doubleOTHEReigen

Section: LAPACK (3)
Updated: Sun Nov 27 2022
Index Return to Main Contents
 

NAME

doubleOTHEReigen - double  

SYNOPSIS


 

Functions


subroutine dbdsvdx (UPLO, JOBZ, RANGE, N, D, E, VL, VU, IL, IU, NS, S, Z, LDZ, WORK, IWORK, INFO)
DBDSVDX
subroutine dggglm (N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, INFO)
DGGGLM
subroutine dsbev (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, INFO)
DSBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dsbev_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, INFO)
DSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dsbevd (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dsbevd_2stage (JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dsbevx (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dsbevx_2stage (JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
DSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dsbgv (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, INFO)
DSBGV
subroutine dsbgvd (JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSBGVD
subroutine dsbgvx (JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSBGVX
subroutine dspev (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO)
DSPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dspevd (JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dspevx (JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dspgv (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, INFO)
DSPGV
subroutine dspgvd (ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSPGVD
subroutine dspgvx (ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSPGVX
subroutine dstev (JOBZ, N, D, E, Z, LDZ, WORK, INFO)
DSTEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dstevd (JOBZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSTEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dstevr (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine dstevx (JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices  

Detailed Description

This is the group of double Other Eigenvalue routines  

Function Documentation

 

subroutine dbdsvdx (character UPLO, 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 NS, double precision, dimension( * ) S, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

DBDSVDX

Purpose:

  DBDSVDX computes the singular value decomposition (SVD) of a real
  N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT,
  where S is a diagonal matrix with non-negative diagonal elements
  (the singular values of B), and U and VT are orthogonal matrices
  of left and right singular vectors, respectively.

  Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ]
  and superdiagonal E = [ e_1 e_2 ... e_N-1 ], DBDSVDX computes the
  singular value decompositon of B through the eigenvalues and
  eigenvectors of the N*2-by-N*2 tridiagonal matrix

        |  0  d_1                |
        | d_1  0  e_1            |
  TGK = |     e_1  0  d_2        |
        |         d_2  .   .     |
        |              .   .   . |

  If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then
  (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) /
  sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and
  P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ].

  Given a TGK matrix, one can either a) compute -s,-v and change signs
  so that the singular values (and corresponding vectors) are already in
  descending order (as in DGESVD/DGESDD) or b) compute s,v and reorder
  the values (and corresponding vectors). DBDSVDX implements a) by
  calling DSTEVX (bisection plus inverse iteration, to be replaced
  with a version of the Multiple Relative Robust Representation
  algorithm. (See P. Willems and B. Lang, A framework for the MR^3
  algorithm: theory and implementation, SIAM J. Sci. Comput.,
  35:740-766, 2013.)


 

Parameters

UPLO

          UPLO is CHARACTER*1
          = 'U':  B is upper bidiagonal;
          = 'L':  B is lower bidiagonal.


JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute singular values only;
          = 'V':  Compute singular values and singular vectors.


RANGE

          RANGE is CHARACTER*1
          = 'A': all singular values will be found.
          = 'V': all singular values in the half-open interval [VL,VU)
                 will be found.
          = 'I': the IL-th through IU-th singular values will be found.


N

          N is INTEGER
          The order of the bidiagonal matrix.  N >= 0.


D

          D is DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the bidiagonal matrix B.


E

          E is DOUBLE PRECISION array, dimension (max(1,N-1))
          The (n-1) superdiagonal elements of the bidiagonal matrix
          B in elements 1 to N-1.


VL

         VL is DOUBLE PRECISION
          If RANGE='V', the lower bound of the interval to
          be searched for singular values. VU > VL.
          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 singular values. VU > VL.
          Not referenced if RANGE = 'A' or 'I'.


IL

          IL is INTEGER
          If RANGE='I', the index of the
          smallest singular value to be returned.
          1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
          Not referenced if RANGE = 'A' or 'V'.


IU

          IU is INTEGER
          If RANGE='I', the index of the
          largest singular value to be returned.
          1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
          Not referenced if RANGE = 'A' or 'V'.


NS

          NS is INTEGER
          The total number of singular values found.  0 <= NS <= N.
          If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1.


S

          S is DOUBLE PRECISION array, dimension (N)
          The first NS elements contain the selected singular values in
          ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (2*N,K)
          If JOBZ = 'V', then if INFO = 0 the first NS columns of Z
          contain the singular vectors of the matrix B corresponding to
          the selected singular values, with U in rows 1 to N and V
          in rows N+1 to N*2, i.e.
          Z = [ U ]
              [ V ]
          If JOBZ = 'N', then Z is not referenced.
          Note: The user must ensure that at least K = NS+1 columns are
          supplied in the array Z; if RANGE = 'V', the exact value of
          NS is not known in advance and an upper bound must be used.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z. LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(2,N*2).


