SRC/sggev3.f(3)

✨ NAME ✨

SRC/sggev3.f — okay so bestie, this is the Library Functions Manual page for our main-character LAPACK subroutine. No cap, it’s the sggev3 page, and we’re gonna cover every single detail. 💖

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🎀 SYNOPSIS 🎀

🧚 Functions/Subroutines 🧚

       subroutine sggev3 (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai,
           beta, vl, ldvl, vr, ldvr, work, lwork, info)
            SGGEV3 computes the eigenvalues and, optionally, the left and/or
           right eigenvectors for GE matrices (blocked algorithm)

🦋 Function/Subroutine Documentation 🦋

   subroutine sggev3 (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)
        SGGEV3 computes the eigenvalues and, optionally, the left and/or right
       eigenvectors for GE matrices (blocked algorithm)

💖 Purpose 💖

Okay so the whole point: SGGEV3 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. Periodt. 💅

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 way that’s literally the setup, fr fr ✨

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). 🦋

🌸 Parameters 🌸

• JOBVL — JOBVL is CHARACTER*1. = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors. iykyk ✨
• 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. (yes, it gets clobbered, no cap) 💅
• 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. 🫶
• 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). 😭 (do not just divide, bestie)
• 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 should 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. workspace query arc 🫶
• 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 SLAQZ0. =N+2: error return from STGEVC. 💀

👑 Author 👑

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

Definition at line 225 of file sggev3.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

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okay that’s the whole page, bestie! You’re looking at LAPACK Version 3.12.0, and this is the SRC/sggev3.f(3) page. slay, bye 💅✨