DSNHEPTS#

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

Notes#

Two related problems are solved, \(AX = X\Lambda\) and \(BY = Y\Lambda^*\), 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#

DS: Direct Solver (or Dense System), 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