DGEGV(3)

📦 version 3.12.0, in opensuse-leap 16.0

👑 DGEGV(3) 👑 — Library Functions Manual

okay so we are in the Library Functions Manual, and this is the page for DGEGV(3). Header energy: /home/abui...TED/dgegv.f(3) · Library Functions Manual · /home/abui...TED/dgegv.f(3). Bestie, this routine is a whole deprecated moment, but the facts are still facts. 💅

🌸 NAME 🌸

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

That's the file. Yes, the whole path. We keep receipts. 🧾✨

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

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

DGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B). Main character energy, honestly. 👑

🦋 Function/Subroutine Documentation 🦋

subroutine DGEGV (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)

DGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B). (Yes, same line again, because the source says it twice. Consistency slay. 💖)

💖 Purpose 💖

🚨 Deprecated alert: this routine is deprecated and has been replaced by routine DGGEV.

DGEGV 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.

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).

The source also has this literal line: Note: this routine performs on A and B — yes, with the blank spot, bestie. I'm not hiding it. 🙈

🎀 Parameters 🎀

  • JOBVL — CHARACTER*1.
    • = 'N': do not compute the left generalized eigenvectors;
    • = 'V': compute the left generalized eigenvectors (returned in VL).
  • JOBVR — CHARACTER*1.
    • = 'N': do not compute the right generalized eigenvectors;
    • = 'V': compute the right generalized eigenvectors (returned in VR).
  • N — INTEGER. The order of the matrices A, B, VL, and VR. N >= 0.
  • A — DOUBLE PRECISION 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 DGGHRD and DHGEQZ for details.
  • LDA — INTEGER. The leading dimension of A. LDA >= max(1,N).
  • B — DOUBLE PRECISION 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 DGGHRD and DHGEQZ for details.
  • LDB — INTEGER. The leading dimension of B. LDB >= max(1,N).
  • ALPHAR — DOUBLE PRECISION array, dimension (N). The real parts of each scalar alpha defining an eigenvalue of GNEP.
  • ALPHAI — DOUBLE PRECISION 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).
  • BETA — DOUBLE PRECISION 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. 😭

  • VL — DOUBLE PRECISION 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 — 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 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 — 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.
  • LWORK — 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 DGEQRF, DORMQR, and DORGQR.) Then compute: NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR; 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.

  • 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: errors that usually indicate LAPACK problems:
      • =N+1: error return from DGGBAL
      • =N+2: error return from DGEQRF
      • =N+3: error return from DORMQR
      • =N+4: error return from DORGQR
      • =N+5: error return from DGGHRD
      • =N+6: error return from DHGEQZ (other than failed iteration)
      • =N+7: error return from DTGEVC
      • =N+8: error return from DGGBAK (computing VL)
      • =N+9: error return from DGGBAK (computing VR)
      • =N+10: error return from DLASCL (various calls)

💜 Author 💜

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

🌈 Further Details 🌈

Further Details: — literally nothing after this in the source, so that's the whole vibe. 🫶

🌸 Balancing 🌸

ok so first things first, this driver hits up DGGBAL to permute AND scale the rows and columns of A and B. two-for-one, we love to see it 💅

the permutations PL and PR are picked so that PL*A*PR and PL*B*R come out upper triangular — except for the diagonal blocks A(i:j,i:j) and B(i:j,i:j), where i and j are squished as close together as humanly (algorithmically?) possible 🫶

then the diagonal scaling matrices DL and DR get chosen so that the pair

DL*PL*A*PR*DR, DL*PL*B*PR*DR

has elements close to one. (except, bestie, the elements that started out zero — those stay zero, no cap.) ✨

and like, once the eigenvalues and eigenvectors of the balanced matrices are computed, DGGBAK swoops in and transforms the eigenvectors back to what they'd have been in perfect arithmetic if nobody had balanced anything at all. it's giving restoration arc 😭👑

🦋 Contents of A and B on Exit 🦋

real talk, here's the tea on what you're left holding:

  • if any eigenvectors are computed (so either JOBVL='V', or JOBVR='V', or both — go off), 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. that's it. that's the deal 🙈

[*] see DHGEQZ, DGEGS, or go read the book, by Golub & van Loan, pub. by Johns Hopkins U. Press. 📚 (iykyk)

lowkey important: Definition at line 304 of file dgegv.f. — that's where this whole vibe lives, fr fr.

👑 Author 👑

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and that's a wrap on DGEGV(3) 💖 this icon is LAPACK Version 3.12.0, living its best life at /home/abui...TED/dgegv.f(3) — slay, periodt, go compute something gorgeous 🌈⭐🎀

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