taylor expansion of multivariate function

Hello everyone,

Would you please help me to find first-order taylor expansion of this non-linear multivariate function of 3*M variables. I try matlab tylor function for multi variable functions but I got the error that "not enough input arguments".

matlab code of this function is here:

w            = [1 0 0 0];
M            = length(w);
x            = sym('x', [1 M]);
y            = sym('y', [1 M]);
z            = sym('z', [1 M]);
p0           = zeros(3, M);

a_f = @(theta, phi) 0;
for m = 1:M
    f = @(theta,phi, x, y, z) (exp(1j*rho*(x(m).*sin(theta).*cos(phi) + ...
        y(m).*sin(theta).*sin(phi) + z(m).*cos(theta)))).*w(m);

    a_f       = @(theta, phi, x, y, z) a_f(theta, phi) + f(theta, phi);
end

% Define the integrand
integrand       = @(theta,phi, x, y, z) abs(a_f(theta,phi)).^2 .* sin(theta);

% Define the integration limits
theta_limits    = [0,pi];
phi_limits      = [0,2*pi];

% Evaluate the integral using Matlab's 'integral2' function
Denominator_phi = @(phi, x, y, z) int(@(theta) integrand(theta, phi, x, y, z), theta_limits(1), theta_limits(2));
Denominator     = @(x, y, z) int(@(phi) Denominator_phi(phi, x, y, z), phi_limits(1), phi_limits(2));

% Calculate the maximum array factor
Numerator       = @(x, y, z) abs(a_f(theta_max, phi_max)).^2;

% Calculate the directivity
D     = @(x, y, z) 4*pi*Numerator(x,y,z) ./ Denominator(x,y,z);

L_D = taylor(D, ...
    [x(1) x(2) x(3) x(4) y(1) y(2) y(3) y(4) z(1) z(2) z(3) z(4)],...
    reshape(p0', 1, []), 'order', 1)

[Torsten]

There are so many problems with this code - I think it will be better if you present the function D in a mathematical notation.

[nasim mh]

D function is:

where w(theta, phi) = 1.

Im*exp(jBm) is considered as w in the code.

[Torsten]

What is D a function of ? It appears all variables are integrated out so that D is only a single number.

[nasim mh]

actually D is function of x, y and z vectors with length of [1 M] for each.

[Torsten]

syms theta phi rho theta_max phi_max real
w = [1 0 0 0];
M = length(w);
x = sym('x', [1 M]);
y = sym('y', [1 M]);
z = sym('z', [1 M]);
assume(x,'real')
assume(y,'real')
assume(z,'real')
p0 = zeros(3, M);
AF(theta,phi) = sum((exp(1j*rho*(x(1:M).*sin(theta).*cos(phi) + ...
                     y(1:M).*sin(theta).*sin(phi) + z(1:M).*cos(theta)))).*w(1:M))

AF(theta, phi) = 

integrand = AF(theta,phi)*AF(theta,phi)'.*sin(theta)

integrand = 

% Define the integration limits
theta_limits    = [0,pi];
phi_limits      = [0,2*pi];
Denominator = int(int(integrand,theta,theta_limits(1),theta_limits(2)),phi,phi_limits(1),phi_limits(2))

Denominator = 

Numerator = 4*pi*AF(theta_max,phi_max)*AF(theta_max,phi_max)'

Numerator = 

D = Numerator/Denominator

D = 

1

[nasim mh]

Thanks @Torsten.