# ?lahrd

Reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

## Syntax

call slahrd( n, k, nb, a, lda, tau, t, ldt, y, ldy )

call dlahrd( n, k, nb, a, lda, tau, t, ldt, y, ldy )

call clahrd( n, k, nb, a, lda, tau, t, ldt, y, ldy )

call zlahrd( n, k, nb, a, lda, tau, t, ldt, y, ldy )

• mkl.fi

## Description

The routine reduces the first nb columns of a real/complex general n-by-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal/unitary similarity transformation QT*A*Q for real flavors, or QH*A*Q for complex flavors. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*VT (for real flavors) or I - V*T*VH (for complex flavors), and also the matrix Y = A*V*T.

The matrix Q is represented as products of nb elementary reflectors:

`Q = H(1)*H(2)*... *H(nb)`

Each `H(i)` has the form

`H(i) = I - tau*v*vT` for real flavors, or

`H(i) = I - tau*v*vH` for complex flavors, or

where tau is a real/complex scalar, and v is a real/complex vector.

This is an obsolete auxiliary routine. Please use the new routine ?lahr2 instead.

## Input Parameters

n

INTEGER. The order of the matrix A (`n ≥ 0`).

k

INTEGER. The offset for the reduction. Elements below the k-th subdiagonal in the first nb columns are reduced to zero.

nb

INTEGER. The number of columns to be reduced.

a

REAL for slahrd

DOUBLE PRECISION for dlahrd

COMPLEX for clahrd

DOUBLE COMPLEX for zlahrd.

Array a(lda, n-k+1) contains the n-by-(n-k+1) general matrix A to be reduced.

lda

INTEGER. The leading dimension of a; at least `max(1, n)`.

ldt

INTEGER. The leading dimension of the output array t; must be at least `max(1, nb)`.

ldy

INTEGER. The leading dimension of the output array y; must be at least `max(1, n)`.

## Output Parameters

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 Application Notes below.

tau

REAL for slahrd

DOUBLE PRECISION for dlahrd

COMPLEX for clahrd

DOUBLE COMPLEX for zlahrd.

Array, DIMENSION (nb).

Contains scalar factors of the elementary reflectors.

t, y

REAL for slahrd

DOUBLE PRECISION for dlahrd

COMPLEX for clahrd

DOUBLE COMPLEX for zlahrd.

Arrays, dimension t(ldt, nb), y(ldy, nb).

The array t contains upper triangular matrix T.

The array y contains the n-by-nb matrix Y .

## Application Notes

For the elementary reflector H(i),

`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 is stored 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*VT) * (A - Y*VT)` for real flavors, or

`A := (I - V*T*VH) * (A - Y*VH)` for complex flavors.

The contents of A on exit are illustrated by the following example with `n = 7`, `k = 3` and `nb = 2`:

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).