taylor expansion of multivariate function

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nasim mh 2023-6-2

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https://ww2.mathworks.cn/matlabcentral/answers/1977209-taylor-expansion-of-multivariate-function

移动：Torsten 2023-6-5

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)