table of contents
| hetf2_rk(3) | LAPACK | hetf2_rk(3) |
NAME¶
hetf2_rk - {he,sy}tf2_rk: triangular factor, level 2
SYNOPSIS¶
Functions¶
subroutine chetf2_rk (uplo, n, a, lda, e, ipiv, info)
CHETF2_RK computes the factorization of a complex Hermitian indefinite
matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method
(BLAS2 unblocked algorithm). subroutine csytf2_rk (uplo, n, a, lda,
e, ipiv, info)
CSYTF2_RK computes the factorization of a complex symmetric indefinite
matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method
(BLAS2 unblocked algorithm). subroutine dsytf2_rk (uplo, n, a, lda,
e, ipiv, info)
DSYTF2_RK computes the factorization of a real symmetric indefinite
matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method
(BLAS2 unblocked algorithm). subroutine ssytf2_rk (uplo, n, a, lda,
e, ipiv, info)
SSYTF2_RK computes the factorization of a real symmetric indefinite
matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method
(BLAS2 unblocked algorithm). subroutine zhetf2_rk (uplo, n, a, lda,
e, ipiv, info)
ZHETF2_RK computes the factorization of a complex Hermitian indefinite
matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method
(BLAS2 unblocked algorithm). subroutine zsytf2_rk (uplo, n, a, lda,
e, ipiv, info)
ZSYTF2_RK computes the factorization of a complex symmetric indefinite
matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method
(BLAS2 unblocked algorithm).
Detailed Description¶
Function Documentation¶
subroutine chetf2_rk (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, integer info)¶
CHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
Purpose:
CHETF2_RK computes the factorization of a complex Hermitian matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
For more information see Further Details section.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step. For more info see Further
Details section.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
INFO
INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
TODO: put further details
Contributors:
December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept.,
Univ. of Tenn., Knoxville abd , USA
subroutine csytf2_rk (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, integer info)¶
CSYTF2_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
Purpose:
CSYTF2_RK computes the factorization of a complex symmetric matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
For more information see Further Details section.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step. For more info see Further
Details section.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
INFO
INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
TODO: put further details
Contributors:
December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept.,
Univ. of Tenn., Knoxville abd , USA
subroutine dsytf2_rk (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, integer, dimension( * ) ipiv, integer info)¶
DSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
Purpose:
DSYTF2_RK computes the factorization of a real symmetric matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
For more information see Further Details section.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is DOUBLE PRECISION array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step. For more info see Further
Details section.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
INFO
INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
TODO: put further details
Contributors:
December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept.,
Univ. of Tenn., Knoxville abd , USA
subroutine ssytf2_rk (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer, dimension( * ) ipiv, integer info)¶
SSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
Purpose:
SSYTF2_RK computes the factorization of a real symmetric matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
For more information see Further Details section.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is REAL array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is REAL array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step. For more info see Further
Details section.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
INFO
INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
TODO: put further details
Contributors:
December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept.,
Univ. of Tenn., Knoxville abd , USA
subroutine zhetf2_rk (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, integer info)¶
ZHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
Purpose:
ZHETF2_RK computes the factorization of a complex Hermitian matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
For more information see Further Details section.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the Hermitian matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX*16 array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step. For more info see Further
Details section.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
INFO
INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
TODO: put further details
Contributors:
December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept.,
Univ. of Tenn., Knoxville abd , USA
subroutine zsytf2_rk (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, integer info)¶
ZSYTF2_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
Purpose:
ZSYTF2_RK computes the factorization of a complex symmetric matrix A
using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.
This is the unblocked version of the algorithm, calling Level 2 BLAS.
For more information see Further Details section.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the symmetric matrix A.
If UPLO = 'U': the leading N-by-N upper triangular part
of A contains the upper triangular part of the matrix A,
and the strictly lower triangular part of A is not
referenced.
If UPLO = 'L': the leading N-by-N lower triangular part
of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not
referenced.
On exit, contains:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX*16 array, dimension (N)
On exit, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is set to 0 in both
UPLO = 'U' or UPLO = 'L' cases.
IPIV
IPIV is INTEGER array, dimension (N)
IPIV describes the permutation matrix P in the factorization
of matrix A as follows. The absolute value of IPIV(k)
represents the index of row and column that were
interchanged with the k-th row and column. The value of UPLO
describes the order in which the interchanges were applied.
Also, the sign of IPIV represents the block structure of
the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
diagonal blocks which correspond to 1 or 2 interchanges
at each factorization step. For more info see Further
Details section.
If UPLO = 'U',
( in factorization order, k decreases from N to 1 ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N);
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k-1) < 0 means:
D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k-1) != k-1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L',
( in factorization order, k increases from 1 to N ):
a) A single positive entry IPIV(k) > 0 means:
D(k,k) is a 1-by-1 diagonal block.
If IPIV(k) != k, rows and columns k and IPIV(k) were
interchanged in the matrix A(1:N,1:N).
If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries
IPIV(k) < 0 and IPIV(k+1) < 0 means:
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
(NOTE: negative entries in IPIV appear ONLY in pairs).
1) If -IPIV(k) != k, rows and columns
k and -IPIV(k) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k) = k, no interchange occurred.
2) If -IPIV(k+1) != k+1, rows and columns
k-1 and -IPIV(k-1) were interchanged
in the matrix A(1:N,1:N).
If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.
INFO
INFO is INTEGER
= 0: successful exit
< 0: If INFO = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
If UPLO = 'U': column k in the upper
triangular part of A contains all zeros.
If UPLO = 'L': column k in the lower
triangular part of A contains all zeros.
Therefore D(k,k) is exactly zero, and superdiagonal
elements of column k of U (or subdiagonal elements of
column k of L ) are all zeros. The factorization has
been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if
it is used to solve a system of equations.
NOTE: INFO only stores the first occurrence of
a singularity, any subsequent occurrence of singularity
is not stored in INFO even though the factorization
always completes.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
TODO: put further details
Contributors:
December 2016, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester
01-01-96 - Based on modifications by
J. Lewis, Boeing Computer Services Company
A. Petitet, Computer Science Dept.,
Univ. of Tenn., Knoxville abd , USA
Author¶
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