ZHPEVX(3)

💅 ZHPEVX(3) — Library Functions Manual 💅

Bestie, welcome to the packed-storage Hermitian eigenvalue selector page. It’s giving “LAPACK but make it cute,” and we are keeping every single detail. No cap. 🫶

✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zhpevx.f

Yes, that’s the NAME section: a whole path. Iconic in a deeply unhinged way. 💀

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🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine ZHPEVX (jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m,
    w, z, ldz, work, rwork, iwork, ifail, info)

Officially: ZHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

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📚 Function/Subroutine Documentation 📚

subroutine ZHPEVX (character jobz, character range, character uplo, integer
    n, complex*16, dimension( * ) ap, 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, because the page said it twice and we are not skipping: ZHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

🎯 Purpose

ZHPEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in packed storage. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Slay. 💅

🧩 Parameters

• JOBZ — CHARACTER*1. The “what are we computing” switch:
  • = 'N': Compute eigenvalues only;
  • = 'V': Compute eigenvalues and eigenvectors.
• RANGE — CHARACTER*1. Tells it which eigenvalues to find:
  • = '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 — CHARACTER*1. Which triangle of A is stored:
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — INTEGER. The order of the matrix A. N >= 0.
• AP — COMPLEX*16 array, dimension (N*(N+1)/2). On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:
  • if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
  • if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.

👀 Bestie, that means AP gets rewritten on exit. Not a drill. 🫶

• VL — 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 — 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 — 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 — 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 — 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').

There’s also this reference line, exactly as the page gives it: See by Demmel and Kahan, LAPACK Working Note #3.

• M — INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
• W — DOUBLE PRECISION array, dimension (N). If INFO = 0, the selected eigenvalues in ascending order.
• Z — 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. This is the “bring enough columns” warning, bestie. 🎀

• LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
• WORK — COMPLEX*16 array, dimension (2*N).
• RWORK — DOUBLE PRECISION array, dimension (7*N).
• IWORK — INTEGER array, dimension (5*N).
• IFAIL — 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 — 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 237 of file zhpevx.f.

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👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

That’s the whole page, bestie. Source/version vibes: LAPACK Version 3.12.0, /home/abui...RC/zhpevx.f(3). Periodt. 💖✨