PSSYGVX - compute all the eigenvalues, and optionally, the eigenvectors of a real generalized SY-definite eigenproblem

SUBROUTINE PSSYGVX(

IBTYPE, JOBZ, RANGE, UPLO, N, A, IA, JA, DESCA, B, IB, JB, DESCB, 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, IB, IBTYPE, IL, INFO, IU, IZ, JA, JB, JZ, LIWORK, LWORK, M, N, NZ

REAL ABSTOL, ORFAC, VL, VU

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

REAL A( * ), B( * ), 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 ONE

PARAMETER ( ONE = 1.0E+0 )

REAL FIVE, ZERO

PARAMETER ( FIVE = 5.0E+0, ZERO = 0.0E+0 )

INTEGER IERRNPD

PARAMETER ( IERRNPD = 16 )

LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ

CHARACTER TRANS

INTEGER IACOL, IAROW, IBCOL, IBROW, ICOFFA, ICOFFB, ICTXT, IROFFA, IROFFB, LIWMIN, LWMIN, MQ0, MYCOL, MYROW, NB, NEIG, NN, NP0, NPCOL, NPROW

REAL EPS, SCALE

INTEGER IDUM1( 5 ), IDUM2( 5 )

LOGICAL LSAME

INTEGER ICEIL, INDXG2P, NUMROC

REAL PSLAMCH

EXTERNAL LSAME, ICEIL, INDXG2P, NUMROC, PSLAMCH

EXTERNAL BLACS_GRIDINFO, CHK1MAT, PCHK1MAT, PCHK2MAT, PSPOTRF, PSSYEVX, PSSYGST, PSTRMM, PSTRSM, PXERBLA, SGEBR2D, SGEBS2D, SSCAL

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

PSSYGVX computes all the eigenvalues, and optionally, the eigenvectors of a real generalized SY-definite eigenproblem, of the form sub( A )*x=(lambda)*sub( B )*x, sub( A )*sub( B )x=(lambda)*x, or sub( B )*sub( A )*x=(lambda)*x. Here sub( A ) denoting A( IA:IA+N-1, JA:JA+N-1 ) is assumed to be SY, and sub( B ) denoting B( IB:IB+N-1, JB:JB+N-1 ) is assumed to be symmetric positive definite.

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

IBTYPE (global input) INTEGER Specifies the problem type to be solved: = 1: sub( A )*x = (lambda)*sub( B )*x = 2: sub( A )*sub( B )*x = (lambda)*x = 3: sub( B )*sub( A )*x = (lambda)*x

JOBZ (global input) CHARACTER*1 = '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 = 'U': Upper triangles of sub( A ) and sub( B ) are stored; = 'L': Lower triangles of sub( A ) and sub( B ) are stored.

N (global input) INTEGER The order of the matrices sub( A ) and sub( B ). N >= 0.

A (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( A ) contains the upper triangular part of the matrix. If UPLO = 'L', the leading N-by-N lower triangular part of sub( A ) contains the lower triangular part of the matrix.

On exit, if JOBZ = 'V', then if INFO = 0, sub( A ) contains the distributed matrix Z of eigenvectors. The eigenvectors are normalized as follows: if IBTYPE = 1 or 2, Z**T*sub( B )*Z = I; if IBTYPE = 3, Z**T*inv( sub( B ) )*Z = I. If JOBZ = 'N', then on exit the upper triangle (if UPLO='U') or the lower triangle (if UPLO='L') of sub( A ), including the diagonal, is destroyed.

IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ).

JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ).

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

B (local input/local output) REAL pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+N-1)). On entry, this array contains the local pieces of the N-by-N symmetric distributed matrix sub( B ). If UPLO = 'U', the leading N-by-N upper triangular part of sub( B ) contains the upper triangular part of the matrix. If UPLO = 'L', the leading N-by-N lower triangular part of sub( B ) contains the lower triangular part of the matrix.

On exit, if INFO <= N, the part of sub( B ) containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization sub( B ) = U**T*U or sub( B ) = L*L**T.

IB (global input) INTEGER The row index in the global array B indicating the first row of sub( B ).

JB (global input) INTEGER The column index in the global array B indicating the first column of sub( B ).

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

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 PSSYGVX 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.) PSSYGVX 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 The row index in the global array Z indicating the first row of sub( Z ).

JZ (global input) INTEGER The column index in the global array Z indicating the first column of sub( Z ).

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) 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 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, PSSYGVX 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', PSSYGVX 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 PSSYGVX to compute the eigenvalues, PSSYGVX 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, NSYGST_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 NSYGST_LWOPT = 2*NP0*NB + NQ0*NB + NB*NB

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 ) NB = DESCA( MB_ ) NP0 = NUMROC( N, NB, 0, 0, NPROW ) NQ0 = NUMROC( N, NB, 0, 0, NPCOL )

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 on 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.

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 (output) INTEGER array, dimension (N) IFAIL provides additional information when INFO .NE. 0 If (MOD(INFO/16,2).NE.0) then IFAIL(1) indicates the order of the smallest minor which is not positive definite. If (MOD(INFO,2).NE.0) on exit, then IFAIL contains the indices of the eigenvectors that failed to converge.

If neither of the above error conditions hold and JOBZ = 'V', then the first M elements of IFAIL are set to zero.

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. 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 PSSYGVX 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. Send e-mail to scalapack@cs.utk.edu if (MOD(INFO/16,2).NE.0), then B was not positive definite. IFAIL(1) indicates the order of the smallest minor which is not positive definite.