?hesv_rk

Description

?hesv_rk computes the solution to a complex system of linear equations A * X = B, where A is an n-by-n Hermitian 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^H)*(P^T), if uplo = 'U', or A = P*L*D*(L^H)*(P^T), if uplo = 'L', where U (or L) is unit upper (or lower) triangular matrix, U^H (or L^H) is the conjugate of U (or L), P is a permutation matrix, P^T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

?hetrf_rk is called to compute the factorization of a complex Hermitian matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine ?hetrs_3.

Input Parameters

uplo

CHARACTER*1

Specifies whether the upper or lower triangular part of the Hermitian 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

COMPLEX for chesv_rk

COMPLEX*16 for zhesv_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).

B

COMPLEX for chesv_rk

COMPLEX*16 for zhesv_rk

On entry, the n-by-nrhs right-hand side matrix B.

The second dimension of B must be at least max(1, nrhs).

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 the 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

COMPLEX for chesv_rk

COMPLEX*16 for zhesv_rk

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

• 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.

For more information, see the description of the ?hetrf_rk routine.

e

COMPLEX for chesv_rk

COMPLEX*16 for zhesv_rk

Array, dimension (n). On exit, contains the output computed by the factorization routine ?hetrf_rk; that is, 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, 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 a 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 ?hetrf_rk routine.

ipiv

INTEGER

Array, dimension (n). Details of the interchanges and the block structure of D, as determined by ?hetrf_rk.

B

COMPLEX for chesv_rk

COMPLEX*16 for zhesv_rk

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

work

COMPLEX for chesv_rk

COMPLEX*16 for zhesv_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.