table of contents
PDLATRD(l) | LAPACK auxiliary routine (version 1.5) | PDLATRD(l) |
NAME¶
PDLATRD - reduce NB rows and columns of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to symmetric tridiagonal form by an orthogonal similarity transformation Q' * sub( A ) * Q,SYNOPSIS¶
- SUBROUTINE PDLATRD(
- UPLO, N, NB, A, IA, JA, DESCA, D, E, TAU, W,IW, JW, DESCW, WORK )
PURPOSE¶
PDLATRD reduces NB rows and columns of a real symmetric distributed matrix sub( A ) = A(IA:IA+N-1,JA:JA+N-1) to symmetric tridiagonal form by an orthogonal 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 ).DTYPE_A = 1.
the BLACS process grid A is distribu-
ted over. The context itself is glo-
bal, but the handle (the integer
value) may vary.
array A.
array A.
the rows of the array.
the columns of the array.
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.
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( 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 thesymmetric
matrix sub( A ) is stored:
- N (global input) INTEGER
- The number of rows and columns to be operated on, i.e. theorder 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) DOUBLE PRECISION 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 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, 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 orthogonal 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 orthogonal 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 thefirst 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension LOCc(JA+N-1)
- if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonalelements 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) DOUBLE PRECISION, array, dimension
- LOCc(JA+N-1). This array contains the scalar factors TAU ofthe elementary reflectors. TAU is tied to the distributed matrix A.
- W (local output) DOUBLE PRECISION pointer into the local memory
- to an array of dimension (LLD_W,NB_W), This array containsthe 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 firstrow of sub( W ).
- JW (global input) INTEGER
- The column index in the global array W indicating thefirst 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) DOUBLE PRECISION array, dimension (NB_A)
FURTHERDETAILS¶
If UPLO = 'U', the matrix Q is represented as a product of elementary reflectorsQ = H(n) H(n-1) . . . H(n-nb+1).
H(i) = I - tau * v * v'
Q = H(1) H(2) . . . H(nb).
H(i) = I - tau * v * v'
( 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).
12 May 1997 | LAPACK version 1.5 |