SGGEV3(3)

💅 SGGEV3(3) — like, the generalized eigenvalue icon

Okay bestie, so this page lives in the LAPACK manual (that's the manual name, iykyk) and it's literally all about SGGEV3. Buckle up 🎀

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✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sggev3.f

Yes queen, the NAME section is just the whole file path. That's the vibe. 💀

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🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine SGGEV3 (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai,
    beta, vl, ldvl, vr, ldvr, work, lwork, info)

SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm) 🔥

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🦋 Function/Subroutine Documentation 🦋

subroutine SGGEV3 (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)

SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm) ✨

🎀 Purpose 🎀

So, what does it actually do? SGGEV3 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. Main character energy fr fr 👑

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, both zero is allowed 🫶

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 the moment, honestly 💅

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💖 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 REAL array, dimension (LDA, N)

On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. She's gone, swapped out, tyvm 🙈

LDA

LDA is INTEGER

The leading dimension of A. LDA >= max(1,N).

B

B is REAL array, dimension (LDB, N)

On entry, the matrix B in the pair (A,B). On exit, B has been overwritten too. Both of them got a makeover 💀

LDB

LDB is INTEGER

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

ALPHAR

ALPHAR is REAL array, dimension (N)

(That's it, that's the whole description. She's mysterious 💜)

ALPHAI

ALPHAI is REAL array, dimension (N)

(Also mysterious. Iconic, honestly.)

BETA

BETA is 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. Slay, conjugate pairs 💅

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).

So like… don't just blindly divide, bestie. That ratio can go up in flames 😭

VL

VL is 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

LDVL is INTEGER

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

VR

VR is 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

LDVR is INTEGER

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

WORK

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. She's telling you how much room she really needed 🫶

LWORK

LWORK is INTEGER

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 LWORK = -1 is the "just tell me how much space you need, no drama" mode ✨

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 ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N.
• > N: =N+1: other than QZ iteration failed in SLAQZ0. =N+2: error return from STGEVC.

Yeah, non-zero INFO is giving "something went sideways" 💀 Read it carefully, bestie.

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👑 Author 👑

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

Definition at line 223 of file sggev3.f.

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🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code.

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And that's the whole page, bestie — no details left behind, periodt. 🎀 This was LAPACK, Version 3.12.0, page /home/abui...RC/sggev3.f(3). Mwah 💋