SRC/cggev3.f(3)

✨ NAME ✨

okay so bestie, this is the Library Functions Manual page for SRC/cggev3.f — yes, the cggev3 docs. It’s giving “blocked-algorithm generalized eigenvalues” energy, and we’re gonna go through every single detail. 💅

🦋 SYNOPSIS 🦋

Functions/Subroutines

• subroutine cggev3 (...) — CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

subroutine cggev3 (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl,
    ldvl, vr, ldvr, work, lwork, rwork, info)

💖 Function/Subroutine Documentation 💖

subroutine cggev3

subroutine cggev3 (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)

CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

🌸 Purpose 🌸

CGGEV3 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.

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.
• 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: 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 cggev3.f.

💫 Author 💫

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🫶 Final sign-off, bestie: this page is from LAPACK, Version 3.12.0, and the page is SRC/cggev3.f(3). Periodt. 💖