?sysv_rk

Description

?sysv_rk computes the solution to a real or complex system of linear equations A * X = B, where A is an n-by-n symmetric matrix and X and B are n-by-nrhs matrices.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as A= P*U*D*(U^T)*(P^T), if uplo = 'U', or A= P*L*D*(L^T)*(P^T), if uplo = 'L', 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.

?sytrf_rk is called to compute the factorization of a real or complex symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine ?sytrs_3.

Input Parameters

uplo

CHARACTER*1

Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:

• = 'U': The upper triangle of A is stored.
• = 'L': The lower triangle of A is stored.

n

INTEGER

The number of linear equations; that is, the order of the matrix A. n ≥ 0.

nrhs

INTEGER

The number of right-hand sides; that is, the number of columns of the matrix B. nrhs ≥ 0.

A

REAL for ssysv_rk

DOUBLE PRECISION for dsysv_rk

COMPLEX for csysv_rk

COMPLEX*16 for zsysv_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).

B

REAL for ssysv_rk

DOUBLE PRECISION for dsysv_rk

COMPLEX for csysv_rk

COMPLEX*16 for zsysv_rk

Array, dimension (ldb,nrhs). On entry, the n-by-nrhs right-hand side matrix B.

ldb

INTEGER

The leading dimension of the array B. ldb ≥ 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 for factorization stage 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

REAL for ssysv_rk

DOUBLE PRECISION for dsysv_rk

COMPLEX for csysv_rk

COMPLEX*16 for zsysv_rk

On exit, if info = 0, the diagonal of the block diagonal matrix D and factors U or L as computed by ?sytrf_rk:

• 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. For more information, see the description of the ?sytrf_rk routine.

e

REAL for ssysv_rk

DOUBLE PRECISION for dsysv_rk

COMPLEX for csysv_rk

COMPLEX*16 for zsysv_rk

Array, dimension (n). On exit, contains the output computed by the factorization routine ?sytrf_rk; that is, 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=1:N-1, 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. For more information, see the description of the?sytrf_rk routine.

ipiv

INTEGER

Array, dimension (n). Details of the interchanges and the block structure of D, as determined by ?sytrf_rk. For more information, see the description of the ?sytrf_rk routine.

B

REAL for ssysv_rk

DOUBLE PRECISION for dsysv_rk

COMPLEX for csysv_rk

COMPLEX*16 for zsysv_rk

On exit, if info = 0, the n-by-nrhs solution matrix X.

work

REAL for ssysv_rk

DOUBLE PRECISION for dsysv_rk

COMPLEX for csysv_rk

COMPLEX*16 for zsysv_rk

Array, dimension ( MAX(1,lwork) ). Work array used in the factorization stage. 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.