SRC/sggev.f(3)

SRC/sggev.f(3) 💅

Library Functions Manual — folding the manual name right in here so you have the whole vibe up top, bestie. ✨

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🌸 NAME 🌸

SRC/sggev.f — like, that’s literally the file, no cap. 🎀

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✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

• subroutine sggev (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info) — SGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices. Slay. 💖

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🌈 Function/Subroutine Documentation 🌈

👑 subroutine sggev 👑

subroutine sggev (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)

SGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices. Iconic behavior, honestly. 🔥

🫶 Purpose:

Okay so bestie, SGGEV computes, for a pair of N-by-N real 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 eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies:

A * v(j) = lambda(j) * B * v(j).

The left eigenvector u(j) corresponding to the eigenvalue 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). Yes, that little **H is doing main character work. 👑

🎀 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 REAL array, dimension (LDA, N)
  • On entry, the matrix A in the pair (A,B).
  • On exit, A has been overwritten. 😭 gone but not forgotten.
• 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 in the pair (A,B).
  • On exit, B has been overwritten. Same energy as A, lowkey. 💀
• LDB — LDB is INTEGER
  • The leading dimension of B. LDB >= max(1,N).
• ALPHAR — ALPHAR is REAL array, dimension (N)
• ALPHAI — ALPHAI is REAL array, dimension (N)
• BETA — BETA is REAL array, dimension (N)
  • On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. 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) negative. 🌀
  • Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). This is important, bestie — do not just divide all willy-nilly. 🫶
• VL — VL is REAL array, dimension (LDVL,N)
  • If JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. If the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th and (j+1)-th 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 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 REAL array, dimension (LDVR,N)
  • If JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. If the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). 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 REAL 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(1,8*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. Sneaky little workspace check, iykyk. 👀
• 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: =N+1: other than QZ iteration failed in SHGEQZ. =N+2: error return from STGEVC. 💀

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👑 Author 👑

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

Definition at line 224 of file sggev.f. 📌

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💖 Author 💖

Generated automatically by Doxygen for LAPACK from the source code. ✨

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Final cute sign-off: this is LAPACK Version 3.12.0, from SRC/sggev.f(3) — footer source/version/date vibes folded into this sparkly goodbye instead of copied. 💖✨🎀