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.

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).

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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

    (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