SRC/dstevr.f(3)

📦
version 3.12.1 · 4 versions
, in arch rolling

🦋 SRC/dstevr.f — like, the manpage, but make it make sense 💅

okay bestie, buckle up. this one's from the Library Functions Manual and it's a whole situation — a Fortran subroutine that finds eigenvalues and (if you ask nicely) eigenvectors of a real symmetric tridiagonal matrix. yes i said tridiagonal. no i will not calm down. 🫶

✨ NAME ✨

SRC/dstevr.f — no cap, that's literally the name. it's a file, it's a subroutine, it's iconic.

🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine dstevr (jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w,
    z, ldz, isuppz, work, lwork, iwork, liwork, info)
DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

(the "OTHER matrices" thing is the official description and i'm keeping it because it's giving mysterious main character energy tbh 💀)

👑 Function/Subroutine Documentation 👑

subroutine dstevr (character jobz, character range, integer n, double
    precision, dimension( * ) d, double precision, dimension( * ) e, 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)

DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices — yes, again, it's that girl. 🎀

💜 Purpose:

so here's the tea, straight from the source:

DSTEVR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
Whenever possible, DSTEVR 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 the i-th unreduced block of T,
  1. (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation,
  2. (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm,
  3. (c) If there is a cluster of close eigenvalues, sigma_i close to the cluster, and go to step (a),
  4. (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, compute the corresponding eigenvector by forming a rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the input parameter ABSTOL.
For more details, see , by Inderjit Dhillon, Computer Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May 1997.

(yes the "see ," bit is exactly how it's written in the source — love that for us 🙈)

Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. DSTEVR 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. 💀

🌈 Parameters

  • JOBZ — JOBZ is CHARACTER*1. Your vibe selector: = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.
  • RANGE — RANGE is CHARACTER*1. Which eigenvalues are we pulling up: = '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.
  • N — N is INTEGER. The order of the matrix. N >= 0. (zero is allowed, she's flexible ✨)
  • D — D is DOUBLE PRECISION array, dimension (N). On entry, the n diagonal elements of the tridiagonal matrix A. On exit, D may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
  • E — E is DOUBLE PRECISION array, dimension (max(1,N-1)). On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A in elements 1 to N-1 of E. On exit, E may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
  • 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. (ascending! she's organized 💅)
  • 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).
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. bestie, allocate generously, i'm begging 🥺
  • 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 ). 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 (and minimal) LWORK.
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= max(1,20*N).

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA. (the -1 trick is such a slay, iykyk ⭐)

  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal (and minimal) LIWORK.
  • LIWORK — LIWORK is INTEGER. The dimension of the array IWORK. LIWORK >= max(1,10*N).

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

  • INFO — INFO is INTEGER. The receipts: = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: Internal error. (internal error is giving "it's not me, it's the algorithm" 😭)

🧚 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

Definition at line 301 of file dstevr.f. 📍

🌟 Author (again, yes, one more time)

Generated automatically by Doxygen for LAPACK from the source code. (the robots wrote the docs, we just vibe with them 🫡)

and that's the whole page, bestie! this little beauty comes to you from LAPACK, Version 3.12.0 — she's from SRC/dstevr.f(3) and she ate, no crumbs. ✨💅🎀

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