# p?larzt

Forms the triangular factor T of a block reflector H=I-V*T*VH as returned by p?tzrzf.

## Syntax

call pslarzt(direct, storev, n, k, v, iv, jv, descv, tau, t, work)

call pdlarzt(direct, storev, n, k, v, iv, jv, descv, tau, t, work)

call pclarzt(direct, storev, n, k, v, iv, jv, descv, tau, t, work)

call pzlarzt(direct, storev, n, k, v, iv, jv, descv, tau, t, work)

## Description

This routine forms the triangular factor T of a real/complex block reflector H of order greater than n, which is defined as a product of k elementary reflectors as returned by p?tzrzf.

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`

Currently, only storev = 'R' and direct = 'B' are supported.

## Input Parameters

direct

  (global) CHARACTER.

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

  if `direct = 'F': H = H(1)*H(2)*...*H(k)` (Forward, not supported)

  if `direct = 'B': H = H(k)*...*H(2)*H(1)` (Backward).

storev

  (global) CHARACTER.

  Specifies how the vectors which defines the elementary reflectors are stored:

  if `storev = 'C'`: columnwise (not supported);

  if `storev = 'R'`: rowwise.

n

  (global). INTEGER.

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

k

  (global). INTEGER.

  The order of the triangular factor T (= the number of elementary reflectors).

  `1≤k≤mb_v(= nb_v)`.

v

  REAL for pslarzt

  DOUBLE PRECISION for pdlarzt

  COMPLEX for pclarzt

  COMPLEX*16 for pzlarzt.

  Pointer into the local memory to an array of local DIMENSION`(LOCr(iv+k-1)`, `LOCc(jv+n-1))`.

  The distributed matrix V contains the Householder vectors. See Application Notes below.

iv, jv

  (global) INTEGER. The row and column indices in the global array V indicating the first row and the first column of the submatrix sub(V), respectively.

descv

  (global and local) INTEGER array, DIMENSION (dlen_). The array descriptor for the distributed matrix V.

tau

  (local)

  REAL for pslarzt

  DOUBLE PRECISION for pdlarzt

  COMPLEX for pclarzt

  COMPLEX*16 for pzlarzt.

  Array, DIMENSION `LOCr(iv+k-1)` if `incv = m_v`, and LOCc(jv+k-1) otherwise. This array contains the Householder scalars related to the Householder vectors.

  tau is tied to the distributed matrix V.

work

  (local).

  REAL for pslarzt

  DOUBLE PRECISION for pdlarzt

  COMPLEX for pclarzt

  COMPLEX*16 for pzlarzt.

  Workspace array, DIMENSION`(k*(k-1)/2)`.

## Output Parameters

v

  REAL for pslarzt

  DOUBLE PRECISION for pdlarzt

  COMPLEX for pclarzt

  COMPLEX*16 for pzlarzt.

t

  (local)

  REAL for pslarzt

  DOUBLE PRECISION for pdlarzt

  COMPLEX for pclarzt

  COMPLEX*16 for pzlarzt.

  Array, DIMENSION (mb_v, mb_v). It contains the k-by-k triangular factor of the block reflector associated with v. t is lower triangular.

## 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