# ?larft

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

## 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 )

## 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*V'` .

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 - V'*T*V`.

## Input Parameters

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

  COMPLEX*16 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

  COMPLEX*16 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

  COMPLEX*16 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.

Parent topic: Auxiliary Routines