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# dlarft

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,
* )

PURPOSE
DLARFT forms the triangular factor T of a real block reflec-
tor H of order n, which is defined as a product of k elemen-
tary 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 reflec-
tors 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 elemen-
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 elemen-
tary reflector H(i).

T       (output) DOUBLE PRECISION array, dimension (LDT,K)
The k by k triangular factor T of the block reflec-
tor.  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 )