dlarzt(3S)

NAME
     DLARZT - 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 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, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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, 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 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 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 )


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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