stemr(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💅 stemr(3) 🦋

okay bestie, welcome to the Library Functions Manual entry for stemr — yes, that's the manual name, we're doing this properly 💖

✨ NAME ✨

stemr — eig, relatively robust representation (RRR)

🎀 SYNOPSIS 🎀

functions

  • subroutine cstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info) — CSTEMR
  • subroutine dstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info) — DSTEMR
  • subroutine sstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info) — SSTEMR
  • subroutine zstemr (jobz, range, n, d, e, vl, vu, il, iu, m, w, z, ldz, nzc, isuppz, tryrac, work, lwork, iwork, liwork, info) — ZSTEMR

🌸 Detailed Description 🌸

💅 Function Documentation 💅

subroutine cstemr 🌈

subroutine cstemr (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, complex, dimension( ldz,
    * ) z, integer ldz, integer nzc, integer, dimension( * ) isuppz, logical
    tryrac, real, dimension( * ) work, integer lwork, integer, dimension( *
    ) iwork, integer liwork, integer info)

CSTEMR

Purpose 💜

okay so here's the whole vibe, bestie:

CSTEMR 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. no cap ✨

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:

  1. 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.
  2. 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).
  3. 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.
  4. 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 🦋

  1. CSTEMR 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. 🫶
  2. LAPACK routines can be used to reduce a complex Hermitean matrix to real symmetric tridiagonal form.

(Any complex Hermitean tridiagonal matrix has real values on its diagonal and potentially complex numbers on its off-diagonals. By applying a similarity transform with an appropriate diagonal matrix diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean matrix can be transformed into a real symmetric matrix and complex arithmetic can be entirely avoided.)

While the eigenvectors of the real symmetric tridiagonal matrix are real, the eigenvectors of original complex Hermitean matrix have complex entries in general. Since LAPACK drivers overwrite the matrix data with the eigenvectors, CSTEMR accepts complex workspace to facilitate interoperability with CUNMTR or CUPMTR. 😭✨

🍓 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 COMPLEX 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 — okay bestie, LWORK is an INTEGER and it’s the dimension of the WORK array. The rules are LWORK >= max(1,18*N) if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. If LWORK = -1, it’s a workspace query: 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 an INTEGER array, dimension (LIWORK). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 🎀
  • LIWORK — LIWORK is an INTEGER and it’s the dimension of the IWORK array. 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, it’s a workspace query: 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 is an INTEGER. On exit, INFO = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; > 0: if INFO = 1X, internal error in SLARRE; if INFO = 2X, internal error in CLARRV. Here, the digit X = ABS( IINFO ) < 10, where IINFO is the nonzero error code returned by SLARRE or CLARRV, respectively. 😭

👑 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 336 of file cstemr.f. 🦋

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

🌟 DSTEMR 🌟

🎯 Purpose 🎯

okay so DSTEMR 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. No cap. 👑

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 🔍

1. DSTEMR 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 a DOUBLE PRECISION array, dimension (N). On entry, the N diagonal elements of the tridiagonal matrix T. On exit, D is overwritten. 🌸
  • E — E is a DOUBLE PRECISION 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 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. 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 a DOUBLE PRECISION array, dimension (N). The first M elements contain the selected eigenvalues in ascending order. ⭐
  • Z — Z is a DOUBLE PRECISION 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 an 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 — TRYRAC is LOGICAL. Okay bestie, if TRYRAC = .TRUE., you’re telling the code to check whether the tridiagonal matrix defines its eigenvalues to high relative accuracy. If it does, 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 use 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 DOUBLE PRECISION array, dimension (LWORK). On exit, if INFO = 0, WORK(1) returns the optimal (and minimal) LWORK.
  • LWORK — LWORK is INTEGER. It’s 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 is INTEGER. On exit, INFO is:
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = 1X, internal error in DLARRE; if INFO = 2X, internal error in DLARRV. Here, the digit X = ABS( IINFO ) < 10, where IINFO is the nonzero error code returned by DLARRE or DLARRV, respectively.

👑 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 dstemr.f.

🦋 subroutine sstemr 🦋

   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

✨ Purpose ✨

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.

