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C++ API

: [u, h, nu] = krylov (A, V, k, eps1, pflg)

Construct an orthogonal basis u of block Krylov subspace

[v a*v a^2*v … a^(k+1)*v]

using Householder reflections to guard against loss of orthogonality.

If V is a vector, then h contains the Hessenberg matrix such that a*u == u*h+rk*ek', in which rk = a*u(:,k)-u*h(:,k), and ek' is the vector [0, 0, …, 1] of length k. Otherwise, h is meaningless.

If V is a vector and k is greater than length (A) - 1, then h contains the Hessenberg matrix such that a*u == u*h.

The value of nu is the dimension of the span of the Krylov subspace (based on eps1).

If b is a vector and k is greater than m-1, then h contains the Hessenberg decomposition of A.

The optional parameter eps1 is the threshold for zero. The default value is 1e-12.

If the optional parameter pflg is nonzero, row pivoting is used to improve numerical behavior. The default value is 0.

Reference: A. Hodel, P. Misra, Partial Pivoting in the Computation of Krylov Subspaces of Large Sparse Systems, Proceedings of the 42nd IEEE Conference on Decision and Control, December 2003.

Package: octave