hetrd_he2hb(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

✨ NAME ✨

hetrd_he2hb — {he,sy}trd_he2hb: full to band (1st stage). 💅 this is the hetrd_he2hb(3) page, bestie, and it's giving "first stage of the glow-up from full to band" energy.

✨ SYNOPSIS ✨

💖 Functions 💖

subroutine chetrd_he2hb (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info) — CHETRD_HE2HB
subroutine dsytrd_sy2sb (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info) — DSYTRD_SY2SB
subroutine ssytrd_sy2sb (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info) — SSYTRD_SY2SB
subroutine zhetrd_he2hb (uplo, n, kd, a, lda, ab, ldab, tau, work, lwork, info) — ZHETRD_HE2HB

✨ Detailed Description ✨

okay so this is where the detailed function documentation starts, and yes, the routines are about to get their main character moment. 🫶

💫 Function Documentation 💫

🦋 subroutine chetrd_he2hb 🦋

subroutine chetrd_he2hb (character uplo, integer n, integer kd, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( * ) tau, complex, dimension( * ) work, integer lwork, integer info)

CHETRD_HE2HB

Purpose:

CHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian band-diagonal form AB by a unitary similarity transformation: Q**H * A * Q = AB. 🔥

Parameters

UPLO — CHARACTER*1. 'U' means the upper triangle of A is stored; 'L' means the lower triangle of A is stored. 👑
N — INTEGER. The order of the matrix A. N >= 0. ✨
KD — INTEGER. The number of superdiagonals of the reduced matrix if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. The reduced matrix is stored in the array AB. 💜
A — 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, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details. 🌸
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💫
AB — COMPLEX array, dimension (LDAB,N). On exit, 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). 🎀
LDAB — INTEGER. The leading dimension of the array AB. LDAB >= KD+1. 🦋
TAU — COMPLEX array, dimension (N-KD). The scalar factors of the elementary reflectors (see Further Details). 🌈
WORK — COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, or if LWORK = -1, WORK(1) returns the size of LWORK. ⭐
LWORK — INTEGER. The dimension of the array WORK which should be calculated by a workspace query. If N <= KD+1, LWORK >= 1, else LWORK = MAX(1, LWORK_QUERY).

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.

LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD

where FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice otherwise putting LWORK=-1 will provide the size of WORK. 💀

INFO — INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. 🙈

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Further Details:

Implemented by Azzam Haidar. 👑

All details are available on technical report, SC11, SC13 papers. 📚

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

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors:

Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd.

Each H(i) has the form:

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in A(i,i+kd+1:n), and tau in TAU(i). 💖

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors:

Q = H(1) H(2) . . . H(k), where k = n-kd.

Each H(i) has the form:

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in A(i+kd+2:n,i), and tau in TAU(i). 🦋

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U':                       if UPLO = 'L':

  (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
  (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
  (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
  (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
  (                            ab    )              (  v1     v2     v3     ab/v4 ab )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). 🫶

📌 Definition at line 243 of file chetrd_he2hb.f.

🌈 subroutine dsytrd_sy2sb 🌈

subroutine dsytrd_sy2sb (character uplo, integer n, integer kd, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DSYTRD_SY2SB

Purpose:

DSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric band-diagonal form AB by a orthogonal similarity transformation: Q**T * A * Q = AB. 🔥

Parameters

UPLO — CHARACTER*1. 'U' means the upper triangle of A is stored; 'L' means the lower triangle of A is stored. 👑
N — INTEGER. The order of the matrix A. N >= 0. ✨
KD — INTEGER. The number of superdiagonals of the reduced matrix if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. The reduced matrix is stored in the array AB. 💜
A

A is DOUBLE PRECISION array, dimension (LDA,N) — and bestie, this is the one you actually need to pay attention to. 👑

On entry, it's your symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A holds the upper triangular part of the matrix, and the strictly lower triangular part is not referenced (so don't stress about what's down there, nobody's looking 🙈). If UPLO = 'L', it flips: the leading N-by-N lower triangular part holds the lower triangular part of A, and the strictly upper triangular part is not referenced.

On exit it gets reworked, so heads up:

If UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, together with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors.
If UPLO = 'L', the diagonal and first subdiagonal of A are over-written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, together with TAU, represent the orthogonal matrix Q as a product of elementary reflectors.

