table of contents
PDSYTRD(l) | LAPACK routine (version 1.5) | PDSYTRD(l) |
NAME¶
PDSYTRD - reduce a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformationSYNOPSIS¶
- SUBROUTINE PDSYTRD(
- UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK, INFO )
PURPOSE¶
PDSYTRD reduces a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal similarity transformation: Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1). NotesDTYPE_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 the symmetric
matrix sub( A ) is stored:
- 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) 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 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 orthogonal 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 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 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) 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-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) DOUBLE PRECISION, 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) DOUBLE PRECISION 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 >= MAX( NB * ( NP +1 ), 3 * NB ) where NB = MB_A = NB_A, NP = NUMROC( N, NB, MYROW, IAROW, NPROW ), IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ). INDXG2P and NUMROC are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO. 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 (global output) INTEGER
- = 0: successful exit
FURTHER DETAILS¶
If UPLO = 'U', the matrix Q is represented as a product of elementary reflectorsQ = H(n-1) . . . H(2) H(1).
H(i) = I - tau * v * v'
Q = H(1) H(2) . . . H(n-1).
H(i) = I - tau * v * v'
( 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).
12 May 1997 | LAPACK version 1.5 |