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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 ssyevr | ( | character | jobz, |
| character | range, | ||
| character | uplo, | ||
| integer | n, | ||
| real, dimension( lda, * ) | a, | ||
| integer | lda, | ||
| 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 ) |
SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
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!> !> SSYEVR computes selected eigenvalues and, optionally, eigenvectors !> of a real symmetric matrix A. Eigenvalues and eigenvectors can be !> selected by specifying either a range of values or a range of indices !> for the desired eigenvalues. Invocations with different choices for !> these parameters may result in the computation of slightly different !> eigenvalues and/or eigenvectors for the same matrix. The reason for !> this behavior is that there exists a variety of algorithms (each !> performing best for a particular set of options) with SSYEVR !> attempting to select the best based on the various parameters. In all !> cases, the computed values are accurate within the limits of finite !> precision arithmetic. !> !> SSYEVR first reduces the matrix A to tridiagonal form T with a call !> to SSYTRD. Then, whenever possible, SSYEVR 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 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. !> !> The desired accuracy of the output can be specified by the input !> parameter ABSTOL. !> !> For more details, see SSTEMR's documentation and: !> - 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. !> !> !> Note 1 : SSYEVR calls SSTEMR when the full spectrum is requested !> on machines which conform to the ieee-754 floating point standard. !> SSYEVR 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. !>
| [in] | JOBZ | !> JOBZ is CHARACTER*1 !> = 'N': Compute eigenvalues only; !> = 'V': Compute eigenvalues and eigenvectors. !> !> This parameter influences the choice of the algorithm and !> may alter the computed values. !> |
| [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. !> For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and !> SSTEIN are called !> !> This parameter influences the choice of the algorithm and !> may alter the computed values. !> |
| [in] | UPLO | !> UPLO is CHARACTER*1 !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !> |
| [in] | N | !> N is INTEGER !> The order of the matrix A. N >= 0. !> |
| [in,out] | A | !> A is REAL array, dimension (LDA, N) !> On entry, the symmetric matrix A. If UPLO = 'U', the !> leading N-by-N upper triangular part of A contains the !> upper triangular part of the matrix A. If UPLO = 'L', !> the leading N-by-N lower triangular part of A contains !> the lower triangular part of the matrix A. !> On exit, the lower triangle (if UPLO='L') or the upper !> triangle (if UPLO='U') of A, including the diagonal, is !> destroyed. !> |
| [in] | LDA | !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !> |
| [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; IL = 1 and IU = 0 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; IL = 1 and IU = 0 if N = 0. !> Not referenced if RANGE = 'A' or 'V'. !> |
| [in] | 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. !> |
| [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 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). !> 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 an upper bound must be used. !> Supplying N columns is always safe. !> |
| [in] | LDZ | !> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= 1, and if !> JOBZ = 'V', LDZ >= max(1,N). !> |
| [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 eigenvector !> is nonzero only in elements ISUPPZ( 2*i-1 ) through !> ISUPPZ( 2*i ). This is an output of SSTEMR (tridiagonal !> matrix). The support of the eigenvectors of A is typically !> 1:N because of the orthogonal transformations applied by SORMTR. !> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 !> |
| [out] | WORK | !> WORK is REAL array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !> |
| [in] | LWORK | !> LWORK is INTEGER !> The dimension of the array WORK. !> If N <= 1, LWORK >= 1, else LWORK >= 26*N. !> For optimal efficiency, LWORK >= (NB+6)*N, !> where NB is the max of the blocksize for SSYTRD and SORMTR !> returned by ILAENV. !> !> 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. !> |
| [out] | IWORK | !> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) !> On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. !> |
| [in] | LIWORK | !> LIWORK is INTEGER !> The dimension of the array IWORK. !> If N <= 1, LIWORK >= 1, else 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. !> |
| [out] | INFO | !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: Internal error !> |
Definition at line 346 of file ssyevr.f.
1.11.0