PSSYEVX - compute selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A by calling the recommended sequence of ScaLAPACK routines

SUBROUTINE PSSYEVX(

JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, VL, VU, IL, IU, ABSTOL, M, NZ, W, ORFAC, Z, IZ, JZ, DESCZ, WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR, GAP, INFO )

CHARACTER JOBZ, RANGE, UPLO

INTEGER IA, IL, INFO, IU, IZ, JA, JZ, LIWORK, LWORK, M, N, NZ

REAL ABSTOL, ORFAC, VL, VU

INTEGER DESCA( * ), DESCZ( * ), ICLUSTR( * ), IFAIL( * ), IWORK( * )

REAL A( * ), GAP( * ), W( * ), WORK( * ), Z( * )

INTEGER BLOCK_CYCLIC_2D, DLEN_, DTYPE_, CTXT_, M_, N_, MB_, NB_, RSRC_, CSRC_, LLD_

PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1, CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6, RSRC_ = 7, CSRC_ = 8, LLD_ = 9 )

REAL ZERO, ONE, TEN, FIVE

PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TEN = 10.0E+0, FIVE = 5.0E+0 )

INTEGER IERREIN, IERRCLS, IERRSPC, IERREBZ

PARAMETER ( IERREIN = 1, IERRCLS = 2, IERRSPC = 4, IERREBZ = 8 )

LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, QUICKRETURN, VALEIG, WANTZ

CHARACTER ORDER

INTEGER CSRC_A, I, IACOL, IAROW, ICOFFA, IINFO, INDD, INDD2, INDE, INDE2, INDIBL, INDISP, INDTAU, INDWORK, IROFFA, IROFFZ, ISCALE, ISIZESTEBZ, ISIZESTEIN, IZROW, LALLWORK, LIWMIN, LLWORK, LWMIN, MAXEIGS, MB_A, MB_Z, MQ0, MYCOL, MYROW, NB, NB_A, NB_Z, NEIG, NN, NNP, NP0, NPCOL, NPROCS, NPROW, NSPLIT, NZZ, OFFSET, RSRC_A, RSRC_Z, SIZEORMTR, SIZESTEIN, SIZESYEVX

REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM, VLL, VUU

INTEGER IDUM1( 4 ), IDUM2( 4 )

LOGICAL LSAME

INTEGER ICEIL, INDXG2P, NUMROC

REAL PSLAMCH, PSLANSY

EXTERNAL LSAME, ICEIL, INDXG2P, NUMROC, PSLAMCH, PSLANSY

EXTERNAL BLACS_GRIDINFO, CHK1MAT, IGAMN2D, PCHK2MAT, PSELGET, PSLARED1D, PSLASCL, PSORMTR, PSSTEBZ, PSSTEIN, PSSYTRD, PXERBLA, SGEBR2D, SGEBS2D, SLASRT, SSCAL

INTRINSIC ABS, ICHAR, MAX, MIN, MOD, REAL, SQRT

PSSYEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A by calling the recommended sequence of ScaLAPACK routines. Eigenvalues/vectors can be selected by specifying a range of values or a range of indices for the desired eigenvalues.

Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location.

Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array".

NOTATION STORED IN EXPLANATION

--------------- -------------- --------------------------------------

DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case, DTYPE_A = 1.

CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating the BLACS process grid A is distribu- ted over. The context itself is glo- bal, but the handle (the integer value) may vary.

M_A (global) DESCA( M_ ) The number of rows in the global array A.

N_A (global) DESCA( N_ ) The number of columns in the global array A.

MB_A (global) DESCA( MB_ ) The blocking factor used to distribute the rows of the array.

NB_A (global) DESCA( NB_ ) The blocking factor used to distribute the columns of the array.

RSRC_A (global) DESCA( RSRC_ ) The process row over which the first row of the array A is distributed.

CSRC_A (global) DESCA( CSRC_ ) The process column over which the first column of the array A is distributed.

LLD_A (local) DESCA( LLD_ ) The leading dimension of the local array. LLD_A >= MAX(1,LOCr(M_A)).

Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q. LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column.

Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row.

The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC:

LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),

LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).

An upper bound for these quantities may be computed by:

LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A

LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

PSSYEVX assumes IEEE 754 standard compliant arithmetic. To port to a system which does not have IEEE 754 arithmetic, modify the appropriate SLmake.inc file to include the compiler switch -DNO_IEEE. This switch only affects the compilation of pslaiect.c.

NP = the number of rows local to a given process. NQ = the number of columns local to a given process.

JOBZ (global input) CHARACTER*1 Specifies whether or not to compute the eigenvectors: = 'N': Compute eigenvalues only. = 'V': Compute eigenvalues and eigenvectors.

RANGE (global input) CHARACTER*1 = 'A': all eigenvalues will be found. = 'V': all eigenvalues in the interval [VL,VU] will be found. = 'I': the IL-th through IU-th eigenvalues will be found.

UPLO (global input) CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular

N (global input) INTEGER The number of rows and columns of the matrix A. N >= 0.

A (local input/workspace) block cyclic REAL array, global dimension (N, N), local dimension ( LLD_A, LOCc(JA+N-1) )

On entry, the symmetric matrix A. If UPLO = 'U', only the upper triangular part of A is used to define the elements of the symmetric matrix. If UPLO = 'L', only the lower triangular part of A is used to define the elements of the symmetric matrix.

On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed.

IA (global input) INTEGER A's global row index, which points to the beginning of the submatrix which is to be operated on.

JA (global input) INTEGER A's global column index, which points to the beginning of the submatrix which is to be operated on.

DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. If DESCA( CTXT_ ) is incorrect, PSSYEVX cannot guarantee correct error reporting.

VL (global input) REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Not referenced if RANGE = 'A' or 'I'.

VU (global input) REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. Not referenced if RANGE = 'A' or 'I'.

IL (global input) INTEGER If RANGE='I', the index (from smallest to largest) of the smallest eigenvalue to be returned. IL >= 1. Not referenced if RANGE = 'A' or 'V'.

IU (global input) INTEGER If RANGE='I', the index (from smallest to largest) of the largest eigenvalue to be returned. min(IL,N) <= IU <= N. Not referenced if RANGE = 'A' or 'V'.

ABSTOL (global input) REAL If JOBZ='V', setting ABSTOL to PSLAMCH( CONTEXT, 'U') yields the most orthogonal eigenvectors.

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*norm(T) will be used in its place, where norm(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*PSLAMCH('S') not zero. If this routine returns with ((MOD(INFO,2).NE.0) .OR. (MOD(INFO/8,2).NE.0)), indicating that some eigenvalues or eigenvectors did not converge, try setting ABSTOL to 2*PSLAMCH('S').

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

See "On the correctness of Parallel Bisection in Floating Point" by Demmel, Dhillon and Ren, LAPACK Working Note #70

M (global output) INTEGER Total number of eigenvalues found. 0 <= M <= N.

NZ (global output) INTEGER Total number of eigenvectors computed. 0 <= NZ <= M. The number of columns of Z that are filled. If JOBZ .NE. 'V', NZ is not referenced. If JOBZ .EQ. 'V', NZ = M unless the user supplies insufficient space and PSSYEVX is not able to detect this before beginning computation. To get all the eigenvectors requested, the user must supply both sufficient space to hold the eigenvectors in Z (M .LE. DESCZ(N_)) and sufficient workspace to compute them. (See LWORK below.) PSSYEVX is always able to detect insufficient space without computation unless RANGE .EQ. 'V'.

W (global output) REAL array, dimension (N) On normal exit, the first M entries contain the selected eigenvalues in ascending order.

ORFAC (global input) REAL Specifies which eigenvectors should be reorthogonalized. Eigenvectors that correspond to eigenvalues which are within tol=ORFAC*norm(A) of each other are to be reorthogonalized. However, if the workspace is insufficient (see LWORK), tol may be decreased until all eigenvectors to be reorthogonalized can be stored in one process. No reorthogonalization will be done if ORFAC equals zero. A default value of 10^-3 is used if ORFAC is negative. ORFAC should be identical on all processes.

