CGGES3(3)

💅 CGGES3 💅

okay so real talk, this page is straight out of the Library Functions Manual, and the file it documents is this whole path situation: /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/cgges3.f. yes that's the whole NAME section, it's just the path, we love a minimal queen 💖

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✨ SYNOPSIS ✨

🎀 Functions/Subroutines

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) 🦋

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🌸 Function/Subroutine Documentation 🌸

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) — that's the one-line vibe check, and honestly it ate 👑

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🌈 Purpose

okay bestie, here's the whole story. 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 the left and/or right Schur vectors (VSL and VSR). this gives you the generalized Schur factorization:

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

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

optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues shows up 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). main character energy for real 🌟

if you only need the generalized eigenvalues, use the driver CGGEV instead, which is faster. no cap, it's the shortcut ✨

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. iconic behaviour tbh 🙈

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 rule, don't fight it 🎀

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💜 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.

⚠️ heads up: 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). literally read the fine print bestie 😭

🫶 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. it's giving transformation 💫

🫶 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). please don't just divide, i'm begging 😭

🫶 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.

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. the classic "just tell me how much room i need" move 💅

🫶 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.

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👑 Author

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

Definition at line 266 of file cgges3.f.

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🌟 Author

generated automatically by Doxygen for LAPACK from the source code. yes bestie, the robots wrote this one 🤖✨

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and that's the whole page! this beauty is from LAPACK, Version 3.12.0, and the file is /home/abui...RC/cgges3.f(3) — mwah, slay, go compute some Schur forms 💖🎀💅