zlarfb_gett.f(3)

NAME
zlarfb_gett.f

SYNOPSIS

  Functions/Subroutines
subroutine zlarfb_gett (IDENT, M, N, K, T, LDT, A, LDA, B, LDB, WORK, LDWORK)
ZLARFB_GETT

Function/Subroutine Documentation

  subroutine zlarfb_gett (character IDENT, integer M, integer N, integer K, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldwork, * ) WORK, integer LDWORK)

Purpose:

     ZLARFB_GETT applies a complex Householder block reflector H from the
     left to a complex (K+M)-by-N  'triangular-pentagonal' matrix
     composed of two block matrices: an upper trapezoidal K-by-N matrix A
     stored in the array A, and a rectangular M-by-(N-K) matrix B, stored
     in the array B. The block reflector H is stored in a compact
     WY-representation, where the elementary reflectors are in the
     arrays A, B and T. See Further Details section.

Parameters
    IDENT

              IDENT is CHARACTER*1
              If IDENT = not 'I', or not 'i', then V1 is unit
                 lower-triangular and stored in the left K-by-K block of
                 the input matrix A,
              If IDENT = 'I' or 'i', then  V1 is an identity matrix and
                 not stored.
              See Further Details section.

    M

              M is INTEGER
              The number of rows of the matrix B.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrices A and B.
              N >= 0.

    K

              K is INTEGER
              The number or rows of the matrix A.
              K is also order of the matrix T, i.e. the number of
              elementary reflectors whose product defines the block
              reflector. 0 <= K <= N.

    T

              T is COMPLEX*16 array, dimension (LDT,K)
              The upper-triangular K-by-K matrix T in the representation
              of the block reflector.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= K.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry:
               a) In the K-by-N upper-trapezoidal part A: input matrix A.
               b) In the columns below the diagonal: columns of V1
                  (ones are not stored on the diagonal).
              On exit:
                A is overwritten by rectangular K-by-N product H*A.
              See Further Details section.

    LDA

              LDB is INTEGER
              The leading dimension of the array A. LDA >= max(1,K).

    B

              B is COMPLEX*16 array, dimension (LDB,N)
              On entry:
                a) In the M-by-(N-K) right block: input matrix B.
                b) In the M-by-N left block: columns of V2.
              On exit:
                B is overwritten by rectangular M-by-N product H*B.
              See Further Details section.

    LDB

              LDB is INTEGER
              The leading dimension of the array B. LDB >= max(1,M).

    WORK

              WORK is COMPLEX*16 array,
              dimension (LDWORK,max(K,N-K))

    LDWORK

              LDWORK is INTEGER
              The leading dimension of the array WORK. LDWORK>=max(1,K).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:

     November 2020, Igor Kozachenko,
                    Computer Science Division,
                    University of California, Berkeley

Further Details:

