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## Is the following algorithm to solve non-linear equations in complex variable consistent?

Asked by Michele

### Michele (view profile)

on 11 Feb 2013
Accepted Answer by Matt J

### Matt J (view profile)

Dear users,

In order to solve the following equation((Z1/(ro_l*sqrt(G)))*((1-R).^2)* sqrt(1+0.25*(((ro_l*sqrt(G))/Z1)-(Z1/(ro_l*sqrt(G)))).^2*(sin(k*L)).^2) - 1=0) with respect to the complex variable G I have implemented this simple algorithm: function solveeqs()

guess=[1+i];

options = optimset('MaxFunEvals',1000000,'MaxIter',100000);

[result,fval,exit,output]=fsolve(@eqns,guess,options);

result

fval

eqns(guess)

output

end

function fcns=eqns(z)

G=z(1);

ro_s=2700; %insert density of the solid in kg/m3

cs= 3000;

Z1= ro_s*cs;

R= 0.995+i;

ro_l=1000;

om=10*10^6;

k=om/cs;

L=100*10^-6;

Z2=ro_l*sqrt(G);

fcns(1)= (Z1/Z2)*((1-R).^2) sqrt(1+0.25((Z2/Z1)-(Z1/Z2)).^2*(sin(k*L)).^2) - 1;

end

the result is the following:

Optimizer appears to be converging to a minimum that is not a root: Sum of squares of the function values is > sqrt(options.TolFun). Try again with a new starting point.

result =

`   2.7429e+004 -3.8961e+003i`

fval =

```-0.0182 - 0.0192i %result at last iteration
```

ans =

` -5.2902e+003 +2.1171e+003i % first iteration result with guess value`

output =

`       iterations: 78`
`        funcCount: 158`
`        algorithm: 'trust-region dogleg'`
`    firstorderopt: 8.8405e-007`
```          message: [1x169 char]
The message error rises because the solution obtained at the last iteration (the 78th) is of the order of magnitude 1 while this algorithm is consistent with a solution of the order of 10^-6. The doubt I have is: do you think this algorithm is consistent? working with complex number in fact I fear that matlab is not able to find a solution closer to the 0.
Is there a better way to threat this problem? (any other function or algorithm to threat with complex variable would be really usefull).```

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## 1 Answer

Answer by Matt J

on 12 Feb 2013
Edited by Matt J

### Matt J (view profile)

on 12 Feb 2013
Accepted answer

I don't think FSOLVE supports complex-valued functions/variables and am amazed it ran without errors. This recent thread looks closely related

http://www.mathworks.com/matlabcentral/answers/62875-solve-nonlinear-complex-equations

and suggests an alternative.

Matt J

### Matt J (view profile)

on 12 Feb 2013

The second line was

`    c=@(x) complex(x(1),x(2));`

This just creates a single-argument alias for the COMPLEX command. c([a,b]) converts the vector [a,b] to the complex number a+b*i

For a system of equations, I would modify as follows

```g =@(z) [real(z);imag(z)];
```
```sol=c( fsolve( @(x) g(c(f(x))), x0)  )
```
kaust

### kaust (view profile)

on 20 Mar 2013

Dear Matt,

I think there is small problem with the last line of the code. "sol=c( fsolve( @(x) g(c(f(x))), x0) )"

It should rather be sol=fsolve( @(x) g(c(f(x))), x0) can you please check to confirm.

Thanks

Matt J

### Matt J (view profile)

on 20 Mar 2013

kaust,

If you don't feed the output of fsolve to c(), then your final result will not be a complex variable.

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