hbevx_2stage(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

💖 hbevx_2stage(3) — Library Functions Manual 💖

okay so this is the hbevx_2stage(3) page — manual name folded near the top, bestie. It's {hb,sb}evx_2stage: eig, bisection, and we're in part 1 of 5. 💅

🌸 NAME 🌸

hbevx_2stage — {hb,sb}evx_2stage: eig, bisection

✨ SYNOPSIS ✨

🫶 Functions 🫶

chbevx_2stage — CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

subroutine chbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)

dsbevx_2stage — DSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

subroutine dsbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info)

ssbevx_2stage — SSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

subroutine ssbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info)

zhbevx_2stage — ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

subroutine zhbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)

🦋 Detailed Description 🦋

(The original page doesn't add extra paragraphs here, bestie — it heads straight into the function documentation.)

👑 Function Documentation 👑

🌟 subroutine `chbevx_2stage` 🌟

subroutine chbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, rwork, iwork, ifail, info)

CHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices.

🎯 Purpose 🎯

CHBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A using the 2stage technique for the reduction to tridiagonal. 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. Not available in this release.
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.
KD — KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 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 KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

On exit, AB is overwritten by values generated during the reduction to tridiagonal form.

LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KD + 1.
Q — Q is COMPLEX array, dimension (LDQ, N). If JOBZ = 'V', the N-by-N unitary matrix used in the reduction 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 = 'V', then 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| ) ,

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 AB to tridiagonal form.

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').

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 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 = '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 array, dimension (LWORK).
LWORK — LWORK is INTEGER. The length of the array WORK. LWORK >= 1, when N <= 1; otherwise: If JOBZ = 'N' and N > 1, LWORK must be queried.

LWORK = MAX(1, dimension) where
dimension = (2KD+1)*N + KD*NTHREADS
where KD is the size of the band.
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.

If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK 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: 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.

🔍 Further Details 🔍

and before we move on, here are the papers that explain all the 2stage tea ☕️📚 (all details about the 2stage techniques are available in:)

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394
A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292
A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

Definition at line 323 of file chbevx_2stage.f.

okay bestie, next up is dsbevx_2stage — she computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices, and she's literally iconic 💅🔥

subroutine dsbevx_2stage (character jobz, character range, character uplo,
    integer n, integer kd, double precision, dimension( ldab, * ) ab,
    integer ldab, 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 lwork, integer, dimension( * ) iwork,
    integer, dimension( * ) ifail, integer info)

Purpose

DSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. It's giving customizable eigenvalue realness — you pick what you want, she finds it ✨

Parameters

JOBZ — JOBZ is CHARACTER*1
- = 'N': compute eigenvalues only;
- = 'V': compute eigenvalues and eigenvectors. Not available in this release 🙈
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.
KD — KD is INTEGER
- The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB — AB is DOUBLE PRECISION array, dimension (LDAB, N)
- On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
  - if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
- On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.
LDAB — LDAB is INTEGER
- The leading dimension of the array AB. LDAB >= KD + 1.
Q — Q is DOUBLE PRECISION array, dimension (LDQ, N)
- If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduction 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 = 'V', then 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 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 AB 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)
- The first M elements contain the selected eigenvalues in ascending order.
Z — Z is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LWORK)
LWORK — LWORK is INTEGER
- The length of the array WORK. LWORK >= 1, when N <= 1; otherwise:
  - If JOBZ = 'N' and N > 1, LWORK must be queried.

LWORK = MAX(1, 7*N, dimension) where
dimension = (2KD+1)*N + KD*NTHREADS + 2*N
where KD is the size of the band.
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.

If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

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.

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.

Further Details

(this section is empty in the original, but we're leaving it here for completeness 💅)

okay so before we jump into the next subroutine, the authors left us a whole reading list — the ~lore~ — all about the 2stage techniques. 📚✨ No cap, these are the receipts:

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8, 11 pages. http://doi.acm.org/10.1145/2063384.2063394
A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292
A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

Definition at line 319 of file dsbevx_2stage.f.

💜 subroutine `ssbevx_2stage` — the single-precision sibling

subroutine ssbevx_2stage (character jobz, character range, character uplo,
    integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab,
    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
    lwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail,
    integer info)

SSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 💅

🎯 Purpose

SSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Translating from manpage-ese, bestie: it's the real symmetric band matrix girlie, and she uses the two-stage pipeline to get down to tridiagonal form. You can pick your eigenvalues by value range or by index range, your call 🌸

🌸 Parameters

JOBZ is CHARACTER*1
- = 'N': compute eigenvalues only;
- = 'V': compute eigenvalues and eigenvectors. Not available in this release. 😭
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 is CHARACTER*1
- = 'U': upper triangle of A is stored;
- = 'L': lower triangle of A is stored.
N is INTEGER — the order of the matrix A. N >= 0.
KD is INTEGER — the number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. (Think of it as how thicc the band is 🎀)
AB is REAL array, dimension (LDAB, N)
- On entry: the upper or lower triangle of the symmetric band matrix A, stored in the first KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
- if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
- On exit: AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB. (So yeah — your input gets yeeted, plan accordingly 🙈)
LDAB is INTEGER — the leading dimension of the array AB. LDAB >= KD + 1.
Q is REAL array, dimension (LDQ, N)
- If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduction to tridiagonal form.
- If JOBZ = 'N', the array Q is not referenced.
LDQ is INTEGER — the leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).
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 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 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 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 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'). 🔥
See by Demmel and Kahan, LAPACK Working Note #3.

