SRC/zheevx.f(3)

🌸 NAME 🌸

SRC/zheevx.f — okay bestie, this is the LAPACK Library Functions Manual page for zheevx, and it's giving complex Hermitian matrix main-character energy 💅. The subroutine computes eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices.

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine zheevx (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
    m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)

ZHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices 🫶.

🎀 Function/Subroutine Documentation 🎀

subroutine zheevx

subroutine zheevx (character jobz, character range, character uplo, integer
    n, complex*16, dimension( lda, * ) a, integer lda, 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, integer lwork,
    double precision, dimension( * ) rwork, integer, dimension( * ) iwork,
    integer, dimension( * ) ifail, integer info)

ZHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices. No cap ✨.

Purpose:

ZHEEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying 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.
• A — A is COMPLEX*16 array, dimension (LDA, N). On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.

On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed 😭. That part is canon, bestie.

• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= 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 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').

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). On normal exit, 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 (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — LWORK is INTEGER. The length of the array WORK. LWORK >= 1, when N <= 1; otherwise 2*N. For optimal efficiency, LWORK >= (NB+1)*N, where NB is the max of the blocksize for ZHETRD and for ZUNMTR as returned by ILAENV.

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.

• 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 256 of file zheevx.f. 👑

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code. 💖

---

Okay that's the whole tea, bestie: LAPACK Version 3.12.0, page SRC/zheevx.f(3) — go slay those Hermitian eigenvalues ✨💅🦋