ZGGEVX(3)

🦋 zggevx.f(3) — the LAPACK girlies' fave 💅

okay so bestie, you found the zggevx.f page from Library Functions Manual and honestly? iconic. this is the part 1 of the glow-up and there's more coming, so don't go anywhere 🫶

✨ NAME ✨

the whole vibe of this file, literally:

/home/abuild/rpmbuild/BUILD/lapack-
static-3.12.0-build/lapack-3.12.0/SRC/zggevx.f

🌸 SYNOPSIS 🌸

Functions/Subroutines

here's the cast, main character energy of course:

subroutine ZGGEVX (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb,
    alpha, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm,
    bbnrm, rconde, rcondv, work, lwork, rwork, iwork, bwork, info)

and what does she do? she computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices ✨

🎀 Function/Subroutine Documentation 🎀

the full signature, no cap

subroutine ZGGEVX (character balanc, character jobvl, character jobvr,
    character sense, 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, integer ilo, integer ihi, double precision, dimension(
    * ) lscale, double precision, dimension( * ) rscale, double precision
    abnrm, double precision bbnrm, double precision, dimension( * ) rconde,
    double precision, dimension( * ) rcondv, complex*16, dimension( * )
    work, integer lwork, double precision, dimension( * ) rwork, integer,
    dimension( * ) iwork, logical, dimension( * ) bwork, integer info)

ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 💖 (yes, she said it twice, because she ate and left no crumbs)

🌈 Purpose 🌈

okay, so picture this 💭 — ZGGEVX takes a pair of N-by-N complex nonsymmetric matrices (A,B) and computes the generalized eigenvalues for them, plus, if you want, the left and/or right generalized eigenvectors. obsessed.

optionally she also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors — that's ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM right there 😌 — plus reciprocal condition numbers for the eigenvalues (RCONDE) and reciprocal condition numbers for the right eigenvectors (RCONDV).

so what even is a generalized eigenvalue for a pair of matrices (A,B)? bestie it's a scalar lambda, or a ratio alpha/beta = lambda, such that A - lambda*B is singular. she's usually represented as the pair (alpha,beta) because there's a reasonable interpretation for beta=0, and even for both being zero. fr fr, both zero is valid here 💀

the right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

A * v(j) = lambda(j) * B * v(j) .

the left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

u(j)**H * A  = lambda(j) * u(j)**H * B.

where u(j)**H is the conjugate-transpose of u(j). conjugate-transpose is basically the glow-up of a matrix, iykyk 🦋

💜 Parameters 💜

every single one, in order, no skips because we don't do that here 🫶

BALANC

BALANC is CHARACTER*1

specifies the balance option to be performed:

'N': do not diagonally scale or permute;
'P': permute only;
'S': scale only;
'B': both permute and scale.

computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. permuting does not change condition numbers (in exact arithmetic), but balancing does. so like, keep that in mind 👀

JOBVL

JOBVL is CHARACTER*1

'N': do not compute the left generalized eigenvectors;
'V': compute the left generalized eigenvectors.

JOBVR

JOBVR is CHARACTER*1

'N': do not compute the right generalized eigenvectors;
'V': compute the right generalized eigenvectors.

SENSE

SENSE is CHARACTER*1

determines which reciprocal condition numbers are computed:

'N': none are computed;
'E': computed for eigenvalues only;
'V': computed for eigenvectors only;
'B': computed for eigenvalues and eigenvectors.

N

N is INTEGER

the order of the matrices A, B, VL, and VR. N >= 0 ⭐

A

A is COMPLEX*16 array, dimension (LDA, N)

on entry, the matrix A in the pair (A,B).

on exit, A has been overwritten. if JOBVL='V' or JOBVR='V' or both, then A contains the first part of the complex Schur form of the versions of the input A and B. 🦋

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 in the pair (A,B).

on exit, B has been overwritten. if JOBVL='V' or JOBVR='V' or both, then B contains the second part of the complex Schur form of the versions of the input A and B. 💫

LDB

LDB is INTEGER

the leading dimension of B. LDB >= max(1,N).

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.

Note: the quotient 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). 😭 pls don't just divide them, bestie, that's the whole point

VL

VL is COMPLEX*16 array, dimension (LDVL,N)

if JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues.

each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1.

not referenced if JOBVL = 'N'.

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 generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues.

each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1.

not referenced if JOBVR = 'N'.

