NAME

clarzt - form 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 * 1 DIRECT, STOREV
  COMPLEX V(LDV,*), TAU(*), T(LDT,*)
  INTEGER N, K, LDV, LDT

  SUBROUTINE CLARZT_64( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
  CHARACTER * 1 DIRECT, STOREV
  COMPLEX V(LDV,*), TAU(*), T(LDT,*)
  INTEGER*8 N, K, LDV, LDT

F95 INTERFACE

  SUBROUTINE LARZT( DIRECT, STOREV, N, K, V, [LDV], TAU, T, [LDT])
  CHARACTER(LEN=1) :: DIRECT, STOREV
  COMPLEX, DIMENSION(:) :: TAU
  COMPLEX, DIMENSION(:,:) :: V, T
  INTEGER :: N, K, LDV, LDT

  SUBROUTINE LARZT_64( DIRECT, STOREV, N, K, V, [LDV], TAU, T, [LDT])
  CHARACTER(LEN=1) :: DIRECT, STOREV
  COMPLEX, DIMENSION(:) :: TAU
  COMPLEX, DIMENSION(:,:) :: V, T
  INTEGER(8) :: N, K, LDV, LDT

C INTERFACE

#include <sunperf.h>

void clarzt(char direct, char storev, int n, int k, complex *v, int ldv, complex *tau, complex *t, int ldt);

void clarzt_64(char direct, char storev, long n, long k, complex *v, long ldv, complex *tau, complex *t, long ldt);

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)
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)
Specifies how the vectors which define the elementary reflectors are stored (see also Further Details):
```
 = 'R': rowwise
```
N (input)
The order of the block reflector H. N > = 0.
K (input)
The order of the triangular factor T ( = the number of elementary reflectors). K > = 1.
V (input)
(LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R' The matrix V. See further details.
LDV (input)
The leading dimension of the array V. If STOREV = 'C', LDV > = max(1,N); if STOREV = 'R', LDV > = K.
TAU (input)
TAU(i) must contain the scalar factor of the elementary reflector H(i).
T (output)
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)
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  .

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 )