Rosenbrock's function is a standard test function for optimization. Therosenbrock.mhelper function computes the function value and uses automatic differentiation to compute its gradient.

typerosenbrock.m

function [y,dydx] = rosenbrock(x) y = 100*(x(2) - x(1).^2).^2 + (1 - x(1)).^2; dydx = dlgradient(y,x); end

To evaluate Rosenbrock's function and its gradient at the point[–1,2], create adlarrayof the point and then calldlfeval在function handle@rosenbrock.

x0 = dlarray([-1,2]); [fval,gradval] = dlfeval(@rosenbrock,x0)

fval = 1x1 dlarray 104

gradval = 1x2 dlarray 396 200

Alternatively, define Rosenbrock's function as a function of two inputs,x1and x2.

typerosenbrock2.m

function [y,dydx1,dydx2] = rosenbrock2(x1,x2) y = 100*(x2 - x1.^2).^2 + (1 - x1).^2; [dydx1,dydx2] = dlgradient(y,x1,x2); end

Calldlfevalto evaluaterosenbrock2on twodlarrayarguments representing the inputs–1and2.

x1 = dlarray (-1); x2 = dlarray(2); [fval,dydx1,dydx2] = dlfeval(@rosenbrock2,x1,x2)

fval = 1x1 dlarray 104

dydx1 = 1x1 dlarray 396

dydx2 = 1x1 dlarray 200

Plot the gradient of Rosenbrock's function for several points in the unit square. First, initialize the arrays representing the evaluation points and the output of the function.

[X1 X2] = meshgrid(linspace(0,1,10)); X1 = dlarray(X1(:)); X2 = dlarray(X2(:)); Y = dlarray(zeros(size(X1))); DYDX1 = Y; DYDX2 = Y;

Evaluate the function in a loop. Plot the result usingquiver.

fori = 1:length(X1) [Y(i),DYDX1(i),DYDX2(i)] = dlfeval(@rosenbrock2,X1(i),X2(i));endquiver(extractdata(X1),extractdata(X2),extractdata(DYDX1),extractdata(DYDX2)) xlabel('x1') ylabel('x2')

使用dlgradientanddlfevalto compute the value and gradient of a function that involves complex numbers. You can compute complex gradients, or restrict the gradients to real numbers only.

Define the functioncomplexFun, listed at the end of this example. This function implements the following complex formula:

Define the functiongradFun, listed at the end of this example. This function callscomplexFunand usesdlgradientto calculate the gradient of the result with respect to the input. For automatic differentiation, the value to differentiate — i.e., the value of the function calculated from the input — must be a real scalar, so the function takes the sum of the real part of the result before calculating the gradient. The function returns the real part of the function value and the gradient, which can be complex.

Define the sample points over the complex plane between -2 and 2 and -2 $\mathit{i}$and 2 $\mathit{i}$and convert todlarray.

functionRes = linspace(-2,2,100); x = functionRes + 1i*functionRes.'; x = dlarray(x);

Calculate the function value and gradient at each sample point.

[y, grad] = dlfeval(@gradFun,x); y = extractdata(y);

Define the sample points at which to display the gradient.

gradientRes = linspace(-2,2,11); xGrad = gradientRes + 1i*gradientRes.';

Extract the gradient values at these sample points.

[~,gradPlot] = dlfeval(@gradFun,dlarray(xGrad)); gradPlot = extractdata(gradPlot);

Plot the results. Useimagescto show the value of the function over the complex plane. Usequiverto show the direction and magnitude of the gradient.

imagesc([-2,2],[-2,2],y); axisxycolorbar holdonquiver(real(xGrad),imag(xGrad),real(gradPlot),imag(gradPlot),"k"); xlabel("Real") ylabel("Imaginary") title("Real Value and Gradient","Re$(f(x)) = $ Re$((2+3i)x)$","interpreter","latex")

The gradient of the function is the same across the entire complex plane. Extract the value of the gradient calculated by automatic differentiation.

grad(1,1)

ans = 1×1 dlarray 2.0000 - 3.0000i

By inspection, the complex derivative of the function has the value

However, the function Re( $\mathit{f}\left(\mathit{x}\right)$) is not analytic, and therefore no complex derivative is defined. For automatic differentiation in MATLAB, the value to differentiate must always be real, and therefore the function can never be complex analytic. Instead, the derivative is computed such that the returned gradient points in the direction of steepest ascent, as seen in the plot. This is done by interpreting the function Re $\left(\mathit{f}\left(\mathit{x}\right)\right)$:C $\to$Ras a function Re $\left(\mathit{f}\left({\mathit{x}}_{\mathit{R}}+\mathit{i}{\mathit{x}}_{\mathit{I}}\right)\right)$:R $\times$R $\to$R.

functiony = complexFun(x) y = (2+3i)*x;endfunction[y,grad] = gradFun(x) y = complexFun(x); y = real(y); grad = dlgradient(sum(y,"all"),x);end