Visible to Intel only — GUID: GUID-3F601047-5DCE-4FCF-9158-73E238FE7F5E
Visible to Intel only — GUID: GUID-3F601047-5DCE-4FCF-9158-73E238FE7F5E
?hetrf_rk
Computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
call chetrf_rk(uplo, n, A, lda, e, ipiv, work, lwork, info)
call zhetrf_rk(uplo, n, A, lda, e, ipiv, work, lwork, info)
Description
?hetrf_rk computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: A = P*U*D*(UH)*(PT) or A = P*L*D*(LH)*(PT), where U (or L) is unit upper (or lower) triangular matrix, UH (or LH) is the conjugate of U (or L), P is a permutation matrix, PT 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 blocked version of the algorithm, calling Level 3 BLAS.
Input Parameters
- uplo
-
CHARACTER*1
Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
- = 'U': Upper triangular.
- = 'L': Lower triangular.
- n
-
INTEGER
The order of the matrix A. n ≥ 0.
- A
-
COMPLEX for chetrf_rk
COMPLEX*16 for zhetrf_rk
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.
- lda
-
INTEGER
The leading dimension of the array A.lda ≥ max(1, n).
- lwork
-
INTEGER
The length of the array work.
If lwork = -1, a workspace query is assumed; the routine calculates only the optimal size of the work array and returns this value as the first entry of the work array, and no error message related to lwork is issued by XERBLA.
Output Parameters
- A
-
COMPLEX for chetrf_rk
COMPLEX*16 for zhetrf_rk
On exit, contains:
Only diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A; that is, D(k,k) = A(k,k). Superdiagonal (or subdiagonal) elements of D are stored on exit in array e.
—and—
If uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L', factor L in the subdiagonal part of A.
- e
-
COMPLEX for chetrf_rk
COMPLEX*16 for zhetrf_rk
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. If uplo = 'U', e(i) = D(i-1,i), i=2:N, and e(1) is set to 0. If uplo = 'L', e(i) = D(i+1,i), i=1:N-1, and 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 the uplo = 'U' and uplo = 'L' cases. - ipiv
-
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 kth 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 that correspond to 1 or 2 interchanges at each factorization step. If uplo = 'U' (in factorization order, k decreases from n to 1):
A single positive entry ipiv(k) > 0 means that 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.
A pair of consecutive negative entries ipiv(k) < 0 and ipiv(k-1) < 0 means that D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (Note that negative entries in ipiv appear only in pairs.)
- 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.
- 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.
In both cases 1 and 2, always ABS( ipiv(k) ) ≤ k.
NOTE:Any entry ipiv(k) is always nonzero on output.If uplo = 'L' (in factorization order, k increases from 1 to n):
A single positive entry ipiv(k) > 0 means that 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.
A pair of consecutive negative entries ipiv(k) < 0 and ipiv(k+1) < 0 means that D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (Note that negative entries in ipiv appear only in pairs.)
- 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.
- 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.
In both cases 1 and 2, always ABS( ipiv(k) ) ≥ k.
NOTE:Any entry ipiv(k) is always nonzero on output. - work
-
COMPLEX for chetrf_rk
COMPLEX*16 for zhetrf_rk
Array, dimension ( MAX(1,lwork) ). On exit, if info = 0, work(1) returns the optimal lwork.
- info
-
INTEGER
- = 0: Successful exit.
- < 0: If info = -k, the kth argument had an illegal value.
- > 0: If info = k, the matrix A is singular. If uplo = 'U', the column k in the upper triangular part of A contains all zeros. If uplo = 'L', the 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 stores only the first occurrence of a singularity; any subsequent occurrence of a singularity is not stored in info even though the factorization always completes.