syevr_2stage#

ssyevr_2stage

Functions

void ssyevr_2stage(
    const char* jobz,
    const char* range,
    const char* uplo,
    const INT   n,
          f32*  A,
    const INT   lda,
    const f32   vl,
    const f32   vu,
    const INT   il,
    const INT   iu,
    const f32   abstol,
          INT*  m,
          f32*  W,
          f32*  Z,
    const INT   ldz,
          INT*  isuppz,
          f32*  work,
    const INT   lwork,
          INT*  iwork,
    const INT   liwork,
          INT*  info
);

void ssyevr_2stage(const char *jobz, const char *range, const char *uplo, const INT n, f32 *A, const INT lda, const f32 vl, const f32 vu, const INT il, const INT iu, const f32 abstol, INT *m, f32 *W, f32 *Z, const INT ldz, INT *isuppz, f32 *work, const INT lwork, INT *iwork, const INT liwork, INT *info)#

SSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric 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.

SSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call to SSYTRD. Then, whenever possible, SSYEVR_2STAGE calls SSTEMR to compute the eigenspectrum using Relatively Robust Representations. SSTEMR 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 each unreduced block (submatrix) of T,

(a) Compute T - sigma I  = L D L^T, so that L and D
    define all the wanted eigenvalues to high relative accuracy.
    This means that small relative changes in the entries of D and L
    cause only small relative changes in the eigenvalues and
    eigenvectors. The standard (unfactored) representation of the
    tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy.
    If the eigenvectors are desired, the algorithm attains full
    accuracy of the computed eigenvalues only right before
    the corresponding vectors have to be computed, see steps c) and d).
(c) For each cluster of close eigenvalues, select a new
    shift close to the cluster, find a new factorization, and refine
    the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation compute
    the corresponding eigenvector by forming a rank revealing twisted
    factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input parameter abstol.

For more details, see SSTEMR’s documentation and:

Inderjit S. Dhillon and Beresford N. Parlett: “Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices,” Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.

Inderjit Dhillon and Beresford Parlett: “Orthogonal Eigenvectors and Relative Gaps,” SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.

Inderjit Dhillon: “A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem”, Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

Note 1 : SSYEVR_2STAGE calls SSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. SSYEVR_2STAGE calls SSTEBZ and SSTEIN on non-ieee machines and when partial spectrum requests are made.

Normal execution of SSTEMR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner.

Contributors:

Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of California at Berkeley, USA
Jason Riedy, Computer Science Division, University of California at Berkeley, USA

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. https://doi.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. https://doi.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. https://doi.org/10.1177/1094342013502097

Parameters

in

jobz

jobz='N': Compute eigenvalues only; jobz='V': Compute eigenvalues and eigenvectors. Not available in this release.

in

range

range='A': all eigenvalues will be found. range='V': all eigenvalues in the half-open interval (vl,vu] will be found. range='I': the il-th through iu-th eigenvalues will be found. For range='V' or range='I' and iu-il<n-1, SSTEBZ and SSTEIN are called.

in

uplo

uplo='U': Upper triangle of A is stored; uplo='L': Lower triangle of A is stored.

in

n

The order of the matrix A. n>=0.

inout

A

Array of dimension (lda, n). On entry, the symmetric matrix A. If uplo='U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A. If uplo='L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if uplo='L') or the upper triangle (if uplo='U') of A, including the diagonal, is destroyed.

in

lda

The leading dimension of the array A. lda>=max(1,n).

in

vl

If range='V', the lower bound of the interval to be searched for eigenvalues. vl<vu. Not referenced if range='A' or range='I'.

in

vu

If range='V', the upper bound of the interval to be searched for eigenvalues. vl<vu. Not referenced if range='A' or range='I'.

in

il

If range='I', the index of the smallest eigenvalue to be returned. 0<=il<=iu<=n-1, if n>0; il=0 and iu=-1 if n=0. Not referenced if range='A' or range='V'.

