Visible to Intel only — GUID: GUID-C4AD3948-66B9-4EEB-B36B-30C1BF66B665
Visible to Intel only — GUID: GUID-C4AD3948-66B9-4EEB-B36B-30C1BF66B665
?sytrf_rk
Computes the factorization of a real or complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
call ssytrf_rk(uplo, n, A, lda, e, ipiv, work, lwork, info)
call dsytrf_rk(uplo, n, A, lda, e, ipiv, work, lwork, info)
call csytrf_rk(uplo, n, A, lda, e, ipiv, work, lwork, info)
call zsytrf_rk(uplo, n, A, lda, e, ipiv, work, lwork, info)
Description
?sytrf_rk computes the factorization of a real or complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: A= P*U*D*(UT)*(PT) or A = P*L*D*(LT)*(PT), where U (or L) is unit upper (or lower) triangular matrix, UT (or LT) is the transpose of U (or L), P is a permutation matrix, PT 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 blocked version of the algorithm, calling Level-3 BLAS.
Input Parameters
- uplo
-
CHARACTER*1
Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- = 'U': Upper triangular
- = 'L': Lower triangular
- n
-
INTEGER
The order of the matrix A. n ≥ 0.
- A
-
REAL for ssytrf_rk
DOUBLE PRECISION for dsytrf_rk
COMPLEX for csytrf_rk
COMPLEX*16 for zsytrf_rk
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.
- lda
-
INTEGER
The leading dimension of the array A.lda ≥ max(1, n).
- lwork
-
INTEGER
The length of the array work.
If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, 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
-
REAL for ssytrf_rk
DOUBLE PRECISION for dsytrf_rk
COMPLEX for csytrf_rk
COMPLEX*16 for zsytrf_rk
On exit, contains:
- Only diagonal elements of the symmetric 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).
- If uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L', factor L in the subdiagonal part of A.
- e
-
REAL for ssytrf_rk
DOUBLE PRECISION for dsytrf_rk
COMPLEX for csytrf_rk
COMPLEX*16 for zsytrf_rk
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. 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 the 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 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 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
-
REAL for ssytrf_rk
DOUBLE PRECISION for dsytrf_rk
COMPLEX for csytrf_rk
COMPLEX*16 for zsytrf_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', 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 stores only the first occurrence of a singularity; any subsequent occurrence of singularity is not stored in info even though the factorization always completes.