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| unhr_col(3) | LAPACK | unhr_col(3) |
NAME¶
unhr_col - {un,or}hr_col: Householder reconstruction
SYNOPSIS¶
Functions¶
subroutine cunhr_col (m, n, nb, a, lda, t, ldt, d, info)
CUNHR_COL subroutine dorhr_col (m, n, nb, a, lda, t, ldt, d,
info)
DORHR_COL subroutine sorhr_col (m, n, nb, a, lda, t, ldt, d,
info)
SORHR_COL subroutine zunhr_col (m, n, nb, a, lda, t, ldt, d,
info)
ZUNHR_COL
Detailed Description¶
Function Documentation¶
subroutine cunhr_col (integer m, integer n, integer nb, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( * ) d, integer info)¶
CUNHR_COL
Purpose:
CUNHR_COL takes an M-by-N complex matrix Q_in with orthonormal columns
as input, stored in A, and performs Householder Reconstruction (HR),
i.e. reconstructs Householder vectors V(i) implicitly representing
another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
where S is an N-by-N diagonal matrix with diagonal entries
equal to +1 or -1. The Householder vectors (columns V(i) of V) are
stored in A on output, and the diagonal entries of S are stored in D.
Block reflectors are also returned in T
(same output format as CGEQRT).
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. M >= N >= 0.
NB
NB is INTEGER
The column block size to be used in the reconstruction
of Householder column vector blocks in the array A and
corresponding block reflectors in the array T. NB >= 1.
(Note that if NB > N, then N is used instead of NB
as the column block size.)
A
A is COMPLEX array, dimension (LDA,N)
On entry:
The array A contains an M-by-N orthonormal matrix Q_in,
i.e the columns of A are orthogonal unit vectors.
On exit:
The elements below the diagonal of A represent the unit
lower-trapezoidal matrix V of Householder column vectors
V(i). The unit diagonal entries of V are not stored
(same format as the output below the diagonal in A from
CGEQRT). The matrix T and the matrix V stored on output
in A implicitly define Q_out.
The elements above the diagonal contain the factor U
of the 'modified' LU-decomposition:
Q_in - ( S ) = V * U
( 0 )
where 0 is a (M-N)-by-(M-N) zero matrix.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T
T is COMPLEX array,
dimension (LDT, N)
Let NOCB = Number_of_output_col_blocks
= CEIL(N/NB)
On exit, T(1:NB, 1:N) contains NOCB upper-triangular
block reflectors used to define Q_out stored in compact
form as a sequence of upper-triangular NB-by-NB column
blocks (same format as the output T in CGEQRT).
The matrix T and the matrix V stored on output in A
implicitly define Q_out. NOTE: The lower triangles
below the upper-triangular blocks will be filled with
zeros. See Further Details.
LDT
LDT is INTEGER
The leading dimension of the array T.
LDT >= max(1,min(NB,N)).
D
D is COMPLEX array, dimension min(M,N).
The elements can be only plus or minus one.
D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
1 <= i <= min(M,N), and Q_in_i is Q_in after performing
i-1 steps of “modified” Gaussian elimination.
See Further Details.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Further Details:
The computed M-by-M unitary factor Q_out is defined implicitly as
a product of unitary matrices Q_out(i). Each Q_out(i) is stored in
the compact WY-representation format in the corresponding blocks of
matrices V (stored in A) and T.
The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
matrix A contains the column vectors V(i) in NB-size column
blocks VB(j). For example, VB(1) contains the columns
V(1), V(2), ... V(NB). NOTE: The unit entries on
the diagonal of Y are not stored in A.
The number of column blocks is
NOCB = Number_of_output_col_blocks = CEIL(N/NB)
where each block is of order NB except for the last block, which
is of order LAST_NB = N - (NOCB-1)*NB.
