table of contents
| tplqt2(3) | LAPACK | tplqt2(3) |
NAME¶
tplqt2 - tplqt2: QR factor, level 2
SYNOPSIS¶
Functions¶
subroutine ctplqt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
CTPLQT2 subroutine dtplqt2 (m, n, l, a, lda, b, ldb, t, ldt,
info)
DTPLQT2 computes a LQ factorization of a real or complex
'triangular-pentagonal' matrix, which is composed of a triangular block and
a pentagonal block, using the compact WY representation for Q. subroutine
stplqt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
STPLQT2 computes a LQ factorization of a real or complex
'triangular-pentagonal' matrix, which is composed of a triangular block and
a pentagonal block, using the compact WY representation for Q. subroutine
ztplqt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
ZTPLQT2 computes a LQ factorization of a real or complex
'triangular-pentagonal' matrix, which is composed of a triangular block and
a pentagonal block, using the compact WY representation for Q.
Detailed Description¶
Function Documentation¶
subroutine ctplqt2 (integer m, integer n, integer l, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldt, * ) t, integer ldt, integer info)¶
CTPLQT2
Purpose:
CTPLQT2 computes a LQ a factorization of a complex 'triangular-pentagonal'
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.
Parameters
M
M is INTEGER
The total number of rows of the matrix B.
M >= 0.
N
N is INTEGER
The number of columns of the matrix B, and the order of
the triangular matrix A.
N >= 0.
L
L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.
A
A is COMPLEX array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is COMPLEX array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).
T
T is COMPLEX array, dimension (LDT,M)
The N-by-N upper triangular factor T of the block reflector.
See Further Details.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,M)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The input matrix C is a M-by-(M+N) matrix
C = [ A ][ B ]
where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
upper trapezoidal matrix B2:
B = [ B1 ][ B2 ]
[ B1 ] <- M-by-(N-L) rectangular
[ B2 ] <- M-by-L lower trapezoidal.
The lower trapezoidal matrix B2 consists of the first L columns of a
N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.
The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C
C = [ A ][ B ]
[ A ] <- lower triangular M-by-M
[ B ] <- M-by-N pentagonal
so that W can be represented as
W = [ I ][ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.
Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,
W = [ V1 ][ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <- M-by-L lower trapezoidal.
The rows of V represent the vectors which define the H(i)'s.
The (M+N)-by-(M+N) block reflector H is then given by
H = I - W**T * T * W
where W^H is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.
subroutine dtplqt2 (integer m, integer n, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldt, * ) t, integer ldt, integer info)¶
DTPLQT2 computes a LQ factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
Purpose:
DTPLQT2 computes a LQ a factorization of a real 'triangular-pentagonal'
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.
Parameters
M
M is INTEGER
The total number of rows of the matrix B.
M >= 0.
N
N is INTEGER
The number of columns of the matrix B, and the order of
the triangular matrix A.
N >= 0.
L
L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.
A
A is DOUBLE PRECISION array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).
T
T is DOUBLE PRECISION array, dimension (LDT,M)
The N-by-N upper triangular factor T of the block reflector.
See Further Details.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,M)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The input matrix C is a M-by-(M+N) matrix
C = [ A ][ B ]
where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
upper trapezoidal matrix B2:
B = [ B1 ][ B2 ]
[ B1 ] <- M-by-(N-L) rectangular
[ B2 ] <- M-by-L lower trapezoidal.
The lower trapezoidal matrix B2 consists of the first L columns of a
N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.
The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C
C = [ A ][ B ]
[ A ] <- lower triangular M-by-M
[ B ] <- M-by-N pentagonal
so that W can be represented as
W = [ I ][ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.
Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,
W = [ V1 ][ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <- M-by-L lower trapezoidal.
The rows of V represent the vectors which define the H(i)'s.
The (M+N)-by-(M+N) block reflector H is then given by
H = I - W**T * T * W
where W^H is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.
subroutine stplqt2 (integer m, integer n, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldt, * ) t, integer ldt, integer info)¶
STPLQT2 computes a LQ factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
Purpose:
STPLQT2 computes a LQ a factorization of a real 'triangular-pentagonal'
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.
Parameters
M
M is INTEGER
The total number of rows of the matrix B.
M >= 0.
N
N is INTEGER
The number of columns of the matrix B, and the order of
the triangular matrix A.
N >= 0.
L
L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.
A
A is REAL array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is REAL array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).
T
T is REAL array, dimension (LDT,M)
The N-by-N upper triangular factor T of the block reflector.
See Further Details.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,M)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The input matrix C is a M-by-(M+N) matrix
C = [ A ][ B ]
where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
upper trapezoidal matrix B2:
B = [ B1 ][ B2 ]
[ B1 ] <- M-by-(N-L) rectangular
[ B2 ] <- M-by-L lower trapezoidal.
The lower trapezoidal matrix B2 consists of the first L columns of a
N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.
The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C
C = [ A ][ B ]
[ A ] <- lower triangular M-by-M
[ B ] <- M-by-N pentagonal
so that W can be represented as
W = [ I ][ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.
Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,
W = [ V1 ][ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <- M-by-L lower trapezoidal.
The rows of V represent the vectors which define the H(i)'s.
The (M+N)-by-(M+N) block reflector H is then given by
H = I - W**T * T * W
where W^H is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.
subroutine ztplqt2 (integer m, integer n, integer l, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldt, * ) t, integer ldt, integer info)¶
ZTPLQT2 computes a LQ factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
Purpose:
ZTPLQT2 computes a LQ a factorization of a complex 'triangular-pentagonal'
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.
Parameters
M
M is INTEGER
The total number of rows of the matrix B.
M >= 0.
N
N is INTEGER
The number of columns of the matrix B, and the order of
the triangular matrix A.
N >= 0.
L
L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0. See Further Details.
A
A is COMPLEX*16 array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is COMPLEX*16 array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B. The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V. See Further Details.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).
T
T is COMPLEX*16 array, dimension (LDT,M)
The N-by-N upper triangular factor T of the block reflector.
See Further Details.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,M)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The input matrix C is a M-by-(M+N) matrix
C = [ A ][ B ]
where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
upper trapezoidal matrix B2:
B = [ B1 ][ B2 ]
[ B1 ] <- M-by-(N-L) rectangular
[ B2 ] <- M-by-L lower trapezoidal.
The lower trapezoidal matrix B2 consists of the first L columns of a
N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.
The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C
C = [ A ][ B ]
[ A ] <- lower triangular M-by-M
[ B ] <- M-by-N pentagonal
so that W can be represented as
W = [ I ][ V ]
[ I ] <- identity, M-by-M
[ V ] <- M-by-N, same form as B.
Thus, all of information needed for W is contained on exit in B, which
we call V above. Note that V has the same form as B; that is,
W = [ V1 ][ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <- M-by-L lower trapezoidal.
The rows of V represent the vectors which define the H(i)'s.
The (M+N)-by-(M+N) block reflector H is then given by
H = I - W**T * T * W
where W^H is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.
Author¶
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| Wed Feb 7 2024 11:30:40 | Version 3.12.0 |