Your company here ā click to reach over 10,000 unique daily visitors

# zlarfb_gett.f - Man Page

SRC/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)

ZLARFB_GETT Ā

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

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

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zlarfb_gett(3) is an alias of zlarfb_gett.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK