SRC/zggevx.f(3)

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🌸 SRC/zggevx.f — Library Functions Manual 🌸

okay bestie, welcome to the zggevx page 💖 this one is giving generalized eigenvalues with main character energy ✨

🎀 NAME 🎀

SRC/zggevx.f

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine zggevx (...) — ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 🦋

       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)

ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 💅

💅 Function/Subroutine Documentation 💅

🌟 subroutine zggevx 🌟

   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 🫶

🌈 Purpose 🌈

Bestie, here's the whole vibe: ZGGEVX computes, for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally the left and/or right generalized eigenvectors 💖

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

A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*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 👑

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) 🦋

💖 Parameters 💖

`BALANC`

BALANC is CHARACTER*1. It 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.

Heads up, bestie: 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 👀

`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. It 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) 😭

`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.

`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.

`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.

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).

Important vibe check: 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 — DOUBLE PRECISION array, dimension (lrwork). And bestie, listen: lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', and at least max(1,2*N) otherwise. It's the real workspace, doing the real work 🫶
IWORK — INTEGER array, dimension (N+2). Fun fact: if SENSE = 'E', IWORK is literally not referenced. She's off the clock 💅
BWORK — LOGICAL array, dimension (N). Same energy: if SENSE = 'N', BWORK is not referenced. Iconic of her honestly ✨
INFO — INTEGER, and this one's the vibe check at the end, so pay attention 👑
- = 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. So like, partial credit fr fr 💀
- > 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

Okay so this is the part where it gets a little unhinged but stay with me 🌸

Balancing a matrix pair (A,B) includes, first, permuting rows and columns to isolate eigenvalues, and second, applying a 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 — so yeah, that's a real difference, no cap. 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. Go read it, bestie, it's giving required reading 🧚

Definition at line 370 of file zggevx.f.

🎀 Author

Generated automatically by Doxygen for LAPACK from the source code.

And that's the whole page, bestie — you made it to the end 🥳 This little masterpiece is LAPACK Version 3.12.0, from SRC/zggevx.f(3), and it slayed every single line. Periodt 💜🦋

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms