SRC/dggevx.f(3)

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version 3.12.1 · 4 versions

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NAME

SRC/dggevx.f — that's the name, bestie. The page itself is SRC/dggevx.f(3) from the Library Functions Manual, and she's ready to spill the whole routine. 💅

SYNOPSIS

Functions/Subroutines

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

DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices. Periodt. ✨

Function/Subroutine Documentation

   subroutine dggevx (character balanc, character jobvl, character jobvr,
       character sense, integer n, double precision, dimension( lda, * ) a,
       integer lda, double precision, dimension( ldb, * ) b, integer ldb,
       double precision, dimension( * ) alphar, double precision, dimension( *
       ) alphai, double precision, dimension( * ) beta, double precision,
       dimension( ldvl, * ) vl, integer ldvl, double precision, 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, double precision,
       dimension( * ) work, integer lwork, integer, dimension( * ) iwork,
       logical, dimension( * ) bwork, integer info)

DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices. That's the whole assignment, and she ate. 💖

Purpose

Okay so bestie, here's the vibe: DGGEVX computes, for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally the left and/or right generalized eigenvectors. Iconic behavior, no cap. 🦋

Optionally it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV). That's a lot of accessories, but she carries them all. 🎀

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. Yes, even both zero — she said what she said. 💀

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). Math but make it fashion. 💅

Parameters

BALANC

BALANC is CHARACTER*1 — it specifies the balance option to be performed. Choose your fighter:

'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 yeah, balancing is the one with main character energy here. 👑

JOBVL

JOBVL is CHARACTER*1:

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

Left eigenvector girlies, this one's for you. 💜

JOBVR

JOBVR is CHARACTER*1:

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

Right eigenvector stans, rise. 🌸

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.

Pick your condition-number combo, bestie. ✨

N

N is INTEGER — the order of the matrices A, B, VL, and VR. N >= 0. No negative order shenanigans, okay? 🚫

A

A is DOUBLE PRECISION 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 real Schur form of the versions of the input A and B. She changes outfits, but it's still her. 🎀

LDA

LDA is INTEGER — the leading dimension of A. LDA >= max(1,N). Respect the leading dimension, periodt. 💅

B

B is DOUBLE PRECISION 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 real Schur form of the versions of the input A and B. Matching set with A, honestly. 🦋

LDB

LDB is INTEGER — the leading dimension of B. LDB >= max(1,N). Same rule, different matrix. ✨

ALPHAR

ALPHAR is DOUBLE PRECISION array, dimension (N). That's the whole note — short and iconic. 💖

ALPHAI

ALPHAI is DOUBLE PRECISION array, dimension (N). Same energy, bestie. 🦋

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. 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. Complex-conjugate besties holding hands. 🫶

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 ALPHA/BETA. This is not a drill, bestie — do not just divide and hope for the best. 😭

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). So the pieces themselves are well-behaved; it's the naive ratio that's giving chaos. 💀

VL

VL is DOUBLE PRECISION array, dimension (LDVL,N).

If JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. If the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). Each eigenvector will be scaled so the largest component 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. Check it before you wreck it. ✨

VR

VR is DOUBLE PRECISION array, dimension (LDVR,N).

If JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. If the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). Each eigenvector will be scaled so the largest component 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. Leading dimension said what she said. 💅

ILO

ILO is INTEGER. Cute little integer, but she matters. 🎀

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. So those are the bounds, bestie — write them down. ✨

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. Order matters, okay? 💜

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 order drama, right-side edition. 🦋

ABNRM

ABNRM is DOUBLE PRECISION — the one-norm of the balanced matrix A. Norm girlie has entered the chat. ✨

BBNRM

BBNRM is DOUBLE PRECISION — the one-norm of the balanced matrix B. Matching norm, matching energy. 🎀

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. For a complex conjugate pair of eigenvalues two consecutive elements of RCONDE are set to the same value. Thus RCONDE(j), RCONDV(j), and the j-th columns of VL and VR all correspond to the j-th eigenpair. Everything lines up, bestie — it's giving coordinated. 💅

If SENSE = 'N' or 'V', RCONDE is not referenced. She's off duty in that case. 🙈

RCONDV

RCONDV — DOUBLE PRECISION array, dimension (N). If SENSE = 'V' or 'B', this is where the estimated reciprocal condition numbers of the eigenvectors live — stored in consecutive elements of the array, bestie. For a complex eigenvector, two consecutive elements of RCONDV get set to the same value. If the eigenvalues can’t be reordered to compute RCONDV(j), then RCONDV(j) is set to 0; this can only happen when the true value would be very small anyway. If SENSE = 'N' or 'E', RCONDV is literally not referenced at all. 💅
WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. Slay. ✨
LWORK — INTEGER. The dimension of the array WORK.
- LWORK >= max(1,2*N).
- If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', then LWORK >= max(1,6*N).
- If SENSE = 'E' or 'B', then LWORK >= max(1,10*N).
- If SENSE = 'V' or 'B', then LWORK >= 2*N*N+8*N+16.

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. It’s giving “just checking the vibe, not doing the work yet.” 🫶

IWORK — INTEGER array, dimension (N+6). If SENSE = 'E', IWORK is not referenced. Bye bestie. 👋
BWORK — LOGICAL array, dimension (N). If SENSE = 'N', BWORK is not referenced. Not me ignoring BWORK when SENSE = 'N' 💀
INFO — INTEGER. The tea on what happened:
- = 0: successful exit. We love a happy ending. 🎀
- < 0: if INFO = -i, the i-th argument had an illegal value. Rude. 😭
- = 1,...,N: The QZ iteration failed. No eigenvectors have been calculated, 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 DHGEQZ. =N+2: error return from DTGEVC.

👑 Author 👑

The absolute legends behind this:

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

Periodt. 💜

🔍 Further Details: 🔍

Balancing a matrix pair (A,B) includes, first, permuting rows and columns to isolate eigenvalues, and 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, okay? 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. No cap. 📚

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. Iconic documentation behavior. 🌈

Definition at line 387 of file dggevx.f.

Author: Generated automatically by Doxygen for LAPACK from the source code. Yes, the robots wrote this one — we’re just here to make it cute. 🤖💖

And that’s the whole page, bestie! This masterpiece is from LAPACK Version 3.12.0, source file SRC/dggevx.f(3). Go off, queen. 👑✨💅

📦 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