SRC/dggev3.f(3)

🌸 NAME 🌸

SRC/dggev3.f — bestie, this is the page from the Library Functions Manual, and it's the whole glow-up on DGGEV3. ✨

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✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

       subroutine dggev3 (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai,
           beta, vl, ldvl, vr, ldvr, work, lwork, info)
            DGGEV3 computes the eigenvalues and, optionally, the left and/or
           right eigenvectors for GE matrices (blocked algorithm)

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💖 Function/Subroutine Documentation 💖

   subroutine dggev3 (character jobvl, character jobvr, integer n, double
       precision, dimension( lda, * ) a, integer lda, double precision,
       dimension( ldb, * ) b, integer ldb, double precision, dimension( * )
       alphar, double precision, dimension( * ) alphai, double precision,
       dimension( * ) beta, double precision, dimension( ldvl, * ) vl, integer
       ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double
       precision, dimension( * ) work, integer lwork, integer info)
        DGGEV3 computes the eigenvalues and, optionally, the left and/or right
       eigenvectors for GE matrices (blocked algorithm)

🎀 Purpose 🎀

Okay so DGGEV3 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. iykyk 🦋

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). slay. 🌈

🔥 Parameters 🔥

• JOBVL — CHARACTER*1
  • = 'N': do not compute the left generalized eigenvectors;
  • = 'V': compute the left generalized eigenvectors.
• JOBVR — CHARACTER*1
  • = 'N': do not compute the right generalized eigenvectors;
  • = 'V': compute the right generalized eigenvectors.
• N — INTEGER
  • The order of the matrices A, B, VL, and VR. N >= 0.
• A — DOUBLE PRECISION array, dimension (LDA, N)
  • On entry, the matrix A in the pair (A,B).
  • On exit, A has been overwritten.
• LDA — INTEGER
  • The leading dimension of A. LDA >= max(1,N).
• B — DOUBLE PRECISION array, dimension (LDB, N)
  • On entry, the matrix B in the pair (A,B).
  • On exit, B has been overwritten.
• LDB — INTEGER
  • The leading dimension of B. LDB >= max(1,N).
• ALPHAR — DOUBLE PRECISION array, dimension (N)
• ALPHAI — DOUBLE PRECISION array, dimension (N)
• BETA — DOUBLE PRECISION 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).
• VL — DOUBLE PRECISION 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 — INTEGER
  • The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.
• VR — DOUBLE PRECISION 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 — INTEGER
  • The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.
• WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK))
  • On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — 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.
• 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 ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N.
  • > N: =N+1: other than QZ iteration failed in DLAQZ0. =N+2: error return from DTGEVC.

🦋 Author 🦋

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

Definition at line 225 of file dggev3.f.

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💀 Author 💀

Generated automatically by Doxygen for LAPACK from the source code.

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byeee bestie 💖 this was SRC/dggev3.f(3) from LAPACK Version 3.12.0 — mwah! ✨🎀