SRC/dsyevr.f(3)

📦

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Okay so welcome to SRC/dsyevr.f(3) from the Library Functions Manual — yes bestie, we're in the manual-page cinematic universe and this one is all about the dsyevr subroutine. It's giving linear algebra main character energy, no cap. 💖

💖 NAME 💖

SRC/dsyevr.f — that's the page, and the subroutine in the spotlight is dsyevr.

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

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

DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices.

🌸 Function/Subroutine Documentation 🌸

subroutine dsyevr (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 computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices.

🦋 Purpose 🦋

Okay so here's the tea: DSYEVR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Invocations with different choices for these parameters may result in the computation of slightly different eigenvalues and/or eigenvectors for the same matrix. Why? Because there exists a variety of algorithms — each performing best for a particular set of options — with DSYEVR attempting to select the best based on the various parameters. In all cases, the computed values are accurate within the limits of finite precision arithmetic. Periodt.

DSYEVR first reduces the matrix A to tridiagonal form T with a call to DSYTRD. Then, whenever possible, DSYEVR 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 calls DSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. DSYEVR calls DSTEBZ and DSTEIN 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. Fr fr, keep that in mind.

🌈 Parameters 🌈

Now the parameter lineup. We're doing every single one because detail is the vibe. 💅

JOBZ

JOBZ is CHARACTER*1.

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.

This parameter influences the choice of the algorithm and may alter the computed values.

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.

This parameter influences the choice of the algorithm and may alter the computed values.

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. Not me gasping — it's literally in the manual.

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`

ISUPPZ is an INTEGER array, dimension ( 2*max(1,M) ). It’s the support of the eigenvectors in Z — like the indices that show which elements in Z are actually nonzero, bestie. 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 a DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. That’s the workspace receipt, basically.

🧚 `LWORK`

LWORK is INTEGER. It’s the dimension of the WORK array. If N <= 1, then LWORK >= 1; otherwise LWORK >= 26*N. For optimal efficiency, LWORK >= (NB+6)*N, where NB is the max of the blocksize for DSYTRD and DORMTR returned by ILAENV.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. So -1 is the “just tell me the size” vibe, no cap.

🎀 `IWORK`

IWORK is an INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. The integer-workspace receipt, slay.

🦋 `LIWORK`

LIWORK is INTEGER. It’s the dimension of the IWORK array. If N <= 1, then LIWORK >= 1; otherwise 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.

🌈 `INFO`

INFO is INTEGER, and it’s the vibe check:

= 0: successful exit — we love a clean run ✨
< 0: if INFO = -i, the i-th argument had an illegal value — not the move, bestie 😭
> 0: Internal error — the math said “girl, something’s off” 💀

👑 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

Definition at line 346 of file dsyevr.f.

🦋 Author

Generated automatically by Doxygen for LAPACK from the source code.

That’s the whole page, bestie — source is SRC/dsyevr.f(3), LAPACK Version 3.12.0, and the footer didn’t give us a date, so we’re just gonna keep it cute and move on 💖🫶💅

📦 where this page comes from

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