The reason is that eigs uses very different algorithms for the two versions. Here's what each algorithm does
'lm': This is the simplest case: eigs does a smarter version of the power iteration, which is just
for i=1:N
x = A*x;
x = x / norm(x);
end
The trick is that the component of x pointing in the direction of the largest eigenvector will grow. The smarter algorithm used in eigs keeps the vectors x in memory, and returns the best linear combination of vectors instead.
'sm', numerical shift sigma: This algorithm uses the shift-and-invert method, which consists in doing (A - simga*I)\x instead of A*x. For 'sm', sigma = 0.
'sa', 'la', 'sr', 'lr', 'si', 'li': These methods do not solve a linear system with A. Instead, they use the power iteration, but choose the linear combination of all vectors that best satisfies the condition given by the input flag.
For your symmetric positive definite problem, the algorithm used in 'sm' will often have faster convergence, while the algorithm used in 'sa' does not require factorization of the matrix A. Which is faster depends on the problem, my personal favorite would be 'sm'.
Best Answer