/home/abui...RC/zggev.f(3)

💅 ZGGEV 💅

okay so bestie, this is a page from the LAPACK manual — specifically the /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zggev.f file, and it's literally the glow-up era of linear algebra. sit down, grab a strawberry drink 🍓, and let me tell you about the queen herself: ZGGEV. she computes eigenvalues and (if you ask nicely) the left and/or right generalized eigenvectors for GE matrices. iconic behavior only 👑

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✨ NAME ✨

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

like, that's the whole address. she lives there. that's her home 🏠💖

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✨ SYNOPSIS ✨

Functions/Subroutines

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

she computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 💫

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✨ Function/Subroutine Documentation ✨

subroutine ZGGEV (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)

she computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices. no cap, that's the whole vibe 🦋

🎀 Purpose 🎀

!> !> ZGGEV 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). !>

so basically: lambda can be written as alpha/beta, and that matters because when beta is zero (or even when both are zero) there's still a reasonable interpretation — she said what she said 💅. the right eigenvector satisfies A * v(j) = lambda(j) * B * v(j) and the left one satisfies u(j)**H * A = lambda(j) * u(j)**H * B, where u(j)**H is the conjugate-transpose of u(j). math is mathing 🔥

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🌸 Parameters 🌸

• JOBVL — CHARACTER*1
  • = 'N': do not compute the left generalized eigenvectors;
  • = 'V': compute the left generalized eigenvectors.

like, do you want the left ones or not, bestie? that's the question 🙈

• JOBVR — CHARACTER*1
  • = 'N': do not compute the right generalized eigenvectors;
  • = 'V': compute the right generalized eigenvectors.

same energy, but for the right side 💫

• N — INTEGER
  • The order of the matrices A, B, VL, and VR. N >= 0.

she's the size of the whole squad 👑

• A — COMPLEX*16 array, dimension (LDA, N)
  • On entry, the matrix A in the pair (A,B).
  • On exit, A has been overwritten.

heads up: she gets overwritten on the way out, so don't get attached 😭

• LDA — INTEGER
  • The leading dimension of A. LDA >= max(1,N).
• B — COMPLEX*16 array, dimension (LDB, N)
  • On entry, the matrix B in the pair (A,B).
  • On exit, B has been overwritten.

also overwritten. also don't get attached. fr fr 💀

• LDB — INTEGER
  • The leading dimension of B. LDB >= max(1,N).
• ALPHA — COMPLEX*16 array, dimension (N)
• BETA — COMPLEX*16 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).

so like, don't just divide them all willy-nilly, bestie — the quotient can over- or underflow and BETA(j) can straight up be zero. but good news: ALPHA is always less than (and usually comparable with) norm(A) in magnitude, and BETA is always less than (and usually comparable with) norm(B) 🫶

• VL — COMPLEX*16 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'.

cute little scaling detail: biggest component always has abs(real part) + abs(imag. part) = 1 ✨

• LDVL — INTEGER
  • The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.
• VR — COMPLEX*16 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'.

same deal, right side this time 🦋

• LDVR — INTEGER
  • The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.
• WORK — COMPLEX*16 array, dimension (MAX(1,LWORK))
  • On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

she's the workspace, and on the way out she's gonna tell you the optimal LWORK if INFO = 0 💜

• LWORK — 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.

so if you set LWORK = -1 she's just doing a little reconnaissance mission 🕵️‍♀️ — calculates the optimal size of WORK, drops it in the first entry, and XERBLA stays quiet about it. no drama.

• RWORK — DOUBLE PRECISION array, dimension (8*N)
• 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 ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N.
  • > N: =N+1: other then QZ iteration failed in ZHGEQZ,
    • =N+2: error return from ZTGEVC.

read the INFO values carefully bestie, she's telling you exactly what went down 😤 — 0 is a slay, negative means bad argument, 1 through N means the QZ iteration failed (no eigenvectors, but ALPHA(j) and BETA(j) should still be correct for j=INFO+1,...,N), and anything > N is extra drama: =N+1 means other then QZ iteration failed in ZHGEQZ, and =N+2 means error return from ZTGEVC 💀

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👑 Author 👑

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

the whole academic friend group co-wrote this 💖

Definition at line 215 of file zggev.f.

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🎀 Author (of the page itself) 🎀

Generated automatically by Doxygen for LAPACK from the source code. she was built by the machines, and honestly? she ate 🤖✨

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and that's the page bestie! 🫶 this whole thing is from LAPACK Version 3.12.0 — signed, sealed, and delivered with love from /home/abui...RC/zggev.f(3). go forth and compute those generalized eigenvalues, you absolute main character 🌈👑💅