stevr(3)

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stevr(3) — Library Functions Manual 💅✨

okay so this is stevr(3) from the Library Functions Manual, and she is literally giving main-character LAPACK energy. no cap.

🌸 NAME 🌸

stevr - stevr: eig, MRRR

short, mysterious, iconic. we love a concise queen 🫶

✨ SYNOPSIS ✨

Functions

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

💅 Detailed Description 💅

bestie this section is literally the detailed description zone — keep scrolling and we get into the function docs, fr fr 🦋

🎀 Function Documentation 🎀

subroutine dstevr

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

Purpose

okay bestie, here's the tea ☕:

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. 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. Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm,
  3. If there is a cluster of close eigenvalues, sigma_i close to the cluster, and go to step (a),
  4. 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.

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

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

pick your vibe, bestie 💅

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

N — N is INTEGER

The order of the matrix. N >= 0.

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.

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.

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.

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

  • = 0: successful exit
  • < 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.

Also for the dstevr side of the story, huge credit to these contributor legends 💖:

  • Inderjit Dhillon, IBM Almaden, USA
  • Osni Marques, LBNL/NERSC, USA
  • Ken Stanley, Computer Science Division, University of California at Berkeley, USA

dstevr is defined at line 301 of file dstevr.f.

subroutine sstevr (...) — tridiagonal eigen-slay, but make it fashion 💅

subroutine sstevr (character jobz, character range, integer n, real,
    dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il,
    integer iu, real abstol, integer m, real, dimension( * ) w, real,
    dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz,
    real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork,
    integer liwork, integer info)

SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose 🎯

Okay so bestie, SSTEVR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T. You can pick which eigenvalues and eigenvectors you want by giving either a range of values or a range of indices for the eigenvalues you’re obsessed with.

Whenever possible, SSTEVR calls SSTEMR to compute the eigenspectrum using Relatively Robust Representations. SSTEMR gets eigenvalues via the dqds algorithm, while the orthogonal eigenvectors come from various L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as much as possible. More specifically, the algorithm steps for the i-th unreduced block of T are:

  1. 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. Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm.
  3. If there is a cluster of close eigenvalues, sigma_i close to the cluster, and go to step (a).
  4. 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.

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

Normal execution of SSTEMR 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. So like… be aware, bestie 😭.

Parameters 📋

  • JOBZ — JOBZ is CHARACTER*1, aka the “do we want eigenvectors too?” switch.
    • = 'N': compute eigenvalues only;
    • = 'V': compute eigenvalues and eigenvectors.
  • RANGE — RANGE is CHARACTER*1, and it controls the selection vibe.
    • = '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, SSTEBZ and SSTEIN are called.
  • N — N is INTEGER; it’s the order of the matrix. Constraint: N >= 0.
  • D — D is REAL 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 REAL 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 REAL.
    • 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 REAL.
    • 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 REAL; it’s 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| ) ,
  • Here 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 SLAMCH( '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. Constraint: 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
  • W — W is REAL array, dimension (N). The first M elements contain the selected eigenvalues in ascending order.
  • Z — Z is REAL 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.
  • LDZ — LDZ is INTEGER; the leading dimension of the array Z. Constraint: LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).
  • ISUPPZ — ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ).
    • This is 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 REAL 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. Constraint: LWORK >= 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.
  • 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. Constraint: LIWORK >= 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.
    • = 0: successful exit.
    • < 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

Definition 📍

SSTEVR is defined at line 303 of file sstevr.f.

Author 🤖

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