SRC/dsbgvx.f(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

Okay bestie, this is SRC/dsbgvx.f(3) from the Library Functions Manual — yes, the manual name is folded in right here at the top like a little name tag. It’s a LAPACK page for the dsbgvx subroutine, and it’s lowkey iconic. 🌈💅

🌸 NAME 🌸

SRC/dsbgvx.f — that’s the name. It’s the manual page for the dsbgvx subroutine. Periodt.

🫶 SYNOPSIS 🫶

🧚 Functions/Subroutines 🧚

       subroutine dsbgvx (jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q,
           ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)

DSBGVX

💖 Function/Subroutine Documentation 💖

🌟 subroutine dsbgvx 🌟

   subroutine dsbgvx (character jobz, character range, character uplo, integer
       n, integer ka, integer kb, double precision, dimension( ldab, * ) ab,
       integer ldab, double precision, dimension( ldbb, * ) bb, integer ldbb,
       double precision, dimension( ldq, * ) q, integer ldq, double precision
       vl, double precision vu, integer il, integer iu, double precision
       abstol, integer m, double precision, dimension( * ) w, double precision,
       dimension( ldz, * ) z, integer ldz, double precision, dimension( * )
       work, integer, dimension( * ) iwork, integer, dimension( * ) ifail,
       integer info)
       DSBGVX

🎯 Purpose: 🎯

Okay so DSBGVX computes selected eigenvalues, and optionally eigenvectors, of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values, or a range of indices for the desired eigenvalues. It’s giving “pick your fave eigenvalues” energy, no cap. ✨

🌸 Parameters 🌸

JOBZ

JOBZ is CHARACTER*1. It’s the “what do you want back?” switch:

  • = 'N': Compute eigenvalues only;
  • = 'V': Compute eigenvalues and eigenvectors.

RANGE

RANGE is CHARACTER*1. It says which eigenvalues we’re hunting for:

  • = 'A': all eigenvalues will be found.
  • = 'V': all eigenvalues in the half-open interval (VL,VU] will be found.
  • = 'I': the IL-th through IU-th eigenvalues will be found.

UPLO

UPLO is CHARACTER*1. It tells us which triangle got invited to the party:

  • = 'U': Upper triangles of A and B are stored;
  • = 'L': Lower triangles of A and B are stored.

N

N is INTEGER. It’s the order of the matrices A and B. N >= 0.

KA

KA is INTEGER. It’s the number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.

KB

KB is INTEGER. It’s the number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.

AB

AB is DOUBLE PRECISION array, dimension (LDAB, N). On entry, it holds the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:

if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

On exit, the contents of AB are destroyed. So yeah, it’s gone-gone. 💀

LDAB

LDAB is INTEGER. It’s the leading dimension of the array AB. LDAB >= KA+1.

BB

BB is DOUBLE PRECISION array, dimension (LDBB, N). On entry, it holds the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows:

if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

On exit, BB holds the factor S from the split Cholesky factorization B = S**T*S, as returned by DPBSTF.

LDBB

LDBB is INTEGER. It’s the leading dimension of the array BB. LDBB >= KB+1.

Q

Q is DOUBLE PRECISION array, dimension (LDQ, N). If JOBZ = 'V', it’s the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.

LDQ

LDQ is INTEGER. It’s the leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).

VL

VL is DOUBLE PRECISION. If RANGE='V', it’s the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.

VU

VU is DOUBLE PRECISION. If RANGE='V', it’s the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.

IL

IL is INTEGER. If RANGE='I', it’s the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.

IU

IU is INTEGER. If RANGE='I', it’s the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.

ABSTOL

ABSTOL is DOUBLE PRECISION. It’s the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to

ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.

Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S').

M

M is INTEGER. It’s the total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W

W is DOUBLE PRECISION array, dimension (N). If INFO = 0, it holds the eigenvalues in ascending order.

Z

Z is DOUBLE PRECISION array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.

LDZ

LDZ is INTEGER. It’s the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

WORK

WORK is DOUBLE PRECISION array, dimension (7*N).

IWORK

IWORK is INTEGER array, dimension (5*N).

IFAIL

IFAIL is INTEGER array, dimension (M). If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvalues that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.

INFO

INFO is INTEGER. It’s the little status report at the end:

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • <= N: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in IFAIL.
  • > N: DPBSTF returned an error code; i.e., if INFO = N + i, for 1 <= i <= N, then the leading principal minor of order i of B is not positive. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

👑 Author 👑

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

🦋 Contributors: 🦋

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 291 of file dsbgvx.f.

Okay that’s the whole page, bestie. This is LAPACK Version 3.12.0, and the footer says it was Generated automatically by Doxygen for LAPACK from the source code. No date is shown, so that’s the entire footer tea — bye and slay your banded eigenproblem. 💖✨

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