For example, if M=6, N=5 and NB=2, the matrix V is
V = ( VB(1), VB(2), VB(3) ) =
= ( 1 )
( v21 1 )
( v31 v32 1 )
( v41 v42 v43 1 )
( v51 v52 v53 v54 1 )
( v61 v62 v63 v54 v65 )
For each of the column blocks VB(i), an upper-triangular block
reflector TB(i) is computed. These blocks are stored as
a sequence of upper-triangular column blocks in the NB-by-N
matrix T. The size of each TB(i) block is NB-by-NB, except
for the last block, whose size is LAST_NB-by-LAST_NB.
For example, if M=6, N=5 and NB=2, the matrix T is
T = ( TB(1), TB(2), TB(3) ) =
= ( t11 t12 t13 t14 t15 )
( t22 t24 )
The M-by-M factor Q_out is given as a product of NOCB
unitary M-by-M matrices Q_out(i).
Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),
where each matrix Q_out(i) is given by the WY-representation
using corresponding blocks from the matrices V and T:
Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,
where I is the identity matrix. Here is the formula with matrix
dimensions:
Q(i){M-by-M} = I{M-by-M} -
VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},
where INB = NB, except for the last block NOCB
for which INB=LAST_NB.
=====
NOTE:
=====
If Q_in is the result of doing a QR factorization
B = Q_in * R_in, then:
B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.
So if one wants to interpret Q_out as the result
of the QR factorization of B, then the corresponding R_out
should be equal to R_out = S * R_in, i.e. some rows of R_in
should be multiplied by -1.
For the details of the algorithm, see [1].
[1] 'Reconstructing Householder vectors from tall-skinny QR',
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
subroutine dorhr_col (integer m, integer n, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) d, integer info)¶
DORHR_COL
Purpose:
DORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns
as input, stored in A, and performs Householder Reconstruction (HR),
i.e. reconstructs Householder vectors V(i) implicitly representing
another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
where S is an N-by-N diagonal matrix with diagonal entries
equal to +1 or -1. The Householder vectors (columns V(i) of V) are
stored in A on output, and the diagonal entries of S are stored in D.
Block reflectors are also returned in T
(same output format as DGEQRT).
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. M >= N >= 0.
NB
NB is INTEGER
The column block size to be used in the reconstruction
of Householder column vector blocks in the array A and
corresponding block reflectors in the array T. NB >= 1.
(Note that if NB > N, then N is used instead of NB
as the column block size.)
A
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry:
The array A contains an M-by-N orthonormal matrix Q_in,
i.e the columns of A are orthogonal unit vectors.
On exit:
The elements below the diagonal of A represent the unit
lower-trapezoidal matrix V of Householder column vectors
V(i). The unit diagonal entries of V are not stored
(same format as the output below the diagonal in A from
DGEQRT). The matrix T and the matrix V stored on output
in A implicitly define Q_out.
The elements above the diagonal contain the factor U
of the 'modified' LU-decomposition:
Q_in - ( S ) = V * U
( 0 )
where 0 is a (M-N)-by-(M-N) zero matrix.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T
T is DOUBLE PRECISION array,
dimension (LDT, N)
Let NOCB = Number_of_output_col_blocks
= CEIL(N/NB)
On exit, T(1:NB, 1:N) contains NOCB upper-triangular
block reflectors used to define Q_out stored in compact
form as a sequence of upper-triangular NB-by-NB column
blocks (same format as the output T in DGEQRT).
The matrix T and the matrix V stored on output in A
implicitly define Q_out. NOTE: The lower triangles
below the upper-triangular blocks will be filled with
zeros. See Further Details.
LDT
LDT is INTEGER
The leading dimension of the array T.
LDT >= max(1,min(NB,N)).
D
D is DOUBLE PRECISION array, dimension min(M,N).
The elements can be only plus or minus one.
D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
1 <= i <= min(M,N), and Q_in_i is Q_in after performing
i-1 steps of “modified” Gaussian elimination.
