1

I've did it, it says "All functions in a script must be closed with an 'end'."

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Solving a nonlinear equation with numerical integration

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