in

iu

If range='I', the index of the largest eigenvalue to be returned. 0<=il<=iu<=n-1, if n>0; il=0 and iu=-1 if n=0. Not referenced if range='A' or range='V'.

in

abstol

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 slamch("Safe minimum")

. Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will. See J. Barlow and J. Demmel, “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.

out

m

The total number of eigenvalues found. 0<=m<=n. If range='A', m=n, and if range='I', m=iu-il+1.

out

W

Array of dimension (n). The first m elements contain the selected eigenvalues in ascending order.

out

Z

Array of dimension (ldz, max(1,m)). If jobz='V', then if info=0, the first m columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W[i]. If jobz='N', then Z is not referenced. Note: the user must ensure that at least max(1,m) columns are supplied in the array Z; if range='V', the exact value of m is not known in advance and an upper bound must be used. Supplying n columns is always safe.

in

ldz

The leading dimension of the array Z. ldz>=1, and if jobz='V', ldz>=max(1,n).

out

isuppz

Integer array of 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] through isuppz[2*i+1]. This is an output of SSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 0:n-1 because of the orthogonal transformations applied by SORMTR. Implemented only for range='A' or range='I' and iu-il=n-1.

out

work

Array of dimension (max(1,lwork)). On exit, if info=0, work[0] returns the optimal lwork.

in

lwork

The dimension of the array work. lwork>=1, when n<=1; otherwise: If jobz='N' and n>1, lwork must be queried.

lwork = MAX(1, 26*n, dimension) where
dimension = max(stage1,stage2) + (kd+1)*n + 5*n
          = n*kd + n*max(kd+1,FACTOPTNB)
            + max(2*kd*kd, kd*NTHREADS)
            + (kd+1)*n + 5*n

where kd is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice, 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.

out

iwork

Integer array of dimension (max(1,liwork)). On exit, if info=0, iwork[0] returns the optimal liwork.

in

liwork

The dimension of the array iwork. If n<=1, liwork>=1, else liwork>=10*n. If liwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the iwork array, returns this value as the first entry of the iwork array, and no error message related to liwork is issued.

out

info

info=0: successful exit
info<0: if info=-i, the i-th argument had an illegal value
info>0: Internal error

dsyevr_2stage

Functions

void dsyevr_2stage(
    const char* jobz,
    const char* range,
    const char* uplo,
    const INT   n,
          f64*  A,
    const INT   lda,
    const f64   vl,
    const f64   vu,
    const INT   il,
    const INT   iu,
    const f64   abstol,
          INT*  m,
          f64*  W,
          f64*  Z,
    const INT   ldz,
          INT*  isuppz,
          f64*  work,
    const INT   lwork,
          INT*  iwork,
    const INT   liwork,
          INT*  info
);

void dsyevr_2stage(const char *jobz, const char *range, const char *uplo, const INT n, f64 *A, const INT lda, const f64 vl, const f64 vu, const INT il, const INT iu, const f64 abstol, INT *m, f64 *W, f64 *Z, const INT ldz, INT *isuppz, f64 *work, const INT lwork, INT *iwork, const INT liwork, INT *info)#

DSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric 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.

DSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call to DSYTRD. Then, whenever possible, DSYEVR_2STAGE 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 each unreduced block (submatrix) of T,

(a) Compute T - sigma I  = L D L^T, so that L and D
    define all the wanted eigenvalues to high relative accuracy.
    This means that small relative changes in the entries of D and L
    cause only small relative changes in the eigenvalues and
    eigenvectors. The standard (unfactored) representation of the
    tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy.
    If the eigenvectors are desired, the algorithm attains full
    accuracy of the computed eigenvalues only right before
    the corresponding vectors have to be computed, see steps c) and d).
(c) For each cluster of close eigenvalues, select a new
    shift close to the cluster, find a new factorization, and refine
    the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation compute
    the corresponding eigenvector by forming a rank revealing twisted
    factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input parameter abstol.

For more details, see DSTEMR’s documentation and:

Inderjit S. Dhillon and Beresford N. Parlett: “Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices,” Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.

