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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: forms the triangular factor T of a complex block reflector H of order &gt; n, which is defined as a product of k

```NAME
LAPACK-3 - 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 ZLARZT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )

CHARACTER      DIRECT, STOREV

INTEGER        K, LDT, LDV, N

COMPLEX*16     T( LDT, * ), TAU( * ), V( LDV, * )

PURPOSE
ZLARZT 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
= '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*16 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*16 array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i).

T       (output) COMPLEX*16 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 )

LAPACK routine (version 3.2)     March 2011                       LAPACK-3(3)
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