SRC/sstemr.f(3)

📦

version 3.12.1 · 7 versions

3.12.1 · arch rolling (latest)
3.9.0 · centos-stream 9
3.10.0 · ubuntu jammy
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.11.0 · debian bookworm

🎀 SRC/sstemr.f 🎀

Library Functions Manual — okay bestie, this is the manual page for SRC/sstemr.f, and we are gonna get through it together. 💅

🌸 NAME 🌸

SRC/sstemr.f

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine sstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz,
           nzc, isuppz, tryrac, work, lwork, iwork, liwork, info)
           SSTEMR

🦋 Function/Subroutine Documentation 🦋

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

SSTEMR

🔥 Purpose 🔥

Okay so SSTEMR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T. Any such unreduced matrix has a well-defined set of pairwise different real eigenvalues, and the corresponding real eigenvectors are pairwise orthogonal. That is the vibe. ✨

The spectrum may be computed either completely or partially by specifying either an interval (VL,VU] or a range of indices IL:IU for the desired eigenvalues.

Depending on the number of desired eigenvalues, these are computed either by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are computed by the use of various suitable L D L^T factorizations near clusters of close eigenvalues — referred to as RRRs, Relatively Robust Representations. An informal sketch of the algorithm 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.

For more details, see:

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.

Further Details

SSTEMR works only on machines which follow IEEE-754 floating-point standard in their handling of infinities and NaNs. This permits the use of efficient inner loops avoiding a check for zero divisors.

💖 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.
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 T. On exit, D is overwritten.
E — E is REAL array, dimension (N). On entry, the (N-1) subdiagonal elements of the tridiagonal matrix T in elements 1 to N-1 of E. E(N) need not be set on input, but is used internally as workspace. On exit, E is overwritten.
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. 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. Not referenced if RANGE = 'A' or 'V'.
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', and if INFO = 0, then the first M columns of Z contain the orthonormal eigenvectors of the matrix T 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 can be computed with a workspace query by setting NZC = -1, see below.
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', then LDZ >= max(1,N).
NZC — NZC is INTEGER. The number of eigenvectors to be held in the array Z. If RANGE = 'A', then NZC >= max(1,N). If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]. If RANGE = 'I', then NZC >= IU-IL+1. If NZC = -1, then a workspace query is assumed; the routine calculates the number of columns of the array Z that are needed to hold the eigenvectors. This value is returned as the first entry of the Z array, and no error message related to NZC is issued by XERBLA.
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 computed eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is relevant in the case when the matrix is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
TRYRAC — TRYRAC is LOGICAL. If TRYRAC = .TRUE., indicates that the code should check whether the tridiagonal matrix defines its eigenvalues to high relative accuracy. If so, the code uses relative-accuracy preserving algorithms that might be (a bit) slower depending on the matrix. If the matrix does not define its eigenvalues to high relative accuracy, the code can uses possibly faster algorithms. If TRYRAC = .FALSE., the code is not required to guarantee relatively accurate eigenvalues and can use the fastest possible techniques. On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix does not define its eigenvalues to high relative accuracy.
WORK — WORK is REAL array, dimension (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(1,18*N) if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. 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.
IWORK — IWORK is INTEGER array, dimension (LIWORK). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK — LIWORK is INTEGER. The dimension of the array IWORK. LIWORK >= max(1,10*N) if the eigenvectors are desired, and LIWORK >= max(1,8*N) if only the eigenvalues are to be computed. 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 — INTEGER. this is the ✨vibe check✨ of the whole call, bestie. On exit, INFO tells u exactly what went down:
- = 0 — 🎉 successful exit. we ate. no crumbs.
- < 0 — if INFO = -i, that means the i-th argument had an illegal value 🙈 like u passed something cursed and the routine is calling u out.
- > 0 — if INFO = 1X, that's an internal error in SLARRE. if INFO = 2X, that's an internal error in SLARRV. and here, the digit X = ABS( IINFO ) < 10, where IINFO is the nonzero error code that SLARRE or SLARRV returned, respectively. 💀 so like the tens digit is who fumbled and the ones digit is how they fumbled. iykyk.

💖 Author

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

🫶 Contributors:

Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA
Aravindh Krishnamoorthy, FAU, Erlangen, Germany

Definition at line 319 of file sstemr.f.

👑 Author

okay so this Author section is the meta one — it's not a person, it's the machine:

Generated automatically by Doxygen for LAPACK from the source code.

no human wrote this bit, it's literally Doxygen doing its little autopilot thing 🧚 slay for it honestly.

🌸 and that's the whole page, bestie! this beauty is from SRC/sstemr.f, running LAPACK Version 3.12.0, and the date is just whatever lil timestamp Doxygen stamped into the footer when it built this. go forth and compute ur eigenvalues, periodt. 💅✨🎀

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms