DSNHEPTS#

Dense Non-Hermitian Eigenvalue Problem (special variant intended for two-sided Krylov solvers).

Notes#

Two related problems are solved, AX = XLambda and BY = YLambda’, where A and B are supposed to come from the Arnoldi factorizations of a certain matrix and its (conjugate) transpose, respectively. Hence, in exact arithmetic the columns of Y are equal to the left eigenvectors of A. Lambda is a diagonal matrix whose diagonal elements are the arguments of DSSolve(). After solve, A is overwritten with the upper quasi-triangular matrix T of the (real) Schur form, AQ = QT, and similarly another (real) Schur relation is computed, BZ = ZS, overwriting B.

In the intermediate state A and B are reduced to upper Hessenberg form.

When left eigenvectors DS_MAT_Y are requested, right eigenvectors of B are returned, while DS_MAT_X contains right eigenvectors of A.

Used DS matrices#

  • DS_MAT_A - first problem matrix obtained from Arnoldi

  • DS_MAT_B - second problem matrix obtained from Arnoldi on the transpose

  • DS_MAT_Q - orthogonal/unitary transformation that reduces A to Hessenberg form (intermediate step) or matrix of orthogonal Schur vectors of A

  • DS_MAT_Z - orthogonal/unitary transformation that reduces B to Hessenberg form (intermediate step) or matrix of orthogonal Schur vectors of B

Implemented methods#

  • 0 - Implicit QR (_hseqr)

See Also#

DSCreate(), DSSetType(), DSType

Level#

beginner

Location#

src/sys/classes/ds/impls/nhepts/dsnhepts.c

Index of all DS routines Table of Contents for all manual pages Index of all manual pages