hbgvx(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

💖 hbgvx(3) — Library Functions Manual 💖

okay so bestie, gather round, because this is the hbgvx page from the Library Functions Manual and it's giving big LAPACK energy. We're talking {hb,sb}gvx: eig, bisection — that's the NAME, periodt. It's a whole family of routines for generalized Hermitian/symmetric-definite banded eigenproblems, and yes, we're gonna get through all of it together. No skipping, no summarizing, we keep every single detail because that's just how we roll. 🫶

✨ SYNOPSIS ✨

🌸 Functions

chbgvx — the complex one 💜

subroutine chbgvx (jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q,
    ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork,
    ifail, info)

CHBGVX

dsbgvx — the double precision one 🔥

subroutine dsbgvx (jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q,
    ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)

DSBGVX

ssbgvx — the single precision one ⭐

subroutine ssbgvx (jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q,
    ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)

SSBGVX

zhbgvx — the double complex one 👑

subroutine zhbgvx (jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q,
    ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, rwork, iwork,
    ifail, info)

ZHBGVX

🦋 Detailed Description

like, this page is literally the "detailed description" hub — it's where all the function docs live. It's the main character of the manual, fr fr. Keep scrolling and we'll meet each routine one at a time. 💅

🎀 Function Documentation 🎀

🌈 `subroutine chbgvx`

okay so here's the full glow-up of the signature, bestie — every type, every dimension, all of it, no cap:

subroutine chbgvx (character jobz, character range, character uplo, integer
    n, integer ka, integer kb, complex, dimension( ldab, * ) ab, integer
    ldab, complex, dimension( ldbb, * ) bb, integer ldbb, complex,
    dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il,
    integer iu, real abstol, integer m, real, dimension( * ) w, complex,
    dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, real,
    dimension( * ) rwork, integer, dimension( * ) iwork, integer, dimension(
    * ) ifail, integer info)

CHBGVX

✨ Purpose

CHBGVX computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.

so yeah — eigenvalues, eigenvectors if you want 'em, and you get to pick which ones. iconic. 🦋

💜 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 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.
KA — KA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.
KB — KB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.
AB — AB is COMPLEX array, dimension (LDAB, N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
- if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).

On exit, the contents of AB are destroyed. 💀 (like, gone gone — she's not coming back)

LDAB — LDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.
BB — BB is COMPLEX array, dimension (LDBB, N) On entry, the upper or lower triangle of the Hermitian band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows:
- if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
- if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).

On exit, the factor S from the split Cholesky factorization B = S**H*S, as returned by CPBSTF.

LDBB — LDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.
Q — Q is COMPLEX array, dimension (LDQ, N) If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced. (she's off duty 🙈)
LDQ — LDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= 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| ) ,

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'). 🎯

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) If INFO = 0, the eigenvalues in ascending order.
Z — Z is COMPLEX array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.
LDZ — LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N.
WORK — WORK is COMPLEX array, dimension (N)
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: if INFO = i, and i is:
  - <= N: then i eigenvectors failed to converge. Their indices are stored in array IFAIL.
  - > N: if INFO = N + i, for 1 <= i <= N, then CPBSTF returned INFO = i: B is not positive definite. 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 297 of file chbgvx.f.

🧚 `subroutine dsbgvx`

and now the double precision sibling steps in — same vibe, double precision everywhere, and heads up: her parameter list is the real flavor (no rwork, abstol is double precision). here's the whole signature, verbatim:

subroutine dsbgvx (character jobz, character range, character uplo, integer
    n, integer ka, integer kb, double precision, dimension( ldab, * ) ab,
    integer ldab, double precision, dimension( ldbb, * ) bb, integer ldbb,
    double precision, dimension( ldq, * ) q, integer ldq, 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, dimension( * ) iwork, integer, dimension( * ) ifail,
    integer info)

DSBGVX

✨ Purpose

DSBGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.

