dlarft(3P)

NAME

dlarft - form the triangular factor T of a real block reflector H of order n, which is defined as a product of k elementary reflectors

SYNOPSIS

SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )

CHARACTER DIRECT, STOREV

INTEGER K, LDT, LDV, N

DOUBLE PRECISION T( LDT, ∗ ), TAU( ∗ ), V( LDV, ∗ )

#include <sunperf.h>

void dlarft(char direct, char storev, int n, int k, double ∗v, int ldv, double ∗tau, double ∗t, int ldt) ;

PURPOSE

DLARFT forms the triangular factor T of a real 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

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)
= ’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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary reflector H(i).

T (output) DOUBLE PRECISION 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

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 = ( 1       )                 V = ( 1 v1 v1 v1 v1 )
                 ( v1 1    )                     (     1 v2 v2 v2 )
                 ( v1 v2 1 )                     (        1 v3 v3 )
                 ( v1 v2 v3 )
                 ( v1 v2 v3 )

DIRECT = ’B’ and STOREV = ’C’:         DIRECT = ’B’ and STOREV = ’R’:

             V = ( v1 v2 v3 )                 V = ( v1 v1 1       )
                 ( v1 v2 v3 )                     ( v2 v2 v2 1    )
                 ( 1 v2 v3 )                     ( v3 v3 v3 v3 1 )
                 (     1 v3 )
                 (        1 )