Z (local output) REAL array, global dimension (N, N), local dimension ( LLD_Z, LOCc(JZ+N-1) ) If JOBZ = 'V', then on normal exit the first M columns of Z contain the orthonormal eigenvectors of the matrix corresponding to the selected eigenvalues. 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.

IZ (global input) INTEGER Z's global row index, which points to the beginning of the submatrix which is to be operated on.

JZ (global input) INTEGER Z's global column index, which points to the beginning of the submatrix which is to be operated on.

DESCZ (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Z. DESCZ( CTXT_ ) must equal DESCA( CTXT_ )

WORK (local workspace/output) REAL array, dimension (LWORK) On return, WROK(1) contains the optimal amount of workspace required for efficient execution. if JOBZ='N' WORK(1) = optimal amount of workspace required to compute eigenvalues efficiently if JOBZ='V' WORK(1) = optimal amount of workspace required to compute eigenvalues and eigenvectors efficiently with no guarantee on orthogonality. If RANGE='V', it is assumed that all eigenvectors may be required.

LWORK (local input) INTEGER Size of WORK See below for definitions of variables used to define LWORK. If no eigenvectors are requested (JOBZ = 'N') then LWORK >= 5 * N + MAX( 5 * NN, NB * ( NP0 + 1 ) ) If eigenvectors are requested (JOBZ = 'V' ) then the amount of workspace required to guarantee that all eigenvectors are computed is: LWORK >= 5*N + MAX( 5*NN, NP0 * MQ0 + 2 * NB * NB ) + ICEIL( NEIG, NPROW*NPCOL)*NN

The computed eigenvectors may not be orthogonal if the minimal workspace is supplied and ORFAC is too small. If you want to guarantee orthogonality (at the cost of potentially poor performance) you should add the following to LWORK: (CLUSTERSIZE-1)*N where CLUSTERSIZE is the number of eigenvalues in the largest cluster, where a cluster is defined as a set of close eigenvalues: { W(K),...,W(K+CLUSTERSIZE-1) | W(J+1) <= W(J) + ORFAC*2*norm(A) } Variable definitions: NEIG = number of eigenvectors requested NB = DESCA( MB_ ) = DESCA( NB_ ) = DESCZ( MB_ ) = DESCZ( NB_ ) NN = MAX( N, NB, 2 ) DESCA( RSRC_ ) = DESCA( NB_ ) = DESCZ( RSRC_ ) = DESCZ( CSRC_ ) = 0 NP0 = NUMROC( NN, NB, 0, 0, NPROW ) MQ0 = NUMROC( MAX( NEIG, NB, 2 ), NB, 0, 0, NPCOL ) ICEIL( X, Y ) is a ScaLAPACK function returning ceiling(X/Y)

When LWORK is too small: If LWORK is too small to guarantee orthogonality, PSSYEVX attempts to maintain orthogonality in the clusters with the smallest spacing between the eigenvalues. If LWORK is too small to compute all the eigenvectors requested, no computation is performed and INFO=-23 is returned. Note that when RANGE='V', PSSYEVX does not know how many eigenvectors are requested until the eigenvalues are computed. Therefore, when RANGE='V' and as long as LWORK is large enough to allow PSSYEVX to compute the eigenvalues, PSSYEVX will compute the eigenvalues and as many eigenvectors as it can.

