SYNOPSIS
- SUBROUTINE DLARZT(
- 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
DLARZT 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, andH = 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
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
Based on contributions byA. 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 )