ZGEEVX(3)

📦 version 3.12.0, in opensuse-leap 16.0

✨ ZGEEVX(3) ✨ — the zgeevx.f era 💅

okay so this is a page from the Library Functions Manual for /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zgeevx.f, and bestie, if you've ever looked at a non-symmetric complex matrix and thought "hmm who even are your eigenvalues", this is the routine for you. 🦋

🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine ZGEEVX (balanc, jobvl, jobvr, sense, n, a, lda, w, vl, ldvl,
    vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork,
    rwork, info)

ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 💖

👑 Function/Subroutine Documentation 👑

subroutine ZGEEVX (character balanc, character jobvl, character jobvr,
    character sense, integer n, complex*16, dimension( lda, * ) a, integer
    lda, complex*16, dimension( * ) w, complex*16, dimension( ldvl, * ) vl,
    integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, integer
    ilo, integer ihi, double precision, dimension( * ) scale, double
    precision abnrm, double precision, dimension( * ) rconde, double
    precision, dimension( * ) rcondv, complex*16, dimension( * ) work,
    integer lwork, double precision, dimension( * ) rwork, integer info)

ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices — yep, the whole package, no cap 🎀

🔥 Purpose 🔥

!> !> ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the !> eigenvalues and, optionally, the left and/or right eigenvectors. !> !> Optionally also, it computes a balancing transformation to improve !> the conditioning of the eigenvalues and eigenvectors (ILO, IHI, !> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues !> (RCONDE), and reciprocal condition numbers for the right !> eigenvectors (RCONDV). !> !> The right eigenvector v(j) of A satisfies !> A * v(j) = lambda(j) * v(j) !> where lambda(j) is its eigenvalue. !> The left eigenvector u(j) of A satisfies !> u(j)H * A = lambda(j) * u(j)H !> where u(j)H denotes the conjugate transpose of u(j). !> !> The computed eigenvectors are normalized to have Euclidean norm !> equal to 1 and largest component real. !> !> Balancing a matrix means permuting the rows and columns to make it !> more nearly upper triangular, and applying a diagonal similarity !> transformation D * A * D(-1), where D is a diagonal matrix, to !> make its rows and columns closer in norm and the condition numbers !> of its eigenvalues and eigenvectors smaller. The computed !> reciprocal condition numbers correspond to the balanced matrix. !> Permuting rows and columns will not change the condition numbers !> (in exact arithmetic) but diagonal scaling will. For further !> explanation of balancing, see section 4.10.2 of the LAPACK !> Users' Guide. !>

so like… it's giving "eigen-everything" and honestly? obsessed. 💜

🧚 Parameters 🧚

BALANC

!> BALANC is CHARACTER*1 !> Indicates how the input matrix should be diagonally scaled !> and/or permuted to improve the conditioning of its !> eigenvalues. !> = 'N': Do not diagonally scale or permute; !> = 'P': Perform permutations to make the matrix more nearly !> upper triangular. Do not diagonally scale; !> = 'S': Diagonally scale the matrix, ie. replace A by !> D*A*D**(-1), where D is a diagonal matrix chosen !> to make the rows and columns of A more equal in !> norm. Do not permute; !> = 'B': Both diagonally scale and permute A. !> !> Computed reciprocal condition numbers will be for the matrix !> after balancing and/or permuting. Permuting does not change !> condition numbers (in exact arithmetic), but balancing does. !>
  • 'N' — leave it alone, no scaling, no permuting, we're rawdogging the matrix 🙈
  • 'P' — permute the rows and columns so it's closer to upper triangular, but no diagonal scaling
  • 'S' — diagonal scaling only, i.e. replace A by D*A*D**(-1) with D picked so rows and columns are more equal in norm; no permuting
  • 'B' — both, bestie, we're doing the whole glow-up ✨
heads up: the computed reciprocal condition numbers are for the matrix after balancing and/or permuting. Permuting doesn't change condition numbers (in exact arithmetic) but balancing absolutely does. iykyk.

JOBVL

!> JOBVL is CHARACTER*1 !> = 'N': left eigenvectors of A are not computed; !> = 'V': left eigenvectors of A are computed. !> If SENSE = 'E' or 'B', JOBVL must = 'V'. !>
  • 'N' — no left eigenvectors, we're skipping
  • 'V' — yes please, compute the left eigenvectors
  • ⚠️ if SENSE = 'E' or 'B', then JOBVL must = 'V' — non-negotiable fr fr

JOBVR

!> JOBVR is CHARACTER*1 !> = 'N': right eigenvectors of A are not computed; !> = 'V': right eigenvectors of A are computed. !> If SENSE = 'E' or 'B', JOBVR must = 'V'. !>
  • 'N' — no right eigenvectors
  • 'V' — yes, compute the right eigenvectors
  • ⚠️ if SENSE = 'E' or 'B', then JOBVR must = 'V' too. main character rules 💅

SENSE

!> SENSE is CHARACTER*1 !> Determines which reciprocal condition numbers are computed. !> = 'N': None are computed; !> = 'E': Computed for eigenvalues only; !> = 'V': Computed for right eigenvectors only; !> = 'B': Computed for eigenvalues and right eigenvectors. !> !> If SENSE = 'E' or 'B', both left and right eigenvectors !> must also be computed (JOBVL = 'V' and JOBVR = 'V'). !>
  • 'N' — none computed, we're keeping it light
  • 'E' — eigenvalues only
  • 'V' — right eigenvectors only
  • 'B' — both eigenvalues and right eigenvectors 🫶
  • ⚠️ if SENSE = 'E' or 'B', both the left and right eigenvectors must also be computed (JOBVL = 'V' and JOBVR = 'V'). it's a package deal.

N

!> N is INTEGER !> The order of the matrix A. N >= 0. !>
  • N is INTEGER, the order of matrix A. Constraint: N >= 0. that's it that's the vibe.

A

!> A is COMPLEX*16 array, dimension (LDA,N) !> On entry, the N-by-N matrix A. !> On exit, A has been overwritten. If JOBVL = 'V' or !> JOBVR = 'V', A contains the Schur form of the balanced !> version of the matrix A. !>
  • A is COMPLEX*16 array, dimension (LDA,N)
  • on entry: the N-by-N matrix A
  • on exit: A has been overwritten 😭 and if JOBVL = 'V' or JOBVR = 'V', A contains the Schur form of the balanced version of the matrix A. it ate and left no crumbs.

LDA

!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !>
  • LDA is INTEGER, the leading dimension of array A. Constraint: LDA >= max(1,N).

W

!> W is COMPLEX*16 array, dimension (N) !> W contains the computed eigenvalues. !>
  • W is COMPLEX*16 array, dimension (N), and it contains the computed eigenvalues 🎀 the whole reason we're here.

VL

!> VL is COMPLEX*16 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 JOBVL = 'N', VL is not referenced. !> u(j) = VL(:,j), the j-th column of VL. !>
  • VL is COMPLEX*16 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 JOBVL = 'N', VL is not referenced — literally ignored, not even a vibe check
  • u(j) = VL(:,j), the j-th column of VL

LDVL

!> LDVL is INTEGER !> The leading dimension of the array VL. LDVL >= 1; if !> JOBVL = 'V', LDVL >= N. !>
  • LDVL is INTEGER, leading dimension of VL. Constraint: LDVL >= 1; and if JOBVL = 'V', then LDVL >= N.

VR

!> VR is COMPLEX*16 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 JOBVR = 'N', VR is not referenced. !> v(j) = VR(:,j), the j-th column of VR. !>
  • VR is COMPLEX*16 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 JOBVR = 'N', VR is not referenced
  • v(j) = VR(:,j), the j-th column of VR

LDVR

!> LDVR is INTEGER !> The leading dimension of the array VR. LDVR >= 1; if !> JOBVR = 'V', LDVR >= N. !>
  • LDVR is INTEGER, leading dimension of VR. Constraint: LDVR >= 1; and if JOBVR = 'V', then LDVR >= N.

