Newton algorithm - couln't calculate Hessian

Question

I'm trying to write an implementation of Newton algorithm in Matlab.

When I call up my function using formula:

result = NewtonMethod(x(1).^2 - 2.1*x(1).^4 + (x(1).^6)/3 + x(1)*x(2) - 4*x(2).^2 + 4*x(2).^4, [0 0], 0.1, 10)

I've got an error message:

??? Undefined function or method 'hessian' for input arguments of type 'double'.

Error in ==> NewtonMethod at 13
    H = hessian(f, x0);

I've got no idea what is wrong. Maybe someone more familiar with Matlab can help me.

Below it's my code:

function xnext = NewtonMethod(f, x0, eps, maxSteps)

% x0        -   starting point (2 – dimensional  vector)
% H         -   matrix of second derivatives (Hessian)
% eps       -   required  tolerance of calculations
% maxSteps  -   length of step

x = x0;

for n=1:maxSteps

    % determine the hessian H at the starting point x0,
    H = hessian(f, x0);

    % determine the gradient of the goal function gradf at the point x,
    gradF = gradient(f, x);

    % determine next point
    xnext = x - inv(H) * x * gradF;

    if abs(xnext - x) < eps
        return                  %found
    else
        x = xnext;              %update
    end
end

It's my first contact with Matlab.

Update:

Now I've got an error:

??? Error using ==> mupadmex
Error in MuPAD command: Index exceeds matrix dimensions.

Error in ==> sym.sym>sym.subsref at 1381
            B = mupadmex('symobj::subsref',A.s,inds{:});

I typed:

syms x
result = NewtonMethod(x(1).^2 - 2.1*x(1).^4 + (x(1).^6)/3 + x(1)*x(2) - 4*x(2).^2 + 4*x(2).^4, [0 0], 0.1, 10)

The hessian function is not defined for double input (which you pass into the function). — George, May 13 '14 at 16:49

jedwards · Answer 1 · 2014-05-13T17:39:39.097

x(1).^2 - 2.1*x(1).^4 + (x(1).^6)/3 + x(1)*x(2) - 4*x(2).^2 + 4*x(2).^4

Is reduced to a double before the NewtonMethod function is called, so when your code reaches hessian(f, x0), you're passing it two double arguments, which is not a supported syntax.

Review the notes on properly specifying a symbolic function, and pass that into NewtonMethod.

It's been a long time since I've done numerical optimization, but take a look at the following:

function xn = NewtonMethod(f, x0, eps, maxSteps)

% x0        -   starting point (2 – dimensional  vector)
% H         -   matrix of second derivatives (Hessian)
% eps       -   required  tolerance of calculations
% maxSteps  -   length of step

syms x y

H = hessian(f);
gradF = gradient(f);

xi = x0;

for i=1:maxSteps

    % evaluate f at xi
    zi = subs(f, [x,y], xi);

    % determine the hessian H at the starting point x0,
    hi = subs(H, [x,y], xi);

    % determine the gradient of the goal function gradf at the point x,
    gi = subs(gradF, [x,y], xi);

    % determine next point
    ss = 0.5;  % step size
    xn = xi - ss.* (inv(hi) * gi);

    % evaluate f at xn
    zn = subs(f, [x,y], xn);

    % some debugging spam
    zd = zn - zi;                          % the change in the value of the
    si = sprintf('[%6.3f, %6.3f]', xi);    %   function from xi -> xn
    sn = sprintf('[%6.3f, %6.3f]', xn);
    printf('Step %3d: %s=%9.4f -> %s=%9.4f  :  zd=%9.4f\n', i, si, zi, sn, zn, zd);

    % stopping condition
    if abs(xi - xn) < eps
        return               %found
    else
        xi = xn;             %update
    end
end

And called with

result = NewtonMethod(f, [0; 1], 0.001, 100)

@user3633449 I updated my answer -- not sure it's correct, but it might give you some insight on how to use the functions correctly. — jedwards, May 13 '14 at 17:40
@jedwards seriously I really can't understand how matlab works. I defined f as x.^2 - 2.1*x.^4 + (x.^6)/3 + x*y - 4*y.^2 + 4*y.^4 so I got an error about undefined x and y - OK. so I typed x = 0 and y = 1 and then called up function as you write above. Result: ??? Undefined function or method 'hessian' for input arguments of type 'double'. If instead of defining variables I typed syms x y I've got: ??? Undefined function or method 'hessian' for input arguments of type 'sym — user3633449, May 13 '14 at 18:04

Newton algorithm - couln't calculate Hessian

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