See Further Details.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Further Details:
The computed M-by-M orthogonal factor Q_out is defined implicitly as
a product of orthogonal matrices Q_out(i). Each Q_out(i) is stored in
the compact WY-representation format in the corresponding blocks of
matrices V (stored in A) and T.
The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
matrix A contains the column vectors V(i) in NB-size column
blocks VB(j). For example, VB(1) contains the columns
V(1), V(2), ... V(NB). NOTE: The unit entries on
the diagonal of Y are not stored in A.
The number of column blocks is
NOCB = Number_of_output_col_blocks = CEIL(N/NB)
where each block is of order NB except for the last block, which
is of order LAST_NB = N - (NOCB-1)*NB.
For example, if M=6, N=5 and NB=2, the matrix V is
V = ( VB(1), VB(2), VB(3) ) =
= ( 1 )
( v21 1 )
( v31 v32 1 )
( v41 v42 v43 1 )
( v51 v52 v53 v54 1 )
( v61 v62 v63 v54 v65 )
For each of the column blocks VB(i), an upper-triangular block
reflector TB(i) is computed. These blocks are stored as
a sequence of upper-triangular column blocks in the NB-by-N
matrix T. The size of each TB(i) block is NB-by-NB, except
for the last block, whose size is LAST_NB-by-LAST_NB.
For example, if M=6, N=5 and NB=2, the matrix T is
T = ( TB(1), TB(2), TB(3) ) =
= ( t11 t12 t13 t14 t15 )
( t22 t24 )
The M-by-M factor Q_out is given as a product of NOCB
orthogonal M-by-M matrices Q_out(i).
Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),
where each matrix Q_out(i) is given by the WY-representation
using corresponding blocks from the matrices V and T:
Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,
where I is the identity matrix. Here is the formula with matrix
dimensions:
Q(i){M-by-M} = I{M-by-M} -
VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},
where INB = NB, except for the last block NOCB
for which INB=LAST_NB.
=====
NOTE:
=====
If Q_in is the result of doing a QR factorization
B = Q_in * R_in, then:
B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.
So if one wants to interpret Q_out as the result
of the QR factorization of B, then the corresponding R_out
should be equal to R_out = S * R_in, i.e. some rows of R_in
should be multiplied by -1.
For the details of the algorithm, see [1].
[1] 'Reconstructing Householder vectors from tall-skinny QR',
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
subroutine sorhr_col (integer m, integer n, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( ldt, * ) t, integer ldt, real, dimension( * ) d, integer info)¶
SORHR_COL
Purpose:
SORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns
as input, stored in A, and performs Householder Reconstruction (HR),
i.e. reconstructs Householder vectors V(i) implicitly representing
another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
where S is an N-by-N diagonal matrix with diagonal entries
equal to +1 or -1. The Householder vectors (columns V(i) of V) are
stored in A on output, and the diagonal entries of S are stored in D.
Block reflectors are also returned in T
(same output format as SGEQRT).
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. M >= N >= 0.
NB
NB is INTEGER
The column block size to be used in the reconstruction
of Householder column vector blocks in the array A and
corresponding block reflectors in the array T. NB >= 1.
(Note that if NB > N, then N is used instead of NB
as the column block size.)
A
A is REAL array, dimension (LDA,N)
On entry:
The array A contains an M-by-N orthonormal matrix Q_in,
i.e the columns of A are orthogonal unit vectors.
On exit:
The elements below the diagonal of A represent the unit
lower-trapezoidal matrix V of Householder column vectors
V(i). The unit diagonal entries of V are not stored
(same format as the output below the diagonal in A from
SGEQRT). The matrix T and the matrix V stored on output
in A implicitly define Q_out.
The elements above the diagonal contain the factor U
of the 'modified' LU-decomposition:
Q_in - ( S ) = V * U
( 0 )
where 0 is a (M-N)-by-(M-N) zero matrix.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T
T is REAL array,
dimension (LDT, N)
Let NOCB = Number_of_output_col_blocks
= CEIL(N/NB)
On exit, T(1:NB, 1:N) contains NOCB upper-triangular
block reflectors used to define Q_out stored in compact
form as a sequence of upper-triangular NB-by-NB column
blocks (same format as the output T in SGEQRT).