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 is REAL array, dimension (N) — the first M elements contain the selected eigenvalues in ascending order. ⭐
Z is REAL 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. (iykyk — guess big, bestie 🫶)
LDZ is INTEGER — the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK is REAL array, dimension (LWORK).
LWORK is INTEGER — the length of the array WORK. LWORK >= 1, when N <= 1; otherwise:
- If JOBZ = 'N' and N > 1, LWORK must be queried.

LWORK = MAX(1, 7*N, dimension) where
dimension = (2KD+1)*N + KD*NTHREADS + 2*N
where KD is the size of the band.
NTHREADS is the number of threads used when
openMP compilation is enabled, otherwise =1.

If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available 😭

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. (the classic "just tell me how much room you need" move 👑

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

🧚 Further Details:

All details about the 2stage techniques are available in these papers, bestie — here are the receipts: 🧾✨

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394
A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292
A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

Definition at line 319 of file ssbevx_2stage.f.

subroutine zhbevx_2stage (character jobz, character range, character uplo,
    integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer
    ldab, 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,
    integer lwork, double precision, dimension( * ) rwork, integer,
    dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

ZHBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices 👑🔥

Purpose 💖

Okay so ZHBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A using the 2stage technique for the reduction to tridiagonal. You can pick which eigenvalues you want by specifying either a range of values or a range of indices for the desired eigenvalues — iconic flexibility, no cap. ✨

Parameters 📋

JOBZ — CHARACTER*1. = 'N': compute eigenvalues only; = 'V': compute eigenvalues and eigenvectors. Not available in this release.
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 triangle of A is stored; = 'L': lower triangle of A is stored.
N — INTEGER. The order of the matrix A. N >= 0.
KD — INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
AB — COMPLEX*16 array, dimension (LDAB, N). On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, AB is overwritten by values generated during the reduction to tridiagonal form. 🌀
LDAB — INTEGER. The leading dimension of the array AB. LDAB >= KD + 1.
Q — COMPLEX*16 array, dimension (LDQ, N). If JOBZ = 'V', the N-by-N unitary matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.
LDQ — INTEGER. The leading dimension of the array Q. If JOBZ = 'V', then 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| ) , 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 AB 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 the Demmel and Kahan reference, 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). The first M elements contain 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. 💅
LDZ — INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
WORK — COMPLEX*16 array, dimension (LWORK).
LWORK — INTEGER. The length of the array WORK. LWORK >= 1, when N <= 1; otherwise if JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = (2KD+1)*N + KD*NTHREADS where KD is the size of the band. NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. 😭
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.

Further Details 🫶

This section is empty in the source — no extra lore, no hidden tea, just a blank little moment. 💅

Okay so real quick — the docs are like "if you wanna go DEEP, here's the reading list," and honestly? iconic of them. 📚✨ All the receipts about how the 2stage techniques actually work are in these three papers:

📚 the reading list (aka the receipts)

Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8, 11 pages. http://doi.acm.org/10.1145/2063384.2063394
A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292
A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

Like, three whole papers of lore, bestie. That's the deep dive if the vibes alone aren't enough for you 🫶

And where does this beautiful little block live? Definition at line 323 of file zhbevx_2stage.f. 💅

👑 Author

Generated automatically by Doxygen for LAPACK from the source code. So like, nobody wrote this by hand, it's giving "the machine did my homework and honestly it ate." 🤖✨

lowkey a slay that you made it all the way to the very last line of this page — main character energy fr fr 🎀

Okay that's the whole thing, bestie — this was LAPACK, Version 3.12.0, and the page is literally called hbevx_2stage(3) so that's a wrap. Bye, love you, go call some routines 💖🌈🦋

📦 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

hbevx_2stage(3)

💖 hbevx_2stage(3) — Library Functions Manual 💖

🌸 NAME 🌸

✨ SYNOPSIS ✨

🫶 Functions 🫶

🦋 Detailed Description 🦋

👑 Function Documentation 👑

🌟 subroutine chbevx_2stage 🌟

🎯 Purpose 🎯

💅 Parameters 💅

🌈 Author 🌈

🔍 Further Details 🔍

Purpose

Parameters

Author

Further Details

💜 subroutine ssbevx_2stage — the single-precision sibling

🎯 Purpose

🌸 Parameters

👑 Author

🧚 Further Details:

Purpose 💖

Parameters 📋

Author 🌟

Further Details 🫶

📚 the reading list (aka the receipts)

👑 Author

🌟 subroutine `chbevx_2stage` 🌟

💜 subroutine `ssbevx_2stage` — the single-precision sibling