SRC/dgges.f(3)

💅 SRC/dgges.f(3) 💅

okay so this page is from the Library Functions Manual, and it’s literally all about the dgges subroutine, bestie. The dgges subroutine is the main character here, iykyk. 🦋

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🌸 NAME 🌸

SRC/dgges.f — the source file, periodt. This page documents the dgges subroutine and its whole vibe.

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

subroutine dgges (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim,
    alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, bwork,
    info)

DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices. Slay. 💖

🦋 Function/Subroutine Documentation 🦋

subroutine dgges (character jobvsl, character jobvsr, character sort,
     external selctg, 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( *
     ) work, integer lwork, logical, dimension( * ) bwork, integer info)

DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices. Yes, same iconic line again, because that’s the whole point. 💅

✨ Purpose ✨

Okay so DGGES computes, for a pair of N-by-N real nonsymmetric matrices (A,B):

• the generalized eigenvalues,
• the generalized real Schur form (S,T),
• 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. The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces). 🫶

If you only need the generalized eigenvalues, use the driver DGGEV instead, which is faster. That’s the “bestie, don’t do extra work for no reason” tip. 💅

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’s usually represented as the pair (alpha,beta), because there’s a reasonable interpretation for beta=0 or 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 happen 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 💖

• JOBVSL — JOBVSL is CHARACTER*1
  • = 'N': do not compute the left Schur vectors;
  • = 'V': compute the left Schur vectors.
• JOBVSR — JOBVSR is CHARACTER*1
  • = 'N': do not compute the right Schur vectors;
  • = 'V': compute the right Schur vectors.
• SORT — SORT is 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 — SELCTG is a 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 in the ill-conditioned case, a selected complex eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 in this case. That’s the “oop, roundoff said no” situation, bestie. 😭

• N — N is INTEGER. The order of the matrices A, B, VSL, and VSR. N >= 0.
• A — A is 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 — LDA is INTEGER. The leading dimension of A. LDA >= max(1,N).
• B — B is 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 — LDB is INTEGER. The leading dimension of B. LDB >= max(1,N).
• SDIM — SDIM is 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 — ALPHAR is DOUBLE PRECISION array, dimension (N).
• ALPHAI — ALPHAI is DOUBLE PRECISION array, dimension (N).
• BETA — BETA is 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). So like, do not just divide and pray, okay? 🙈

• VSL — VSL is DOUBLE PRECISION array, dimension (LDVSL,N). If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'.
• LDVSL — LDVSL is INTEGER. The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >= N.
• VSR — VSR is DOUBLE PRECISION array, dimension (LDVSR,N). If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'.
• LDVSR — LDVSR is INTEGER. The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N.
• WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — LWORK is INTEGER. The dimension of the array WORK. If N = 0, LWORK >= 1, else LWORK >= MAX(8*N,6*N+16). For good performance, LWORK must generally be larger. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
• BWORK — BWORK is LOGICAL array, dimension (N). Not referenced if SORT = 'N'.
• INFO — INFO is INTEGER
  • = 0: successful exit.
  • < 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.
  • > 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. =N+3: reordering failed in DTGSEN.

👑 Author 👑

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

Definition at line 281 of file dgges.f.

🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code.

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Source/version vibe check: LAPACK Version 3.12.0, from SRC/dgges.f(3). The footer didn’t give a separate date, so I’m not inventing one — no cap, bestie. 💖✨🎀