SUBROUTINE PCHENTRD( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, $ LWORK, RWORK, LRWORK, INFO ) * * -- ScaLAPACK routine (version 1.7) -- * University of Tennessee, Knoxville, Oak Ridge National Laboratory, * and University of California, Berkeley. * October 15, 1999 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER IA, INFO, JA, LRWORK, LWORK, N * .. * .. Array Arguments .. INTEGER DESCA( * ) REAL D( * ), E( * ), RWORK( * ) COMPLEX A( * ), TAU( * ), WORK( * ) * .. * Bugs * ==== * * * Support for UPLO='U' is limited to calling the old, slow, PCHETRD * code. * * * Purpose * * ======= * * PCHENTRD is a prototype version of PCHETRD which uses tailored * codes (either the serial, CHETRD, or the parallel code, PCHETTRD) * when the workspace provided by the user is adequate. * * * PCHENTRD reduces a complex Hermitian matrix sub( A ) to Hermitian * tridiagonal form T by an unitary similarity transformation: * Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1). * * Features * ======== * * PCHENTRD is faster than PCHETRD on almost all matrices, * particularly small ones (i.e. N < 500 * sqrt(P) ), provided that * enough workspace is available to use the tailored codes. * * The tailored codes provide performance that is essentially * independent of the input data layout. * * The tailored codes place no restrictions on IA, JA, MB or NB. * At present, IA, JA, MB and NB are restricted to those values allowed * by PCHETRD to keep the interface simple. These restrictions are * documented below. (Search for "restrictions".) * * Notes * ===== * * * 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 * * * Arguments * ========= * * UPLO (global input) CHARACTER * Specifies whether the upper or lower triangular part of the * Hermitian matrix sub( A ) is stored: * = 'U': Upper triangular * = 'L': Lower triangular * * N (global input) INTEGER * The number of rows and columns to be operated on, i.e. the * order of the distributed submatrix sub( A ). N >= 0. * * A (local input/local output) COMPLEX 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 * Hermitian 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, and its strictly * lower triangular part is not referenced. If UPLO = 'L', the * leading N-by-N lower triangular part of sub( A ) contains the * lower triangular part of the matrix, and its strictly upper * triangular part is not referenced. On exit, if UPLO = 'U', * the diagonal and first superdiagonal of sub( A ) are over- * written by the corresponding elements of the tridiagonal * matrix T, and the elements above the first superdiagonal, * with the array TAU, represent the unitary matrix Q as a * product of elementary reflectors; if UPLO = 'L', the diagonal * and first subdiagonal of sub( A ) are overwritten by the * corresponding elements of the tridiagonal matrix T, and the * elements below the first subdiagonal, with the array TAU, * represent the unitary matrix Q as a product of elementary * reflectors. See Further Details. * * 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. * * D (local output) REAL array, dimension LOCc(JA+N-1) * The diagonal elements of the tridiagonal matrix T: * D(i) = A(i,i). D is tied to the distributed matrix A. * * E (local output) REAL array, dimension LOCc(JA+N-1) * if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal * elements of the tridiagonal matrix T: E(i) = A(i,i+1) if * UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the * distributed matrix A. * * TAU (local output) COMPLEX, array, dimension * LOCc(JA+N-1). This array contains the scalar factors TAU of * the elementary reflectors. TAU is tied to the distributed * matrix A. * * WORK (local workspace/local output) COMPLEX array, * dimension (LWORK) * On exit, WORK( 1 ) returns the optimal LWORK. * * LWORK (local or global input) INTEGER * The dimension of the array WORK. * LWORK is local input and must be at least * LWORK >= MAX( NB * ( NP +1 ), 3 * NB ) * * For optimal performance, greater workspace is needed, i.e. * LWORK >= 2*( ANB+1 )*( 4*NPS+2 ) + ( NPS + 4 ) * NPS * ICTXT = DESCA( CTXT_ ) * ANB = PJLAENV( ICTXT, 3, 'PCHETTRD', 'L', 0, 0, 0, 0 ) * SQNPC = INT( SQRT( REAL( 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. * * * RWORK (local workspace/local output) COMPLEX array, * dimension (LRWORK) * On exit, RWORK( 1 ) returns the optimal LRWORK. * * LRWORK (local or global input) INTEGER * The dimension of the array RWORK. * LRWORK is local input and must be at least * LRWORK >= 1 * * For optimal performance, greater workspace is needed, i.e. * LRWORK >= MAX( 2 * N ) * * * 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. * * Further Details * =============== * * If UPLO = 'U', the matrix Q is represented as a product of elementary * reflectors * * Q = H(n-1) . . . H(2) H(1). * * Each H(i) has the form * * H(i) = I - tau * v * v' * * where tau is a complex scalar, and v is a complex vector with * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in * A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1). * * If UPLO = 'L', the matrix Q is represented as a product of elementary * reflectors * * Q = H(1) H(2) . . . H(n-1). * * Each H(i) has the form * * H(i) = I - tau * v * v' * * where tau is a complex scalar, and v is a complex vector with * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in * A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1). * * The contents of sub( A ) on exit are illustrated by the following * examples with n = 5: * * if UPLO = 'U': if UPLO = 'L': * * ( d e v2 v3 v4 ) ( d ) * ( d e v3 v4 ) ( e d ) * ( d e v4 ) ( v1 e d ) * ( d e ) ( v1 v2 e d ) * ( d ) ( v1 v2 v3 e d ) * * where d and e denote diagonal and off-diagonal elements of T, and vi * denotes an element of the vector defining H(i). * * Alignment requirements * ====================== * * The distributed submatrix sub( A ) must verify some alignment proper- * ties, namely the following expression should be true: * ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA .AND. IROFFA.EQ.0 ) with * IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ). * * ===================================================================== * * .. Parameters .. 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 ) COMPLEX CONE PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. LOGICAL LQUERY, UPPER CHARACTER COLCTOP, ROWCTOP INTEGER ANB, CTXTB, I, IACOL, IAROW, ICOFFA, ICTXT, $ IINFO, INDB, INDRD, INDRE, INDTAU, INDW, IPW, $ IROFFA, J, JB, JX, K, KK, LLRWORK, LLWORK, $ LRWMIN, LWMIN, MINSZ, MYCOL, MYCOLB, MYROW, $ MYROWB, NB, NP, NPCOL, NPCOLB, NPROW, NPROWB, $ NPS, NQ, ONEPMIN, ONEPRMIN, SQNPC, TTLRWMIN, $ TTLWMIN * .. * .. Local Arrays .. INTEGER DESCB( DLEN_ ), DESCW( DLEN_ ), IDUM1( 3 ), $ IDUM2( 3 ) * .. * .. External Subroutines .. EXTERNAL BLACS_GET, BLACS_GRIDEXIT, BLACS_GRIDINFO, $ BLACS_GRIDINIT, CHETRD, CHK1MAT, DESCSET, $ IGAMN2D, PCELSET, PCHER2K, PCHETD2, PCHETTRD, $ PCHK1MAT, PCLAMR1D, PCLATRD, PCTRMR2D, $ PSLAMR1D, PB_TOPGET, PB_TOPSET, PXERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER INDXG2L, INDXG2P, NUMROC, PJLAENV EXTERNAL LSAME, INDXG2L, INDXG2P, NUMROC, PJLAENV * .. * .. Intrinsic Functions .. INTRINSIC CMPLX, ICHAR, INT, MAX, MIN, MOD, REAL, SQRT * .. * .. Executable Statements .. * * This is just to keep ftnchek and toolpack/1 happy IF( BLOCK_CYCLIC_2D*CSRC_*CTXT_*DLEN_*DTYPE_*LLD_*MB_*M_*NB_*N_* $ RSRC_.LT.0 )RETURN * Get grid parameters * ICTXT = DESCA( CTXT_ ) CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL ) * * Test the input parameters * INFO = 0 IF( NPROW.EQ.-1 ) THEN INFO = -( 600+CTXT_ ) ELSE CALL CHK1MAT( N, 2, N, 2, IA, JA, DESCA, 6, INFO ) UPPER = LSAME( UPLO, 'U' ) IF( INFO.EQ.0 ) THEN NB = DESCA( NB_ ) IROFFA = MOD( IA-1, DESCA( MB_ ) ) ICOFFA = MOD( JA-1, DESCA( NB_ ) ) IAROW = INDXG2P( IA, NB, MYROW, DESCA( RSRC_ ), NPROW ) IACOL = INDXG2P( JA, NB, MYCOL, DESCA( CSRC_ ), NPCOL ) NP = NUMROC( N, NB, MYROW, IAROW, NPROW ) NQ = MAX( 1, NUMROC( N+JA-1, NB, MYCOL, DESCA( CSRC_ ), $ NPCOL ) ) LWMIN = MAX( ( NP+1 )*NB, 3*NB ) ANB = PJLAENV( ICTXT, 3, 'PCHETTRD', 'L', 0, 0, 0, 0 ) MINSZ = PJLAENV( ICTXT, 5, 'PCHETTRD', 'L', 0, 0, 0, 0 ) SQNPC = INT( SQRT( REAL( NPROW*NPCOL ) ) ) NPS = MAX( NUMROC( N, 1, 0, 0, SQNPC ), 2*ANB ) TTLWMIN = 2*( ANB+1 )*( 4*NPS+2 ) + ( NPS+2 )*NPS LRWMIN = 1 TTLRWMIN = 2*NPS * WORK( 1 ) = CMPLX( REAL( TTLWMIN ) ) RWORK( 1 ) = REAL( TTLRWMIN ) LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 * * The following two restrictions are not necessary provided * that either of the tailored codes are used. * ELSE IF( IROFFA.NE.ICOFFA .OR. ICOFFA.NE.0 ) THEN INFO = -5 ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN INFO = -( 600+NB_ ) ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -11 ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN INFO = -13 END IF END IF IF( UPPER ) THEN IDUM1( 1 ) = ICHAR( 'U' ) ELSE IDUM1( 1 ) = ICHAR( 'L' ) END IF IDUM2( 1 ) = 1 IF( LWORK.EQ.-1 ) THEN IDUM1( 2 ) = -1 ELSE IDUM1( 2 ) = 1 END IF IDUM2( 2 ) = 11 IF( LRWORK.EQ.-1 ) THEN IDUM1( 3 ) = -1 ELSE IDUM1( 3 ) = 1 END IF IDUM2( 3 ) = 13 CALL PCHK1MAT( N, 2, N, 2, IA, JA, DESCA, 6, 3, IDUM1, IDUM2, $ INFO ) END IF * IF( INFO.NE.0 ) THEN CALL PXERBLA( ICTXT, 'PCHENTRD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * ONEPMIN = N*N + 3*N + 1 LLWORK = LWORK CALL IGAMN2D( ICTXT, 'A', ' ', 1, 1, LLWORK, 1, 1, -1, -1, -1, $ -1 ) * ONEPRMIN = 2*N LLRWORK = LRWORK CALL IGAMN2D( ICTXT, 'A', ' ', 1, 1, LLRWORK, 1, 1, -1, -1, -1, $ -1 ) * * * Use the serial, LAPACK, code: CTRD on small matrices if we * we have enough space. * NPROWB = 0 IF( ( N.LT.MINSZ .OR. SQNPC.EQ.1 ) .AND. LLWORK.GE.ONEPMIN .AND. $ LLRWORK.GE.ONEPRMIN .AND. .NOT.UPPER ) THEN NPROWB = 1 NPS = N ELSE IF( LLWORK.GE.TTLWMIN .AND. LLRWORK.GE.TTLRWMIN .AND. .NOT. $ UPPER ) THEN NPROWB = SQNPC END IF END IF * IF( NPROWB.GE.1 ) THEN NPCOLB = NPROWB SQNPC = NPROWB INDB = 1 INDRD = 1 INDRE = INDRD + NPS INDTAU = INDB + NPS*NPS INDW = INDTAU + NPS LLWORK = LLWORK - INDW + 1 * CALL BLACS_GET( ICTXT, 10, CTXTB ) CALL BLACS_GRIDINIT( CTXTB, 'Row major', SQNPC, SQNPC ) CALL BLACS_GRIDINFO( CTXTB, NPROWB, NPCOLB, MYROWB, MYCOLB ) CALL DESCSET( DESCB, N, N, 1, 1, 0, 0, CTXTB, NPS ) * CALL PCTRMR2D( UPLO, 'N', N, N, A, IA, JA, DESCA, WORK( INDB ), $ 1, 1, DESCB, ICTXT ) * * * Only those processors in context CTXTB are needed for a while * IF( NPROWB.GT.0 ) THEN * IF( NPROWB.EQ.1 ) THEN CALL CHETRD( UPLO, N, WORK( INDB ), NPS, RWORK( INDRD ), $ RWORK( INDRE ), WORK( INDTAU ), $ WORK( INDW ), LLWORK, INFO ) ELSE * CALL PCHETTRD( 'L', N, WORK( INDB ), 1, 1, DESCB, $ RWORK( INDRD ), RWORK( INDRE ), $ WORK( INDTAU ), WORK( INDW ), LLWORK, $ INFO ) * END IF END IF * * All processors participate in moving the data back to the * way that PCHENTRD expects it. * CALL PSLAMR1D( N-1, RWORK( INDRE ), 1, 1, DESCB, E, 1, JA, $ DESCA ) * CALL PSLAMR1D( N, RWORK( INDRD ), 1, 1, DESCB, D, 1, JA, $ DESCA ) * CALL PCLAMR1D( N, WORK( INDTAU ), 1, 1, DESCB, TAU, 1, JA, $ DESCA ) * CALL PCTRMR2D( UPLO, 'N', N, N, WORK( INDB ), 1, 1, DESCB, A, $ IA, JA, DESCA, ICTXT ) * IF( MYROWB.GE.0 ) $ CALL BLACS_GRIDEXIT( CTXTB ) * ELSE * CALL PB_TOPGET( ICTXT, 'Combine', 'Columnwise', COLCTOP ) CALL PB_TOPGET( ICTXT, 'Combine', 'Rowwise', ROWCTOP ) CALL PB_TOPSET( ICTXT, 'Combine', 'Columnwise', '1-tree' ) CALL PB_TOPSET( ICTXT, 'Combine', 'Rowwise', '1-tree' ) * IPW = NP*NB + 1 * IF( UPPER ) THEN * * Reduce the upper triangle of sub( A ). * KK = MOD( JA+N-1, NB ) IF( KK.EQ.0 ) $ KK = NB CALL DESCSET( DESCW, N, NB, NB, NB, IAROW, $ INDXG2P( JA+N-KK, NB, MYCOL, DESCA( CSRC_ ), $ NPCOL ), ICTXT, MAX( 1, NP ) ) * DO 10 K = N - KK + 1, NB + 1, -NB JB = MIN( N-K+1, NB ) I = IA + K - 1 J = JA + K - 1 * * Reduce columns I:I+NB-1 to tridiagonal form and form * the matrix W which is needed to update the unreduced part of * the matrix * CALL PCLATRD( UPLO, K+JB-1, JB, A, IA, JA, DESCA, D, E, $ TAU, WORK, 1, 1, DESCW, WORK( IPW ) ) * * Update the unreduced submatrix A(IA:I-1,JA:J-1), using an * update of the form: * A(IA:I-1,JA:J-1) := A(IA:I-1,JA:J-1) - V*W' - W*V' * CALL PCHER2K( UPLO, 'No transpose', K-1, JB, -CONE, A, $ IA, J, DESCA, WORK, 1, 1, DESCW, ONE, A, $ IA, JA, DESCA ) * * Copy last superdiagonal element back into sub( A ) * JX = MIN( INDXG2L( J, NB, 0, IACOL, NPCOL ), NQ ) CALL PCELSET( A, I-1, J, DESCA, CMPLX( E( JX ) ) ) * DESCW( CSRC_ ) = MOD( DESCW( CSRC_ )+NPCOL-1, NPCOL ) * 10 CONTINUE * * Use unblocked code to reduce the last or only block * CALL PCHETD2( UPLO, MIN( N, NB ), A, IA, JA, DESCA, D, E, $ TAU, WORK, LWORK, IINFO ) * ELSE * * Reduce the lower triangle of sub( A ) * KK = MOD( JA+N-1, NB ) IF( KK.EQ.0 ) $ KK = NB CALL DESCSET( DESCW, N, NB, NB, NB, IAROW, IACOL, ICTXT, $ MAX( 1, NP ) ) * DO 20 K = 1, N - NB, NB I = IA + K - 1 J = JA + K - 1 * * Reduce columns I:I+NB-1 to tridiagonal form and form * the matrix W which is needed to update the unreduced part * of the matrix * CALL PCLATRD( UPLO, N-K+1, NB, A, I, J, DESCA, D, E, TAU, $ WORK, K, 1, DESCW, WORK( IPW ) ) * * Update the unreduced submatrix A(I+NB:IA+N-1,I+NB:IA+N-1), * using an update of the form: A(I+NB:IA+N-1,I+NB:IA+N-1) := * A(I+NB:IA+N-1,I+NB:IA+N-1) - V*W' - W*V' * CALL PCHER2K( UPLO, 'No transpose', N-K-NB+1, NB, -CONE, $ A, I+NB, J, DESCA, WORK, K+NB, 1, DESCW, $ ONE, A, I+NB, J+NB, DESCA ) * * Copy last subdiagonal element back into sub( A ) * JX = MIN( INDXG2L( J+NB-1, NB, 0, IACOL, NPCOL ), NQ ) CALL PCELSET( A, I+NB, J+NB-1, DESCA, CMPLX( E( JX ) ) ) * DESCW( CSRC_ ) = MOD( DESCW( CSRC_ )+1, NPCOL ) * 20 CONTINUE * * Use unblocked code to reduce the last or only block * CALL PCHETD2( UPLO, KK, A, IA+K-1, JA+K-1, DESCA, D, E, TAU, $ WORK, LWORK, IINFO ) END IF * CALL PB_TOPSET( ICTXT, 'Combine', 'Columnwise', COLCTOP ) CALL PB_TOPSET( ICTXT, 'Combine', 'Rowwise', ROWCTOP ) * END IF * WORK( 1 ) = CMPLX( REAL( TTLWMIN ) ) RWORK( 1 ) = REAL( TTLRWMIN ) * RETURN * * End of PCHENTRD * END