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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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| 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
Download CSTEMR + dependencies [TGZ] [ZIP] [TXT]
!>
!> 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, 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.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.
!> | [in] | JOBZ | !> JOBZ is CHARACTER*1 !> = 'N': Compute eigenvalues only; !> = 'V': Compute eigenvalues and eigenvectors. !> |
| [in] | 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. !> |
| [in] | N | !> N is INTEGER !> The order of the matrix. N >= 0. !> |
| [in,out] | D | !> D is REAL array, dimension (N) !> On entry, the N diagonal elements of the tridiagonal matrix !> T. On exit, D is overwritten. !> |
| [in,out] | 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. !> |
| [in] | 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'. !> |
| [in] | 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'. !> |
| [in] | 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'. !> |
| [in] | 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'. !> |
| [out] | 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. !> |
| [out] | W | !> W is REAL array, dimension (N) !> The first M elements contain the selected eigenvalues in !> ascending order. !> |
| [out] | 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. !> |
| [in] | LDZ | !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= 1, and if !> JOBZ = 'V', then LDZ >= max(1,N). !> |
| [in] | 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. !> |
| [out] | 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. !> |
| [in,out] | 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. !> |
| [out] | WORK | !> WORK is REAL array, dimension (LWORK) !> On exit, if INFO = 0, WORK(1) returns the optimal !> (and minimal) LWORK. !> |
| [in] | 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. !> |
| [out] | IWORK | !> IWORK is INTEGER array, dimension (LIWORK) !> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. !> |
| [in] | 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. !> |
| [out] | 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 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. !> |
Definition at line 334 of file cstemr.f.
1.11.0