SRC/dsyevr_2stage.f(3)

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🌸 `SRC/dsyevr_2stage.f(3)` 🌸

okay bestie, this is the SRC/dsyevr_2stage.f(3) page from the Library Functions Manual — manual name folded in right here, as promised. It’s giving LAPACK source-file realness, and honestly? Iconic. 💅

✨ NAME ✨

SRC/dsyevr_2stage.f — that’s the file, bestie. Main character energy from the jump. 🎀

🦋 SYNOPSIS 🦋

🧚 Functions/Subroutines

subroutine dsyevr_2stage — DSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices. ✨

subroutine dsyevr_2stage (jobz, range, uplo, n, a, lda, vl, vu, il, iu,
    abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)

💖 Function/Subroutine Documentation 💖

🫶 subroutine `dsyevr_2stage` 🫶

subroutine dsyevr_2stage (character jobz, character range, character uplo,
    integer n, double precision, dimension( lda, * ) a, integer lda, 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, integer, dimension( * )
    isuppz, double precision, dimension( * ) work, integer lwork, integer,
    dimension( * ) iwork, integer liwork, integer info)

DSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices. That’s the whole vibe: symmetric-matrix eigensolver, two-stage edition. 👑

💡 Purpose 💡

okay so here’s the lore, bestie. DSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric 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. 🦋

DSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call to DSYTRD. Then, whenever possible, DSYEVR_2STAGE calls DSTEMR to compute the eigenspectrum using Relatively Robust Representations. DSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. ✨

For each unreduced block (submatrix) of T:

(a) Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d).
(c) For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input parameter ABSTOL. 🎀

For more details, see DSTEMR's documentation and:

Inderjit S. Dhillon and Beresford N. Parlett: Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
Inderjit Dhillon and Beresford Parlett: SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.
Inderjit Dhillon: , Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

Note 1 : DSYEVR_2STAGE calls DSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. DSYEVR_2STAGE calls DSTEBZ and SSTEIN on non-ieee machines and when partial spectrum requests are made. 💀

Normal execution of DSTEMR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner. 😭

🎀 Parameters 🎀

`JOBZ`

JOBZ is CHARACTER*1.

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors. Not available in this release.

bestie, do not miss that 'V' caveat — “Not available in this release” is right there. 💅

`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.

For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and DSTEIN are called. 👑

`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. ✨

`A`

A is 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 yeah, she’s gone. 💀

`LDA`

LDA is INTEGER.

The leading dimension of the array A. LDA >= 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 A to tridiagonal form.

See by Demmel and Kahan, LAPACK Working Note #3.

If high relative accuracy is important, set ABSTOL to DLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will. See J. Barlow and J. Demmel, , LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy. 💜

`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 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. Supplying N columns is always safe. 🙈

`LDZ`

LDZ is INTEGER.

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

`ISUPPZ`

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ).

The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is an output of DSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the orthogonal transformations applied by DORMTR. Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1. ⭐

`WORK`

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)).

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🧚

`LWORK`

LWORK — INTEGER, bestie, it's the dimension of the WORK array 💖
- If N <= 1, then LWORK must be at least 1.
- If JOBZ = 'N' and N > 1, LWORK must be queried ✨
- 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. No cap, that's just how it goes 🙈

And here's the exact formula, copied with love (do not lose a single character, iykyk):

LWORK = MAX(1, 26*N, dimension) where
        dimension = max(stage1,stage2) + (KD+1)*N + 5*N
                  = N*KD + N*max(KD+1,FACTOPTNB)
                    + max(2*KD*KD, KD*NTHREADS)
                    + (KD+1)*N + 5*N

where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm — usually FACTOPTNB=128 is a good choice — and NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1.

🚨 Also: if JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available — like, it's giving "coming soon to a LAPACK near you", periodt.

IWORK — INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK 👑
LIWORK — INTEGER, the dimension of the array IWORK 🎀
- If N <= 1, LIWORK >= 1, else LIWORK >= 10*N.
- If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA. Same vibe check as LWORK, fr fr 💅
INFO — INTEGER, the receipts 📋
- = 0: successful exit ✨ slay
- < 0: if INFO = -i, the i-th argument had an illegal value 😭
- > 0: Internal error 💀

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🫶 Contributors:

Inderjit Dhillon, IBM Almaden, USA

Osni Marques, LBNL/NERSC, USA

Ken Stanley, Computer Science Division, University of California at Berkeley, USA

Jason Riedy, Computer Science Division, University of California at Berkeley, USA

🌈 Further Details:

All details about the 2stage techniques are available in — bestie, the lore is documented, go read it 📚:

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 380 of file dsyevr_2stage.f.

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