DSYEVR(3)

💖 DSYEVR(3) — Library Functions Manual 💖

✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dsyevr.f — that's the file, bestie. This page is DSYEVR(3), and it's giving major symmetric-matrix eigensolver energy. 💅

🌸 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 💫

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 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. iconic. 👑

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.

⚠️ Heads up, bestie: 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.

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. 💀

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 the Demmel and Kahan reference, 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

• LWORK is INTEGER
• It's the dimension of the array WORK, and the vibe check says LWORK >= max(1,26*N) — that's the floor, bestie, no going below it 😤
• For optimal efficiency though? You want LWORK >= (NB+6)*N, where NB is the max of the blocksize for DSYTRD and DORMTR returned by ILAENV. Like, the minimum works but the optimal is where the real main character energy lives ✨
• And okay, special case: 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. It's giving "just asking, not committing" 💅

---

🔢 IWORK

• IWORK is INTEGER array, dimension (MAX(1,LIWORK))
• On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. Yup, LWORK — that's what it says, we don't make the rules 🫶

---

🧵 LIWORK

• LIWORK is INTEGER
• It's the dimension of the array IWORK, and LIWORK >= max(1,10*N) — same deal, that's the minimum, respect it 🎀
• 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. Again: window shopping only, no purchase 😌

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🚨 INFO

• INFO is INTEGER, and it's literally the receipts:
  • = 0: successful exit — slay, we ate 🔥
  • < 0: if INFO = -i, the i-th argument had an illegal value — not me calling out your bad argument 🙈
  • > 0: Internal error 💀 something went sideways inside, not your fault (probably)

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👑 Author

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

Legends, honestly. The whole squad 🫶

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🌟 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

Shoutout to the girlies (gender-neutral) who built this 🎀💖

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Definition at line 331 of file dsyevr.f. That's where the magic lives, iykyk ✨

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📜 Author

Generated automatically by Doxygen for LAPACK from the source code. Yes, a robot wrote this part. Respect the robot 🤖💅

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And that's a wrap, bestie — this page came to you from LAPACK Version 3.12.0 out of /home/abui...RC/dsyevr.f(3). Go forth, compute those eigenvalues, and remember: query your workspace first 🦋🌸💜