Developer Reference for Intel® oneAPI Math Kernel Library for C
?hesv_rk
?hesv_rk computes the solution to a system of linear equations A * X = B for Hermitian matrices.
Syntax
lapack_intLAPACKE_chesv_rk ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , lapack_complex_float*A , lapack_intlda , lapack_complex_float*e , lapack_int*ipiv , lapack_complex_float*B , lapack_intldb );
lapack_intLAPACKE_zhesv_rk ( intmatrix_layout , charuplo , lapack_intn , lapack_intnrhs , lapack_complex_double*A , lapack_intlda , lapack_complex_double*e , lapack_int*ipiv , lapack_complex_double*B , lapack_intldb );
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
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
- uplo
-
CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix 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 of size max(1, 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 .
- 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 ). The size of B is max(1, ldb * nrhs ) for column-major layout and max(1, ldb * n ) for row-major layout.
- ldb
-
INTEGER
The leading dimension of the array B . ldb ≥ max(1, n ) for column-major layout and ldb ≥ nrhs for row-major layout.
- 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 : For more information, see the description of the ?hetrf_rk routine.
- e
-
COMPLEX for chesv_rk COMPLEX*16 for zhesv_rk
Array of size 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:
For more information, see the description of the ?hetrf_rk routine.
- ipiv
-
INTEGER
Array of size 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 .
Return Values
This function returns a value info .
If info = 0, the execution is successful.
If info = -k , the kth argument had an illegal value.
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.