Relationship between workspace, orthogonality & performance: Greater performance can be achieved if adequate workspace is provided. On the other hand, in some situations, performance can decrease as the workspace provided increases above the workspace amount shown below:

For optimal performance, greater workspace may be needed, i.e. LWORK >= MAX( LWORK, 5*N + NSYTRD_LWOPT ) Where: LWORK, as defined previously, depends upon the number of eigenvectors requested, and NSYTRD_LWOPT = N + 2*( ANB+1 )*( 4*NPS+2 ) + ( NPS + 3 ) * NPS

ANB = PJLAENV( DESCA( CTXT_), 3, 'PSSYTTRD', 'L', 0, 0, 0, 0) SQNPC = INT( SQRT( DBLE( NPROW * NPCOL ) ) ) NPS = MAX( NUMROC( N, 1, 0, 0, SQNPC ), 2*ANB )

NUMROC is a ScaLAPACK tool functions; PJLAENV is a ScaLAPACK envionmental inquiry function MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

For large N, no extra workspace is needed, however the biggest boost in performance comes for small N, so it is wise to provide the extra workspace (typically less than a Megabyte per process).

If CLUSTERSIZE >= N/SQRT(NPROW*NPCOL), then providing enough space to compute all the eigenvectors orthogonally will cause serious degradation in performance. In the limit (i.e. CLUSTERSIZE = N-1) PSSTEIN will perform no better than SSTEIN on 1 processor. For CLUSTERSIZE = N/SQRT(NPROW*NPCOL) reorthogonalizing all eigenvectors will increase the total execution time by a factor of 2 or more. For CLUSTERSIZE > N/SQRT(NPROW*NPCOL) execution time will grow as the square of the cluster size, all other factors remaining equal and assuming enough workspace. Less workspace means less reorthogonalization but faster execution.

If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the size required for optimal performance for all work arrays. Each of these values is returned in the first entry of the corresponding work arrays, and no error message is issued by PXERBLA.

IWORK (local workspace) INTEGER array On return, IWORK(1) contains the amount of integer workspace required.

LIWORK (local input) INTEGER size of IWORK LIWORK >= 6 * NNP Where: NNP = MAX( N, NPROW*NPCOL + 1, 4 ) If LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA.

IFAIL (global output) INTEGER array, dimension (N) If JOBZ = 'V', then on normal exit, the first M elements of IFAIL are zero. If (MOD(INFO,2).NE.0) on exit, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.

ICLUSTR (global output) integer array, dimension (2*NPROW*NPCOL) This array contains indices of eigenvectors corresponding to a cluster of eigenvalues that could not be reorthogonalized due to insufficient workspace (see LWORK, ORFAC and INFO). Eigenvectors corresponding to clusters of eigenvalues indexed ICLUSTR(2*I-1) to ICLUSTR(2*I), could not be reorthogonalized due to lack of workspace. Hence the eigenvectors corresponding to these clusters may not be orthogonal. ICLUSTR() is a zero terminated array. (ICLUSTR(2*K).NE.0 .AND. ICLUSTR(2*K+1).EQ.0) if and only if K is the number of clusters ICLUSTR is not referenced if JOBZ = 'N'

GAP (global output) REAL array, dimension (NPROW*NPCOL) This array contains the gap between eigenvalues whose eigenvectors could not be reorthogonalized. The output values in this array correspond to the clusters indicated by the array ICLUSTR. As a result, the dot product between eigenvectors correspoding to the I^th cluster may be as high as ( C * n ) / GAP(I) where C is a small constant.

INFO (global output) INTEGER = 0: successful exit < 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i. > 0: if (MOD(INFO,2).NE.0), then one or more eigenvectors failed to converge. Their indices are stored in IFAIL. Ensure ABSTOL=2.0*PSLAMCH( 'U' ) Send e-mail to scalapack@cs.utk.edu if (MOD(INFO/2,2).NE.0),then eigenvectors corresponding to one or more clusters of eigenvalues could not be reorthogonalized because of insufficient workspace. The indices of the clusters are stored in the array ICLUSTR. if (MOD(INFO/4,2).NE.0), then space limit prevented PSSYEVX from computing all of the eigenvectors between VL and VU. The number of eigenvectors computed is returned in NZ. if (MOD(INFO/8,2).NE.0), then PSSTEBZ failed to compute eigenvalues. Ensure ABSTOL=2.0*PSLAMCH( 'U' ) Send e-mail to scalapack@cs.utk.edu