real, symmetric, banded, and B positive definite — the whole package, bestie. 🌸

💜 Parameters

JOBZ

(part 1 stops right here mid-parameter, because the page continues — more coming in part 2! don't go anywhere 🫶)

JOBZ — CHARACTER*1. Pick your vibe: = 'N': compute eigenvalues only; = 'V': compute eigenvalues and eigenvectors. 💖
RANGE — CHARACTER*1. Which eigenvalues are we hunting? = '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. = 'U': upper triangles of A and B are stored; = 'L': lower triangles of A and B are stored. 🫶
N — INTEGER. The order of the matrices A and B. N >= 0. 👑
KA — INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.
KB — INTEGER. The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.
AB — DOUBLE PRECISION array, dimension (LDAB, N). On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

On exit, the contents of AB are destroyed 💀 — no cap, they're gone.

LDAB — INTEGER. The leading dimension of the array AB. LDAB >= KA+1.
BB — DOUBLE PRECISION array, dimension (LDBB, N). On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows:

if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by DPBSTF. 🦋

LDBB — INTEGER. The leading dimension of the array BB. LDBB >= KB+1.
Q — DOUBLE PRECISION array, dimension (LDQ, N). If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.
LDQ — INTEGER. The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= 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| ) ,

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'). 🔥

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 eigenvalues in ascending order.
Z — DOUBLE PRECISION array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.
LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK — DOUBLE PRECISION array, dimension (7*N).
IWORK — INTEGER array, dimension (5*N).
IFAIL — INTEGER array, dimension (M). 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 eigenvalues 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
- <= N: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in IFAIL.
- > N: DPBSTF returned an error code; i.e., 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

Definition at line 291 of file dsbgvx.f.

🌟 subroutine `ssbgvx`

subroutine ssbgvx (character jobz, character range, character uplo, integer
    n, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab,
    real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( ldq, * )
    q, integer ldq, 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, dimension( * ) iwork, integer,
    dimension( * ) ifail, integer info)
    SSBGVX

🎯 Purpose

SSBGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues. 🌈

💅 Parameters

JOBZ — CHARACTER*1. = 'N': compute eigenvalues only; = 'V': compute eigenvalues and eigenvectors. 💖
RANGE — 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 — CHARACTER*1. = 'U': upper triangles of A and B are stored; = 'L': lower triangles of A and B are stored. 🫶
N — INTEGER. The order of the matrices A and B. N >= 0. 👑
KA — INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.
KB — INTEGER. The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.
AB — REAL array, dimension (LDAB, N). On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+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(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

On exit, the contents of AB are destroyed. 💀

LDAB — INTEGER. The leading dimension of the array AB. LDAB >= KA+1.
BB

BB — bestie, this is a REAL array, dimension (LDBB, N). On entry, it's the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of BB like this:

if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

On exit, it's the factor S from the split Cholesky factorization B = S**T*S, as returned by SPBSTF. Iconic 💅.

LDBB — LDBB is INTEGER. It's the leading dimension of the array BB. The rule is LDBB >= KB+1, no cap.

Q — Q is REAL array, dimension (LDQ, N). If JOBZ = 'V', this is the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced at all — it's giving background character energy 🙈.

LDQ — LDQ is INTEGER. It's the leading dimension of the array Q. If JOBZ = 'N', then LDQ >= 1. If JOBZ = 'V', then LDQ >= 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'. Periodt.

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', this is 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', this is 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. 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| ) ,

M — M is INTEGER. It's the total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', then M = N, and if RANGE = 'I', then M = IU-IL+1.

W — W is REAL array, dimension (N). If INFO = 0, this holds the eigenvalues in ascending order. Sorted and slaying 💖.

Z — Z is REAL array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.

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 REAL array, dimension (7*N). It's the workspace, doing the behind-the-scenes heavy lifting 🫶.

IWORK — IWORK is INTEGER array, dimension (5*N). Integer workspace, also clocking in.

IFAIL — IFAIL is INTEGER array, dimension (M). 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 eigenvalues that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.

