ggev(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🎀 NAME 🎀

This is ggev(3) from the Library Functions Manual, bestie — and it's giving generalized eigenvalue core. Part 1, let's go. 💖

ggev — ggev: eig, unblocked

🌈 SYNOPSIS 🌈

✨ Functions ✨

  • cggev
subroutine cggev (character jobvl, character jobvr, integer n, complex,
       dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b,
       integer ldb, complex, dimension( * ) alpha, complex, dimension( * )
       beta, complex, dimension( ldvl, * ) vl, integer ldvl, complex,
       dimension( ldvr, * ) vr, integer ldvr, complex, dimension( * ) work,
       integer lwork, real, dimension( * ) rwork, integer info)

CGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices.

  • dggev
subroutine dggev (character jobvl, character jobvr, 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, double
       precision, dimension( * ) work, integer lwork, integer info)

DGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices.

  • sggev
subroutine sggev (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta,
       vl, ldvl, vr, ldvr, work, lwork, info)

SGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices.

  • zggev
subroutine zggev (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl,
       ldvl, vr, ldvr, work, lwork, rwork, info)

ZGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices.

💫 Detailed Description 💫

This section is lowkey just a landing pad — the real function tea is right below. 🫶

🔥 Function Documentation 🔥

🧚 subroutine cggev

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

CGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices.

🌸 Purpose:

Okay so CGGEV 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. 💅

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. No cap.

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

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

The left generalized eigenvector u(j) corresponding to the generalized eigenvalues 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

  • 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.
  • N — N is INTEGER
    • The order of the matrices A, B, VL, and VR. N >= 0.
  • A — A is COMPLEX array, dimension (LDA, N)
    • On entry, the matrix A in the pair (A,B).
    • On exit, A has been overwritten. So don't expect your original A to survive, bestie. 😭
  • LDA — LDA is INTEGER
    • The leading dimension of A. LDA >= max(1,N).
  • B — B is COMPLEX array, dimension (LDB, N)
    • On entry, the matrix B in the pair (A,B).
    • On exit, B has been overwritten.
  • LDB — LDB is INTEGER
    • The leading dimension of B. LDB >= max(1,N).
  • 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.
Note, bestie: 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).
  • VL — VL is COMPLEX 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 is scaled so the largest component has 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 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 is scaled so the largest component has 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.
  • 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. LWORK >= max(1,2*N).
    • For good performance, LWORK must generally be larger.
    • 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 is REAL 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: =N+1: other then QZ iteration failed in CHGEQZ, =N+2: error return from CTGEVC.

👑 Author

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

📍 Definition at line 215 of file cggev.f.

🦋 subroutine dggev

subroutine dggev (character jobvl, character jobvr, 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, double
       precision, dimension( * ) work, integer lwork, integer info)

DGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices.

🌸 Purpose:

DGGEV 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. ✨

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

  • 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.
  • N — N is INTEGER
    • The order of the matrices A, B, VL, and VR. N >= 0.
  • 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.
  • LDA — LDA is INTEGER
    • The leading dimension of A. LDA >= max(1,N).
  • 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.
  • LDB — LDB is INTEGER
    • The leading dimension of B. LDB >= max(1,N).
  • ALPHAR — ALPHAR is DOUBLE PRECISION array, dimension (N)
  • ALPHAI — ALPHAI is DOUBLE PRECISION array, dimension (N)
  • BETA
  • 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.
    • ⚠️ note note note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. So do not go naively computing the ratio alpha/beta, bestie — that's a trap and it will not slay. 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 like, they're well-behaved, it's fine 💅
  • VL — 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) — literally just the j-th column of VL, iconic.
    • 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 is scaled so the largest component has abs(real part)+abs(imag. part)=1.
    • Not referenced if JOBVL = 'N' — she's off duty, don't bother her.
  • LDVL — INTEGER
    • The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.
  • VR — 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 is scaled so the largest component has abs(real part)+abs(imag. part)=1.
    • Not referenced if JOBVR = 'N'.
  • LDVR — INTEGER
    • The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.
  • WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    • On exit, if INFO = 0, WORK(1) returns the optimal LWORK. she's telling you what she wants, listen to her 🫶
  • LWORK — INTEGER
    • The dimension of the array WORK. LWORK >= max(1,8*N).
    • For good performance, LWORK must generally be larger. (bigger workspace = better vibes, fr fr)
    • 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.
  • INFO — 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 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 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 224 of file dggev.f. ⭐

