zlarzt (l) - Linux Manuals

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

NAME

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

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 = aqFaq, H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = aqBaq, H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = aqCaq, 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 * Vaq
If STOREV = aqRaq, the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and

H  =  I - Vaq * T * V
Currently, only STOREV = aqRaq and DIRECT = aqBaq are supported.

ARGUMENTS

DIRECT (input) CHARACTER*1

Specifies the order in which the elementary reflectors are multiplied to form the block reflector:
= aqFaq: H = H(1) H(2) . . . H(k) (Forward, not supported yet)
= aqBaq: 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):
= aqRaq: 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 = aqCaq (LDV,N) if STOREV = aqRaq The matrix V. See further details.

LDV (input) INTEGER

The leading dimension of the array V. If STOREV = aqCaq, LDV >= max(1,N); if STOREV = aqRaq, 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 = aqFaq, T is upper triangular; if DIRECT = aqBaq, 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 = aqFaq and STOREV = aqCaq: DIRECT = aqFaq and STOREV = aqRaq:
                                      ______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 = aqBaq and STOREV = aqCaq: DIRECT = aqBaq and STOREV = aqRaq:
                                                ______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 )