DGGESX(3)

💅 DGGESX(3) — generalized Schur form, but make it a vibe 💅

From the Library Functions Manual, bestie. Yes, the manual name is folded in right here like a little aesthetic bookmark. 🎀

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dggesx.f — that's the file this page is about, periodt. It lives in the Library Functions Manual, and the star of the show is DGGESX. 🫶

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine DGGESX (jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, iwork, liwork, bwork, info) — DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices. Slay. 💅

subroutine DGGESX (jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b,
           ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde,
           rcondv, work, lwork, iwork, liwork, bwork, info)

🦋 Function/Subroutine Documentation 🦋

subroutine DGGESX (character jobvsl, character jobvsr, character sort, external selctg, character sense, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, integer sdim, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvsl, * ) vsl, integer ldvsl, double precision, dimension( ldvsr, * ) vsr, integer ldvsr, double precision, dimension( 2 ) rconde, double precision, dimension( 2 ) rcondv, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices. No cap, that's the whole assignment. 🌈

subroutine DGGESX (character jobvsl, character jobvsr, character sort,
       external selctg, character sense, integer n, double precision,
       dimension( lda, * ) a, integer lda, double precision, dimension( ldb, *
       ) b, integer ldb, integer sdim, double precision, dimension( * ) alphar,
       double precision, dimension( * ) alphai, double precision, dimension( *
       ) beta, double precision, dimension( ldvsl, * ) vsl, integer ldvsl,
       double precision, dimension( ldvsr, * ) vsr, integer ldvsr, double
       precision, dimension( 2 ) rconde, double precision, dimension( 2 )
       rcondv, double precision, dimension( * ) work, integer lwork, integer,
       dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork,
       integer info)

Purpose

Okay so DGGESX computes, for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the real Schur form (S,T), and, optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization:

(A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces). Iconic behavior, honestly. ✨

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0 or for both being zero.

A pair of matrices (S,T) is in generalized real Schur form if T is upper triangular with non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S will be by making the corresponding elements of T have the form:

[  a  0  ]
[  0  b  ]

and the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair of generalized eigenvalues.

Parameters

Bestie, here's the parameter lineup. Every flag, dimension, and condition is exactly as listed — no cap. 🧚

JOBVSL — CHARACTER*1.
- = 'N': do not compute the left Schur vectors;
- = 'V': compute the left Schur vectors.
JOBVSR — CHARACTER*1.
- = 'N': do not compute the right Schur vectors;
- = 'V': compute the right Schur vectors.
SORT — CHARACTER*1. Specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
- = 'N': Eigenvalues are not ordered;
- = 'S': Eigenvalues are ordered (see SELCTG).
SELCTG — LOGICAL FUNCTION of three DOUBLE PRECISION arguments. SELCTG must be declared EXTERNAL in the calling subroutine.
- If SORT = 'N', SELCTG is not referenced.
- If SORT = 'S', SELCTG is used to select eigenvalues to sort to the top left of the Schur form.
- An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either one of a complex conjugate pair of eigenvalues is selected, then both complex eigenvalues are selected.
- Note that a selected complex eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned), in this case INFO is set to N+3.
SENSE — CHARACTER*1. Determines which reciprocal condition numbers are computed.
- = 'N': None are computed;
- = 'E': Computed for average of selected eigenvalues only;
- = 'V': Computed for selected deflating subspaces only;
- = 'B': Computed for both.
- If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
N — INTEGER. The order of the matrices A, B, VSL, and VSR. N >= 0.
A — DOUBLE PRECISION array, dimension (LDA, N).
- On entry, the first of the pair of matrices.
- On exit, A has been overwritten by its generalized Schur form S.
LDA — INTEGER. The leading dimension of A. LDA >= max(1,N).
B — DOUBLE PRECISION array, dimension (LDB, N).
- On entry, the second of the pair of matrices.
- On exit, B has been overwritten by its generalized Schur form T.
LDB — INTEGER. The leading dimension of B. LDB >= max(1,N).
SDIM — INTEGER.
- If SORT = 'N', SDIM = 0.
- If SORT = 'S', SDIM = number of eigenvalues (after sorting) for which SELCTG is true. (Complex conjugate pairs for which SELCTG is true for either eigenvalue count as 2.)
ALPHAR — DOUBLE PRECISION array, dimension (N).
ALPHAI — DOUBLE PRECISION array, dimension (N).
BETA — DOUBLE PRECISION array, dimension (N).
- On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i and BETA(j), j=1,...,N are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 complex unitary transformations.
- If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative.
- > Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio. However, ALPHAR and ALPHAI will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B).
VSL — DOUBLE PRECISION array, dimension (LDVSL,N).
- If JOBVSL = 'V', VSL will contain the left Schur vectors.
- Not referenced if JOBVSL = 'N'.
LDVSL — INTEGER. The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >= N.
VSR — DOUBLE PRECISION array, dimension (LDVSR,N).
- If JOBVSR = 'V', VSR will contain the right Schur vectors.
- Not referenced if JOBVSR = 'N'.
LDVSR — INTEGER. The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N.
RCONDE — DOUBLE PRECISION array, dimension ( 2 ).
- If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenvalues.
- Not referenced if SENSE = 'N' or 'V'.
RCONDV — DOUBLE PRECISION array, dimension ( 2 ).
- If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the reciprocal condition numbers for the selected deflating subspaces.
- Not referenced if SENSE = 'N' or 'E'.
WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
- On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — INTEGER. The dimension of the array WORK.
- If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else LWORK >= max( 8*N, 6*N+16 ).
- Note that 2*SDIM*(N-SDIM) <= N*N/2.
- Note also that an error is only returned if LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B' this may not be large enough.
- If LWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.
IWORK — INTEGER array, dimension (MAX(1,LIWORK)).
- On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
LIWORK — INTEGER. (Part 2 has the rest of this parameter — the page is long, bestie, but we're mid-page, not done.)

LIWORK

LIWORK is INTEGER — the dimension of the array IWORK.
If SENSE = 'N' or N = 0, then LIWORK >= 1, otherwise LIWORK >= N+6.

And the workspace-query tea ☕: if LIWORK = -1, then a workspace query is assumed. The routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA. So like, don't come for XERBLA, it's literally off duty for this one 🫶

BWORK

BWORK is a LOGICAL array, dimension (N).

Not referenced if SORT = 'N' — so if you're not sorting, BWORK is just vibing in the background, unbothered 💅

INFO

INFO is INTEGER — the receipts of the whole operation 📋

= 0: successful exit ✨ (we love a clean run)
< 0: if INFO = -i, the i-th argument had an illegal value 😬
= 1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N. So like, partial credit, iykyk.
> N:
- =N+1: other than QZ iteration failed in DHGEQZ
- =N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy SELCTG=.TRUE.. This could also be caused due to scaling. So yeah, that's a thing that happens 🙈
- =N+3: reordering failed in DTGSEN

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

✨ Further Details:

Okay bestie, error-bound storytime — these are the approximations, so don't @ me:

An approximate (asymptotic) bound on the average absolute error of the selected eigenvalues is

EPS * norm((A, B)) / RCONDE( 1 ).

An approximate (asymptotic) bound on the maximum angular error in the computed deflating subspaces is

EPS * norm((A, B)) / RCONDV( 2 ).

See LAPACK User's Guide, section 4.11 for more information. Go read it, it's giving academic icon behavior 📚✨

📌 Definition

Definition at line 361 of file dggesx.f.

🎀 Author (the Doxygen one, yes there are two, don't ask)

Generated automatically by Doxygen for LAPACK from the source code.

And that's the whole page, bestie — you made it to the end, I'm so proud 🥹💖 Basically everything here is from LAPACK, Version 3.12.0, sourced from /home/abui...RC/dggesx.f(3). Slay, go forth and generalize those Schur forms responsibly 👑🌸

📦 where this page comes from

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opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
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