!>
!> SLARFB_GETT applies a real Householder block reflector H from the
!> left to a real (K+M)-by-N   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 REAL 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 REAL 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 REAL 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 REAL 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**T ) * ( A_in ) =
!>       ( B_out )        ( B_in )                          ( B_in )
!>                  = ( I - ( V1 ) * T * ( V1**T, V2**T ) ) * ( 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**T ) * ( A1_in )
!>       ( B1_out )        (     0 )                          (     0 )
!>
!>       ( A2_out ) := H * ( A2_in ) = ( I - V * T * V**T ) * ( A2_in )
!>       ( B2_out )        ( B2_in )                          ( B2_in )
!>
!>    If IDENT != 'I':
!>
!>       The computation for column block 1:
!>
!>       A1_out: = A1_in - V1*T*(V1**T)*A1_in
!>
!>       B1_out: = - V2*T*(V1**T)*A1_in
!>
!>       The computation for column block 2, which exists if N > K:
!>
!>       A2_out: = A2_in - V1*T*( (V1**T)*A2_in + (V2**T)*B2_in )
!>
!>       B2_out: = B2_in - V2*T*( (V1**T)*A2_in + (V2**T)*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**T)*B2_in )
!>
!>       B2_out: = B2_in - V2*T*( A2_in + (V2**T)*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**T) * W2 = (unit_lower_tr_of_(A1)**T) * W2
!>       col2_(3) W2: = W2 + (V2**T) * B2 = W2 + (B1**T) * 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**T) * W1 = (unit_lower_tr_of_(A1)**T) * 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**T) * B2 = W2 + (B1**T) * 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**T) * W2
!>                      = (unit_lower_tr_of_(A1)**T) * W2
!>      end if
!>      col2_(3)  W2: = W2 + (V2**T) * B2 = W2 + (B1**T) * 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**T) * W1
!>                    = (unit_lower_tr_of_(A1)**T) * 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
!>
!>