hegvx(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

okay so bestie, welcome to hegvx(3) from the Library Functions Manual 💅 this is the part where we meet the {he,sy}gvx crew and get into the chegvx docs. let’s go 🫶

🌸 NAME 🌸

hegvx - {he,sy}gvx: eig, bisection

🎀 SYNOPSIS 🎀

💫 Functions 💫

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

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

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

       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

🦋 Detailed Description 🦋

okay so this is the detailed-description zone, and then the actual function docs roll in. no extra lore, just the map 💖

💖 Function Documentation 💖

✨ subroutine `chegvx` ✨

   subroutine chegvx (integer itype, character jobz, character range, character
       uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex,
       dimension( ldb, * ) b, integer ldb, real vl, real vu, integer il,
       integer iu, real abstol, integer m, real, dimension( * ) w, complex,
       dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work,
       integer lwork, real, dimension( * ) rwork, integer, dimension( * )
       iwork, integer, dimension( * ) ifail, integer info)
       CHEGVX

🌟 Purpose 🌟

CHEGVX computes selected eigenvalues, and optionally, eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form 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. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. iconic behavior, honestly 👑

💅 Parameters 💅

ITYPE — ITYPE is INTEGER. 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. The order of the matrices A and B. N >= 0.
A — A is COMPLEX 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. yes bestie, destroyed — say bye to that triangle 💀

LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
B — B is COMPLEX array, dimension (LDB, N). On entry, 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. The leading dimension of the array B. LDB >= max(1,N).
VL — VL is REAL. 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 REAL. 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 REAL. 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 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*SLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S'). ngl, this is the “read the manual twice” parameter 😭

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 REAL array, dimension (N). The first M elements contain the selected eigenvalues in ascending order.
Z — Z is COMPLEX 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. bestie, this is a “you gotta budget your columns” situation 🎀

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 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 >= max(1,2*N). For optimal efficiency, LWORK >= (NB+1)*N, where NB is the blocksize for CHETRD 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 REAL 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: CPOTRF or CHEEVX returned an error code:
  - <= N: if INFO = i, CHEEVX 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.

👑 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 chegvx.f.

💜 `dsygvx` — the double precision icon

   subroutine dsygvx (integer itype, character jobz, character range, character
       uplo, integer n, double precision, dimension( lda, * ) a, integer lda,
       double precision, dimension( ldb, * ) b, integer ldb, double precision
       vl, double precision vu, integer il, integer iu, double precision
       abstol, integer m, double precision, dimension( * ) w, double precision,
       dimension( ldz, * ) z, integer ldz, double precision, dimension( * )
       work, integer lwork, integer, dimension( * ) iwork, integer, dimension(
       * ) ifail, integer info)
       DSYGVX

🦋 Purpose

okay so bestie, DSYGVX computes selected eigenvalues — and optionally eigenvectors too, like it's flexible — of a real generalized symmetric-definite eigenproblem. Three possible forms, pick your fighter:

A*x=(lambda)*B*x
A*Bx=(lambda)*x
B*A*x=(lambda)*x

Here A and B are assumed to be symmetric, and B is also positive definite (that part is non-negotiable, it's giving main character requirement ✨). And the way you get to pick which eigenvalues you want? By specifying either a range of values or a range of indices for the desired eigenvalues. Obsessed honestly.

🎀 Parameters

ITYPE — INTEGER, tells the routine which problem type to solve:
- = 1: A*x = (lambda)*B*x
- = 2: A*B*x = (lambda)*x
- = 3: B*A*x = (lambda)*x
JOBZ — CHARACTER*1, the "do I want the eigenvectors" switch:
- = 'N': compute eigenvalues only
- = 'V': compute eigenvalues and eigenvectors
RANGE — CHARACTER*1, controls the selection vibe:
- = '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 is the one storing the tea:
- = 'U': upper triangle of A and B are stored
- = 'L': lower triangle of A and B are stored
N — INTEGER, the order of the matrix pencil (A,B). Must be N >= 0.
A — DOUBLE PRECISION array, dimension (LDA, N). On entry, the symmetric 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. 😭 so if you needed that copy, you should've copied it, bestie.

LDA — INTEGER, the leading dimension of the array A. Must be LDA >= max(1,N).
B — DOUBLE PRECISION array, dimension (LDB, N). On entry, the symmetric 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**T*U or B = L*L**T. Yup, B gets a whole glow-up too 🦋.

LDB — INTEGER, the leading dimension of the array B. Must be LDB >= max(1,N).
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 C to tridiagonal form, where C is the symmetric matrix of the standard symmetric problem to which the generalized problem is transformed. (iykyk 🧚)

Real talk though: eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. And if this routine returns with INFO>0, meaning some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S'). Not me telling you twice 💀

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). On normal exit, the first M elements contain the selected eigenvalues in ascending order. Slay, sorted ✨
Z — DOUBLE PRECISION array, dimension (LDZ, max(1,M)). If JOBZ = 'N', then Z is not referenced at all, ignore her. 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.

