     Next: is NOT Hermitian Positive Up: Shift and Invert Spectral Previous: Shift and Invert Spectral is Hermitian Positive Definite

If is Hermitian positive definite and well conditioned ( is of modest size), then computing the Cholesky factorization   and converting equation (3.2.1) to

provides a transformation to a standard eigenvalue problem.   In this case, a request for a matrix vector product would be satisfied with the following three steps:
1.
Solve for
2.
Matrix-vector multiply
3.
Solve for
Upon convergence, a computed eigenvector for is converted to an eigenvector of the original problem by solving the the triangular system This transformation is most appropriate when is Hermitian, is Hermitian positive definite and extremal eigenvalues are sought. This is because will be Hermitian when is.

If is Hermitian positive definite and the smallest eigenvalues are sought, then it would be best to reverse the roles of and in the above description and ask for the largest algebraic eigenvalues or those of largest magnitude. Upon convergence, a computed eigenvalue would then be converted to an eigenvalue of the original problem by the relation     Next: is NOT Hermitian Positive Up: Shift and Invert Spectral Previous: Shift and Invert Spectral
Chao Yang
11/7/1997