See Further Details if you want the full lore ✨

LDA — INTEGER, the leading dimension of the array A. The rule is LDA >= max(1,N) and it is not a suggestion, it's a vibe check you must pass 💅

AB — DOUBLE PRECISION array, dimension (LDAB,N). On exit, this bad boy holds 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 lives in the j-th column of AB like so:

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).

LDAB — INTEGER, the leading dimension of the array AB. Needs LDAB >= KD+1. Simple as. 🎀

TAU — DOUBLE PRECISION array, dimension (N-KD). These are the scalar factors of the elementary reflectors (peep Further Details for the receipts).

WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, or if LWORK = -1, then WORK(1) returns the size of LWORK. Iconic behaviour honestly.

LWORK — INTEGER. This is the dimension of the array WORK, and it should be calculated by a workspace query. Listen:

If N <= KD+1, then LWORK >= 1, else LWORK = MAX(1, LWORK_QUERY).
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.

And the formula, copy it exactly or perish:

LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD

where FACTOPTNB is the blocking used by the QR or LQ algorithm — usually FACTOPTNB=128 is a good choice, otherwise putting LWORK=-1 will provide the size of WORK. Literally just ask the routine, it knows 🫶

INFO — INTEGER, the reporting bestie:

= 0: successful exit 🎉
< 0: if INFO = -i, the i-th argument had an illegal value 💀

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

✨ Further Details: ✨

Implemented by Azzam Haidar. Legend behaviour, no cap 👑

All details are available on technical report, SC11, SC13 papers. Here's the full reading list if you want to go full academic girly:

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

Now the actual math part, and honestly it's giving main character energy 🦋

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors

Q = H(k)**T . . . H(2)**T H(1)**T, where k = n-kd.

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in A(i,i+kd+1:n), and tau in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = n-kd.

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in A(i+kd+2:n,i), and tau in TAU(i).

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U':                       if UPLO = 'L':

  (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
  (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
  (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
  (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
  (                            ab    )              (  v1     v2     v3     ab/v4 ab )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). 🧚

Definition at line 243 of file dsytrd_sy2sb.f.

💜 `subroutine ssytrd_sy2sb` 💜

Okay bestie, now the REAL version — same energy, single precision, still slaying:

subroutine ssytrd_sy2sb (character uplo, integer n, integer kd, real,
    dimension( lda, * ) a, integer lda, real, dimension( ldab, * ) ab,
    integer ldab, real, dimension( * ) tau, real, dimension( * ) work,
    integer lwork, integer info)

⭐ Purpose: ⭐

SSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric band-diagonal form AB by a orthogonal similarity transformation:

Q**T * A * Q = AB.

Periodt. That's the whole job 💅

🎀 Parameters 🎀

UPLO — CHARACTER*1, pick your triangle:

= '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 reduced matrix if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. The reduced matrix is stored in the array AB.

A — REAL 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, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over-written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. 🦋

LDA — INTEGER, the leading dimension of the array A. LDA >= max(1,N).

AB — REAL array, dimension (LDAB,N). On exit, 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).

LDAB — INTEGER, the leading dimension of the array AB. LDAB >= KD+1.

TAU — REAL array, dimension (N-KD). The scalar factors of the elementary reflectors (see Further Details).

WORK — REAL array, dimension (LWORK). On exit, if INFO = 0, or if LWORK = -1, WORK(1) returns the size of LWORK.

LWORK —

LWORK — INTEGER. It's the dimension of the WORK array, and bestie, you're supposed to get this number from a workspace query. Rules: if N <= KD+1, then LWORK >= 1; otherwise LWORK = MAX(1, LWORK_QUERY).

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. Iconic, no notes.

LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD

where FACTOPTNB is the blocking used by the QR or LQ algorithm — usually FACTOPTNB=128 is a good choice 💅 — otherwise putting LWORK=-1 will provide the size of WORK. Just do the query, it's giving "trust the process".

INFO — INTEGER, the vibe report:
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value 😭

💖 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌈 Further Details

Implemented by Azzam Haidar. 👑

All details are available on technical report, SC11, SC13 papers — like, the receipts are right there:

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

okay so now the actual tea on how Q gets built 🫖

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors:

Q = H(k)**T . . . H(2)**T H(1)**T, where k = n-kd.

Each H(i) has the form:

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in A(i,i+kd+1:n), and tau in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors:

Q = H(1) H(2) . . . H(k), where k = n-kd.

