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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: reduces the first NB columns of a complex general n-by-(n

```NAME
LAPACK-3  -  reduces the first NB columns of a complex general n-by-(n-
k+1) matrix A so that elements below the k-th subdiagonal are zero

SYNOPSIS
SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )

INTEGER        K, LDA, LDT, LDY, N, NB

COMPLEX*16     A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )

PURPOSE
ZLAHRD reduces the first NB columns of a complex  general  n-by-(n-k+1)
matrix A so that elements below the k-th subdiagonal are zero. The
reduction is performed by a unitary 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 OBSOLETE auxiliary routine.
This routine will be 'deprecated' in a  future release.

ARGUMENTS
N       (input) INTEGER
The order of the matrix A.

K       (input) INTEGER
The offset for the reduction. Elements below the k-th
subdiagonal in the first NB columns are reduced to zero.

NB      (input) INTEGER
The number of columns to be reduced.

A       (input/output) COMPLEX*16 array, dimension (LDA,N-K+1)
On entry, the n-by-(n-k+1) general matrix A.
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 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 are
unchanged. See Further Details.
LDA     (input) INTEGER
The leading dimension of the array A.  LDA >= max(1,N).

TAU     (output) COMPLEX*16 array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.

T       (output) COMPLEX*16 array, dimension (LDT,NB)
The upper triangular matrix T.

LDT     (input) INTEGER
The leading dimension of the array T.  LDT >= NB.

Y       (output) COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y.

LDY     (input) INTEGER
The leading dimension of the array Y. LDY >= max(1,N).

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 complex scalar, and v is a complex vector with
v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
A(i+k+1:n,i), and tau in TAU(i).
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 := (I - V*T*V') * (A - Y*V').
The contents of A 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, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).

LAPACK auxiliary routine (versionMarch 2011                       LAPACK-3(3)
```

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