SRC/clarfb_gett.f(3)

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🌸 SRC/clarfb_gett.f

Okay bestie, you're in the Library Functions Manual, section 3 — this is SRC/clarfb_gett.f(3), and it's giving subroutine main-character energy. 💅 We're talking about clarfb_gett, and yes, every parameter is gonna get its moment. No skipping. Periodt. ✨

✨ NAME ✨

SRC/clarfb_gett.f — that's the file, bestie. This page is all about the subroutine clarfb_gett. 🎀

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

subroutine clarfb_gett (ident, m, n, k, t, ldt, a, lda, b, ldb, work, ldwork) — the whole call, right here. 🫶

subroutine clarfb_gett (ident, m, n, k, t, ldt, a, lda, b, ldb, work,
    ldwork)

And the page labels it: CLARFB_GETT. Iconic. 👑

💅 Function/Subroutine Documentation 💅

Here's the full signature, exactly as written — no cap:

subroutine clarfb_gett (character ident, integer m, integer n, integer k,
    complex, dimension( ldt, * ) t, integer ldt, complex, dimension( lda, *
    ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex,
    dimension( ldwork, * ) work, integer ldwork)

Also labeled: CLARFB_GETT. 💖

🎀 Purpose 🎀

CLARFB_GETT applies a complex Householder block reflector H from the left to a complex (K+M)-by-N matrix.

That matrix is composed of two block matrices:

an upper trapezoidal K-by-N matrix A stored in the array A
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 the Further Details section. 🦋

🍓 Parameters 🍓

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 the Further Details section. 💜
M — is INTEGER. The number of rows of the matrix B. M >= 0. 💖
N — is INTEGER. The number of columns of the matrices A and B. N >= 0. 🌸
K — is INTEGER. The number or rows of the matrix A — yes, the page literally says "or" there, and I'm keeping it exactly as written, bestie. 😭 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 — is COMPLEX array, dimension (LDT,K). The upper-triangular K-by-K matrix T in the representation of the block reflector. ✨
LDT — is INTEGER. The leading dimension of the array T. LDT >= K. 🦋
A — is COMPLEX 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 the Further Details section. 💅
LDA — the page says LDB is INTEGER here — literally, that's what it says — and then describes the leading dimension of the array A. LDA >= max(1,K). 🙈
B — is COMPLEX 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 the Further Details section. 🔥
LDB — is INTEGER. The leading dimension of the array B. LDB >= max(1,M). 🌈
WORK — is COMPLEX array, dimension (LDWORK,max(K,N-K)). 🧚
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.

All four, bestie. We love a credited squad. 🫶

🦋 Contributors 🦋

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

Exactly as listed. 💜

🌈 Further Details

🧮 (1) Description of the Algebraic Operation

Okay bestie, here's the tea on the algebra. We got matrix A which is a K-by-N matrix made of two column block matrices, A1 (K-by-K) and A2 (K-by-(N-K)):

A = ( A1, A2 ).

And matrix B is an M-by-N matrix made of two column block matrices, B1 (M-by-K) and B2 (M-by-(N-K)):

B = ( B1, B2 ).

Then we 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, check it:

a) ( A_in ) over ( B_in ) consists of two block columns:

       ( 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 ) over ( V2 ) where:

if IDENT == 'I', V1 is a K-by-K identity matrix, not stored;
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 ) over ( B_out ) consists of two block columns:

       ( 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 )

That's the algebra, bestie. Now let's get into the algorithm. 💅

🛠️ (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 clarfb_gett.f.

👑 Author

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