subroutine sggev — the REAL one 🦋

subroutine sggev (character jobvl, character jobvr, integer n, real,
    dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer
    ldb, real, dimension( * ) alphar, real, dimension( * ) alphai, real,
    dimension( * ) beta, real, dimension( ldvl, * ) vl, integer ldvl, real,
    dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) work,
    integer lwork, integer info)

SGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices ✨

Purpose 🎀

okay so SGGEV 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. that's her whole job and she ate 💅

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 really, both zero is allowed, no cap.

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 💖

  • JOBVL — CHARACTER*1
    • = 'N': do not compute the left generalized eigenvectors;
    • = 'V': compute the left generalized eigenvectors. 👑
  • JOBVR — CHARACTER*1
    • = 'N': do not compute the right generalized eigenvectors;
    • = 'V': compute the right generalized eigenvectors. 👑
  • N — INTEGER
    • The order of the matrices A, B, VL, and VR. N >= 0.
  • A — REAL array, dimension (LDA, N)
    • On entry, the matrix A in the pair (A,B).
    • On exit, A has been overwritten. 😭 she's gone, mourned, moving on.
  • LDA — INTEGER
    • The leading dimension of A. LDA >= max(1,N).
  • B — REAL array, dimension (LDB, N)
    • On entry, the matrix B in the pair (A,B).
    • On exit, B has been overwritten. also gone. it's giving sacrifice for the greater good 🫶
  • LDB — INTEGER
    • The leading dimension of B. LDB >= max(1,N).
  • ALPHAR — REAL array, dimension (N)
  • ALPHAI — REAL array, dimension (N)
  • BETA — REAL 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.
    • ⚠️ 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. 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). 💅
  • VL — REAL 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 is scaled so the largest component has abs(real part)+abs(imag. part)=1.
    • Not referenced if JOBVL = 'N'.
  • LDVL — INTEGER
    • The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.
  • VR — REAL 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 is scaled so the largest component has abs(real part)+abs(imag. part)=1.
    • Not referenced if JOBVR = 'N'.
  • LDVR — INTEGER
    • The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.
  • WORK — REAL array, dimension (MAX(1,LWORK))
    • On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ⭐
  • LWORK — INTEGER
    • The dimension of the array WORK. LWORK >= max(1,8*N).
    • For good performance, LWORK must generally be larger.
    • 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.
  • INFO — 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 ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N.
    • > N: =N+1: other than QZ iteration failed in SHGEQZ. =N+2: error return from STGEVC. 💀

Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 224 of file sggev.f.

subroutine zggev (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)

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

✨ Purpose ✨

Okay so, here's the tea: ZGGEV 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. 🦋

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. no cap, that pair is doing main character work 💖

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

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

The left generalized eigenvector u(j) corresponding to the generalized eigenvalues 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 literally the moment, iykyk ✨

💅 Parameters 💅

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.

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. rip original A, but it served looks 🫶

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. also getting the overwrite treatment, periodt 💜

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 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). bestie do NOT just divide all willy-nilly, that is the warning 😭

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 is scaled so the largest component has 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 is scaled so the largest component has 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.

WORK

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. slay, workspace said “I got you” ✨

LWORK

LWORK is INTEGER — the dimension of the array WORK. LWORK >= max(1,2*N).

For good performance, LWORK must generally be larger.

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. query mode is lowkey iconic 🎀

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: =N+1: other then QZ iteration failed in ZHGEQZ, =N+2: error return from ZTGEVC. when this happens, it is giving “the math said no” energy, fr fr 💀

💖 Author 💖

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

Definition at line 215 of file zggev.f.

🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

And that's the whole vibe, bestie — signing off from the LAPACK Version 3.12.0 ggev(3) corner with sparkle and respect for the QZ iteration 🫶✨💖

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