The matrix T and the matrix V stored on output in A
implicitly define Q_out. NOTE: The lower triangles
below the upper-triangular blocks will be filled with
zeros. See Further Details.
LDT
LDT is INTEGER
The leading dimension of the array T.
LDT >= max(1,min(NB,N)).
D
D is REAL array, dimension min(M,N).
The elements can be only plus or minus one.
D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
1 <= i <= min(M,N), and Q_in_i is Q_in after performing
i-1 steps of “modified” Gaussian elimination.
See Further Details.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Further Details:
The computed M-by-M orthogonal factor Q_out is defined implicitly as
a product of orthogonal matrices Q_out(i). Each Q_out(i) is stored in
the compact WY-representation format in the corresponding blocks of
matrices V (stored in A) and T.
The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
matrix A contains the column vectors V(i) in NB-size column
blocks VB(j). For example, VB(1) contains the columns
V(1), V(2), ... V(NB). NOTE: The unit entries on
the diagonal of Y are not stored in A.
The number of column blocks is
NOCB = Number_of_output_col_blocks = CEIL(N/NB)
where each block is of order NB except for the last block, which
is of order LAST_NB = N - (NOCB-1)*NB.
For example, if M=6, N=5 and NB=2, the matrix V is
V = ( VB(1), VB(2), VB(3) ) =
= ( 1 )
( v21 1 )
( v31 v32 1 )
( v41 v42 v43 1 )
( v51 v52 v53 v54 1 )
( v61 v62 v63 v54 v65 )
For each of the column blocks VB(i), an upper-triangular block
reflector TB(i) is computed. These blocks are stored as
a sequence of upper-triangular column blocks in the NB-by-N
matrix T. The size of each TB(i) block is NB-by-NB, except
for the last block, whose size is LAST_NB-by-LAST_NB.
For example, if M=6, N=5 and NB=2, the matrix T is
T = ( TB(1), TB(2), TB(3) ) =
= ( t11 t12 t13 t14 t15 )
( t22 t24 )
The M-by-M factor Q_out is given as a product of NOCB
orthogonal M-by-M matrices Q_out(i).
Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),
where each matrix Q_out(i) is given by the WY-representation
using corresponding blocks from the matrices V and T:
Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,
where I is the identity matrix. Here is the formula with matrix
dimensions:
Q(i){M-by-M} = I{M-by-M} -
VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},
where INB = NB, except for the last block NOCB
for which INB=LAST_NB.
=====
NOTE:
=====
If Q_in is the result of doing a QR factorization
B = Q_in * R_in, then:
B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.
So if one wants to interpret Q_out as the result
of the QR factorization of B, then the corresponding R_out
should be equal to R_out = S * R_in, i.e. some rows of R_in
should be multiplied by -1.
For the details of the algorithm, see [1].
[1] 'Reconstructing Householder vectors from tall-skinny QR',
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
subroutine zunhr_col (integer m, integer n, integer nb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( * ) d, integer info)¶
ZUNHR_COL
Purpose:
ZUNHR_COL takes an M-by-N complex matrix Q_in with orthonormal columns
as input, stored in A, and performs Householder Reconstruction (HR),
i.e. reconstructs Householder vectors V(i) implicitly representing
another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
where S is an N-by-N diagonal matrix with diagonal entries
equal to +1 or -1. The Householder vectors (columns V(i) of V) are
stored in A on output, and the diagonal entries of S are stored in D.
Block reflectors are also returned in T
(same output format as ZGEQRT).
Parameters
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. M >= N >= 0.
NB
NB is INTEGER
The column block size to be used in the reconstruction
of Householder column vector blocks in the array A and
corresponding block reflectors in the array T. NB >= 1.
