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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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| subroutine clasyf_rk | ( | character | uplo, |
| integer | n, | ||
| integer | nb, | ||
| integer | kb, | ||
| complex, dimension( lda, * ) | a, | ||
| integer | lda, | ||
| complex, dimension( * ) | e, | ||
| integer, dimension( * ) | ipiv, | ||
| complex, dimension( ldw, * ) | w, | ||
| integer | ldw, | ||
| integer | info ) |
CLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
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!> CLASYF_RK computes a partial factorization of a complex symmetric !> matrix A using the bounded Bunch-Kaufman (rook) diagonal !> pivoting method. The partial factorization has the form: !> !> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: !> ( 0 U22 ) ( 0 D ) ( U12**T U22**T ) !> !> A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L', !> ( L21 I ) ( 0 A22 ) ( 0 I ) !> !> where the order of D is at most NB. The actual order is returned in !> the argument KB, and is either NB or NB-1, or N if N <= NB. !> !> CLASYF_RK is an auxiliary routine called by CSYTRF_RK. It uses !> blocked code (calling Level 3 BLAS) to update the submatrix !> A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). !>
| [in] | 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 !> |
| [in] | N | !> N is INTEGER !> The order of the matrix A. N >= 0. !> |
| [in] | NB | !> NB is INTEGER !> The maximum number of columns of the matrix A that should be !> factored. NB should be at least 2 to allow for 2-by-2 pivot !> blocks. !> |
| [out] | KB | !> KB is INTEGER !> The number of columns of A that were actually factored. !> KB is either NB-1 or NB, or N if N <= NB. !> |
| [in,out] | 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. !> |
| [in] | LDA | !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !> |
| [out] | 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. !> |
| [out] | 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. !> !> 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 submatrix A(1:N,N-KB+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,N-KB+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 submatrix A(1:N,N-KB+1:N). !> If -IPIV(k-1) = k-1, no interchange occurred. !> !> c) In both cases a) and b) is 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 submatrix A(1:N,1:KB). !> 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 submatrix A(1:N,1:KB). !> 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 submatrix A(1:N,1:KB). !> If -IPIV(k+1) = k+1, no interchange occurred. !> !> c) In both cases a) and b) is always ABS( IPIV(k) ) >= k. !> !> d) NOTE: Any entry IPIV(k) is always NONZERO on output. !> |
| [out] | W | !> W is COMPLEX array, dimension (LDW,NB) !> |
| [in] | LDW | !> LDW is INTEGER !> The leading dimension of the array W. LDW >= max(1,N). !> |
| [out] | 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. !> |
!> !> 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 !> !>
Definition at line 258 of file clasyf_rk.f.
1.11.0