# man pslatrd (Fonctions bibliothèques) - 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,

## NAME

PSLATRD - 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 PSLATRD(
- 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 A( * ), D( * ), E( * ), TAU( * ), W( * ), WORK( * )

## PURPOSE

PSLATRD 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 ).

If UPLO = 'U', PSLATRD reduces the last NB rows and columns of a
matrix, of which the upper triangle is supplied;

if UPLO = 'L', PSLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by PSSYTRD.

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
symmetric 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) REAL 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 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) REAL, 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) REAL 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) REAL array, dimension (NB_A)

## FURTHER DETAILS

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 real scalar, and v is a real 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

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real 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 symmetric 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).