Is there a matlab built-in that calculates the quadratic form (x'*A*x)?

Question

Pretty straightforward question: Given an N x N symmetric matrix A and an N-vector x, is there a built-in Matlab function to calculate x'*A*x? i.e., instead of y = x'*A*x, is there a function quadraticform s.t. y = quadraticform(A, x)?

Obviously I can just do y = x'*A*x, but I need performance and it seems like there ought to be a way to take advantage of

1. A is symmetric
2. The left and right multipliers are the same vector

If there's not a single built-in function, is there method that's faster than x'*A*x? OR, is the Matlab parser smart enough to optimize x'*A*x? If so, can you point me to a place in the documentation that verifies the fact?

Answer 1

I couldn't find such a built-in function, and I have an idea why.

y=x'*A*x can be written as a sum of n^2 terms A(i,j)*x(i)*x(j), where i and j runs from 1 to n (where A is an nxn matrix). A is symmetric: A(i,j) = A(j,i) for all i and j. Due to symmetry, every term appears twice in the sum, except for those where i equals j. So we have n*(n+1)/2 different terms. Each has two floating-point multiplications, so a naive method would need n*(n+1) multiplications in total. It is easy to see that the naive calculation of x'*A*x, that is, calculating z=A*x and then y=x'*z, also needs n*(n+1) multiplications. However, there is a faster way to sum our n*(n+1)/2 different terms: for every i, we can factor out x(i), which means that only n*(n-1)/2+3*n multiplications is enough. But this does not really help: the running time of the calculation of y=x'*A*x is still O(n^2).

So, I think that the calculation of quadratic forms cannot be done faster than O(n^2), and since this can also be achieved by the formula y=x'*A*x, there would be no real advantage of a special "quadraticform" function.

=== UPDATE ===

I've written the function "quadraticform" in C, as a Matlab extension:

// y = quadraticform(A, x)
#include "mex.h" 

/* Input Arguments */
#define A_in prhs[0]
#define x_in prhs[1]

/* Output Arguments */
#define y_out plhs[0] 

void mexFunction(int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[])
{
  mwSize mA, nA, n, mx, nx;
  double *A, *x;
  double z, y;
  int i, j, k;

  if (nrhs != 2) { 
      mexErrMsgTxt("Two input arguments required."); 
  } else if (nlhs > 1) {
      mexErrMsgTxt("Too many output arguments."); 
  }

  mA = mxGetM(A_in);
  nA = mxGetN(A_in);
  if (mA != nA)
    mexErrMsgTxt("The first input argument must be a quadratic matrix.");
  n = mA;

  mx = mxGetM(x_in);
  nx = mxGetN(x_in);
  if (mx != n || nx != 1)
    mexErrMsgTxt("The second input argument must be a column vector of proper size.");

  A = mxGetPr(A_in);
  x = mxGetPr(x_in);
  y = 0.0;
  k = 0;
  for (i = 0; i < n; ++i)
  {
    z = 0.0;
    for (j = 0; j < i; ++j)
      z += A[k + j] * x[j];
    z *= x[i];
    y += A[k + i] * x[i] * x[i] + z + z;
    k += n;
  }

  y_out = mxCreateDoubleScalar(y);
}

I saved this code as "quadraticform.c", and compiled it with Matlab:

mex -O quadraticform.c

I wrote a simple performance test to compare this function with x'Ax:

clear all; close all; clc;

sizes = int32(logspace(2, 3, 25));
nsizes = length(sizes);
etimes = zeros(nsizes, 2); % Matlab vs. C
nrepeats = 100;
h = waitbar(0, 'Please wait...');
for i = 1 : nrepeats
  for j = 1 : nsizes
    n = sizes(j);
    A = randn(n); 
    A = (A + A') / 2;
    x = randn(n, 1);
    if randn > 0
      start = tic;
      y1 = x' * A * x;
      etimes(j, 1) = etimes(j, 1) + toc(start);
      start = tic;
      y2 = quadraticform(A, x);
      etimes(j, 2) = etimes(j, 2) + toc(start);      
    else
      start = tic;
      y2 = quadraticform(A, x);
      etimes(j, 2) = etimes(j, 2) + toc(start);      
      start = tic;
      y1 = x' * A * x;
      etimes(j, 1) = etimes(j, 1) + toc(start);
    end;
    if abs((y1 - y2) / y2) > 1e-10
      error('"x'' * A * x" is not equal to "quadraticform(A, x)"');
    end;
    waitbar(((i - 1) * nsizes + j) / (nrepeats * nsizes), h);
  end;
end;
close(h);
clear A x y;
etimes = etimes / nrepeats;

n = double(sizes);
n2 = n .^ 2.0;
i = nsizes - 2 : nsizes;
n2_1 = mean(etimes(i, 1)) * n2 / mean(n2(i));
n2_2 = mean(etimes(i, 2)) * n2 / mean(n2(i));

figure;
loglog(n, etimes(:, 1), 'r.-', 'LineSmoothing', 'on');
hold on;
loglog(n, etimes(:, 2), 'g.-', 'LineSmoothing', 'on');
loglog(n, n2_1, 'k-', 'LineSmoothing', 'on');
loglog(n, n2_2, 'k-', 'LineSmoothing', 'on');
axis([n(1) n(end) 1e-4 1e-2]);
xlabel('Matrix size, n');
ylabel('Running time (a.u.)');
legend('x'' * A * x', 'quadraticform(A, x)', 'O(n^2)', 'Location', 'NorthWest');

W = 16 / 2.54; H = 12 / 2.54; dpi = 100;
set(gcf, 'PaperPosition', [0, 0, W, H]);
set(gcf, 'PaperSize', [W, H]);
print(gcf, sprintf('-r%d',dpi), '-dpng', 'quadraticformtest.png');

The result is very interesting. The running time of both x'*A*x and quadraticform(A,x) converges to O(n^2), but the former has a smaller factor: