?sytrf_rk

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*(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 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 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):

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.

2. 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.

3. 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):

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.

2. 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.

3. 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 k^th 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.