SSTEVR(3)

✨ SSTEVR(3) ✨

okay so bestie, gather round, because this one is a whole LAPACK moment 💅 this page lives in the Library Functions Manual, and it's all about SSTEVR — a Fortran subroutine that computes eigenvalues (and optionally eigenvectors) of a real symmetric tridiagonal matrix, and she is very selective about which ones she gives you. iconic behavior, honestly.

The file itself is chillin' at:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sstevr.f

(shortened to /home/abui...RC/sstevr.f(3) on this page because path names in headers are lowkey extra 💀)

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🌸 SYNOPSIS 🌸

Functions/Subroutines

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 ✨

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🦋 Function/Subroutine Documentation 🦋

subroutine SSTEVR

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

SSTEVR 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. so you get to be picky. main character energy only 🔥

Whenever possible, SSTEVR calls SSTEMR to compute the eigenspectrum using Relatively Robust Representations. SSTEMR 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:

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

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.

big warning bestie 😭: 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. no cap, that's real.

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🎀 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, SSTEBZ and SSTEIN are called
• N — N is INTEGER ✨
  • The order of the matrix. 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 🎯
  • 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 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. 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. so like, plan ahead bestie 🙈
• 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 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. 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. 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

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💜 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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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

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Definition at line 303 of file sstevr.f.

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🎀 Author

Generated automatically by Doxygen for LAPACK from the source code. (yes, the robot wrote this part, not me 💅)

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okay that's the whole thing, you ate that reading session fr fr 🫶 this page is brought to you by LAPACK, Version 3.12.0 and it's chillen at /home/abui...RC/sstevr.f(3) — mwah, go compute some eigenvalues bestie 💖✨