# How can I solve a system in complex variables ?

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Hi everyone,

I am founding lot of problem in solving a system in the form A*a - B*b=0 where A and B are matrices and a and b vector columns, where inside A and B I have got my complex unknowns; the number of unknowns match the number of equations, but the problem is that the unknown are inside the matrices and I am not able to simplify the problem. To call the complex unknown I have used the sym function, and to solve the problem respect to my functions I have used the comand "solve()". I report my system below: as you can see the only unknowns are G and N1, while s B C g are terms that depends upon these unknowns. Once defined these terms I have defined my matrices and vectors (the vectors are compose by known values).

for x1=(0:0.1:1);

for x2=(1.1:0.1:2);

G = sym('G');

N1 = sym('N1');

beta_s= (mi)/ro_s;

beta_l= (-i*om*G)/ro_l;

s_s= N1/beta_s;

s_l= N1/beta_l;

B_s= sqrt(1-beta_s^2*s_s^2);

B_l= sqrt(1-beta_s^2*s_s^2);

C_s= 1-2*beta_s^2*s_s^2;

C_l= 1-2*beta_l^2*s_l^2;

g_s= exp(i*om*B_s*beta_s^-1*x1);

g_l= exp(i*om*B_s*beta_s^-1*x2);

A1=[0

i*om*ro_s*s_s*beta_s^-1*B_s*g_s

0 -i*om*ro_s*s_s*beta_s^-1*B_s*g_s;

0

i*om*ro_s*s_s*beta_s*C_s*g_s^-1

0

-i*om*ro_s*beta_s*C_s*g_s;

0

-B_s*g_s^-1

0

B_s*g_s;...

0

-beta_s*s_s*g_s^-1

0

-beta_s*s_s*g_s];

A2=[0

i*om*ro_l*s_l*beta_l^-1*B_l*g_l

0

-i*om*ro_l*s_l*beta_l^-1*B_l*g_l;

0

i*om*ro_l*s_l*beta_l*C_l*g_l^-1

0

-i*om*ro_l*beta_l*C_l*g_l;

0

-B_l*g_l^-1 0 B_l*g_s;...

0

-beta_l*s_l*g_l^-1

0

-beta_s*s_l*g_l];

V_1=[0; Rs1;0; Ts1];

V_2=[0;0;0;Ts2];

S= solve(A1*V_1-A2*V_2);

end

end

In the end I come out with this error

??? Error using ==> maple at 129

at offset 480, `;` unexpected

Error in ==> sym.findsym at 33

v = maple('indets', sc ,'symbol');

Error in ==> solve at 99

vars = ['[' findsym(sym(eqns),neqns) ']'];

Error in ==> sym.solve at 49

[varargout{1:max(1,nargout)}] = solve(S{:});

Error in ==> xxx at 39

S=solve(A1*V_1-A2*V_2);

So my question is: is there a way to solve this kind of problem in Matlab? Can I obtain an explicit expression for my variable G for a certain value of x in the form G(x)= Re+Im?

Thank you very much for the help

##### 3 Comments

Walter Roberson
on 7 Feb 2013

### Accepted Answer

Miroslav Balda
on 12 Feb 2013

The structure of the MATLAB code without an application of the Symbolic Toolbox should be as follows:

1. All constant parameters not depending on unknown ones should be defined outside the res function, otherwise they would be assigned in each call of res, what would make computations slower.

2. The function LMFnlsq works with real parameters only. It means that also column vector p0 of the initial guess should be real, composed out of components of complex numbers.

3. The code should be kept clear and all visiblewithout hidden parts.

The following code has the right structure:

% MAIN.m

% ~~~~~~

% 12.02.2013 Michele How can I solve asystem of complex variables

%

global p0 ro_s E ni mi ro_l om Ts1 Rs1 Ts2 x1 x2 beta_s V_1 V_2

%

ro_s = 2000;% insert density of the solid in kg/m3

E = 20*10^9;% stiffness module for the solid (= 2e10)

ni = 0.3;

mi = E/(2*(1+ni)); % for the solid

ro_l = 1000; % insert density of the liquid in kg/m3

om = 10*10^6; % insert frequency in Hz (=10e6)

Ts1 = 1; % amplitude of the wave transmitted in medium 1

Rs1 = 0.8 + i*0.1;

Ts2 = 1- Rs1;

V_1 = [0; Rs1;0; Ts1];

V_2 = [0;0;0;Ts2];

x1 = 5*10^-3;

x2 = x1;

beta_s = (mi)/ro_s;

%

%p0 = [10^6;10^6; 1; 1]; % first guess (it gives NaNs)

p0 = [10^1;10^1; 1; 1]; % first guess

%

[p,ssq,cnt] = LMFnlsq(@res, ones(size(p0)), 'Display',-5);

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

% The solution:

p = p.*p0;

G = complex(p(1),p(3));

N1 = complex(p(2),p(4));

%

fprintf('\n Sum of squares = %10.2e\n Function evaluations = %g\n',...

ssq,cnt);

fprintf(' G = %12.4e + %12.4ei\n N1 = %12.4e + %12.4ei\n',...

p(1),p(3), p(2),p(4));

It is clear that the constant terms are evaluated outside the iteration cycle repesented by the function LMFnlsq. The supporting function res.m evaluating residuals of the system of equations has the form

function r = res(p)

%~~~~~~~~~~~~~~~~~

global p0 ro_s E ni mi ro_l om Ts1 Rs1 Ts2 x1 x2 beta_s V_1 V_2

%

p = p.*p0;

G = complex(p(1),p(3));

N1 = complex(p(2),p(4));

beta_l = (-i*om*G)/ro_l;

s_s = N1/beta_s;

s_l = N1/beta_l;

B_s = sqrt(1-beta_s^2*s_s^2);

B_l = sqrt(1-beta_s^2*s_s^2);

C_s = 1-2*beta_s^2*s_s^2;

C_l = 1-2*beta_l^2*s_l^2;

g_s = exp(i*om*B_s*beta_s^-1*x1);

g_l = exp(i*om*B_s*beta_s^-1*x2);

%

A1 = [0, i*om*ro_s*s_s*beta_s^-1*B_s*g_s, 0, -i*om*ro_s*s_s*beta_s^-1*B_s*g_s;

0, i*om*ro_s*s_s*beta_s*C_s*g_s^-1, 0, -i*om*ro_s*beta_s*C_s*g_s;

0, -B_s*g_s^-1, 0, B_s*g_s;

0, -beta_s*s_s*g_s^-1, 0, -beta_s*s_s*g_s];

%

A2 = [0 i*om*ro_l*s_l*beta_l^-1*B_l*g_l 0 -i*om*ro_l*s_l*beta_l^-1*B_l*g_l;

0 i*om*ro_l*s_l*beta_l*C_l*g_l^-1 0 -i*om*ro_l*beta_l*C_l*g_l;

0 -B_l*g_l^-1 0 B_l*g_s;

0 -beta_l*s_l*g_l^-1 0 -beta_s*s_l*g_l];

%

rx = A1*V_1 - A2*V_2; % column vector of complex residuals

r = [real(rx(1));imag(rx(1)); real(rx(2));imag(rx(2))];

The statements remained unchanged but some formal changes leading to better visibility. A1 and A2 could be written better in the complex form. Thus formulated task gave the following results:

>> main

**************************************************

itr nfJ SUM(r^2) x dx

**************************************************

0 1 3.1035e+038 1.0000e+000 0.0000e+000

1.0000e+000 0.0000e+000

1.0000e+000 0.0000e+000

1.0000e+000 0.0000e+000

5 6 4.7887e+024 -1.5385e+002 7.5141e-003

6.4562e-001 1.7133e-009

3.0769e+003 -1.5147e-001

6.5933e-001 1.7496e-009

8 9 3.1034e+011 -1.5385e+002 1.9139e-009

6.4562e-001 -4.0313e-018

3.0769e+003 -3.8455e-008

6.5933e-001 6.1153e-018

Sum of squares = 3.10e+011

Function evaluations = 8

G = -1.5385e+003 + 3.0769e+003i

N1 = 6.4562e+000 + 6.5933e-001i

It is clear that the recommended process operates, nevertheless, the numerical values of results are dependent on the original data and formulas. It has been observed that the original initial guess is very far from stable solution (the resultshave been all NaNs. After changing it, the process converged. While G has been almost insensitive on the guess, the N1 has changed strongly with it.

I regard the presented process as solved.

Good luck!

### More Answers (3)

Miroslav Balda
on 7 Feb 2013

Maybe that you can get a solution in a simpler way just in MATLAB without using symbolic approach. Your equation is complex dependent on a complex vector p composed out of 2 complex unknowns G and N1. The solution can be obtained by the following (pseudo)code:

p0 = [real(G0);real(N10); imag(G0); imag(N10)]; % initial estimates of unknowns

[p,ssq,cnt] = LMFnlsq(@res, ones(size(p0)), 'Display',-10);

% LMFnlsq in FEX at www.mathworks.com/matlabcentral/fileexchange/17534

p = p.*p0;

% The solution:

G = complex(p(1),p(3))

N1 = complex(p(2),p(4))

fprintf('Sum of squares = %10.2e\n...

'Function evaluations %g\n...

'G = %10.2e + %10.2ei\n...

'NI = %10.2e + %10.2ei\n', ssq, cnt, p(1),p(3), p(2),p(4));

It is neccessary to build the function for evaluating differences of the given matrix equation from zero:

function r = res(p)

p = p.*p0;

G = complex(p(1),p(3))

N1 = complex(p(2),p(4))

% Evaluate a complex residual d between the equation and zero as

% d = A(p)*a(p)-B(p)*b(p); % and set the vector of its components:

r = [real(d); imag(d)]; % residuals

As is obvious, it is necessary to solve a system of 2 equations for 2 complex unknowns.

##### 2 Comments

Miroslav Balda
on 7 Feb 2013

More rigorous would be to write d = A(q)*a(q)-B(q)*b(q), where

q = [G;N1]; instead of d = A(p)*a(p)-B(p)*b(p) in the function res.

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