(Note that if NB > N, then N is used instead of NB
as the column block size.)
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry:
The array A contains an M-by-N orthonormal matrix Q_in,
i.e the columns of A are orthogonal unit vectors.
On exit:
The elements below the diagonal of A represent the unit
lower-trapezoidal matrix V of Householder column vectors
V(i). The unit diagonal entries of V are not stored
(same format as the output below the diagonal in A from
ZGEQRT). The matrix T and the matrix V stored on output
in A implicitly define Q_out.
The elements above the diagonal contain the factor U
of the 'modified' LU-decomposition:
Q_in - ( S ) = V * U
( 0 )
where 0 is a (M-N)-by-(M-N) zero matrix.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
T
T is COMPLEX*16 array,
dimension (LDT, N)
Let NOCB = Number_of_output_col_blocks
= CEIL(N/NB)
On exit, T(1:NB, 1:N) contains NOCB upper-triangular
block reflectors used to define Q_out stored in compact
form as a sequence of upper-triangular NB-by-NB column
blocks (same format as the output T in ZGEQRT).
The matrix T and the matrix V stored on output in A
implicitly define Q_out. NOTE: The lower triangles
below the upper-triangular blocks will be filled with
zeros. See Further Details.
LDT
LDT is INTEGER
The leading dimension of the array T.
LDT >= max(1,min(NB,N)).
D
D is COMPLEX*16 array, dimension min(M,N).
The elements can be only plus or minus one.
D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
1 <= i <= min(M,N), and Q_in_i is Q_in after performing
i-1 steps of “modified” Gaussian elimination.
See Further Details.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Further Details:
The computed M-by-M unitary factor Q_out is defined implicitly as
a product of unitary matrices Q_out(i). Each Q_out(i) is stored in
the compact WY-representation format in the corresponding blocks of
matrices V (stored in A) and T.
The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
matrix A contains the column vectors V(i) in NB-size column
blocks VB(j). For example, VB(1) contains the columns
V(1), V(2), ... V(NB). NOTE: The unit entries on
the diagonal of Y are not stored in A.
The number of column blocks is
NOCB = Number_of_output_col_blocks = CEIL(N/NB)
where each block is of order NB except for the last block, which
is of order LAST_NB = N - (NOCB-1)*NB.
For example, if M=6, N=5 and NB=2, the matrix V is
V = ( VB(1), VB(2), VB(3) ) =
= ( 1 )
( v21 1 )
( v31 v32 1 )
( v41 v42 v43 1 )
( v51 v52 v53 v54 1 )
( v61 v62 v63 v54 v65 )
For each of the column blocks VB(i), an upper-triangular block
reflector TB(i) is computed. These blocks are stored as
a sequence of upper-triangular column blocks in the NB-by-N
matrix T. The size of each TB(i) block is NB-by-NB, except
for the last block, whose size is LAST_NB-by-LAST_NB.
For example, if M=6, N=5 and NB=2, the matrix T is
T = ( TB(1), TB(2), TB(3) ) =
= ( t11 t12 t13 t14 t15 )
( t22 t24 )
The M-by-M factor Q_out is given as a product of NOCB
unitary M-by-M matrices Q_out(i).
Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),
where each matrix Q_out(i) is given by the WY-representation
using corresponding blocks from the matrices V and T:
Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,
where I is the identity matrix. Here is the formula with matrix
dimensions:
Q(i){M-by-M} = I{M-by-M} -
VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},
where INB = NB, except for the last block NOCB
for which INB=LAST_NB.
=====
NOTE:
=====
If Q_in is the result of doing a QR factorization
B = Q_in * R_in, then:
B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.
So if one wants to interpret Q_out as the result
of the QR factorization of B, then the corresponding R_out
should be equal to R_out = S * R_in, i.e. some rows of R_in
should be multiplied by -1.
For the details of the algorithm, see [1].
[1] 'Reconstructing Householder vectors from tall-skinny QR',
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
Author¶
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