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DSNHEPTS

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

Notes

Two related problems are solved, A*X = X*Lambda and B*Y = 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, A*Q = Q*T, and similarly another (real) Schur relation is computed, B*Z = Z*S, 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