GradientWe are going to include the concepts in our Derivative function created before, to develop a Matlab function to calculate the gradient of a multidimensional scalar function. The function is going to have the following functionality: % Usage: g = Grad(fun, x0) % fun: name of the multidimensional scalar function % (string). This function takes a vector argument of % length n and returns a scalar. % x0: point of interest (vector of length n) % g: column vector containing the gradient of fun at x0. The % size(g) = size(x) function g = Grad(fun, x0) % |delta(i)| is relative to |x0(i)| delta = x0 / 1000; for i = 1 : length(x0) if x0(i) == 0 % avoids delta(i) = 0 (**arbitrary value**) delta(i) = 1e-12; end % recovers original x0 u = x0; u(i) = x0(i) + delta(i); % fun(x0(i-1), x0(i)+delta(i), x0(i+1), ...) f1 = feval ( fun, u ); u(i) = x0(i) - delta(i); % fun(x0(i-1), x0(i)-delta(i), x0(i+1), ...) f2 = feval ( fun, u ); % partial derivatives in column vector g(i,1) = (f1 - f2) / (2 * delta(i)); end We can try this algorithm, creating a function bowl (which includes two variables) in an separate m-file, as follows: function y = bowl(x) y = (x(1)-6)^2 + (x(2)-4.5)^4 / 25; Then, we can test it from the command window: x = [0 0] f = 'bowl' Grad(f, x) Expected: [-12 -14.58]' Obtained: [-12.0011 -14.5803]' x = [1 1] Grad(f, x) Expected: [-10 -6.86]' Obtained: [-10.0000 -6.8600]' x = [6 4.5] Grad(f, x) Expected: [0 0]' Obtained: [0 0]' x = [2 1.5] Grad(f, x) Expected: [-8 -4.32]' Obtained: [-8.0000 -4.3200]' Now, another test with a different multivariable function: function y = semirosen(x) y = 100 * ( x(2) - x(1)^2 ) + ( 1 - x(1) )^2 ; x = [0 0] f = 'semirosen' Grad(f, x) Expected: [-2 100]' Obtained: [-2.0001 100.0000]' x = [1 1] Grad(f, x) Expected: [-200 100]' Obtained: [-200.0000 100.0000]' x = [9 15] Grad(f, x) Expected: [-1784 100]' Obtained: 1.0e+003 *[-1.7840 0.1000]' From 'Gradient' to home From 'Gradient' to 'Matlab Cookbook'
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