SGEGV(3)

📦 version 3.12.0, in opensuse-leap 16.0

🌸 SGEGV(3) — like, the whole vibe 🌸

okay bestie, so this is a page from the Library Functions Manual, and it's all about sgegv.f — yes the (3) means it's a library function, we love a categorised queen 👑. The page lives at this absolutely unhinged path, and no I'm not making it up:

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

(and yeah, DEPRECATED is literally in the path. we'll get to that energy later, but for now just know it's giving "vintage" 💅)

✨ NAME ✨

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

like... that's the name. that's the whole name. it's a whole file path and honestly it ate 🍓

🦋 SYNOPSIS 🦋

Functions/Subroutines

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

here's the signature, copy it exactly bestie, no typos, we do NOT freehand a Fortran signature in this house 🙈:

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

so what does this icon actually do? SGEGV computes the eigenvalues</p> — and, if you want them, optionally the left and/or right eigenvectors of a real matrix pair (A,B)</p>. that's it. that's the whole sell. eigenvalues? slayed. eigenvectors? only if you ask nicely, which is what the jobvl and jobvr args are for 😌✨

no cap, that's the entire function in one breath, and it's lowkey iconic of it to just say the thing and dip. more of this page is coming in part 2, so don't go anywhere — we're not done yet, bestie 🫶

💅 Function/Subroutine Documentation 💅

okay bestie, here's the actual star of the show — the subroutine herself:

   subroutine SGEGV (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)

SGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B). like, that's her whole job description and she ate 💅

✨ Purpose ✨

🚨 !> This routine is deprecated and has been replaced by routine SGGEV. 🚨

like she still works, but she's the old model and there's a newer one, no cap.

SGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B).

Given two square matrices A and B, the generalized nonsymmetric eigenvalue problem (GNEP) is to find the eigenvalues lambda and corresponding (non-zero) eigenvectors x such that

A*x = lambda*B*x.

An alternate form is to find the eigenvalues mu and corresponding eigenvectors y such that

mu*A*y = B*y.

These two forms are equivalent with mu = 1/lambda and x = y if neither lambda nor mu is zero. In order to deal with the case that lambda or mu is zero or small, two values alpha and beta are returned for each eigenvalue, such that lambda = alpha/beta and mu = beta/alpha. so you're never dividing by something sus 🫶

The vectors x and y in the above equations are right eigenvectors of the matrix pair (A,B). Vectors u and v satisfying

u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B

are left eigenvectors of (A,B).

Note: this routine performs on A and B

(yes that sentence is literally cut off in the source, the double space and everything — she left us on read 😭)

🎀 Parameters 🎀

JOBVL

JOBVL is CHARACTER*1

  • = 'N': do not compute the left generalized eigenvectors;
  • = 'V': compute the left generalized eigenvectors (returned in VL).

JOBVR

JOBVR is CHARACTER*1

  • = 'N': do not compute the right generalized eigenvectors;
  • = 'V': compute the right generalized eigenvectors (returned in VR).

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.

If JOBVL = 'V' or JOBVR = 'V', then on exit A contains the real Schur form of A from the generalized Schur factorization of the pair (A,B) after balancing.

If no eigenvectors were computed, then only the diagonal blocks from the Schur form will be correct. See SGGHRD and SHGEQZ for details. ngl if that's your situation, only part of A is slaying and the rest is just there 💀

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.

If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the upper triangular matrix obtained from B in the generalized Schur factorization of the pair (A,B) after balancing.

If no eigenvectors were computed, then only those elements of B corresponding to the diagonal blocks from the Schur form of A will be correct. See SGGHRD and SHGEQZ for details.

LDB

LDB is INTEGER

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

ALPHAR

ALPHAR is REAL array, dimension (N)

The real parts of each scalar alpha defining an eigenvalue of GNEP. 💖

ALPHAI

ALPHAI is REAL array, dimension (N)

The imaginary parts of each scalar alpha defining an eigenvalue of GNEP.

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) = -ALPHAI(j). they come as a matching set, iconic duo energy 🦋

BETA

BETA is REAL array, dimension (N)

The scalars beta that define the eigenvalues of GNEP.

Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed. 🚨 like fr fr, don't go there.

VL

VL is REAL array, dimension (LDVL,N)

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

If the j-th eigenvalue is real, then u(j) = VL(:,j).

If the j-th and (j+1)-st 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 that its largest component has abs(real part) + abs(imag. part) = 1, except for eigenvectors corresponding to an eigenvalue with alpha = beta = 0, which are set to zero.

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 x(j) are stored in the columns of VR, in the same order as their eigenvalues.

If the j-th eigenvalue is real, then x(j) = VR(:,j).

If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then

  x(j) = VR(:,j) + i*VR(:,j+1)

and

  x(j+1) = VR(:,j) - i*VR(:,j+1).

Each eigenvector is scaled so that its largest component has abs(real part) + abs(imag. part) = 1, except for eigenvalues corresponding to an eigenvalue with alpha = beta = 0, which are set to zero.

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 tells you how much room she actually wanted.

LWORK

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,8*N).

For good performance, LWORK must generally be larger.

To compute the optimal value of LWORK, call ILAENV to get blocksizes (for SGEQRF, SORMQR, and SORGQR.) Then compute:

NB -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR;

The optimal LWORK is:

    2*N + MAX( 6*N, N*(NB+1) ).

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 -1 is the "just tell me how much" cheat code ✨

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: errors that usually indicate LAPACK problems:
    • =N+1: error return from SGGBAL
    • =N+2: error return from SGEQRF
    • =N+3: error return from SORMQR
    • =N+4: error return from SORGQR
    • =N+5: error return from SGGHRD
    • =N+6: error return from SHGEQZ (other than failed iteration)
    • =N+7: error return from STGEVC
    • =N+8: error return from SGGBAK (computing VL)
    • =N+9: error return from SGGBAK (computing VR)
    • =N+10: error return from SLASCL (various calls)

👑 Author 👑

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

the whole squad showed up for this one 💜

🌈 Further Details: 🌈

Balancing

---------

This driver calls SGGBAL to both permute and scale rows and columns of A and B. The permutations PL and PR are chosen so that PL*A*PR and PL*B*R will be upper triangular except for the diagonal blocks A(i:j,i:j) and B(i:j,i:j), with i and j as close together as possible. The diagonal scaling matrices DL and DR are chosen so that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to one (except for the elements that start out zero.) 🧚 bestie she's tidying up the vibes before the real work starts.

After the eigenvalues and eigenvectors of the balanced matrices have been computed, SGGBAK transforms the eigenvectors back to what they would have been (in perfect arithmetic) if they had not been balanced.

Contents of A and B on Exit

-------- -- - --- - -- ----

If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both), then on exit the arrays A and B will contain the real Schur form[*] of the versions of A and B. If no eigenvectors are computed, then only the diagonal blocks will be correct.

[*] See SHGEQZ, SGEGS, or read the book , by Golub & van Loan, pub. by Johns Hopkins U. Press.

(yes the book title is a blank in the source, we're not gonna pretend otherwise 🙈)

Definition at line 304 of file sgegv.f.

Author (the auto-generated one)

Generated automatically by Doxygen for LAPACK from the source code. 💅 a robot wrote this bit, and honestly, respect.

that's the whole page, bestie — she's deprecated but she's still main character energy 👑✨

LAPACK · Version 3.12.0 · source file /home/abui...TED/sgegv.f(3) — that's a wrap, go compute something iconic 🎀🦋

📦 where this page comes from