Heads up bestie: 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. Plan ahead, okay? 💅

LDZ — INTEGER, the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. It's literally telling you the answer 🫶
LWORK — INTEGER, the length of the array WORK. LWORK >= max(1,8*N). For optimal efficiency, LWORK >= (NB+3)*N, where NB is the blocksize for DSYTRD 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. Sneaky but iconic.

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, the whole mood report:
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value 🙈
- > 0: DPOTRF or DSYEVX returned an error code:
  - <= N: if INFO = i, DSYEVX 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. Rough 💀

👑 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 294 of file dsygvx.f.

🍓 `ssygvx` — same energy, single precision

   subroutine ssygvx (integer itype, character jobz, character range, character
       uplo, integer n, real, dimension( lda, * ) a, integer lda, real,
       dimension( ldb, * ) b, integer ldb, real vl, real vu, integer il,
       integer iu, real abstol, integer m, real, dimension( * ) w, real,
       dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer
       lwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail,
       integer info)
       SSYGVX

🌈 Purpose

SSYGVX computes selected eigenvalues — and optionally eigenvectors, because she's generous — of a real generalized symmetric-definite eigenproblem, of the form:

A*x=(lambda)*B*x
A*Bx=(lambda)*x
B*A*x=(lambda)*x

Here A and B are assumed to be symmetric and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Same deal as her double-precision sister, just real instead of double precision — the vibe check passed ✅

💜 Parameters

Okay so the parameter list kicks off with the problem-type selector — first up: ITYPE

ITYPE — bestie, ITYPE is INTEGER, and it picks the whole problem-type vibe:
- = 1: A*x = (lambda)*B*x
- = 2: A*B*x = (lambda)*x
- = 3: B*A*x = (lambda)*x
JOBZ — JOBZ is CHARACTER*1, so it decides how much we're doing:
- = 'N': Compute eigenvalues only;
- = 'V': Compute eigenvalues and eigenvectors.
RANGE — RANGE is CHARACTER*1, and this is the “which eigenvalues, exactly?” selector:
- = '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, and it tells us which triangle got stored:
- = 'U': Upper triangle of A and B are stored;
- = 'L': Lower triangle of A and B are stored.
N — N is INTEGER. It's the order of the matrix pencil (A,B). And N >= 0. obvi.
A — A is REAL array, dimension (LDA, N). On entry, it's the symmetric matrix A, okay?
- 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. Like, gone gone. No cap. 💀
LDA — LDA is INTEGER. It's the leading dimension of the array A, and LDA >= max(1,N).
B — B is REAL array, dimension (LDB, N). On entry, it's the symmetric 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**T*U or B = L*L**T. So yeah, B gets a whole new outfit. 💅
LDB — LDB is INTEGER. Leading dimension of the array B, and LDB >= max(1,N).
VL — VL is REAL. If RANGE='V', this is the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
VU — VU is REAL. If RANGE='V', this is 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 REAL. It's the absolute error tolerance for the eigenvalues, bestie. 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| ) ,

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*SLAMCH('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 REAL array, dimension (N). On normal exit, the first M elements contain the selected eigenvalues in ascending order. Slay. ✨
Z — Z is REAL 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. This is your reminder, not a suggestion. 👑
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK — WORK is REAL 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 >= max(1,8*N). For optimal efficiency, LWORK >= (NB+3)*N, where NB is the blocksize for SSYTRD 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. iykyk.
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, and this is the final vibe check:
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: SPOTRF or SSYEVX returned an error code:
  - <= N: if INFO = i, SSYEVX 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.

👑 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 294 of file ssygvx.f.

   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, of the form 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. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Literally the choose-your-own-adventure of eigenproblems. 💖

🎀 Parameters 🎀

ITYPE — ITYPE is INTEGER, and 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. The order of the matrices A and B. N >= 0.
A —
A — okay so A is COMPLEX*16 array, dimension (LDA, N). On entry, she’s the Hermitian matrix A — main character energy. 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 💀 — like, gone-gone, not okay. This is a warning, bestie.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 🦋
B — B is COMPLEX*16 array, dimension (LDB, N). On entry, 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. slay. ✨
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= 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 C to tridiagonal form, where C is the symmetric matrix of the standard symmetric problem to which the generalized problem is transformed.

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 = '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. 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 >= 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 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: 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. 💀

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 🌈

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hegvx(3)

🌸 NAME 🌸

🎀 SYNOPSIS 🎀

💫 Functions 💫

🦋 Detailed Description 🦋

💖 Function Documentation 💖

✨ subroutine chegvx ✨

🌟 Purpose 🌟

💅 Parameters 💅

👑 Author 👑

🫶 Contributors: 🫶

💜 dsygvx — the double precision icon

🦋 Purpose

🎀 Parameters

👑 Author

🌟 Contributors:

🍓 ssygvx — same energy, single precision

🌈 Purpose

💜 Parameters

👑 Author 👑

🌸 Contributors 🌸

🦋 ZHEGVX 🦋

🌈 Purpose 🌈

🎀 Parameters 🎀

Author 👑

Contributors: 🫶

Author 🌈

✨ subroutine `chegvx` ✨

💜 `dsygvx` — the double precision icon

🍓 `ssygvx` — same energy, single precision