INFO — INFO is INTEGER. The receipts:

= 0: successful exit — we love a clean win 👑.
< 0: if INFO = -i, the i-th argument had an illegal value. Bestie, check your inputs.
<= N: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in IFAIL. Not the moment 😭.
> N: SPBSTF returned an error code; i.e., 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. That's the danger zone, fr fr 💀.

💖 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 291 of file ssbgvx.f.

💅 `subroutine zhbgvx` 💅

subroutine zhbgvx (character jobz, character range, character uplo, integer
n, integer ka, integer kb, complex*16, dimension( ldab, * ) ab, integer
ldab, complex*16, dimension( ldbb, * ) bb, integer ldbb, 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)

ZHBGVX

🦋 Purpose 🦋

Okay so ZHBGVX computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues. Main character energy for a banded eigenproblem, honestly 🌈.

✨ 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 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.

KA — KA is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.

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

On exit, the contents of AB are destroyed. Literally gone, no cap 💀.

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

BB — BB is COMPLEX*16 array, dimension (LDBB, N). On entry, the upper or lower triangle of the Hermitian band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows:

if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

On exit, the factor S from the split Cholesky factorization B = S**H*S, as returned by ZPBSTF. Note the **H — this is the complex bestie version 💜.

LDBB — LDBB is INTEGER. The leading dimension of the array BB. LDBB >= KB+1.

Q — Q is COMPLEX*16 array, dimension (LDQ, N). If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C 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 = 'N', LDQ >= 1. If JOBZ = 'V', 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 — INTEGER. When RANGE='I', this is the index of the smallest eigenvalue you want back. The bounds are 1 <= IL <= IU <= N if N > 0; and if N = 0 then IL = 1 and IU = 0. Not referenced if RANGE = 'A' or 'V'. 💅
IU — INTEGER. When RANGE='I', this is the index of the largest eigenvalue to be returned. Same bounds, bestie: 1 <= IL <= IU <= N if N > 0; if N = 0 then IL = 1 and IU = 0. Not referenced if RANGE = 'A' or 'V'. ✨
ABSTOL — DOUBLE PRECISION. This is 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'). 🫶

M — INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', then M = N, and if RANGE = 'I', then M = IU-IL+1. 🎀
W — DOUBLE PRECISION array, dimension (N). If INFO = 0, this holds the eigenvalues in ascending order. ⭐
Z — COMPLEX*16 array, dimension (LDZ, N). If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced. 🦋
LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N. 👑
WORK — COMPLEX*16 array, dimension (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, and i is:
  - <= N: then i eigenvectors failed to converge. Their indices are stored in array IFAIL. 😭
  - > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF returned INFO = i: B is not positive definite. 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 297 of file zhbgvx.f.

Author 💻 (the auto-generated one)

Generated automatically by Doxygen for LAPACK from the source code.

okay that’s the whole page, bestie — huge shoutout to LAPACK Version 3.12.0 for this hbgvx(3) moment, periodt 💖

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

hbgvx(3)

💖 hbgvx(3) — Library Functions Manual 💖

✨ SYNOPSIS ✨

🌸 Functions

🦋 Detailed Description

🎀 Function Documentation 🎀

🌈 subroutine chbgvx

✨ Purpose

💜 Parameters

👑 Author

🌟 Contributors:

🧚 subroutine dsbgvx

✨ Purpose

💜 Parameters

👑 Author

🫶 Contributors

📍 Definition

🌟 subroutine ssbgvx

🎯 Purpose

💅 Parameters

💖 Author 💖

🎀 Contributors 🎀

💅 subroutine zhbgvx 💅

🦋 Purpose 🦋

✨ Parameters ✨

Author 🫶

Contributors: ✨

Author 💻 (the auto-generated one)

🌈 `subroutine chbgvx`

🧚 `subroutine dsbgvx`

🌟 subroutine `ssbgvx`

💅 `subroutine zhbgvx` 💅