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PCLATRD - reduce NB rows and columns of a complex Hermitian distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to complex tridiagonal form by an unitary similarity transformation Q’ * sub( A ) * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of sub( A )

SUBROUTINE PCLATRD( UPLO, N, NB, A, IA, JA, DESCA, D, E, TAU, W, IW, JW, DESCW, WORK ) CHARACTER UPLO INTEGER IA, IW, JA, JW, N, NB INTEGER DESCA( * ), DESCW( * ) REAL D( * ), E( * ) COMPLEX A( * ), TAU( * ), W( * ), WORK( * )

PCLATRD reduces NB rows and columns of a complex Hermitian distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to complex tridiagonal form by an unitary similarity transformation Q’ * sub( A ) * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of sub( A ). If UPLO = ’U’, PCLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = ’L’, PCLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied. This is an auxiliary routine called by PCHETRD. 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

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. NB (global input) INTEGER The number of rows and columns to be reduced. 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 last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements above the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. If UPLO = ’L’, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of sub( A ); the elements below the diagonal 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. W (local output) COMPLEX pointer into the local memory to an array of dimension (LLD_W,NB_W), This array contains the local pieces of the N-by-NB_W matrix W required to update the unreduced part of sub( A ). IW (global input) INTEGER The row index in the global array W indicating the first row of sub( W ). JW (global input) INTEGER The column index in the global array W indicating the first column of sub( W ). DESCW (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix W. WORK (local workspace) COMPLEX array, dimension (NB_A)

If UPLO = ’U’, the matrix Q is represented as a product of elementary reflectors Q = H(n) H(n-1) . . . H(n-nb+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:n) = 0 and v(i-1) = 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(nb). 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 elements of the vectors v together form the N-by-NB matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: sub( A ) := sub( A ) - V*W’ - W*V’. The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: if UPLO = ’U’: if UPLO = ’L’: ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).