WORK

          WORK is DOUBLE PRECISION array, dimension (14*N)


IWORK

          IWORK is INTEGER array, dimension (12*N)
          If JOBZ = 'V', then if INFO = 0, the first NS elements of
          IWORK are zero. If INFO > 0, then IWORK contains the indices
          of the eigenvectors that failed to converge in DSTEVX.


INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, then i eigenvectors failed to converge
                   in DSTEVX. The indices of the eigenvectors
                   (as returned by DSTEVX) are stored in the
                   array IWORK.
                if INFO = N*2 + 1, an internal error occurred.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dggglm (integer N, integer M, integer P, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) D, double precision, dimension( * ) X, double precision, dimension( * ) Y, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DGGGLM

Purpose:

 DGGGLM solves a general Gauss-Markov linear model (GLM) problem:

         minimize || y ||_2   subject to   d = A*x + B*y
             x

 where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
 given N-vector. It is assumed that M <= N <= M+P, and

            rank(A) = M    and    rank( A B ) = N.

 Under these assumptions, the constrained equation is always
 consistent, and there is a unique solution x and a minimal 2-norm
 solution y, which is obtained using a generalized QR factorization
 of the matrices (A, B) given by

    A = Q*(R),   B = Q*T*Z.
          (0)

 In particular, if matrix B is square nonsingular, then the problem
 GLM is equivalent to the following weighted linear least squares
 problem

              minimize || inv(B)*(d-A*x) ||_2
                  x

 where inv(B) denotes the inverse of B.


 

Parameters

N

          N is INTEGER
          The number of rows of the matrices A and B.  N >= 0.


M

          M is INTEGER
          The number of columns of the matrix A.  0 <= M <= N.


P

          P is INTEGER
          The number of columns of the matrix B.  P >= N-M.


A

          A is DOUBLE PRECISION array, dimension (LDA,M)
          On entry, the N-by-M matrix A.
          On exit, the upper triangular part of the array A contains
          the M-by-M upper triangular matrix R.


LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).


B

          B is DOUBLE PRECISION array, dimension (LDB,P)
          On entry, the N-by-P matrix B.
          On exit, if N <= P, the upper triangle of the subarray
          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
          if N > P, the elements on and above the (N-P)th subdiagonal
          contain the N-by-P upper trapezoidal matrix T.


LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,N).


D

          D is DOUBLE PRECISION array, dimension (N)
          On entry, D is the left hand side of the GLM equation.
          On exit, D is destroyed.


X

          X is DOUBLE PRECISION array, dimension (M)


Y

          Y is DOUBLE PRECISION array, dimension (P)

          On exit, X and Y are the solutions of the GLM problem.


WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.


LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N+M+P).
          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
          where NB is an upper bound for the optimal blocksizes for
          DGEQRF, SGERQF, DORMQR and SORMRQ.

          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.


INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1:  the upper triangular factor R associated with A in the
                generalized QR factorization of the pair (A, B) is
                singular, so that rank(A) < M; the least squares
                solution could not be computed.
          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
                factor T associated with B in the generalized QR
                factorization of the pair (A, B) is singular, so that
                rank( A B ) < N; the least squares solution could not
                be computed.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dsbev (character JOBZ, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)

DSBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSBEV computes all the eigenvalues and, optionally, eigenvectors of
 a real symmetric band matrix A.


 

Parameters

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N

          N is INTEGER
          The order of the matrix A.  N >= 0.


KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.


AB

          AB is DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.


LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.


W

          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (max(1,3*N-2))


INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dsbev_2stage (character JOBZ, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
 a real symmetric band matrix A using the 2stage technique for
 the reduction to tridiagonal.


 

Parameters

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.
                  Not available in this release.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N

          N is INTEGER
          The order of the matrix A.  N >= 0.


KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.


AB

          AB is DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.


LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.


W

          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension LWORK
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.


LWORK

          LWORK is INTEGER
          The length of the array WORK. LWORK >= 1, when N <= 1;
          otherwise  
          If JOBZ = 'N' and N > 1, LWORK must be queried.
                                   LWORK = MAX(1, dimension) where
                                   dimension = (2KD+1)*N + KD*NTHREADS + N
                                   where KD is the size of the band.
                                   NTHREADS is the number of threads used when
                                   openMP compilation is enabled, otherwise =1.
          If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

          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.


INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  All details about the 2stage techniques are available in:

  Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
  Parallel reduction to condensed forms for symmetric eigenvalue problems
  using aggregated fine-grained and memory-aware kernels. In Proceedings
  of 2011 International Conference for High Performance Computing,
  Networking, Storage and Analysis (SC '11), New York, NY, USA,
  Article 8 , 11 pages.
  http://doi.acm.org/10.1145/2063384.2063394

  A. Haidar, J. Kurzak, P. Luszczek, 2013.
  An improved parallel singular value algorithm and its implementation 
  for multicore hardware, In Proceedings of 2013 International Conference
  for High Performance Computing, Networking, Storage and Analysis (SC '13).
  Denver, Colorado, USA, 2013.
  Article 90, 12 pages.
  http://doi.acm.org/10.1145/2503210.2503292

  A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
  A novel hybrid CPU-GPU generalized eigensolver for electronic structure 
  calculations based on fine-grained memory aware tasks.
  International Journal of High Performance Computing Applications.
  Volume 28 Issue 2, Pages 196-209, May 2014.
  http://hpc.sagepub.com/content/28/2/196 


 

 

subroutine dsbevd (character JOBZ, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSBEVD computes all the eigenvalues and, optionally, eigenvectors of
 a real symmetric band matrix A. If eigenvectors are desired, it uses
 a divide and conquer algorithm.

 The divide and conquer algorithm makes very mild assumptions about
 floating point arithmetic. It will work on machines with a guard
 digit in add/subtract, or on those binary machines without guard
 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
 Cray-2. It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.


 

Parameters

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N

          N is INTEGER
          The order of the matrix A.  N >= 0.


KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.


AB

          AB is DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.


LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.


W

          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array,
                                         dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.


LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          IF N <= 1,                LWORK must be at least 1.
          If JOBZ  = 'N' and N > 2, LWORK must be at least 2*N.
          If JOBZ  = 'V' and N > 2, LWORK must be at least
                         ( 1 + 5*N + 2*N**2 ).

          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 LIWORK.


LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 2, LIWORK must be at least 3 + 5*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:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dsbevd_2stage (character JOBZ, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
 a real symmetric band matrix A using the 2stage technique for
 the reduction to tridiagonal. If eigenvectors are desired, it uses
 a divide and conquer algorithm.

 The divide and conquer algorithm makes very mild assumptions about
 floating point arithmetic. It will work on machines with a guard
 digit in add/subtract, or on those binary machines without guard
 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
 Cray-2. It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.


 

Parameters

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.
                  Not available in this release.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N

          N is INTEGER
          The order of the matrix A.  N >= 0.


KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.


AB

          AB is DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.


LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.


W

          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension LWORK
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.


LWORK

          LWORK is INTEGER
          The length of the array WORK. LWORK >= 1, when N <= 1;
          otherwise  
          If JOBZ = 'N' and N > 1, LWORK must be queried.
                                   LWORK = MAX(1, dimension) where
                                   dimension = (2KD+1)*N + KD*NTHREADS + N
                                   where KD is the size of the band.
                                   NTHREADS is the number of threads used when
                                   openMP compilation is enabled, otherwise =1.
          If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.

          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 LIWORK.


LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 2, LIWORK must be at least 3 + 5*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:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  All details about the 2stage techniques are available in:

  Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
  Parallel reduction to condensed forms for symmetric eigenvalue problems
  using aggregated fine-grained and memory-aware kernels. In Proceedings
  of 2011 International Conference for High Performance Computing,
  Networking, Storage and Analysis (SC '11), New York, NY, USA,
  Article 8 , 11 pages.
  http://doi.acm.org/10.1145/2063384.2063394

  A. Haidar, J. Kurzak, P. Luszczek, 2013.
  An improved parallel singular value algorithm and its implementation 
  for multicore hardware, In Proceedings of 2013 International Conference
  for High Performance Computing, Networking, Storage and Analysis (SC '13).
  Denver, Colorado, USA, 2013.
  Article 90, 12 pages.
  http://doi.acm.org/10.1145/2503210.2503292

  A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
  A novel hybrid CPU-GPU generalized eigensolver for electronic structure 
  calculations based on fine-grained memory aware tasks.
  International Journal of High Performance Computing Applications.
  Volume 28 Issue 2, Pages 196-209, May 2014.
  http://hpc.sagepub.com/content/28/2/196 


 

 

subroutine dsbevx (character JOBZ, character RANGE, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)

DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSBEVX computes selected eigenvalues and, optionally, eigenvectors
 of a real symmetric band matrix A.  Eigenvalues and eigenvectors can
 be selected by specifying either a range of values or a range of
 indices for the desired eigenvalues.


