 PSLAHRD(1) reduce the first NB columns of a real general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero

## SYNOPSIS

SUBROUTINE PSLAHRD(
N, K, NB, A, IA, JA, DESCA, TAU, T, Y, IY, JY, DESCY, WORK )

INTEGER IA, IY, JA, JY, K, N, NB

INTEGER DESCA( * ), DESCY( * )

REAL A( * ), T( * ), TAU( * ), WORK( * ), Y( * )

## PURPOSE

PSLAHRD reduces the first NB columns of a real general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogo- nal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.

This is an auxiliary routine called by PSGEHRD. In the following comments sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1).

## ARGUMENTS

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.
K (global input) INTEGER
The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero.
NB (global input) INTEGER
The number of 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-K)). On entry, this array contains the the local pieces of the N-by-(N-K+1) general distributed matrix A(IA:IA+N-1,JA:JA+N-K). On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced distributed matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A(IA:IA+N-1,JA:JA+N-K) are unchanged. 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.
TAU (local output) REAL array, dimension LOCc(JA+N-2)
The scalar factors of the elementary reflectors (see Further Details). TAU is tied to the distributed matrix A.
T (local output) REAL array, dimension (NB_A,NB_A)
The upper triangular matrix T.
Y (local output) REAL pointer into the local memory
to an array of dimension (LLD_Y,NB_A). On exit, this array contains the local pieces of the N-by-NB distributed matrix Y. LLD_Y >= LOCr(IA+N-1).
IY (global input) INTEGER
The row index in the global array Y indicating the first row of sub( Y ).
JY (global input) INTEGER
The column index in the global array Y indicating the first column of sub( Y ).
DESCY (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix Y.
WORK (local workspace) REAL array, dimension (NB)

## FURTHER DETAILS

The matrix Q is represented as a product of nb 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 real scalar, and v is a real vector with
v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(ia+i+k:ia+n-1,ja+i-1), and tau in TAU(ja+i-1).

The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A(ia:ia+n-1,ja:ja+n-k) := (I-V*T*V')*(A(ia:ia+n-1,ja:ja+n-k)-Y*V').

The contents of A(ia:ia+n-1,ja:ja+n-k) on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:

( a   h   a   a   a )

( a   h   a   a   a )

( a   h   a   a   a )

( h   h   a   a   a )

( v1  h   a   a   a )

( v1  v2  a   a   a )

( v1  v2  a   a   a )

where a denotes an element of the original matrix
A(ia:ia+n-1,ja:ja+n-k), h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).