SRC/dggev.f(3)

💅 SRC/dggev.f — from the LAPACK Library Functions Manual, bestie

okay so this is the manual page for SRC/dggev.f, and it's genuinely one of the more iconic ones — generalized eigenvalues AND eigenvectors?? for real nonsymmetric matrix pairs?? the range. let me walk you through it 🫶

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✨ SYNOPSIS ✨

🎀 Functions/Subroutines

subroutine dggev (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta,
    vl, ldvl, vr, ldvr, work, lwork, info)

DGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices — yeah that's the whole bit, and it ate 💅

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💖 Function/Subroutine Documentation 💖

subroutine dggev (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)

DGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices. periodt. ✨

🌸 Purpose 🌸

so the whole thing is written as a doc-comment in the source, which is honestly such a vibe — here it is in full, no edits, no cap:

!>
!> DGGEV 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).
!>
!>

like, the way that it's just casually defining a generalized eigenvalue as a pair (alpha,beta) instead of a single number because beta=0 is a whole valid situation?? main character energy fr fr 👑

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🦋 Parameters 🦋

JOBVL

!>          JOBVL is CHARACTER*1
!>          = 'N':  do not compute the left generalized eigenvectors;
!>          = 'V':  compute the left generalized eigenvectors.
!>

two options. 'N' = leave them, 'V' = give me the left eigenvectors. choose your fighter 💅

JOBVR

!>          JOBVR is CHARACTER*1
!>          = 'N':  do not compute the right generalized eigenvectors;
!>          = 'V':  compute the right generalized eigenvectors.
!>

same deal, but for the right ones. 'N' = nope, 'V' = yes bestie 🌈

N

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

the order of A, B, VL, and VR. and it must be N >= 0 — zero is fine, negative is not 🙈

A

!>          A is DOUBLE PRECISION array, dimension (LDA, N)
!>          On entry, the matrix A in the pair (A,B).
!>          On exit, A has been overwritten.
!>

on the way in it's the A of the pair (A,B). on the way out?? overwritten. she's gone. don't count on your input coming back to you 😭

LDA

!>          LDA is INTEGER
!>          The leading dimension of A.  LDA >= max(1,N).
!>

leading dimension of A, and it's got to be LDA >= max(1,N) ⭐

B

!>          B is DOUBLE PRECISION array, dimension (LDB, N)
!>          On entry, the matrix B in the pair (A,B).
!>          On exit, B has been overwritten.
!>

the B of the pair. also gets overwritten on exit. both of them. gone. the disrespect 💀

LDB

!>          LDB is INTEGER
!>          The leading dimension of B.  LDB >= max(1,N).
!>

leading dimension of B, LDB >= max(1,N).

ALPHAR

!>          ALPHAR is DOUBLE PRECISION array, dimension (N)
!>

that's it. that's the whole description. dimension (N), no commentary. she's mysterious 🎀

ALPHAI

!>          ALPHAI is DOUBLE PRECISION array, dimension (N)
!>

same. dimension (N). the real info is in BETA's description, iykyk.

BETA

!>          BETA is 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).
!>

this is the important one, okay, pay attention 🫶

on exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), for j=1,...,N, will be the generalized eigenvalues. if ALPHAI(j) is zero → the j-th eigenvalue is real. if it's positive → the j-th and (j+1)-st eigenvalues form a complex conjugate pair, and ALPHAI(j+1) is negative.

⚠️ real talk warning: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. so please, please don't just naively compute the ratio alpha/beta. 🙈 however — and this is the comforting part — ALPHAR and ALPHAI will always be less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). so the pieces themselves are well-behaved even when the ratio isn't. chef's kiss ✨

VL

!>          VL is 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'.
!>

dimension (LDVL,N). if JOBVL = 'V', the left eigenvectors u(j) get stored one after another in the columns of VL, in the same order as their eigenvalues. j-th eigenvalue real → u(j) = VL(:,j), literally just the j-th column. j-th and (j+1)-th forming a complex conjugate pair → 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. cute, consistent, normalized 💅

and not referenced if JOBVL = 'N' — she doesn't even show up to the party.

LDVL

!>          LDVL is INTEGER
!>          The leading dimension of the matrix VL. LDVL >= 1, and
!>          if JOBVL = 'V', LDVL >= N.
!>

LDVL >= 1 always, and LDVL >= N if JOBVL = 'V'. conditional requirements, love that for us 🦋

VR

!>          VR is 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'.
!>

dimension (LDVR,N). exact same story as VL but right-side: if JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, same order as their eigenvalues. real j-th eigenvalue → v(j) = VR(:,j). complex conjugate pair → v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). same scaling rule: largest component has abs(real part)+abs(imag. part)=1. and not referenced if JOBVR = 'N' 🌸

LDVR

!>          LDVR is INTEGER
!>          The leading dimension of the matrix VR. LDVR >= 1, and
!>          if JOBVR = 'V', LDVR >= N.
!>

LDVR >= 1, and LDVR >= N when JOBVR = 'V'. symmetric queen ⭐

WORK

!>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>

dimension (MAX(1,LWORK)). on exit, if INFO = 0, WORK(1) returns the optimal LWORK. free intel 🎀

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.
!>

dimension of WORK, and LWORK >= max(1,8*N). for good performance, LWORK must generally be larger — like, technically max(1,8*N) clears the bar but she's not gonna slay at that size, ngl 🔥

and if LWORK = -1 → workspace query mode: 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. it's the "just asking, not committing" of parameters 💅

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 DHGEQZ.
!>                =N+2: error return from DTGEVC.
!>

the status report, and it's giving full spectrum:

• = 0 → successful exit. we love a clean win 👑
• < 0 → if INFO = -i, the i-th argument had an illegal value. somebody passed something cursed.
• = 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. partial credit, bestie 😭
• > N → two sub-cases: =N+1 means other than QZ iteration failed in DHGEQZ; =N+2 means error return from DTGEVC. she's telling you exactly who dropped the ball 💀

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💜 Author 💜

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

the whole collective, slaying together 🫶

Definition at line 224 of file dggev.f.

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🌈 Author (footer edition) 🌈

Generated automatically by Doxygen for LAPACK from the source code. not a human, just a very diligent little bot doing her thing ✨

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and that's the page!! LAPACK — Version 3.12.0 — SRC/dggev.f(3) — mwah 💋