gges3(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

🦋 okay so this is `gges3(3)` 🦋

from the Library Functions Manual — bestie, this is the whole page for gges3(3), and it's a long one, so we're doing it in parts. Part 1, let's go 💅

✨ NAME ✨

gges3 — gges3: Schur form

Like, that's genuinely what it says. Short and iconic. Periodt.

🌸 SYNOPSIS 🌸

Functions

cgges3

subroutine cgges3 (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb,
    sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork,
    bwork, info)

CGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

dgges3

subroutine dgges3 (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb,
    sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork,
    bwork, info)

DGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

sgges3

subroutine sgges3 (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb,
    sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork,
    bwork, info)

SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

zgges3

subroutine zgges3 (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb,
    sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork,
    bwork, info)

ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

🌈 Detailed Description

Okay so, real talk — this heading is in the original and there is literally nothing under it. Empty. Void. No cap. It's just a header sitting there with main character energy and zero content 🫶.

👑 Function Documentation 👑

🎀 subroutine cgges3

subroutine cgges3 (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)

CGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

💫 Purpose:

Okay bestie, here's the whole deal, copied with love:

CGGES3 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 CGGEV instead, which is faster.) — like, fr fr, if you don't need the vectors, go use CGGEV, it's the faster 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 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.

✨ 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 two COMPLEX 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). Like, that's a real thing that can happen, bestie, don't come for me.
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)
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 CGGES3. 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 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. (iykyk, that's the classic query trick ✨)
RWORK — RWORK is REAL array, dimension (8*N)
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 ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N.
- > N:
  - =N+1: other than QZ iteration failed in CLAQZ0
  - =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 cgges3.f.

💜 subroutine dgges3

subroutine dgges3 (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)

DGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

💫 Purpose:

(…the double-precision deets are up next, and that's literally where part 1 dips out — more coming your way 👀)

okay so literally the whole point of DGGES3: give it a pair of N-by-N real nonsymmetric matrices (A,B) and it hands you back the generalized eigenvalues, the generalized real Schur form (S,T), and — if you want them — the left and/or right matrices of Schur vectors (VSL and VSR). That's the whole generalized Schur factorization, bestie:

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

Optionally it also orders the eigenvalues so a selected cluster lands 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 (aka the deflating subspaces). 🦋

If you only need the generalized eigenvalues, use the driver DGGEV instead — it's faster. No cap. 💅

Okay so what even is a generalized eigenvalue? For a pair of matrices (A,B) it's a scalar w, or a ratio alpha/beta = w, such that A - w*B is singular. We usually represent it as the pair (alpha,beta) because there's a reasonable interpretation for beta=0 or both of them being zero. ✨

And 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 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 🌈

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. 😭

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).

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 >= 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 DLAQZ0.
  - =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 dgges3.f.

🦋 sgges3 🦋

okay so the single-precision sibling showed up and it's giving main character energy. 🌸

   subroutine sgges3 (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)

SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

💜 Purpose: 💜

Okay so, continuing the SGGES3 saga 💅 — this one computes, for a pair of N-by-N real nonsymmetric matrices (A,B), literally all the good stuff: the generalized eigenvalues, the generalized real Schur form (S,T), and optionally the left and/or right matrices of Schur vectors (VSL and VSR). It gives you 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 SGGEV 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 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. 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. 🎀

🧮 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; 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 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 >= 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 SLAQZ0. =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 sgges3.f.

🦋 `zgges3`

subroutine zgges3 (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)

ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm).

🎯 Purpose

ZGGES3 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 a 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. 🌈

🧮 Parameters

JOBVSL

`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 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)

`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 ZGGES3. 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

= 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 ZLAQZ0
- =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 zgges3.f.

🧚 Author

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That's the whole gges3(3) page, bestie. Source/version check: LAPACK, Version 3.12.0. Byeee 💖

📦 where this page comes from

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gges3(3)

🦋 okay so this is gges3(3) 🦋

✨ NAME ✨

🌸 SYNOPSIS 🌸

Functions

🌈 Detailed Description

👑 Function Documentation 👑

🎀 subroutine cgges3

💫 Purpose:

✨ Parameters

🫶 Author

💜 subroutine dgges3

💫 Purpose:

🌈 Parameters 🌈

👑 Author 👑

🦋 sgges3 🦋

💜 Purpose: 💜

🧮 Parameters

👑 Author

🦋 zgges3

🎯 Purpose

🧮 Parameters

JOBVSL

JOBVSR

SORT

SELCTG

N

A

LDA

B

LDB

SDIM

ALPHA

BETA

VSL

LDVSL

VSR

LDVSR

WORK

LWORK

RWORK

BWORK

INFO

👑 Author

🧚 Author

🦋 okay so this is `gges3(3)` 🦋

🦋 `zgges3`

`JOBVSL`

`JOBVSR`

`SORT`

`SELCTG`

`N`

`A`

`LDA`

`B`

`LDB`

`SDIM`

`ALPHA`

`BETA`

`VSL`

`LDVSL`

`VSR`

`LDVSR`

`WORK`

`LWORK`

`RWORK`

`BWORK`

`INFO`