Each H(i) has the form:

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in A(i+kd+2:n,i), and tau in TAU(i).

The contents of A on exit are illustrated by the following examples with n = 5 🎀:

if UPLO = 'U':                       if UPLO = 'L':

  (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
  (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
  (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
  (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
  (                            ab    )              (  v1     v2     v3     ab/v4 ab )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).

Definition at line 243 of file ssytrd_sy2sb.f.

🦋 `subroutine zhetrd_he2hb (character uplo, integer n, integer kd, complex16, dimension( lda, ) a, integer lda, complex16, dimension( ldab, ) ab, integer ldab, complex16, dimension( ) tau, complex16, dimension( ) work, integer lwork, integer info)`

ZHETRD_HE2HB

✨ Purpose

ZHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian band-diagonal form AB by a unitary similarity transformation:

Q**H * A * Q = AB.

she's literally transforming the whole situation, no cap 💅

🌸 Parameters

UPLO — CHARACTER*1, the "which half of the matrix are we even looking at" flag:
- = '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 reduced matrix if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0. The reduced matrix is stored in the array AB. 🍓
A — COMPLEX*16 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, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over-written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details. 👀

LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AB — COMPLEX*16 array, dimension (LDAB,N).

On exit, 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).

LDAB — INTEGER. The leading dimension of the array AB. LDAB >= KD+1. 🧚
TAU — COMPLEX*16 array, dimension (N-KD). The scalar factors of the elementary reflectors (see Further Details).
WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, or if LWORK = -1, WORK(1) returns the size of LWORK.
LWORK — INTEGER. The dimension of the array WORK which should be calculated by a workspace query. If N <= KD+1, LWORK >= 1, else LWORK = MAX(1, LWORK_QUERY).

LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD

where FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice otherwise putting LWORK=-1 will provide the size of WORK. yes bestie, just pass -1 🙈

INFO — INTEGER, the receipts:
- = 0: successful exit 🫶
- < 0: if INFO = -i, the i-th argument had an illegal value 💀

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

⭐ Further Details

Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

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

okay so the Q representation is literally this:

If UPLO = 'U' — the matrix Q is represented as a product of elementary reflectors:

Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd.

Each H(i) has the form:

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in A(i,i+kd+1:n), and tau in TAU(i). periodt, that's the storage tea. 💅

If UPLO = 'L' — the matrix Q is represented as a product of elementary reflectors:

Q = H(1) H(2) . . . H(k), where k = n-kd.

Each H(i) has the form:

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in A(i+kd+2:n,i), and tau in TAU(i). no cap, bestie, that's exactly where it goes. ✨

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U':                       if UPLO = 'L':

  (  ab  ab/v1  v1      v1     v1    )              (  ab                            )
  (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )
  (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )
  (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )
  (                            ab    )              (  v1     v2     v3     ab/v4 ab )

and the page's note is literally: where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). iykyk 🦋

Definition at line 243 of file zhetrd_he2hb.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code. not me being auto-generated and still serving looks 💖

okay that's literally the whole hetrd_he2hb(3) page, bestie — the footer's source/version + date line gives us LAPACK Version 3.12.0, and we are so done. mwah! 💖✨🎀

📦 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

hetrd_he2hb(3)

✨ NAME ✨

✨ SYNOPSIS ✨

💖 Functions 💖

✨ Detailed Description ✨

💫 Function Documentation 💫

🦋 subroutine chetrd_he2hb 🦋

🌈 subroutine dsytrd_sy2sb 🌈

🌸 Author 🌸

✨ Further Details: ✨

💜 subroutine ssytrd_sy2sb 💜

⭐ Purpose: ⭐

🎀 Parameters 🎀

💖 Author

🌈 Further Details

🦋 subroutine zhetrd_he2hb (character uplo, integer n, integer kd, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRD_HE2HB

✨ Purpose

🌸 Parameters

👑 Author

⭐ Further Details

👑 Author 👑

💜 `subroutine ssytrd_sy2sb` 💜

🦋 `subroutine zhetrd_he2hb (character uplo, integer n, integer kd, complex16, dimension( lda, ) a, integer lda, complex16, dimension( ldab, ) ab, integer ldab, complex16, dimension( ) tau, complex16, dimension( ) work, integer lwork, integer info)`