SRC/dlarfb_gett.f(3)

📦

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✨ NAME ✨

okay so this is SRC/dlarfb_gett.f(3) from the Library Functions Manual — bestie, we are officially in Fortran linear-algebra territory and it's giving iconic 💅

🎀 SYNOPSIS 🎀

Functions/Subroutines
- subroutine dlarfb_gett (ident, m, n, k, t, ldt, a, lda, b, ldb, work, ldwork)
- DLARFB_GETT

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

🛠️ Function/Subroutine Documentation 🛠️

subroutine dlarfb_gett (character ident, integer m, integer n, integer k,
    double precision, dimension( ldt, * ) t, integer ldt, double precision,
    dimension( lda, * ) a, integer lda, double precision, dimension( ldb, *
    ) b, integer ldb, double precision, dimension( ldwork, * ) work, integer
    ldwork)
DLARFB_GETT

🎯 Purpose 🎯

Okay so DLARFB_GETT applies a real Householder block reflector H from the left to a real (K+M)-by-N matrix. That matrix is made 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 live in the arrays A, B, and T. See the Further Details section for the juicy continuation 💖

🧃 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 the 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 — yes, “or”, we're keeping the original typo energy 🫶 — and K is also the order of the matrix T, i.e. the number of elementary reflectors whose product defines the block reflector. 0 <= K <= N.

`T`

T is DOUBLE PRECISION 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 DOUBLE PRECISION 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`

LDA is INTEGER

(and yes, the source text labels this line LDB — iconic typo moment 💀 — but the leading dimension is for the array A) The leading dimension of the array A. LDA >= max(1,K).

`B`

B is DOUBLE PRECISION 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`

LDB is INTEGER

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

`WORK`

WORK is DOUBLE PRECISION 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 🔍

okay so we're picking up mid-function-docs, bestie — this is where the math gets its full main-character moment 💅

💖 (1) Description of the Algebraic Operation 💖

So A is a K-by-N matrix made of two column-block matrices: A1, which is K-by-K, and A2, which is K-by-(N-K). And B is an M-by-N matrix made of two column-block matrices: B1, which is M-by-K, and B2, which is M-by-(N-K):

A = ( A1, A2 ).
B = ( B1, B2 ).

Here's the operation, no cap:

( 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_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 )),

and 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).

Then for V:

V = ( V1 )
    ( 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).

And T is a K-by-K upper-triangular matrix stored in the array T(1:K,1:K).

On output, ( A_out ) over ( B_out ) consists of two block columns:

( A_out ) = (( A1_out ) ( A2_out ))
( B_out )   (( B1_out ) ( B2_out )),

and 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 is:

A1_out: = A1_in - V1*T*(V1**T)*A1_in

B1_out: = - V2*T*(V1**T)*A1_in

and the computation for column block 2, which exists if N > K, is:

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 is:

A1_out: = A1_in - V1*T**A1_in

B1_out: = - V2*T**A1_in

and the computation for column block 2, which exists if N > K, is:

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): `V1` is unit lower-triangular, i.e. `IDENT != 'I'`

This case has more computations than in 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): `V1` is an identity matrix, i.e. `IDENT == 'I'`

This case has less computations than in 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

🌈 Combined Algorithm

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

📍 Definition 📍

Definition at line 390 of file dlarfb_gett.f.

🫶 Author 🫶

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