ILO

!> ILO is INTEGER !>
  • ILO is INTEGER. that's the whole description bestie, she's mysterious 🦋

IHI

!> IHI is INTEGER !> ILO and IHI are integer values determined when A was !> balanced. The balanced A(i,j) = 0 if I > J and !> J = 1,...,ILO-1 or I = IHI+1,...,N. !>
  • IHI is INTEGER
  • ILO and IHI are integer values determined when A was balanced. The balanced A(i,j) = 0 if I > J and J = 1,...,ILO-1 or I = IHI+1,...,N.

SCALE

!> SCALE is DOUBLE PRECISION array, dimension (N) !> Details of the permutations and scaling factors applied !> when balancing A. If P(j) is the index of the row and column !> interchanged with row and column j, and D(j) is the scaling !> factor applied to row and column j, then !> SCALE(J) = P(J), for J = 1,...,ILO-1 !> = D(J), for J = ILO,...,IHI !> = P(J) for J = IHI+1,...,N. !> The order in which the interchanges are made is N to IHI+1, !> then 1 to ILO-1. !>
  • SCALE is DOUBLE PRECISION array, dimension (N)
  • it holds the details of the permutations and scaling factors applied when balancing A. If P(j) is the index of the row and column interchanged with row and column j, and D(j) is the scaling factor applied to row and column j, then:
SCALE(J) = P(J),    for J = 1,...,ILO-1
         = D(J),    for J = ILO,...,IHI
         = P(J)     for J = IHI+1,...,N.
  • the order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. order matters, periodt.

ABNRM

!> ABNRM is DOUBLE PRECISION !> The one-norm of the balanced matrix (the maximum !> of the sum of absolute values of elements of any column). !>
  • ABNRM is DOUBLE PRECISION, the one-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column) 🔥

RCONDE

!> RCONDE is DOUBLE PRECISION array, dimension (N) !> RCONDE(j) is the reciprocal condition number of the j-th !> eigenvalue. !>
  • RCONDE is DOUBLE PRECISION array, dimension (N), and RCONDE(j) is the reciprocal condition number of the j-th eigenvalue 🌈

RCONDV

!> RCONDV is DOUBLE PRECISION array, dimension (N) !> RCONDV(j) is the reciprocal condition number of the j-th !> right eigenvector. !>
  • RCONDV is DOUBLE PRECISION array, dimension (N), and RCONDV(j) is the reciprocal condition number of the j-th right eigenvector 💫

WORK

!> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
  • WORK is COMPLEX*16 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. If SENSE = 'N' or 'E', !> LWORK >= max(1,2*N), and if SENSE = 'V' or 'B', !> LWORK >= N*N+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. !>
  • LWORK is INTEGER, the dimension of the array WORK
  • if SENSE = 'N' or 'E', then LWORK >= max(1,2*N)
  • if SENSE = 'V' or 'B', then LWORK >= N*N+2*N
  • for good performance, LWORK must generally be larger — big workspace energy, she doesn't cut corners 💅
  • 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 but legit 🧚

RWORK

!> RWORK is DOUBLE PRECISION array, dimension (2*N) !>
  • RWORK is DOUBLE PRECISION array, dimension (2*N). that's it. she's just chilling. 🍓

INFO

!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value. !> > 0: if INFO = i, the QR algorithm failed to compute all the !> eigenvalues, and no eigenvectors or condition numbers !> have been computed; elements 1:ILO-1 and i+1:N of W !> contain eigenvalues which have converged. !>
  • INFO is INTEGER
  • = 0: successful exit, we love to see it 💖
  • < 0: if INFO = -i, the i-th argument had an illegal value 👀
  • > 0: if INFO = i, the QR algorithm failed to compute all the eigenvalues, and no eigenvectors or condition numbers have been computed; elements 1:ILO-1 and i+1:N of W contain eigenvalues which have converged. not ideal but at least you got some of them 😭

🌟 Author 🌟

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

the academic dream team fr fr 🫶

📍 Definition at line 285 of file zgeevx.f. 📍

⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code.

and that's the whole page bestie, no notes 💅

LAPACK — Version 3.12.0 — zgeevx.f(3). mwah, go balance a matrix, ily. 🎀💜
📦 where this page comes from