SRC/ssbgvx.f(3)

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version 3.12.1 · 4 versions
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SRC/ssbgvx.f(3) ✨

okay bestie, welcome to the Library Functions Manual moment for SRC/ssbgvx.f — this is the SSBGVX page, and it's lowkey iconic if you're doing generalized symmetric-definite banded eigenproblems. 🎀

💖 NAME 💖

SRC/ssbgvx.f

that's it, that's the name, no cap. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine ssbgvx (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)
    SSBGVX

🫶 Function/Subroutine Documentation 🫶

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

SSBGVX

🎯 Purpose 🎯

SSBGVX 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. 🦋

🧾 Parameters 🧾

JOBZ

JOBZ is CHARACTER*1

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

RANGE

RANGE is CHARACTER*1

  • '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

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

N

N is INTEGER

The order of the matrices A and B. N >= 0.

KA

KA is INTEGER

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

The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.

AB

AB is REAL array, dimension (LDAB, N)

On entry, 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.

LDAB

LDAB is INTEGER

The leading dimension of the array AB. LDAB >= KA+1.

BB

BB is REAL array, dimension (LDBB, N)

On entry, 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, the factor S from the split Cholesky factorization B = S**T*S, as returned by SPBSTF.

LDBB

LDBB is INTEGER

The leading dimension of the array BB. LDBB >= KB+1.

Q

Q is REAL array, dimension (LDQ, N)

If JOBZ = 'V', 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

The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).

VL

VL is REAL

If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

VU

VU is REAL

If RANGE='V', 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', 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', 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 REAL

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*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S').

M

M is INTEGER

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 REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

Z

Z is REAL 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

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

WORK

WORK is REAL 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

  • = 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: SPBSTF 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.
bestie, that last one is a real "the math said no" moment — if B isn't cooperating, SSBGVX is not about to manifest eigenvalues out of nowhere. 💀

✍️ 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 ssbgvx.f.

🤖 Author (automated) 🤖

Generated automatically by Doxygen for LAPACK from the source code.

that's a wrap from LAPACK Version 3.12.0, page SRC/ssbgvx.f(3) 💅✨ — the footer here has the version and page tag, and there's no date line in this manpage snippet, so I'm not inventing one, bestie. 🫶

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