ZHEGVX(3)

💖 NAME 💖

Manual name: ZHEGVX(3) — yes bestie, this is the ZHEGVX(3) page from the Library Functions Manual. The NAME line points at the source file /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zhegvx.f, so that’s the file we’re talking about. No cap.

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✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

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

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🌸 Function/Subroutine Documentation 🌸

Here’s the full typed-out signature, exactly as it appears, bestie:

subroutine ZHEGVX (integer itype, character jobz, character range, character
    uplo, integer n, complex*16, dimension( lda, * ) a, integer lda,
    complex*16, dimension( ldb, * ) b, integer ldb, 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)
    ZHEGVX

🎀 Purpose 🎀

Okay so ZHEGVX computes selected eigenvalues, and optionally eigenvectors, of a complex generalized Hermitian-definite eigenproblem. The forms are A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, and B is also positive definite. You can select eigenvalues and eigenvectors by specifying either a range of values or a range of indices for the desired eigenvalues. Iconic behavior, honestly.

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🧚 Parameters 🧚

💅 ITYPE

ITYPE is INTEGER. It specifies the problem type to be solved:

• = 1: A*x = (lambda)*B*x
• = 2: A*B*x = (lambda)*x
• = 3: B*A*x = (lambda)*x

🦋 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. It’s the order of the matrices A and B. N >= 0.

💖 A

A is a COMPLEX*16 array, dimension (LDA, N).

On entry, it’s 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. Not me losing the diagonal, but yes.

🌈 LDA

LDA is INTEGER. It’s the leading dimension of the array A. LDA >= max(1,N).

🦋 B

B is a COMPLEX*16 array, dimension (LDB, N).

On entry, it’s the Hermitian matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.

On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**H*U or B = L*L**H.

🌸 LDB

LDB is INTEGER. It’s the leading dimension of the array B. LDB >= max(1,N).

🎀 VL

VL is DOUBLE PRECISION.

If RANGE='V', it’s 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', it’s 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', it’s 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', it’s 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. It’s the absolute error tolerance for the eigenvalues.

Here’s the convergence vibe check: an approximate eigenvalue is accepted as converged when it’s 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 C to tridiagonal form, where C is the symmetric matrix of the standard symmetric problem to which the generalized problem is transformed.

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'). That’s the tea.

🌸 M

M is INTEGER. It’s 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 a DOUBLE PRECISION array, dimension (N). The first M elements contain the selected eigenvalues in ascending order.

✨ Z

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

If JOBZ = 'N', then Z is not referenced.

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).

The eigenvectors are normalized as follows:

if ITYPE = 1 or 2, Z**T*B*Z = I;
if ITYPE = 3, Z**T*inv(B)*Z = 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.

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

🦋 WORK

WORK is a COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

🌈 LWORK

LWORK is INTEGER. It’s the length of the array WORK. LWORK >= max(1,2*N).

For optimal efficiency, LWORK >= (NB+1)*N, where NB is the blocksize for ZHETRD 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 a DOUBLE PRECISION array, dimension (7*N).

🎀 IWORK

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

✨ IFAIL

IFAIL is an 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: ZPOTRF or ZHEEVX returned an error code:
  • <= N: if INFO = i, ZHEEVX failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFAIL.
  • > N: 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.

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👑 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 304 of file zhegvx.f. 📌

🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

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footer tea: this page is from LAPACK Version 3.12.0, and the footer path is /home/abui...RC/zhegvx.f(3). There’s no separate date printed in that footer, so that’s the whole sign-off, periodt. 💅✨