        (1) Description of the Algebraic Operation.
        The matrix A is a K-by-N matrix composed of two column block
        matrices, A1, which is K-by-K, and A2, which is K-by-(N-K):
        A = ( A1, A2 ).
        The matrix B is an M-by-N matrix composed of two column block
        matrices, B1, which is M-by-K, and B2, which is M-by-(N-K):
        B = ( B1, B2 ).
        Perform the operation:
           ( A_out ) := H * ( A_in ) = ( I - V * T * V**H ) * ( A_in ) =
           ( B_out )        ( B_in )                          ( B_in )
                      = ( I - ( V1 ) * T * ( V1**H, V2**H ) ) * ( A_in )
                              ( V2 )                            ( B_in )
         On input:
        a) ( A_in )  consists of two block columns:
           ( B_in )
           ( A_in ) = (( A1_in ) ( A2_in )) = (( A1_in ) ( A2_in ))
           ( B_in )   (( B1_in ) ( B2_in ))   ((     0 ) ( B2_in )),
           where the column blocks are:
           (  A1_in )  is a K-by-K upper-triangular matrix stored in the
                       upper triangular part of the array A(1:K,1:K).
           (  B1_in )  is an M-by-K rectangular ZERO matrix and not stored.
           ( A2_in )  is a K-by-(N-K) rectangular matrix stored
                      in the array A(1:K,K+1:N).
           ( B2_in )  is an M-by-(N-K) rectangular matrix stored
                      in the array B(1:M,K+1:N).
        b) V = ( V1 )
               ( V2 )
           where:
           1) if IDENT == 'I',V1 is a K-by-K identity matrix, not stored;
           2) if IDENT != 'I',V1 is a K-by-K unit lower-triangular matrix,
              stored in the lower-triangular part of the array
              A(1:K,1:K) (ones are not stored),
           and V2 is an M-by-K rectangular stored the array B(1:M,1:K),
                     (because on input B1_in is a rectangular zero
                      matrix that is not stored and the space is
                      used to store V2).
        c) T is a K-by-K upper-triangular matrix stored
           in the array T(1:K,1:K).
        On output:
        a) ( A_out ) consists of two  block columns:
           ( B_out )
           ( A_out ) = (( A1_out ) ( A2_out ))
           ( B_out )   (( B1_out ) ( B2_out )),
           where the column blocks are:
           ( A1_out )  is a K-by-K square matrix, or a K-by-K
                       upper-triangular matrix, if V1 is an
                       identity matrix. AiOut is stored in
                       the array A(1:K,1:K).
           ( B1_out )  is an M-by-K rectangular matrix stored
                       in the array B(1:M,K:N).
           ( A2_out )  is a K-by-(N-K) rectangular matrix stored
                       in the array A(1:K,K+1:N).
           ( B2_out )  is an M-by-(N-K) rectangular matrix stored
                       in the array B(1:M,K+1:N).
        The operation above can be represented as the same operation
        on each block column:
           ( A1_out ) := H * ( A1_in ) = ( I - V * T * V**H ) * ( A1_in )
           ( B1_out )        (     0 )                          (     0 )
           ( A2_out ) := H * ( A2_in ) = ( I - V * T * V**H ) * ( A2_in )
           ( B2_out )        ( B2_in )                          ( B2_in )
        If IDENT != 'I':
           The computation for column block 1:
           A1_out: = A1_in - V1*T*(V1**H)*A1_in
           B1_out: = - V2*T*(V1**H)*A1_in
           The computation for column block 2, which exists if N > K:
           A2_out: = A2_in - V1*T*( (V1**H)*A2_in + (V2**H)*B2_in )
           B2_out: = B2_in - V2*T*( (V1**H)*A2_in + (V2**H)*B2_in )
        If IDENT == 'I':
           The operation for column block 1:
           A1_out: = A1_in - V1*T*A1_in
           B1_out: = - V2*T*A1_in
           The computation for column block 2, which exists if N > K:
           A2_out: = A2_in - T*( A2_in + (V2**H)*B2_in )
           B2_out: = B2_in - V2*T*( A2_in + (V2**H)*B2_in )
        (2) Description of the Algorithmic Computation.
        In the first step, we compute column block 2, i.e. A2 and B2.
        Here, we need to use the K-by-(N-K) rectangular workspace
        matrix W2 that is of the same size as the matrix A2.
        W2 is stored in the array WORK(1:K,1:(N-K)).
        In the second step, we compute column block 1, i.e. A1 and B1.
        Here, we need to use the K-by-K square workspace matrix W1
        that is of the same size as the as the matrix A1.
        W1 is stored in the array WORK(1:K,1:K).
        NOTE: Hence, in this routine, we need the workspace array WORK
        only of size WORK(1:K,1:max(K,N-K)) so it can hold both W2 from
        the first step and W1 from the second step.
        Case (A), when V1 is unit lower-triangular, i.e. IDENT != 'I',
        more computations than in the Case (B).
        if( IDENT != 'I' ) then
         if ( N > K ) then
           (First Step - column block 2)
           col2_(1) W2: = A2
           col2_(2) W2: = (V1**H) * W2 = (unit_lower_tr_of_(A1)**H) * W2
           col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2
           col2_(4) W2: = T * W2
           col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2
           col2_(6) W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2
           col2_(7) A2: = A2 - W2
         else
           (Second Step - column block 1)
           col1_(1) W1: = A1
           col1_(2) W1: = (V1**H) * W1 = (unit_lower_tr_of_(A1)**H) * W1
           col1_(3) W1: = T * W1
           col1_(4) B1: = - V2 * W1 = - B1 * W1
           col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1
           col1_(6) square A1: = A1 - W1
         end if
        end if
        Case (B), when V1 is an identity matrix, i.e. IDENT == 'I',
        less computations than in the Case (A)
        if( IDENT == 'I' ) then
         if ( N > K ) then
           (First Step - column block 2)
           col2_(1) W2: = A2
           col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2
           col2_(4) W2: = T * W2
           col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2
           col2_(7) A2: = A2 - W2
         else
           (Second Step - column block 1)
           col1_(1) W1: = A1
           col1_(3) W1: = T * W1
           col1_(4) B1: = - V2 * W1 = - B1 * W1
           col1_(6) upper-triangular_of_(A1): = A1 - W1
         end if
        end if
        Combine these cases (A) and (B) together, this is the resulting
        algorithm:
        if ( N > K ) then
          (First Step - column block 2)
          col2_(1)  W2: = A2
          if( IDENT != 'I' ) then
            col2_(2)  W2: = (V1**H) * W2
                          = (unit_lower_tr_of_(A1)**H) * W2
          end if
          col2_(3)  W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2]
          col2_(4)  W2: = T * W2
          col2_(5)  B2: = B2 - V2 * W2 = B2 - B1 * W2
          if( IDENT != 'I' ) then
            col2_(6)    W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2
          end if
          col2_(7) A2: = A2 - W2
        else
        (Second Step - column block 1)
          col1_(1) W1: = A1
          if( IDENT != 'I' ) then
            col1_(2) W1: = (V1**H) * W1
                        = (unit_lower_tr_of_(A1)**H) * W1
          end if
          col1_(3) W1: = T * W1
          col1_(4) B1: = - V2 * W1 = - B1 * W1
          if( IDENT != 'I' ) then
            col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1
            col1_(6_a) below_diag_of_(A1): =  - below_diag_of_(W1)
          end if
          col1_(6_b) up_tr_of_(A1): = up_tr_of_(A1) - up_tr_of_(W1)
        end if

Definition at line 390 of file zlarfb_gett.f.

Author

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