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- bookworm 3.11.0-2
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- unstable 3.11.0-2
- experimental 3.12.0-1
| larfb_gett(3) | LAPACK | larfb_gett(3) |
NAME¶
larfb_gett - larfb_gett: step in ungtsqr_row
SYNOPSIS¶
Functions¶
subroutine clarfb_gett (ident, m, n, k, t, ldt, a, lda, b,
ldb, work, ldwork)
CLARFB_GETT subroutine dlarfb_gett (ident, m, n, k, t, ldt, a,
lda, b, ldb, work, ldwork)
DLARFB_GETT subroutine slarfb_gett (ident, m, n, k, t, ldt, a,
lda, b, ldb, work, ldwork)
SLARFB_GETT subroutine zlarfb_gett (ident, m, n, k, t, ldt, a,
lda, b, ldb, work, ldwork)
ZLARFB_GETT
Detailed Description¶
Function Documentation¶
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)¶
CLARFB_GETT
Purpose:
CLARFB_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 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 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 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 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:
(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
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:
DLARFB_GETT applies a real Householder block reflector H from the
left to a real (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 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 Further Details section.
LDA
LDB is INTEGER
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 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:
(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
subroutine slarfb_gett (character ident, integer m, integer n, integer k, real, dimension( ldt, * ) t, integer ldt, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldwork, * ) work, integer ldwork)¶
SLARFB_GETT
Purpose:
SLARFB_GETT applies a real Householder block reflector H from the
left to a real (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 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).
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:
(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
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
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
Univ. of Colorado Denver
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
Author¶
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