DGEEVX(3)

okay so bestie, welcome to DGEEVX(3) from the Library Functions Manual ✨ this is the whole vibe, no cap.

💅 NAME 💅

the main character here is the source file at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dgeevx.f — that’s literally the whole NAME section, periodt. 💖

✨ SYNOPSIS ✨

💫 Functions/Subroutines 💫

subroutine DGEEVX (balanc, jobvl, jobvr, sense, n, a, lda, wr, wi, vl,
    ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork,
    iwork, info)

DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 🦋

🎀 Function/Subroutine Documentation 🎀

subroutine DGEEVX (character balanc, character jobvl, character jobvr,
    character sense, integer n, double precision, dimension( lda, * ) a,
    integer lda, double precision, dimension( * ) wr, double precision,
    dimension( * ) wi, double precision, dimension( ldvl, * ) vl, integer
    ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer
    ilo, integer ihi, double precision, dimension( * ) scale, double
    precision abnrm, double precision, dimension( * ) rconde, double
    precision, dimension( * ) rcondv, double precision, dimension( * ) work,
    integer lwork, integer, dimension( * ) iwork, integer info)

DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices 💜

🌸 Purpose 🌸

okay so DGEEVX computes, for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. iconic. 😌

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. slay. 💅

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. 📚

🦋 Parameters 🦋

• BALANC — BALANC is CHARACTER*1. It 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, i.e. 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.

• 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'.

• 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'.

• SENSE — SENSE is CHARACTER*1. It 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 — N is INTEGER. The order of the matrix A. N >= 0.
• A — A is DOUBLE PRECISION 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 real Schur form of the balanced version of the input matrix A.
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• WR — WR is DOUBLE PRECISION array, dimension (N).
• WI — WI is DOUBLE PRECISION array, dimension (N). WR and WI contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.
• 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 JOBVL = 'N', VL is not referenced. 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)-st 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).
• LDVL — LDVL is INTEGER. The leading dimension of the array VL. LDVL >= 1; if JOBVL = 'V', LDVL >= N.
• 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 JOBVR = 'N', VR is not referenced. 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)-st 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).
• LDVR — LDVR is INTEGER. The leading dimension of the array VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.
• ILO — ILO is INTEGER.
• 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.
• 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.

• 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).
• RCONDE — RCONDE is DOUBLE PRECISION array, dimension (N). 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.
• WORK — WORK is DOUBLE PRECISION 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 JOBVL = 'V' or JOBVR = 'V', LWORK >= 3*N. If SENSE = 'V' or 'B', LWORK >= N*(N+6). 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.

• IWORK — IWORK is INTEGER array, dimension (2*N-2). If SENSE = 'N' or 'E', not referenced.
• 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 WR and WI contain eigenvalues which have converged.

👑 Author 👑

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

Definition at line 303 of file dgeevx.f.

👑 AUTHOR 👑

okay so nobody wrote this page by hand, bestie — it's generated automatically by Doxygen for LAPACK from the source code. 💅 So the "author" is really just... the source code itself, living its main character life, with Doxygen doing the grunt work and turning it into this gorgeous little document for you. 🎀 No mysterious human to thank, no lonely midnight typist — just automation, slaying quietly in the background. ✨

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🫶 fr fr, if you're citing this, cite LAPACK — not the robot that formatted it.

and that's the whole vibe check, babe. 🌈 this page came from dgeevx.f — like, that exact file — and it's serving LAPACK Version 3.12.0 energy, shipped out on the date stamped in the footer of your build. 🌸 so go forth, call your DGEEVX, read those reciprocal condition numbers, and remember: the docs were auto-generated, but the love is real. 💖🦋🧚

— xoxo, your local LAPACK girlie 💜