SUBROUTINE PCHETD2( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, $ LWORK, INFO ) * * -- ScaLAPACK auxiliary routine (version 1.5) -- * University of Tennessee, Knoxville, Oak Ridge National Laboratory, * and University of California, Berkeley. * May 1, 1997 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER IA, INFO, JA, LWORK, N * .. * .. Array Arguments .. INTEGER DESCA( * ) REAL D( * ), E( * ) COMPLEX A( * ), TAU( * ), WORK( * ) * .. * * Purpose * ======= * * PCHETD2 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). * * 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 minimal and optimal LWORK. * * LWORK (local or global input) INTEGER * The dimension of the array WORK. * LWORK is local input and must be at least * LWORK >= 3*N. * * If LWORK = -1, then LWORK 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. * * INFO (local 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 ) with * IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ). * * ===================================================================== * * .. Parameters .. INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_, $ LLD_, MB_, M_, NB_, N_, RSRC_ 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 ) COMPLEX HALF, ONE, ZERO PARAMETER ( HALF = ( 0.5E+0, 0.0E+0 ), $ ONE = ( 1.0E+0, 0.0E+0 ), $ ZERO = ( 0.0E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. LOGICAL LQUERY, UPPER INTEGER IACOL, IAROW, ICOFFA, ICTXT, II, IK, IROFFA, J, $ JJ, JK, JN, LDA, LWMIN, MYCOL, MYROW, NPCOL, $ NPROW COMPLEX ALPHA, TAUI * .. * .. External Subroutines .. EXTERNAL BLACS_ABORT, BLACS_GRIDINFO, CAXPY, CGEBR2D, $ CGEBS2D, CHK1MAT, CHEMV, $ CHER2, CLARFG, INFOG2L, PXERBLA * .. * .. External Functions .. LOGICAL LSAME COMPLEX CDOTC EXTERNAL LSAME, CDOTC * .. * .. Intrinsic Functions .. INTRINSIC CMPLX, REAL * .. * .. Executable Statements .. * * 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 UPPER = LSAME( UPLO, 'U' ) CALL CHK1MAT( N, 2, N, 2, IA, JA, DESCA, 6, INFO ) LWMIN = 3 * N * WORK( 1 ) = CMPLX( REAL( LWMIN ) ) LQUERY = ( LWORK.EQ.-1 ) IF( INFO.EQ.0 ) THEN IROFFA = MOD( IA-1, DESCA( MB_ ) ) ICOFFA = MOD( JA-1, DESCA( NB_ ) ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( IROFFA.NE.ICOFFA ) 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 END IF END IF END IF * IF( INFO.NE.0 ) THEN CALL PXERBLA( ICTXT, 'PCHETD2', -INFO ) CALL BLACS_ABORT( ICTXT, 1 ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.LE.0 ) $ RETURN * * Compute local information * LDA = DESCA( LLD_ ) CALL INFOG2L( IA, JA, DESCA, NPROW, NPCOL, MYROW, MYCOL, II, JJ, $ IAROW, IACOL ) * IF( UPPER ) THEN * * Process(IAROW, IACOL) owns block to be reduced * IF( MYCOL.EQ.IACOL ) THEN IF( MYROW.EQ.IAROW ) THEN * * Reduce the upper triangle of sub( A ) * IK = II+N-1+(JJ+N-2)*LDA A( IK ) = REAL( A( IK ) ) DO 10 J = N-1, 1, -1 IK = II + J - 1 JK = JJ + J - 1 * * Generate elementary reflector H(i) = I - tau * v * v' * to annihilate A(IA:IA+J-1,JA:JA+J-1) * ALPHA = A( IK+JK*LDA ) CALL CLARFG( J, ALPHA, A( II+JK*LDA ), 1, TAUI ) E( JK+1 ) = REAL( ALPHA ) * IF( TAUI.NE.ZERO ) THEN * * Apply H(i) from both sides to * A(IA:IA+J-1,JA:JA+J-1) * A( IK+JK*LDA ) = ONE * * Compute x := tau * A * v storing x in TAU(1:i) * CALL CHEMV( UPLO, J, TAUI, A( II+(JJ-1)*LDA ), $ LDA, A( II+JK*LDA ), 1, ZERO, $ TAU( JJ ), 1 ) * * Compute w := x - 1/2 * tau * (x'*v) * v * ALPHA = -HALF*TAUI*CDOTC( J, TAU( JJ ), 1, $ A( II+JK*LDA ), 1 ) CALL CAXPY( J, ALPHA, A( II+JK*LDA ), 1, $ TAU( JJ ), 1 ) * * Apply the transformation as a rank-2 update: * A := A - v * w' - w * v' * CALL CHER2( UPLO, J, -ONE, A( II+JK*LDA ), 1, $ TAU( JJ ), 1, A( II+(JJ-1)*LDA ), $ LDA ) END IF * * Copy D, E, TAU to broadcast them columnwise. * A( IK+JK*LDA ) = CMPLX( E( JK+1 ) ) D( JK+1 ) = REAL( A( IK+1+JK*LDA ) ) WORK( J+1 ) = CMPLX( D( JK+1 ) ) WORK( N+J+1 ) = CMPLX( E( JK+1 ) ) TAU( JK+1 ) = TAUI WORK( 2*N+J+1 ) = TAU( JK+1 ) * 10 CONTINUE D( JJ ) = REAL( A( II+(JJ-1)*LDA ) ) WORK( 1 ) = CMPLX( D( JJ ) ) WORK( N+1 ) = ZERO WORK( 2*N+1 ) = ZERO * CALL CGEBS2D( ICTXT, 'Columnwise', ' ', 1, 3*N, WORK, 1 ) * ELSE CALL CGEBR2D( ICTXT, 'Columnwise', ' ', 1, 3*N, WORK, 1, $ IAROW, IACOL ) DO 20 J = 2, N JN = JJ + J - 1 D( JN ) = REAL( WORK( J ) ) E( JN ) = REAL( WORK( N+J ) ) TAU( JN ) = WORK( 2*N+J ) 20 CONTINUE D( JJ ) = REAL( WORK( 1 ) ) END IF END IF * ELSE * * Process (IAROW, IACOL) owns block to be factorized * IF( MYCOL.EQ.IACOL ) THEN IF( MYROW.EQ.IAROW ) THEN * * Reduce the lower triangle of sub( A ) * A( II+(JJ-1)*LDA ) = REAL( A( II+(JJ-1)*LDA ) ) DO 30 J = 1, N - 1 IK = II + J - 1 JK = JJ + J - 1 * * Generate elementary reflector H(i) = I - tau * v * v' * to annihilate A(IA+J-JA+2:IA+N-1,JA+J-1) * ALPHA = A( IK+1+(JK-1)*LDA ) CALL CLARFG( N-J, ALPHA, A( IK+2+(JK-1)*LDA ), 1, $ TAUI ) E( JK ) = REAL( ALPHA ) * IF( TAUI.NE.ZERO ) THEN * * Apply H(i) from both sides to * A(IA+J-JA+1:IA+N-1,JA+J+1:JA+N-1) * A( IK+1+(JK-1)*LDA ) = ONE * * Compute x := tau * A * v storing y in TAU(i:n-1) * CALL CHEMV( UPLO, N-J, TAUI, A( IK+1+JK*LDA ), $ LDA, A( IK+1+(JK-1)*LDA ), 1, $ ZERO, TAU( JK ), 1 ) * * Compute w := x - 1/2 * tau * (x'*v) * v * ALPHA = -HALF*TAUI*CDOTC( N-J, TAU( JK ), 1, $ A( IK+1+(JK-1)*LDA ), 1 ) CALL CAXPY( N-J, ALPHA, A( IK+1+(JK-1)*LDA ), $ 1, TAU( JK ), 1 ) * * Apply the transformation as a rank-2 update: * A := A - v * w' - w * v' * CALL CHER2( UPLO, N-J, -ONE, $ A( IK+1+(JK-1)*LDA ), 1, $ TAU( JK ), 1, A( IK+1+JK*LDA ), $ LDA ) END IF * * Copy D(JK), E(JK), TAU(JK) to broadcast them * columnwise. * A( IK+1+(JK-1)*LDA ) = CMPLX( E( JK ) ) D( JK ) = REAL( A( IK+(JK-1)*LDA ) ) WORK( J ) = CMPLX( D( JK ) ) WORK( N+J ) = CMPLX( E( JK ) ) TAU( JK ) = TAUI WORK( 2*N+J ) = TAU( JK ) 30 CONTINUE JN = JJ + N - 1 D( JN ) = REAL( A( II+N-1+(JN-1)*LDA ) ) WORK( N ) = CMPLX( D( JN ) ) TAU( JN ) = ZERO WORK( 2*N ) = ZERO * CALL CGEBS2D( ICTXT, 'Columnwise', ' ', 1, 3*N-1, WORK, $ 1 ) * ELSE CALL CGEBR2D( ICTXT, 'Columnwise', ' ', 1, 3*N-1, WORK, $ 1, IAROW, IACOL ) DO 40 J = 1, N - 1 JN = JJ + J - 1 D( JN ) = REAL( WORK( J ) ) E( JN ) = REAL( WORK( N+J ) ) TAU( JN ) = WORK( 2*N+J ) 40 CONTINUE JN = JJ + N - 1 D( JN ) = REAL( WORK( N ) ) TAU( JN ) = ZERO END IF END IF END IF * WORK( 1 ) = CMPLX( REAL( LWMIN ) ) * RETURN * * End of PCHETD2 * END