# ?larft

Forms the triangular factor T of a block reflector H = I - V*T*V**H.

## Syntax

call slarft( direct, storev, n, k, v, ldv, tau, t, ldt )

call dlarft( direct, storev, n, k, v, ldv, tau, t, ldt )

call clarft( direct, storev, n, k, v, ldv, tau, t, ldt )

call zlarft( direct, storev, n, k, v, ldv, tau, t, ldt )

C:

lapack_int LAPACKE_<?>larft (int matrix_layout, char direct, char storev, lapack_int n, lapack_int k, const <datatype> * v, lapack_int ldv, const <datatype> * tau, <datatype> * t, lapack_int ldt);

## Include Files

• Fortran: mkl.fi
• C: mkl.h

## Description

The routine ?larft forms the triangular factor T of a real/complex block reflector H of order n, which is defined as a product of k elementary reflectors.

If `direct = 'F'`, `H = H(1)*H(2)* . . .*H(k)` and T is upper triangular;

If `direct = 'B'`, `H = H(k)*. . .*H(2)*H(1)` and T is lower triangular.

If `storev = 'C'`, the vector which defines the elementary reflector H(i) is stored in the i-th column of the array v, and `H = I - V*T*VT` (for real flavors) or `H = I - V*T*VH` (for complex flavors) .

If `storev = 'R'`, the vector which defines the elementary reflector H(i) is stored in the i-th row of the array v, and `H = I - VT*T*V` (for real flavors) or `H = I - VH*T*V` (for complex flavors).

## Input Parameters

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

direct

CHARACTER*1.

Specifies the order in which the elementary reflectors are multiplied to form the block reflector:

= 'F': `H = H(1)*H(2)*. . . *H(k)` (forward)

= 'B': `H = H(k)*. . .*H(2)*H(1)` (backward)

storev

CHARACTER*1.

Specifies how the vectors which define the elementary reflectors are stored (see also Application Notes below):

= 'C': column-wise

= 'R': row-wise.

n

INTEGER. The order of the block reflector H. `n ≥ 0`.

k

INTEGER. The order of the triangular factor T (equal to the number of elementary reflectors). `k ≥ 1`.

v

REAL for slarft

DOUBLE PRECISION for dlarft

COMPLEX for clarft

DOUBLE COMPLEX for zlarft

Array, DIMENSION

(ldv, k) if `storev = 'C'` or

(ldv, n) if `storev = 'R'`.

The matrix V.

ldv

INTEGER. The leading dimension of the array v.

If `storev = 'C'`, `ldv ≥ max(1,n)`;

if `storev = 'R'`, `ldv ≥ k`.

tau

REAL for slarft

DOUBLE PRECISION for dlarft

COMPLEX for clarft

DOUBLE COMPLEX for zlarft

Array, DIMENSION (k). tau(i) must contain the scalar factor of the elementary reflector H(i).

ldt

INTEGER. The leading dimension of the output array t. `ldt ≥ k`.

## Output Parameters

t

REAL for slarft

DOUBLE PRECISION for dlarft

COMPLEX for clarft

DOUBLE COMPLEX for zlarft

Array, DIMENSION (ldt,k). The k-by-k triangular factor T of the block reflector. If `direct = 'F'`, T is upper triangular; if `direct = 'B'`, T is lower triangular. The rest of the array is not used.

v

The matrix V.

## Application Notes

The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with `n = 5` and `k = 3`. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used.