SRC/zhbgvx.f(3)

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version 3.12.1 · 7 versions
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🎀 NAME 🎀

SRC/zhbgvx.f — okay bestie, this is the zhbgvx.f(3) page from the Library Functions Manual, and it’s all about the ZHBGVX subroutine. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

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

ZHBGVX — the icon herself. 👑

💻 Function/Subroutine Documentation 💻

subroutine zhbgvx (character jobz, character range, character uplo, integer
    n, integer ka, integer kb, complex*16, dimension( ldab, * ) ab, integer
    ldab, complex*16, dimension( ldbb, * ) bb, integer ldbb, complex*16,
    dimension( ldq, * ) q, integer ldq, double precision vl, double
    precision vu, integer il, integer iu, double precision abstol, integer
    m, double precision, dimension( * ) w, complex*16, dimension( ldz, * )
    z, integer ldz, complex*16, dimension( * ) work, double precision,
    dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension(
    * ) ifail, integer info)

ZHBGVX 💖

🌟 Purpose 🌟

ZHBGVX computes all the eigenvalues, and optionally the eigenvectors, of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian 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 COMPLEX*16 array, dimension (LDAB, N).

On entry, the upper or lower triangle of the Hermitian 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 COMPLEX*16 array, dimension (LDBB, N).

On entry, the upper or lower triangle of the Hermitian 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(kb+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**H*S, as returned by ZPBSTF.

LDBB

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

Q

Q is COMPLEX*16 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 DOUBLE PRECISION.

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 DOUBLE PRECISION.

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 DOUBLE PRECISION. 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 AP 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. 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, the eigenvalues in ascending order. 🌸

Z

Z is COMPLEX*16 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 that Z**H*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 >= N.

WORK

WORK is COMPLEX*16 array, dimension (N).

RWORK

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

IWORK

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

IFAIL

IFAIL is INTEGER array, dimension (N).

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 eigenvectors 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
  • > 0: if INFO = i, and i is:
    • <= N: then i eigenvectors failed to converge. Their indices are stored in array IFAIL.
    • > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF returned INFO = i: B is not positive definite. 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 📍

Definition at line 297 of file zhbgvx.f.

🤖 Author 🤖

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okay that’s the whole page, bestie 🫶 — it’s from LAPACK Version 3.12.0, page SRC/zhbgvx.f(3), and that’s the cute little source/version sign-off at the end. catch u later! 🦋💜

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