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## NAME

```       PDLAHRD  -  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 PDLAHRD( 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( * )

DOUBLE          PRECISION A( * ), T( * ), TAU( * ), WORK( * ), Y( *
)

```

## PURPOSE

```       PDLAHRD  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  PDGEHRD.  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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (NB_A,NB_A)
The upper triangular matrix T.

Y       (local output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (NB)

```

## FURTHERDETAILS

```       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).
```