gges(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💅 NAME 💅

gges — that's gges: Schur form, unblocked, bestie. That's the whole name, no notes. It's the umbrella for the four routines below and honestly the whole family ate. 🎀

✨ SYNOPSIS ✨

🌸 Functions

  • cgges 🦋
subroutine cgges (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim,
    alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork,
    info)

so CGGES is the one computing the eigenvalues, the Schur form, and — if you asked nicely — the matrix of Schur vectors, all of it for GE matrices. periodt.

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

same energy as above but DGGES: eigenvalues, Schur form, and optionally the matrix of Schur vectors for GE matrices. 💖

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

yup, SGGES — eigenvalues, Schur form, optionally the matrix of Schur vectors, GE matrices, you know the drill by now. ✨

  • zgges 💜
subroutine zgges (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim,
    alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork,
    info)

and ZGGES, the last sibling: eigenvalues, Schur form, and — say it with me — optionally the matrix of Schur vectors for GE matrices. 👑

🫶 Detailed Description 🫶

bestie there is literally nothing here. like, the section exists and the content does not. it's giving "I said what I said". moving on 💀

🎀 Function Documentation 🎀

🌟 subroutine cgges

here's the full declaration in all its glory, character for character, do not touch:

subroutine cgges (character jobvsl, character jobvsr, character sort,
    external selctg, integer n, complex, dimension( lda, * ) a, integer lda,
    complex, dimension( ldb, * ) b, integer ldb, integer sdim, complex,
    dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension(
    ldvsl, * ) vsl, integer ldvsl, complex, dimension( ldvsr, * ) vsr,
    integer ldvsr, complex, dimension( * ) work, integer lwork, real,
    dimension( * ) rwork, logical, dimension( * ) bwork, integer info)

CGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices — yes that's the tagline, and yes it goes hard. 🔥

💫 Purpose:

ok so, deep breath, here's what CGGES actually does:

CGGES computes, for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized complex Schur form (S, T), and optionally the left and/or right Schur vectors (VSL and VSR). the way it hands you the generalized Schur factorization:

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

where (VSR)**H is the conjugate-transpose of VSR. conjugate-transpose bestie, not just transpose, do not come for me. 😭

optionally it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T. the leading columns of VSL and VSR then form an unitary basis for the corresponding left and right eigenspaces (deflating subspaces). cute, right? 🦋

if you only need the generalized eigenvalues, use the driver CGGEV instead — it's faster. no cap.

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, and even for both being zero. 🙈

a pair of matrices (S,T) is in generalized complex Schur form if S and T are upper triangular and, in addition, the diagonal elements of T are non-negative real numbers. that's the whole vibe check. ✅

🌸 Parameters

  • JOBVSL — JOBVSL is CHARACTER*1, and it's literally a yes/no for the left Schur vectors:
    • = 'N': do not compute the left Schur vectors;
    • = 'V': compute the left Schur vectors. 💅
  • JOBVSR — JOBVSR is CHARACTER*1, same deal but for the right side:
    • = 'N': do not compute the right Schur vectors;
    • = 'V': compute the right Schur vectors. 💅
  • SORT — SORT is CHARACTER*1 and it 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 two COMPLEX arguments. it must be declared EXTERNAL in the calling subroutine, don't forget, bestie. 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 ALPHA(j)/BETA(j) is selected if SELCTG(ALPHA(j),BETA(j)) is true.
note that a selected complex eigenvalue may no longer satisfy SELCTG(ALPHA(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+2 (see INFO below). lowkey rude of the roundoff but ok. 😤
  • N — N is INTEGER: the order of the matrices A, B, VSL, and VSR. N >= 0. ✨
  • A — A is COMPLEX 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 COMPLEX 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. 🦋
  • ALPHA — ALPHA is COMPLEX array, dimension (N). that's it, that's the bullet. 💜
  • BETA — BETA is COMPLEX array, dimension (N). on exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), j=1,...,N are the diagonals of the complex Schur form (A,B) output by CGGES. the BETA(j) will be non-negative real.
note: the quotients ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. thus, the user should avoid naively computing the ratio alpha/beta. however, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). like, don't just divide, bestie, that's the whole warning. 🍓
  • VSL — VSL is COMPLEX 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 COMPLEX 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 COMPLEX 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. LWORK >= max(1,2*N). 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. sneaky but legal. 🙈
  • RWORK — RWORK is REAL array, dimension (8*N). 💫
  • BWORK — BWORK is LOGICAL array, dimension (N). not referenced if SORT = 'N'. 🎀
  • INFO — INFO is INTEGER, and this is the receipts section, so listen up:
    • = 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 ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N.
    • > N:
      • =N+1: other than QZ iteration failed in CHGEQZ
      • =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 CTGSEN. 💀

🌈 Author

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

Definition at line 267 of file cgges.f. 📍

🌟 subroutine dgges

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, again, and yes, still iconic. 💖

💫 Purpose:

ok so DGGES — and hold that thought, because there's more coming... 🦋

Okay so, picking up the DGGES saga right where we left it: DGGES is the icon that computes, for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized 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. The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces). Slay.

If you only need the generalized eigenvalues, use the driver DGGEV instead — it's faster. Noted, bestie.

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. Iconic.

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. The 1-by-1 blocks correspond to real generalized eigenvalues, while the 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. Periodt.

🎀 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.
Ill-conditioned case alert: 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.
  • 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.
Ratio warning: 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 — 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.

Okay so now the SGGES chapter starts, and her signature is a whole moment:

subroutine sgges (character jobvsl, character jobvsr, character sort,
    external selctg, integer n, real, dimension( lda, * ) a, integer lda,
    real, dimension( ldb, * ) b, integer ldb, integer sdim, real, dimension(
    * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta,
    real, dimension( ldvsl, * ) vsl, integer ldvsl, real, dimension( ldvsr,
    * ) vsr, integer ldvsr, real, dimension( * ) work, integer lwork,
    logical, dimension( * ) bwork, integer info)

SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose

okay so SGGES is out here computing, for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized real Schur form (S,T), and optionally the left and/or right matrices of Schur vectors (VSL and VSR). slay. 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). periodt.

(If only the generalized eigenvalues are needed, use the driver SGGEV instead, which is faster.)

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 both being zero. no cap.

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. it's giving structured, honestly.

🌸 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 REAL 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.
  • N — N is INTEGER; the order of the matrices A, B, VSL, and VSR. N >= 0.
  • A — A is REAL 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 REAL 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 REAL array, dimension (N).
  • ALPHAI — ALPHAI is REAL array, dimension (N).
  • BETA — BETA is REAL 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 — VSL is REAL 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 REAL 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 REAL 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 SHGEQZ. =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 STGSEN.

💖 Author 💖

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

Definition at line 281 of file sgges.f.

🦋 subroutine zgges 🦋

subroutine zgges (character jobvsl, character jobvsr, character sort,
    external selctg, integer n, complex*16, dimension( lda, * ) a, integer
    lda, complex*16, dimension( ldb, * ) b, integer ldb, integer sdim,
    complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta,
    complex*16, dimension( ldvsl, * ) vsl, integer ldvsl, complex*16,
    dimension( ldvsr, * ) vsr, integer ldvsr, complex*16, dimension( * )
    work, integer lwork, double precision, dimension( * ) rwork, logical,
    dimension( * ) bwork, integer info)

ZGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices. iconic.

✨ Purpose ✨

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

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

where (VSR)**H is the conjugate-transpose of VSR.

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T. The leading columns of VSL and VSR then form an unitary basis for the corresponding left and right eigenspaces (deflating subspaces).

(If only the generalized eigenvalues are needed, use the driver ZGGEV instead, which is faster.)

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, and even for both being zero.

A pair of matrices (S,T) is in generalized complex Schur form if S and T are upper triangular and, in addition, the diagonal elements of T are non-negative real numbers. the way that is so specific, honestly.

💅 Parameters 💅

JOBVSL

  • JOBVSL — okay so JOBVSL is a CHARACTER*1, bestie, and here's the whole menu:
    • 'N': do not compute the left Schur vectors;
    • 'V': compute the left Schur vectors.
  • JOBVSR — also CHARACTER*1, same energy but for the right side:
    • 'N': do not compute the right Schur vectors;
    • 'V': compute the right Schur vectors.
  • SORT — CHARACTER*1; this one specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form. Vibe check:
    • 'N': Eigenvalues are not ordered;
    • 'S': Eigenvalues are ordered (see SELCTG).
  • SELCTG — this one's a LOGICAL FUNCTION of two COMPLEX*16 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 ALPHA(j)/BETA(j) is selected if SELCTG(ALPHA(j),BETA(j)) is true.
Note that a selected complex eigenvalue may no longer satisfy SELCTG(ALPHA(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+2 (See INFO below). 😭
  • N — N is INTEGER: the order of the matrices A, B, VSL, and VSR. N >= 0.
  • A — A is COMPLEX*16 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 COMPLEX*16 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.
  • ALPHA — ALPHA is COMPLEX*16 array, dimension (N). That's the whole note, no extra lore.
  • BETA — BETA is COMPLEX*16 array, dimension (N). On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), j=1,...,N are the diagonals of the complex Schur form (A,B) output by ZGGES. The BETA(j) will be non-negative real.
Note: the quotients ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio alpha/beta. However, ALPHA 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 — VSL is COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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. LWORK >= max(1,2*N). 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.

  • RWORK — RWORK is DOUBLE PRECISION array, dimension (8*N).
  • BWORK — BWORK is LOGICAL array, dimension (N). Not referenced if SORT = 'N'.
  • INFO — INFO is INTEGER, and here's the whole status tea:
    • = 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 ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N.
    • > N:
      • =N+1: other than QZ iteration failed in ZHGEQZ
      • =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 ZTGSEN.

👑 Author

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

Definition at line 267 of file zgges.f.

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