ZGEGV(3)

🌸 NAME

okay so this page is all about the file /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/DEPRECATED/zgegv.f ✨ — and yeah, bestie, that DEPRECATED folder is right there in the path, we'll get to that 😭

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✨ SYNOPSIS

Functions/Subroutines

• subroutine ZGEGV (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info) — literally the whole point of this page: ZGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a complex matrix pair (A,B) 💅

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💖 Function/Subroutine Documentation

🎀 The signature, in all its glory

subroutine ZGEGV (character jobvl, character jobvr, integer n, complex*16,
    dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b,
    integer ldb, complex*16, dimension( * ) alpha, complex*16, dimension( *
    ) beta, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16,
    dimension( ldvr, * ) vr, integer ldvr, complex*16, dimension( * ) work,
    integer lwork, double precision, dimension( * ) rwork, integer info)

ZGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a complex matrix pair (A,B). 🦋

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🧚 Purpose

💀 Heads up, bestie: This routine is deprecated and has been replaced by routine ZGGEV. Not me warning you first thing — use ZGGEV if you're starting fresh, iykyk.

ZGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a complex matrix pair (A,B).

Given two square matrices A and B, the generalized nonsymmetric eigenvalue problem (GNEP) is to find the eigenvalues lambda and corresponding (non-zero) eigenvectors x such that

A*x = lambda*B*x.

An alternate form is to find the eigenvalues mu and corresponding eigenvectors y such that

mu*A*y = B*y.

These two forms are equivalent with mu = 1/lambda and x = y if neither lambda nor mu is zero. In order to deal with the case that lambda or mu is zero or small, two values alpha and beta are returned for each eigenvalue, such that lambda = alpha/beta and mu = beta/alpha. ✨

The vectors x and y in the above equations are right eigenvectors of the matrix pair (A,B). Vectors u and v satisfying

u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B

are left eigenvectors of (A,B).

And then the source just drops this line, verbatim, bless it:

!> Note: this routine performs  on A and B

that's it, that's the sentence, no cap it's just like that in the original 🙈

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🌈 Parameters

JOBVL

!>          JOBVL is CHARACTER*1
!>          = 'N':  do not compute the left generalized eigenvectors;
!>          = 'V':  compute the left generalized eigenvectors (returned
!>                  in VL).

so 'N' = nope, skip the left ones; 'V' = yes please, they land in VL 💜

JOBVR

!>          JOBVR is CHARACTER*1
!>          = 'N':  do not compute the right generalized eigenvectors;
!>          = 'V':  compute the right generalized eigenvectors (returned
!>                  in VR).

same vibe, except the right eigenvectors go in VR 🫶

N

!>          N is INTEGER
!>          The order of the matrices A, B, VL, and VR.  N >= 0.

the order of the matrices A, B, VL, and VR — and it's gotta be N >= 0 ⭐

A

!>          A is COMPLEX*16 array, dimension (LDA, N)
!>          On entry, the matrix A.
!>          If JOBVL = 'V' or JOBVR = 'V', then on exit A
!>          contains the Schur form of A from the generalized Schur
!>          factorization of the pair (A,B) after balancing.  If no
!>          eigenvectors were computed, then only the diagonal elements
!>          of the Schur form will be correct.  See ZGGHRD and ZHGEQZ
!>          for details.

main character matrix incoming on entry, and on exit it's giving Schur form energy (after balancing). If you skipped the eigenvectors, only the diagonal elements are correct — see ZGGHRD and ZHGEQZ for the deets 👑

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 matrix B.
!>          If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
!>          upper triangular matrix obtained from B in the generalized
!>          Schur factorization of the pair (A,B) after balancing.
!>          If no eigenvectors were computed, then only the diagonal
!>          elements of B will be correct.  See ZGGHRD and ZHGEQZ for
!>          details.

LDB

!>          LDB is INTEGER
!>          The leading dimension of B.  LDB >= max(1,N).

ALPHA

!>          ALPHA is COMPLEX*16 array, dimension (N)
!>          The complex scalars alpha that define the eigenvalues of
!>          GNEP.

BETA

!>          BETA is COMPLEX*16 array, dimension (N)
!>          The complex scalars beta that define the eigenvalues of GNEP.
!>
!>          Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
!>          represent the j-th eigenvalue of the matrix pair (A,B), in
!>          one of the forms lambda = alpha/beta or mu = beta/alpha.
!>          Since either lambda or mu may overflow, they should not,
!>          in general, be computed.