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 🌟

  1. 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 — INTEGER array, dimension ( 2*max(1,M) ). This is the support of the eigenvectors in Z, i.e. the indices that flag 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. And bestie, listen: ISUPPZ is only accessed when JOBZ is 'V' and N > 0 — otherwise she literally doesn't even look at it 💅
  • TRYRAC — LOGICAL. If TRYRAC = .TRUE., that's you telling the code "hey, check whether the tridiagonal matrix defines its eigenvalues to high relative accuracy." If it does, 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 use 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. So she snitches on the matrix, basically 🫶
  • WORK — REAL array, dimension (LWORK). On exit, if INFO = 0, WORK(1) returns the optimal (and minimal) LWORK. ✨
  • LWORK — 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 — INTEGER array, dimension (LIWORK). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 🌸
  • LIWORK — 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 — INTEGER. On exit, INFO is:
    • = 0: successful exit. slay 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value 🙈
    • > 0: if INFO = 1X, internal error in SLARRE; if INFO = 2X, internal error in SLARRV. Here, the digit X = ABS( IINFO ) < 10, where IINFO is the nonzero error code returned by SLARRE or SLARRV, respectively. 💀

💖 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

🦋 subroutine zstemr

Okay bestie, deep breath — this is the big complex*16 sibling, and the way that she still wants real accuracy out of a complex problem is giving main character energy 👑

subroutine zstemr (character jobz, character range, integer n, double
    precision, dimension( * ) d, double precision, dimension( * ) e, double
    precision vl, double precision vu, integer il, integer iu, integer m,
    double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z,
    integer ldz, integer nzc, integer, dimension( * ) isuppz, logical
    tryrac, double precision, dimension( * ) work, integer lwork, integer,
    dimension( * ) iwork, integer liwork, integer info)
ZSTEMR

Purpose:

ZSTEMR 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.

The spectrum may be computed either completely or partially — you pick, 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 by the dqds algorithm. Numerically orthogonal eigenvectors are computed by the use of various suitable L D L^T factorizations near clusters of close eigenvalues — these are referred to as RRRs, Relatively Robust Representations. Here's the informal sketch of the algorithm, bestie:

For each unreduced block (submatrix) of T,

  1. Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. What that means: 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. 🫠
  2. 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). ⏰
  3. 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. 🎯
  4. 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

  1. ZSTEMR 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. ⚡
  2. LAPACK routines can be used to reduce a complex Hermitean matrix to real symmetric tridiagonal form.
(Any complex Hermitean tridiagonal matrix has real values on its diagonal and potentially complex numbers on its off-diagonals. By applying a similarity transform with an appropriate diagonal matrix diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean matrix can be transformed into a real symmetric matrix and complex arithmetic can be entirely avoided.) 🧠

While the eigenvectors of the real symmetric tridiagonal matrix are real, the eigenvectors of the original complex Hermitean matrix have complex entries in general. Since LAPACK drivers overwrite the matrix data with the eigenvectors, ZSTEMR accepts complex workspace to facilitate interoperability with ZUNMTR or ZUPMTR. 🦋

Parameters

  • JOBZ — CHARACTER*1
    • = 'N': Compute eigenvalues only.
    • = 'V': Compute eigenvalues and eigenvectors.
  • RANGE — 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 — INTEGER. The order of the matrix. N >= 0. 🍓
  • D — DOUBLE PRECISION array, dimension (N). On entry, the N diagonal elements of the tridiagonal matrix T. On exit, D is overwritten. 🙈
  • E — DOUBLE PRECISION 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 — 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 — 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 — 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 — 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 — INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. 🔢
  • W — DOUBLE PRECISION array, dimension (N). The first M elements contain the selected eigenvalues in ascending order. ate and left no crumbs 👑
  • Z — okay so this one's next up, bestie — hold that thought, she's literally right there in the next part 💜

Z 🦋

Z is COMPLEX*16 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 — yes, you, bestie — 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 use 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 DOUBLE PRECISION 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 is INTEGER. On exit, INFO:

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = 1X, internal error in DLARRE, if INFO = 2X, internal error in ZLARRV. Here, the digit X = ABS( IINFO ) < 10, where IINFO is the nonzero error code returned by DLARRE or ZLARRV, respectively.

✨ 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 336 of file zstemr.f.

Author Generated automatically by Doxygen for LAPACK from the source code.

That’s the whole stemr(3) saga, bestie! Footer love: LAPACK version 3.12.0, stemr(3), and the date your generated copy is stamping on it — iconic, no cap. 💅✨👑

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