SRC/DEPRECATED/zgegv.f(3)

💖 NAME 💖

SRC/DEPRECATED/zgegv.f — okay bestie, this is the LAPACK Library Functions Manual page for zgegv, and yes, it lives in the deprecated folder for a reason. 💅

✨ SYNOPSIS ✨

🌀 Functions/Subroutines 🌀

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

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

🧚 Function/Subroutine Documentation 🧚

   subroutine zgegv (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)

ZGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a complex matrix pair (A,B). Iconic behavior, honestly.

🎯 Purpose 🎯

🚨 Deprecated alert: This routine is deprecated and has been replaced by routine ZGGEV.

ZGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a complex 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 literally has this note, blank spot and all: Note: this routine performs on A and B — yes bestie, I am not filling in that blank for you. 😭

🎀 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 COMPLEX*16 array, dimension (LDA, N). On entry, the matrix A. If JOBVL = 'V' or JOBVR = 'V', then on exit A contains the 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 elements of the Schur form will be correct. See ZGGHRD and ZHGEQZ for details.
• 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. 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 the diagonal elements of B will be correct. See ZGGHRD and ZHGEQZ for details.
• LDB — LDB is INTEGER; the leading dimension of B. LDB >= max(1,N).
• ALPHA — ALPHA is COMPLEX*16 array, dimension (N). The complex scalars alpha that define the eigenvalues of GNEP.
• BETA — BETA is COMPLEX*16 array, dimension (N). The complex scalars beta that define the eigenvalues of GNEP.
  • Together, the quantities alpha = ALPHA(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 — VL is COMPLEX*16 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.
  • 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 COMPLEX*16 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.
  • 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 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.
• LWORK — LWORK is INTEGER; the dimension of the array WORK. LWORK >= max(1,2*N).
  • For good performance, LWORK must generally be larger.
  • To compute the optimal value of LWORK, call ILAENV to get blocksizes (for ZGEQRF, ZUNMQR, and ZUNGQR.) Then compute:
    • NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR;
    • The optimal LWORK is MAX( 2*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.
• 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: errors that usually indicate LAPACK problems:
    • =N+1: error return from ZGGBAL
    • =N+2: error return from ZGEQRF
    • =N+3: error return from ZUNMQR
    • =N+4: error return from ZUNGQR
    • =N+5: error return from ZGGHRD
    • =N+6: error return from ZHGEQZ (other than failed iteration)
    • =N+7: error return from ZTGEVC
    • =N+8: error return from ZGGBAK (computing VL)
    • =N+9: error return from ZGGBAK (computing VR)
    • =N+10: error return from ZLASCL (various calls)

👑 Author 👑

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

🌈 Further Details 🌈

⚖️ Balancing ⚖️

This driver calls ZGGBAL 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.)

After the eigenvalues and eigenvectors of the balanced matrices have been computed, ZGGBAK 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 complex Schur form[*] of the versions of A and B. If no eigenvectors are computed, then only the diagonal blocks will be correct.

[*] In other words, upper triangular form.

Definition at line 280 of file zgegv.f.

🤖 Author 🤖

Generated automatically by Doxygen for LAPACK from the source code.

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💖 cute sign-off: this page is from LAPACK Version 3.12.0, page SRC/DEPRECATED/zgegv.f(3) — go forth and maybe use ZGGEV instead, bestie. 💖