Inderjit Dhillon and Beresford Parlett: “Orthogonal Eigenvectors and Relative Gaps,” SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.

Inderjit Dhillon: “A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem”, Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

Note 1 : DSYEVR_2STAGE calls DSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. DSYEVR_2STAGE calls DSTEBZ and SSTEIN 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.

Contributors:

Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of California at Berkeley, USA
Jason Riedy, Computer Science Division, University of California at Berkeley, USA

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. https://doi.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. https://doi.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. https://doi.org/10.1177/1094342013502097

Parameters

in

jobz

jobz='N': Compute eigenvalues only; jobz='V': Compute eigenvalues and eigenvectors. Not available in this release.

in

range

range='A': all eigenvalues will be found. range='V': all eigenvalues in the half-open interval (vl,vu] will be found. range='I': the il-th through iu-th eigenvalues will be found. For range='V' or range='I' and iu-il<n-1, DSTEBZ and DSTEIN are called.

in

uplo

uplo='U': Upper triangle of A is stored; uplo='L': Lower triangle of A is stored.

in

n

The order of the matrix A. n>=0.

inout

A

Array of dimension (lda, n). On entry, the symmetric matrix A. If uplo='U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A. If uplo='L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if uplo='L') or the upper triangle (if uplo='U') of A, including the diagonal, is destroyed.

in

lda

The leading dimension of the array A. lda>=max(1,n).

in

vl

If range='V', the lower bound of the interval to be searched for eigenvalues. vl<vu. Not referenced if range='A' or range='I'.

in

vu

If range='V', the upper bound of the interval to be searched for eigenvalues. vl<vu. Not referenced if range='A' or range='I'.

in

il

If range='I', the index of the smallest eigenvalue to be returned. 0<=il<=iu<=n-1, if n>0; il=0 and iu=-1 if n=0. Not referenced if range='A' or range='V'.

in

iu

If range='I', the index of the largest eigenvalue to be returned. 0<=il<=iu<=n-1, if n>0; il=0 and iu=-1 if n=0. Not referenced if range='A' or range='V'.

in

abstol

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.

out

m

The total number of eigenvalues found. 0<=m<=n. If range='A', m=n, and if range='I', m=iu-il+1.

out

W

Array of dimension (n). The first m elements contain the selected eigenvalues in ascending order.

out

Z

Array of dimension (ldz, max(1,m)). If jobz='V', then if info=0, the first m columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W[i]. If jobz='N', then Z is not referenced. Note: the user must ensure that at least max(1,m) columns are supplied in the array Z; if range='V', the exact value of m is not known in advance and an upper bound must be used. Supplying n columns is always safe.

in

ldz

The leading dimension of the array Z. ldz>=1, and if jobz='V', ldz>=max(1,n).

out

isuppz

Integer array of 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] through isuppz[2*i+1]. This is an output of DSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 0:n-1 because of the orthogonal transformations applied by DORMTR. Implemented only for range='A' or range='I' and iu-il=n-1.

out

work

Array of dimension (max(1,lwork)). On exit, if info=0, work[0] returns the optimal lwork.

in

lwork

The dimension of the array work. lwork>=1, when n<=1; otherwise: If jobz='N' and n>1, lwork must be queried.

lwork = MAX(1, 26*n, dimension) where
dimension = max(stage1,stage2) + (kd+1)*n + 5*n
          = n*kd + n*max(kd+1,FACTOPTNB)
            + max(2*kd*kd, kd*NTHREADS)
            + (kd+1)*n + 5*n

where kd is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice, 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.

out

iwork

Integer array of dimension (max(1,liwork)). On exit, if info=0, iwork[0] returns the optimal liwork.

in

liwork

The dimension of the array iwork. If n<=1, liwork>=1, else liwork>=10*n. If liwork=-1, then a workspace query is assumed; the routine only calculates the optimal size of the iwork array, returns this value as the first entry of the iwork array, and no error message related to liwork is issued.

out

info

info=0: successful exit
info<0: if info=-i, the i-th argument had an illegal value
info>0: Internal error