---
name: "clarzt"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/clarzt/3lapack"
---

# clarzt(3lapack)

forms the triangular factor T of a complex block reflector H of order \> n, which is defined as a product of k elementary reflectors

## NAME

CLARZT - forms the triangular factor T of a complex block reflector H of order \> n, which is defined as a product of k elementary reflectors

## SYNOPSIS

- SUBROUTINE CLARZT(

  DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )

- CHARACTER

  DIRECT, STOREV

- INTEGER

  K, LDT, LDV, N

- COMPLEX

  T( LDT, \* ), TAU( \* ), V( LDV, \* )

## PURPOSE

CLARZT forms the triangular factor T of a 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\
 Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

## ARGUMENTS

- DIRECT (input) 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, not supported yet)\
   \= 'B': H = H(k) . . . H(2) H(1) (Backward)

- STOREV (input) CHARACTER\*1

  Specifies how the vectors which define the elementary reflectors are stored (see also Further Details):\
   \= 'R': rowwise

- N (input) INTEGER

  The order of the block reflector H. N \>= 0.

- K (input) INTEGER

  The order of the triangular factor T (= the number of elementary reflectors). K \>= 1.

- V (input/output) COMPLEX array, dimension

  (LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R' The matrix V. See further details.

- LDV (input) INTEGER

  The leading dimension of the array V. If STOREV = 'C', LDV \>= max(1,N); if STOREV = 'R', LDV \>= K.

- TAU (input) COMPLEX array, dimension (K)

  TAU(i) must contain the scalar factor of the elementary reflector H(i).

- T (output) COMPLEX 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.

- LDT (input) INTEGER

  The leading dimension of the array T. LDT \>= K.

## FURTHER DETAILS

Based on contributions by\
\
   A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA 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.\
 DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':\
                                             \_\_\_\_\_\_V\_\_\_\_\_\
\
        ( v1 v2 v3 )                        /                   ( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )\
    V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )\
        ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )\
        ( v1 v2 v3 )\
\
           .  .  .\
\
           .  .  .\
\
           1  .  .\
\
              1  .\
\
                 1\
 DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':\
                                                       \_\_\_\_\_\_V\_\_\_\_\_\
           1                                          /                      .  1                           ( 1 . . . . v1 v1 v1 v1 v1 )\
           .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )\
           .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )\
           .  .  .\
\
        ( v1 v2 v3 )\
\
        ( v1 v2 v3 )\
\
    V = ( v1 v2 v3 )\
\
        ( v1 v2 v3 )\
\
        ( v1 v2 v3 )\