🚨 real talk: either lambda or mu may overflow, so in general you should not compute them. Keep alpha and beta as-is, bestie, that's the safe route 😭

VL

!>          VL is COMPLEX*16 array, dimension (LDVL,N)
!>          If JOBVL = 'V', the left eigenvectors u(j) are stored
!>          in the columns of VL, in the same order as their eigenvalues.
!>          Each eigenvector is scaled so that its largest component has
!>          abs(real part) + abs(imag. part) = 1, except for eigenvectors
!>          corresponding to an eigenvalue with alpha = beta = 0, which
!>          are set to zero.
!>          Not referenced if JOBVL = 'N'.

scaling tea ☕: biggest component gets abs(real part) + abs(imag. part) = 1, except the alpha = beta = 0 ones — those are just zero. And if JOBVL = 'N', this array isn't even referenced 💅

LDVL

!>          LDVL is INTEGER
!>          The leading dimension of the matrix VL. LDVL >= 1, and
!>          if JOBVL = 'V', LDVL >= N.

VR

!>          VR is COMPLEX*16 array, dimension (LDVR,N)
!>          If JOBVR = 'V', the right eigenvectors x(j) are stored
!>          in the columns of VR, in the same order as their eigenvalues.
!>          Each eigenvector is scaled so that its largest component has
!>          abs(real part) + abs(imag. part) = 1, except for eigenvectors
!>          corresponding to an eigenvalue with alpha = beta = 0, which
!>          are set to zero.
!>          Not referenced if JOBVR = 'N'.

same scaling rules, right-side edition 🦋

LDVR

!>          LDVR is INTEGER
!>          The leading dimension of the matrix VR. LDVR >= 1, and
!>          if JOBVR = 'V', LDVR >= N.

WORK

!>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

if INFO = 0, then WORK(1) hands you the optimal LWORK on the way out 🎁

LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK.  LWORK >= max(1,2*N).
!>          For good performance, LWORK must generally be larger.
!>          To compute the optimal value of LWORK, call ILAENV to get
!>          blocksizes (for ZGEQRF, ZUNMQR, and ZUNGQR.)  Then compute:
!>          NB  -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR;
!>          The optimal LWORK is  MAX( 2*N, N*(NB+1) ).
!>
!>          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.

so: minimum is LWORK >= max(1,2*N), but for good performance you wanna go bigger, fr fr. The whole ILAENV blocksize recipe is right there — NB is the MAX of the blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR, and the optimal LWORK is MAX( 2*N, N*(NB+1) ).

And the iconic move: LWORK = -1 means workspace query — the routine just calculates the optimal size of the WORK array, returns it as the first entry of WORK, and XERBLA stays quiet about LWORK 💀

RWORK

!>          RWORK is DOUBLE PRECISION array, dimension (8*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.  No eigenvectors have been
!>                calculated, but ALPHA(j) and BETA(j) should be
!>                correct for j=INFO+1,...,N.
!>          > N:  errors that usually indicate LAPACK problems:
!>                =N+1: error return from ZGGBAL
!>                =N+2: error return from ZGEQRF
!>                =N+3: error return from ZUNMQR
!>                =N+4: error return from ZUNGQR
!>                =N+5: error return from ZGGHRD
!>                =N+6: error return from ZHGEQZ (other than failed
!>                                               iteration)
!>                =N+7: error return from ZTGEVC
!>                =N+8: error return from ZGGBAK (computing VL)
!>                =N+9: error return from ZGGBAK (computing VR)
!>                =N+10: error return from ZLASCL (various calls)

okay so the vibe check on INFO: 0 = slay, you're done. < 0 = argument -i was illegal. =1,...,N = the QZ iteration failed, no eigenvectors calculated, but ALPHA(j) and BETA(j) should still be correct for j=INFO+1,...,N. And > N is the "usually indicates LAPACK problems" zone with all those =N+1 through =N+10 error returns listed above — keep that list, bestie, it's literally the map 🗺️

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🌸 Author

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

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⭐ Further Details

Balancing

This driver calls ZGGBAL to both permute and scale rows and columns of A and B. The permutations PL and PR are chosen so that PL*A*PR and PL*B*R will be upper triangular except for the diagonal blocks A(i:j,i:j) and B(i:j,i:j), with i and j as close together as possible. The diagonal scaling matrices DL and DR are chosen so that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to one (except for the elements that start out zero.) 🎀

After the eigenvalues and eigenvectors of the balanced matrices have been computed, ZGGBAK transforms the eigenvectors back to what they would have been (in perfect arithmetic) if they had not been balanced. 🧚

Contents of A and B on Exit

If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both), then on exit the arrays A and B will contain the complex Schur form[*] of the versions of A and B. If no eigenvectors are computed, then only the diagonal blocks will be correct.

[*] In other words, upper triangular form. 💅

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📌 Definition

Definition at line 280 of file zgegv.f.

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👑 Author

Generated automatically by Doxygen for LAPACK from the source code. ✨

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that's the whole page, bestie 🫶 — it's LAPACK Version 3.12.0, and the footer path is /home/abui...TED/zgegv.f(3). go forth and compute those eigenvalues 🌈💖