Fastest way to compute spectral norm of a matrix?

Hello

I can't find any mention of the spectral norm in the documentation. I want to calculate

max(|Ax|)/x for any vector x, given a matrix A.

There's a method via the eigenvalues of A'A but that sounds very memory intensive, I was wondering if anyone knows a faster in-built option.

Thanks

Mike

Answer 1

[Matt J]

See the NORM command. It accepts matrix inputs.

[Michael]

Ok, but which gives the spectral norm. Is it just norm(A)?

[Matt J]

Yes. Try it:

    >> A=rand(5);
    >> max(sqrt(eig(A.'*A)))

    ans =

        2.2584

    >> norm(A)

    ans =

        2.2584

[Matt J]

Hmmm. Looks like NORM is not as fast as the direct calculation,

    A=rand(1000);

    tic;
     n1=sqrt( max( eig(A.'*A) ) );
    toc; %Elapsed time is 0.150860 seconds.

    tic;
     n2=norm(A);
    toc; %Elapsed time is 0.319899 seconds.

However, I'm pretty sure it's because NORM uses SVD instead of EIG, which is probably more numerically stable.