MATLAB: Is it possible to use the FSOLVE function in the Optimization Toolbox to solve complex equations

complex variable optimizationOptimization Toolbox

A function of a complex variable "z" is called a complex function "f(z)". I have the following complex function:

 f(z) = z^2 - z + 1

How do I determine the zeros of this complex function?

Best Answer

The FSOLVE function cannot directly handle complex-valued functions or variables. However, a complex-valued function of a complex variable "z" can also be thought of as a system of 2 real-valued equations in 2 real variables.

Consider the following example:

 f(z) = z^2 - z + 1

To solve this complex function using the FSOLVE function, we first create the following function file:

function F = myfun(X)
z = X(1, :) + j*X(2, :); % Create complex value from real and imaginary parts
f = z.^2 - z + 1; % Evaluate complex function
F = [real(f); imag(f)]; % Separate real and imaginary parts

Then, using the initial guess "z = 1 + 2j", the function can be solved by the following command:

[z fval] = fsolve(@myfun, [1; 2])

which produces the results:

 z =
     0.5000
     0.8661
 fval =
   6.2523e-005

Therefore, the complex value that produces a local minimum of "abs(f(z))" is found to be:

 z = 0.5 + i*0.8661

For information on how to optimize complex functions using the Genetic Algorithm and Direct Search Toolbox, see the Related Solution.

Related Solutions

MATLAB: How to optimize real-valued functions of complex variables within the Optimization Toolbox

A function of a complex variable "z" is simply a function of two variables; the real part "Re(z)" and imaginary part "Im(z)". Therefore, a function of a complex variable "z" should be a function of two variables "x1" and "x2" (i.e., "z = x1 + i*x2"). The input argument to the objective function will be the vector:

 X =   [x1]
       [x2]

Consider the example:

 f(z) = |z - (1+i)|^2 + |z - (2-3i)|^2

First, we create the following function:

function f = objfun(X)
z = X(1, :) + i*X(2, :); % Create complex value from real and imaginary parts
f = abs(z - (1 + i)).^2 + abs(z - (2 - 3i)).^2; % Evaluate complex function

Then, using the initial guess "z = 1 + 2i", the function can be optimized via the following command:

[X f] = fminunc(@objfun, [1; 2]); % Optimize function
z = X(1) + i*X(2) % Create optimal complex value

which produces the optimal value:

 z =
   -0.5000 + 2.0000i

Note that a warning message will be returned that FMINUNC is switching from the default algorithm.

MATLAB: How to optimize complex functions within the Genetic Algorithm and Direct Search Toolbox

This bug has been fixed for Release 14 (R14). For previous releases, read below for any possible workarounds:

In general, complex functions "f(z)" evaluate to a complex value. In this situation, you must define a measure of optimality of these complex results that evaluate to a real value (e.g., "|f(z)|").

The real and imaginary parts must be considered separate variables within the objective function. Therefore, the input argument to the objective function will be a 1-by-2 vector including the real and imaginary part. For this example, the objective function is in the form:

function f = objfun(X)
% Create complex value from real and imaginary parts:
z = X(:, 1) + i*X(:, 2); 
f = z.^2 - z + 1; % Evaluate complex function
f = abs(f); %Generate optimality measure

Then, using the initial guess "z = 1 + 2i", the function can be optimized by the following command:

[z f] = ga(@objfun, 2)

which produces the results:

 z =
     0.5000     0.8661
 f =
   6.2523e-005

Therefore, the complex value that produces a local minimum of "abs(f(z))" is found to be:

 z = 0.5 + i*0.8661

Note that this method is the same for the PATTERNSEARCH function, except a 1-by-2 initial guess must be specified in the second input argument.

For information on how to optimize complex functions using the Optimization Toolbox, see the Related Solution.

Best Answer

Related Solutions

MATLAB: How to optimize real-valued functions of complex variables within the Optimization Toolbox

MATLAB: How to optimize complex functions within the Genetic Algorithm and Direct Search Toolbox

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