 

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.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N

          N is INTEGER
          The order of the matrix A.  N >= 0.


KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.


AB

          AB is DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.


LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.


Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)
          If JOBZ = 'V', the N-by-N orthogonal matrix used in the
                         reduction to tridiagonal form.
          If JOBZ = 'N', the array Q is not referenced.


LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.  If JOBZ = 'V', then
          LDQ >= max(1,N).


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; 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 DOUBLE PRECISION
          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 AB to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').

          See 'Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy,' by Demmel and
          Kahan, LAPACK Working Note #3.


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 DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.


Z

          Z is DOUBLE PRECISION 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 an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          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.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (7*N)


IWORK

          IWORK is INTEGER array, dimension (5*N)


IFAIL

          IFAIL is INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.


INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dsbevx_2stage (character JOBZ, character RANGE, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)

DSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
 of a real symmetric band matrix A using the 2stage technique for
 the reduction to tridiagonal. Eigenvalues and eigenvectors can
 be selected by specifying either a range of values or a range of
 indices for the desired eigenvalues.


 

Parameters

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.
                  Not available in this release.


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.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N

          N is INTEGER
          The order of the matrix A.  N >= 0.


KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.


AB

          AB is DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, AB is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the first
          superdiagonal and the diagonal of the tridiagonal matrix T
          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
          the diagonal and first subdiagonal of T are returned in the
          first two rows of AB.


LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD + 1.


Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)
          If JOBZ = 'V', the N-by-N orthogonal matrix used in the
                         reduction to tridiagonal form.
          If JOBZ = 'N', the array Q is not referenced.


LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.  If JOBZ = 'V', then
          LDQ >= max(1,N).


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; 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 DOUBLE PRECISION
          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 AB to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').

          See 'Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy,' by Demmel and
          Kahan, LAPACK Working Note #3.


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 DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.


Z

          Z is DOUBLE PRECISION 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 an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          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.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)


LWORK

          LWORK is INTEGER
          The length of the array WORK. LWORK >= 1, when N <= 1;
          otherwise  
          If JOBZ = 'N' and N > 1, LWORK must be queried.
                                   LWORK = MAX(1, 7*N, dimension) where
                                   dimension = (2KD+1)*N + KD*NTHREADS + 2*N
                                   where KD is the size of the band.
                                   NTHREADS is the number of threads used when
                                   openMP compilation is enabled, otherwise =1.
          If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available

          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 (5*N)


IFAIL

          IFAIL is INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.


INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  All details about the 2stage techniques are available in:

  Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
  Parallel reduction to condensed forms for symmetric eigenvalue problems
  using aggregated fine-grained and memory-aware kernels. In Proceedings
  of 2011 International Conference for High Performance Computing,
  Networking, Storage and Analysis (SC '11), New York, NY, USA,
  Article 8 , 11 pages.
  http://doi.acm.org/10.1145/2063384.2063394

  A. Haidar, J. Kurzak, P. Luszczek, 2013.
  An improved parallel singular value algorithm and its implementation 
  for multicore hardware, In Proceedings of 2013 International Conference
  for High Performance Computing, Networking, Storage and Analysis (SC '13).
  Denver, Colorado, USA, 2013.
  Article 90, 12 pages.
  http://doi.acm.org/10.1145/2503210.2503292

  A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
  A novel hybrid CPU-GPU generalized eigensolver for electronic structure 
  calculations based on fine-grained memory aware tasks.
  International Journal of High Performance Computing Applications.
  Volume 28 Issue 2, Pages 196-209, May 2014.
  http://hpc.sagepub.com/content/28/2/196 


 

 

subroutine dsbgv (character JOBZ, character UPLO, integer N, integer KA, integer KB, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldbb, * ) BB, integer LDBB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)

DSBGV

Purpose:

 DSBGV computes all the eigenvalues, and optionally, the eigenvectors
 of a real generalized symmetric-definite banded eigenproblem, of
 the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric
 and banded, and B is also positive definite.


 

Parameters

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.


N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.


KA

          KA is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'. KA >= 0.


KB

          KB is INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'. KB >= 0.


AB

          AB is DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the contents of AB are destroyed.


LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.


BB

          BB is DOUBLE PRECISION array, dimension (LDBB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix B, stored in the first kb+1 rows of the array.  The
          j-th column of B is stored in the j-th column of the array BB
          as follows:
          if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

          On exit, the factor S from the split Cholesky factorization
          B = S**T*S, as returned by DPBSTF.


LDBB

          LDBB is INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.