LDVR

LDVR is INTEGER

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

ILO

ILO is INTEGER

IHI

IHI is INTEGER

ILO and IHI are integer values such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N.

if BALANC = 'N' or 'S', ILO = 1 and IHI = N. 🎀

LSCALE

LSCALE is DOUBLE PRECISION array, dimension (N)

details of the permutations and scaling factors applied to the left side of A and B. if PL(j) is the index of the row interchanged with row j, and DL(j) is the scaling factor applied to row j, then

LSCALE(j) = PL(j)  for j = 1,...,ILO-1
          = DL(j)  for j = ILO,...,IHI
          = PL(j)  for j = IHI+1,...,N.

the order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. don't come for me, that's just how she rolls 💅

RSCALE

RSCALE is DOUBLE PRECISION array, dimension (N)

details of the permutations and scaling factors applied to the right side of A and B. if PR(j) is the index of the column interchanged with column j, and DR(j) is the scaling factor applied to column j, then

RSCALE(j) = PR(j)  for j = 1,...,ILO-1
          = DR(j)  for j = ILO,...,IHI
          = PR(j)  for j = IHI+1,...,N

the order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. same energy, other side ✨

ABNRM

ABNRM is DOUBLE PRECISION

the one-norm of the balanced matrix A. ⭐

BBNRM

BBNRM is DOUBLE PRECISION

the one-norm of the balanced matrix B. ⭐

RCONDE

RCONDE is DOUBLE PRECISION array, dimension (N)

if SENSE = 'E' or 'B', the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array.

if SENSE = 'N' or 'V', RCONDE is not referenced. 🙈

RCONDV

RCONDV is DOUBLE PRECISION array, dimension (N)

if JOB = 'V' or 'B', the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. if the eigenvalues cannot be reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can only occur when the true value would be very small anyway. so like, it's fine, lowkey 💀

if SENSE = 'N' or 'E', RCONDV is not referenced.

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).

if SENSE = 'E', LWORK >= max(1,4*N).

if SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N).

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 valid 🧚

RWORK

RWORK is a DOUBLE PRECISION array, dimension (lrwork) — and lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', and at least max(1,2*N) otherwise. Real workspace. She's real, she's workspace, she's doing the most. ✨

🦋 `IWORK`

IWORK is an INTEGER array, dimension (N+2). If SENSE = 'E', IWORK is not referenced. So she's literally just vibing offstage for that mode, bestie. 💅

🐛 `BWORK`

BWORK is a LOGICAL array, dimension (N). If SENSE = 'N', BWORK is not referenced. Not even a little bit. 🫶

💀 `INFO`

INFO is an INTEGER. Here's the tea, no cap:

= 0: successful exit. Slay. ✅
< 0: if INFO = -i, the i-th argument had an illegal value. Rude, but okay. 🙈
= 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: =N+1: other than QZ iteration failed in ZHGEQZ. =N+2: error return from ZTGEVC.

👑 Author

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

🌈 Further Details

Balancing a matrix pair (A,B) is a whole two-step moment, okay? First, permuting rows and columns to isolate eigenvalues. Second, applying diagonal similarity transformation to the rows and columns to make the rows and columns as close in norm as possible. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see section 4.11.1.2 of LAPACK Users' Guide. 🦋

An approximate error bound on the chordal distance between the i-th computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is

chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

An approximate error bound for the angle between the i-th computed eigenvector VL(i) or VR(i) is given by

EPS * norm(ABNRM, BBNRM) / DIF(i).

For further explanation of the reciprocal condition numbers RCONDE and RCONDV, see section 4.11 of LAPACK User's Guide. ✨

Definition at line 370 of file zggevx.f.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

That's the whole page, bestie — LAPACK Version 3.12.0, straight from /home/abui...RC/zggevx.f(3), with all the footer energy included. Periodt. 🎀💖✨

📦 where this page comes from

opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

ZGGEVX(3)

🦋 zggevx.f(3) — the LAPACK girlies' fave 💅

✨ NAME ✨

🌸 SYNOPSIS 🌸

Functions/Subroutines

🎀 Function/Subroutine Documentation 🎀

the full signature, no cap

🌈 Purpose 🌈

💜 Parameters 💜

BALANC

JOBVL

JOBVR

SENSE

N

A

LDA

B

LDB

ALPHA

BETA

VL

LDVL

VR

LDVR

ILO

IHI

LSCALE

RSCALE

ABNRM

BBNRM

RCONDE

RCONDV

WORK

LWORK

RWORK

🦋 IWORK

🐛 BWORK

💀 INFO

👑 Author

🌈 Further Details

👑 Author

🦋 `IWORK`

🐛 `BWORK`

💀 `INFO`