SRC/zhbevx.f(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

okay so bestie, welcome to SRC/zhbevx.f(3) from the Library Functions Manual 💅 this is the LAPACK page for zhbevx, and it is giving serious linear-algebra main-character energy. every flag, dimension, formula, warning, and reference is still here — just with better vibes ✨

✨ NAME ✨

SRC/zhbevx.f — that’s the file, periodt 🎀

🌈 SYNOPSIS 🌈

🧚 Functions/Subroutines 🧚

here’s the call, exactly as it appears, no cap:

subroutine zhbevx (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork, ifail, info)

and the one-liner is literally:

ZHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 💖

🛠️ Function/Subroutine Documentation 🛠️

💜 subroutine zhbevx 💜

full signature with all the types and dimensions, copied exactly:

subroutine zhbevx (character jobz, character range, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, 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)

and again, for the people in the back:

ZHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices ✨

🎯 Purpose 🎯

bestie, here’s the point: ZHBEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. you can pick which ones by giving either 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 triangle of A is stored;
  • = 'L': lower triangle of A is stored.

N

N is INTEGER — the order of the matrix A. N >= 0.

KD

KD is INTEGER — the number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 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 KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

on exit, AB is overwritten by values generated during the reduction to tridiagonal form 💀

LDAB

LDAB is INTEGER — the leading dimension of the array AB. LDAB >= KD + 1.

Q

Q is COMPLEX*16 array, dimension (LDQ, N)

  • if JOBZ = 'V', the N-by-N unitary matrix used in the reduction 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 = 'V', then 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 AB 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').

the page also says: See by Demmel and Kahan, LAPACK Working Note #3. 👑

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) — the first M elements contain the selected eigenvalues in ascending order.

Z

Z is COMPLEX*16 array, dimension (LDZ, max(1,M))

  • if JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i).
  • if an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL.
  • if JOBZ = 'N', then Z is not referenced.
  • note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.

LDZ

LDZ is INTEGER — the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,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, then i eigenvectors failed to converge. their indices are stored in array IFAIL.

👑 Author 👑

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

Definition at line 264 of file zhbevx.f.

🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

and that’s the whole page, bestie 🫶 the footer gives LAPACK Version 3.12.0 and SRC/zhbevx.f(3) — no date in the footer, so I’m not inventing one 💀✨

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