W

          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors, with the i-th column of Z holding the
          eigenvector associated with W(i). The eigenvectors are
          normalized so that Z**T*B*Z = I.
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= N.


WORK

          WORK is DOUBLE PRECISION array, dimension (3*N)


INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is:
             <= N:  the algorithm failed to converge:
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then DPBSTF
                    returned INFO = i: B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dsbgvd (character JOBZ, character UPLO, integer N, integer KA, integer KB, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldbb, * ) BB, integer LDBB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DSBGVD

Purpose:

 DSBGVD computes all the eigenvalues, and optionally, the eigenvectors
 of a real generalized symmetric-definite banded eigenproblem, of the
 form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric and
 banded, and B is also positive definite.  If eigenvectors are
 desired, it uses a divide and conquer algorithm.

 The divide and conquer algorithm makes very mild assumptions about
 floating point arithmetic. It will work on machines with a guard
 digit in add/subtract, or on those binary machines without guard
 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
 Cray-2. It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.


 

Parameters

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.


N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.


KA

          KA is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.


KB

          KB is INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.


AB

          AB is DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the contents of AB are destroyed.


LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.


BB

          BB is DOUBLE PRECISION array, dimension (LDBB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix B, stored in the first kb+1 rows of the array.  The
          j-th column of B is stored in the j-th column of the array BB
          as follows:
          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

          On exit, the factor S from the split Cholesky factorization
          B = S**T*S, as returned by DPBSTF.


LDBB

          LDBB is INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.


W

          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors, with the i-th column of Z holding the
          eigenvector associated with W(i).  The eigenvectors are
          normalized so Z**T*B*Z = I.
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.


LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK >= 1.
          If JOBZ = 'N' and N > 1, LWORK >= 2*N.
          If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.

          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 LIWORK > 0, IWORK(1) returns the optimal LIWORK.


LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*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:  if INFO = i, and i is:
             <= N:  the algorithm failed to converge:
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then DPBSTF
                    returned INFO = i: B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

 

subroutine dsbgvx (character JOBZ, character RANGE, character UPLO, integer N, integer KA, integer KB, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldbb, * ) BB, integer LDBB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)

DSBGVX

Purpose:

 DSBGVX computes selected eigenvalues, and optionally, eigenvectors
 of a real generalized symmetric-definite banded eigenproblem, of
 the form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric
 and banded, and B is also positive definite.  Eigenvalues and
 eigenvectors can be selected by specifying either all eigenvalues,
 a range of values or a range of indices for the desired eigenvalues.


 

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.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.


N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.


KA

          KA is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.


KB

          KB is INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KB >= 0.


AB

          AB is DOUBLE PRECISION array, dimension (LDAB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).

          On exit, the contents of AB are destroyed.


LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.


BB

          BB is DOUBLE PRECISION array, dimension (LDBB, N)
          On entry, the upper or lower triangle of the symmetric band
          matrix B, stored in the first kb+1 rows of the array.  The
          j-th column of B is stored in the j-th column of the array BB
          as follows:
          if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
          if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).

          On exit, the factor S from the split Cholesky factorization
          B = S**T*S, as returned by DPBSTF.


LDBB

          LDBB is INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.


Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)
          If JOBZ = 'V', the n-by-n matrix used in the reduction of
          A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
          and consequently C to tridiagonal form.
          If JOBZ = 'N', the array Q is not referenced.


LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.  If JOBZ = 'N',
          LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).


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; 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 DOUBLE PRECISION
          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.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').


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 DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors, with the i-th column of Z holding the
          eigenvector associated with W(i).  The eigenvectors are
          normalized so Z**T*B*Z = I.
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (7*N)


IWORK

          IWORK is INTEGER array, dimension (5*N)


IFAIL

          IFAIL is INTEGER array, dimension (M)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvalues that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.


INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          <= N: if INFO = i, then i eigenvectors failed to converge.
                  Their indices are stored in IFAIL.
          > N:  DPBSTF returned an error code; i.e.,
                if INFO = N + i, for 1 <= i <= N, then the leading
                minor of order i of B is not positive definite.
                The factorization of B could not be completed and
                no eigenvalues or eigenvectors were computed.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

 

subroutine dspev (character JOBZ, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)

DSPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSPEV computes all the eigenvalues and, optionally, eigenvectors of a
 real symmetric matrix A in packed storage.


 

Parameters

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N

          N is INTEGER
          The order of the matrix A.  N >= 0.


AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          On exit, AP is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
          and first superdiagonal of the tridiagonal matrix T overwrite
          the corresponding elements of A, and if UPLO = 'L', the
          diagonal and first subdiagonal of T overwrite the
          corresponding elements of A.


W

          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (3*N)


INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dspevd (character JOBZ, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSPEVD computes all the eigenvalues and, optionally, eigenvectors
 of a real symmetric matrix A in packed storage. If eigenvectors are
 desired, it uses a divide and conquer algorithm.

 The divide and conquer algorithm makes very mild assumptions about
 floating point arithmetic. It will work on machines with a guard
 digit in add/subtract, or on those binary machines without guard
 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
 Cray-2. It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.


 

Parameters

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N

          N is INTEGER
          The order of the matrix A.  N >= 0.


AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          On exit, AP is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
          and first superdiagonal of the tridiagonal matrix T overwrite
          the corresponding elements of A, and if UPLO = 'L', the
          diagonal and first subdiagonal of T overwrite the
          corresponding elements of A.


W

          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the required LWORK.


LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK must be at least 1.
          If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.
          If JOBZ = 'V' and N > 1, LWORK must be at least
                                                 1 + 6*N + N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the required 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 required LIWORK.


LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the required 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:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of an intermediate tridiagonal
                form did not converge to zero.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dspevx (character JOBZ, character RANGE, character UPLO, integer N, double precision, dimension( * ) AP, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)

DSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSPEVX computes selected eigenvalues and, optionally, eigenvectors
 of a real symmetric matrix A in packed storage.  Eigenvalues/vectors
 can be selected by specifying either a range of values or a range of
 indices for the desired eigenvalues.


 

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.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N

          N is INTEGER
          The order of the matrix A.  N >= 0.


AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          On exit, AP is overwritten by values generated during the
          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
          and first superdiagonal of the tridiagonal matrix T overwrite
          the corresponding elements of A, and if UPLO = 'L', the
          diagonal and first subdiagonal of T overwrite the
          corresponding elements of A.


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; 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 DOUBLE PRECISION
          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 AP to tridiagonal form.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').

          See 'Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy,' by Demmel and
          Kahan, LAPACK Working Note #3.


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 DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the selected eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION 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 an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          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.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (8*N)


IWORK

          IWORK is INTEGER array, dimension (5*N)


IFAIL

          IFAIL is INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.


INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dspgv (integer ITYPE, character JOBZ, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) BP, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)

DSPGV

Purpose:

 DSPGV computes all the eigenvalues and, optionally, the eigenvectors
 of a real generalized symmetric-definite eigenproblem, of the form
 A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
 Here A and B are assumed to be symmetric, stored in packed format,
 and B is also positive definite.


 

Parameters

ITYPE

          ITYPE is INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x


JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.


N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.


AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          On exit, the contents of AP are destroyed.


BP

          BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          B, packed columnwise in a linear array.  The j-th column of B
          is stored in the array BP as follows:
          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

          On exit, the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T, in the same storage
          format as B.


W

          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors.  The eigenvectors are normalized as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (3*N)


INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  DPPTRF or DSPEV returned an error code:
             <= N:  if INFO = i, DSPEV failed to converge;
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero.
             > N:   if INFO = n + i, for 1 <= i <= n, then the leading
                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dspgvd (integer ITYPE, character JOBZ, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) BP, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DSPGVD

Purpose:

 DSPGVD computes all the eigenvalues, and optionally, the eigenvectors
 of a real generalized symmetric-definite eigenproblem, of the form
 A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
 B are assumed to be symmetric, stored in packed format, and B is also
 positive definite.
 If eigenvectors are desired, it uses a divide and conquer algorithm.

 The divide and conquer algorithm makes very mild assumptions about
 floating point arithmetic. It will work on machines with a guard
 digit in add/subtract, or on those binary machines without guard
 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
 Cray-2. It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.


 

Parameters

ITYPE

          ITYPE is INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x


JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangles of A and B are stored;
          = 'L':  Lower triangles of A and B are stored.


N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.


AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          On exit, the contents of AP are destroyed.


BP

          BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          B, packed columnwise in a linear array.  The j-th column of B
          is stored in the array BP as follows:
          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

          On exit, the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T, in the same storage
          format as B.


W

          W is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
          eigenvectors.  The eigenvectors are normalized as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the required LWORK.


LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If N <= 1,               LWORK >= 1.
          If JOBZ = 'N' and N > 1, LWORK >= 2*N.
          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the required 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 required LIWORK.


LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
          If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the required 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:  DPPTRF or DSPEVD returned an error code:
             <= N:  if INFO = i, DSPEVD failed to converge;
                    i off-diagonal elements of an intermediate
                    tridiagonal form did not converge to zero;
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

 

subroutine dspgvx (integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) BP, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)

DSPGVX

Purpose:

 DSPGVX computes selected eigenvalues, and optionally, eigenvectors
 of a real generalized symmetric-definite eigenproblem, of the form
 A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
 and B are assumed to be symmetric, stored in packed storage, and B
 is also positive definite.  Eigenvalues and eigenvectors can be
 selected by specifying either a range of values or a range of indices
 for the desired eigenvalues.


 

Parameters

ITYPE

          ITYPE is INTEGER
          Specifies the problem type to be solved:
          = 1:  A*x = (lambda)*B*x
          = 2:  A*B*x = (lambda)*x
          = 3:  B*A*x = (lambda)*x


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.


UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A and B are stored;
          = 'L':  Lower triangle of A and B are stored.


N

          N is INTEGER
          The order of the matrix pencil (A,B).  N >= 0.


AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

          On exit, the contents of AP are destroyed.


BP

          BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          B, packed columnwise in a linear array.  The j-th column of B
          is stored in the array BP as follows:
          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.

          On exit, the triangular factor U or L from the Cholesky
          factorization B = U**T*U or B = L*L**T, in the same storage
          format as B.


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; 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 DOUBLE PRECISION
          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.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').


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 DOUBLE PRECISION array, dimension (N)
          On normal exit, the first M elements contain the selected
          eigenvalues in ascending order.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
          If JOBZ = 'N', then Z is not referenced.
          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).
          The eigenvectors are normalized as follows:
          if ITYPE = 1 or 2, Z**T*B*Z = I;
          if ITYPE = 3, Z**T*inv(B)*Z = I.

          If an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          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.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (8*N)


IWORK

          IWORK is INTEGER array, dimension (5*N)


IFAIL

          IFAIL is INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.


INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  DPPTRF or DSPEVX returned an error code:
             <= N:  if INFO = i, DSPEVX failed to converge;
                    i eigenvectors failed to converge.  Their indices
                    are stored in array IFAIL.
             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

 

subroutine dstev (character JOBZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)

DSTEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSTEV computes all eigenvalues and, optionally, eigenvectors of a
 real symmetric tridiagonal matrix A.


 

Parameters

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.


N

          N is INTEGER
          The order of the matrix.  N >= 0.


D

          D is DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.
          On exit, if INFO = 0, the eigenvalues in ascending order.


E

          E is DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A, stored in elements 1 to N-1 of E.
          On exit, the contents of E are destroyed.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with D(i).
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2))
          If JOBZ = 'N', WORK is not referenced.


INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of E did not converge to zero.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dstevd (character JOBZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DSTEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSTEVD computes all eigenvalues and, optionally, eigenvectors of a
 real symmetric tridiagonal matrix. If eigenvectors are desired, it
 uses a divide and conquer algorithm.

 The divide and conquer algorithm makes very mild assumptions about
 floating point arithmetic. It will work on machines with a guard
 digit in add/subtract, or on those binary machines without guard
 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
 Cray-2. It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.


 

Parameters

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.


N

          N is INTEGER
          The order of the matrix.  N >= 0.


D

          D is DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.
          On exit, if INFO = 0, the eigenvalues in ascending order.


E

          E is DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A, stored in elements 1 to N-1 of E.
          On exit, the contents of E are destroyed.


Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
          eigenvectors of the matrix A, with the i-th column of Z
          holding the eigenvector associated with D(i).
          If JOBZ = 'N', then Z is not referenced.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array,
                                         dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.


LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If JOBZ  = 'N' or N <= 1 then LWORK must be at least 1.
          If JOBZ  = 'V' and N > 1 then LWORK must be at least
                         ( 1 + 4*N + N**2 ).

          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 LIWORK.


LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.
          If JOBZ  = 'N' or N <= 1 then LIWORK must be at least 1.
          If JOBZ  = 'V' and N > 1 then LIWORK must be at least 3+5*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:  if INFO = i, the algorithm failed to converge; i
                off-diagonal elements of E did not converge to zero.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

subroutine dstevr (character JOBZ, character RANGE, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)

DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSTEVR 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.

 Whenever possible, DSTEVR calls DSTEMR to compute the
 eigenspectrum using Relatively Robust Representations.  DSTEMR
 computes eigenvalues by the dqds algorithm, while orthogonal
 eigenvectors are computed from various 'good' 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, 'choose' 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 'A new O(n^2) algorithm for the symmetric
 tridiagonal eigenvalue/eigenvector problem', by Inderjit Dhillon,
 Computer Science Division Technical Report No. UCB//CSD-97-971,
 UC Berkeley, May 1997.


 Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested
 on machines which conform to the ieee-754 floating point standard.
 DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and
 when partial spectrum requests are made.

 Normal execution of DSTEMR 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.


 

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, DSTEBZ and
          DSTEIN are called


N

          N is INTEGER
          The order of the matrix.  N >= 0.


D

          D is DOUBLE PRECISION 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 DOUBLE PRECISION 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 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; 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 DOUBLE PRECISION
          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 'Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy,' by Demmel and
          Kahan, LAPACK Working Note #3.

          If high relative accuracy is important, set ABSTOL to
          DLAMCH( '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,
          'Computing Accurate Eigensystems of Scaled Diagonally
          Dominant Matrices', 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 DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.


Z

          Z is DOUBLE PRECISION 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.


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 DOUBLE PRECISION 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 >= max(1,20*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 >= max(1,10*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


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Inderjit Dhillon, IBM Almaden, USA

 Osni Marques, LBNL/NERSC, USA 

 Ken Stanley, Computer Science Division, University of California at Berkeley, USA 

 

subroutine dstevx (character JOBZ, character RANGE, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)

DSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

 DSTEVX computes selected eigenvalues and, optionally, eigenvectors
 of a real symmetric tridiagonal matrix A.  Eigenvalues and
 eigenvectors can be selected by specifying either a range of values
 or a range of indices for the desired eigenvalues.


 

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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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; 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 DOUBLE PRECISION
          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.

          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
          If this routine returns with INFO>0, indicating that some
          eigenvectors did not converge, try setting ABSTOL to
          2*DLAMCH('S').

          See 'Computing Small Singular Values of Bidiagonal Matrices
          with Guaranteed High Relative Accuracy,' by Demmel and
          Kahan, LAPACK Working Note #3.


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 DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.


Z

          Z is DOUBLE PRECISION 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 an eigenvector fails to converge (INFO > 0), then that
          column of Z contains the latest approximation to the
          eigenvector, and the index of the eigenvector is returned
          in IFAIL.  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.


LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).


WORK

          WORK is DOUBLE PRECISION array, dimension (5*N)


IWORK

          IWORK is INTEGER array, dimension (5*N)


IFAIL

          IFAIL is INTEGER array, dimension (N)
          If JOBZ = 'V', then if INFO = 0, the first M elements of
          IFAIL are zero.  If INFO > 0, then IFAIL contains the
          indices of the eigenvectors that failed to converge.
          If JOBZ = 'N', then IFAIL is not referenced.


INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, then i eigenvectors failed to converge.
                Their indices are stored in array IFAIL.


 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 

Author

Generated automatically by Doxygen for LAPACK from the source code.


 

Index

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine dbdsvdx (character UPLO, 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 NS, double precision, dimension( * ) S, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
subroutine dggglm (integer N, integer M, integer P, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) D, double precision, dimension( * ) X, double precision, dimension( * ) Y, double precision, dimension( * ) WORK, integer LWORK, integer INFO)
subroutine dsbev (character JOBZ, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)
subroutine dsbev_2stage (character JOBZ, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer INFO)
subroutine dsbevd (character JOBZ, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
subroutine dsbevd_2stage (character JOBZ, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
subroutine dsbevx (character JOBZ, character RANGE, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
subroutine dsbevx_2stage (character JOBZ, character RANGE, character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
subroutine dsbgv (character JOBZ, character UPLO, integer N, integer KA, integer KB, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldbb, * ) BB, integer LDBB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)
subroutine dsbgvd (character JOBZ, character UPLO, integer N, integer KA, integer KB, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldbb, * ) BB, integer LDBB, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
subroutine dsbgvx (character JOBZ, character RANGE, character UPLO, integer N, integer KA, integer KB, double precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( ldbb, * ) BB, integer LDBB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
subroutine dspev (character JOBZ, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)
subroutine dspevd (character JOBZ, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
subroutine dspevx (character JOBZ, character RANGE, character UPLO, integer N, double precision, dimension( * ) AP, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
subroutine dspgv (integer ITYPE, character JOBZ, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) BP, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)
subroutine dspgvd (integer ITYPE, character JOBZ, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) BP, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
subroutine dspgvx (integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N, double precision, dimension( * ) AP, double precision, dimension( * ) BP, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
subroutine dstev (character JOBZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer INFO)
subroutine dstevd (character JOBZ, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
subroutine dstevr (character JOBZ, character RANGE, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO)
subroutine dstevx (character JOBZ, character RANGE, integer N, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision VL, double precision VU, integer IL, integer IU, double precision ABSTOL, integer M, double precision, dimension( * ) W, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO)
Author

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