Developer Reference for Intel® oneAPI Math Kernel Library for C
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?hpsvx
Uses the diagonal pivoting factorization to compute the solution to the system of linear equations with a Hermitian coefficient matrix A stored in packed format, and provides error bounds on the solution.
lapack_int LAPACKE_chpsvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_float* ap, lapack_complex_float* afp, lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );
lapack_int LAPACKE_zhpsvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_double* ap, lapack_complex_double* afp, lapack_int* ipiv, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );
- mkl.h
The routine uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations A*X = B, where A is a n-by-n Hermitian matrix stored in packed format, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.
Error bounds on the solution and a condition estimate are also provided.
The routine ?hpsvx performs the following steps:
If fact = 'N', the diagonal pivoting method is used to factor the matrix A. The form of the factorization is A = U*D*UH or A = L*D*LH, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is a Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
If some di,i = 0, so that D is exactly singular, then the routine returns with info = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, info = n+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
matrix_layout |
Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR). |
fact |
Must be 'F' or 'N'. Specifies whether or not the factored form of the matrix A has been supplied on entry. If fact = 'F': on entry, afp and ipiv contain the factored form of A. Arrays ap, afp, and ipiv are not modified. If fact = 'N', the matrix A is copied to afp and factored. |
uplo |
Must be 'U' or 'L'. Indicates whether the upper or lower triangular part of A is stored and how A is factored: If uplo = 'U', the array ap stores the upper triangular part of the Hermitian matrix A, and A is factored as U*D*UH. If uplo = 'L', the array ap stores the lower triangular part of the Hermitian matrix A, and A is factored as L*D*LH. |
n |
The order of matrix A; n≥ 0. |
nrhs |
The number of right-hand sides, the number of columns in B; nrhs≥ 0. |
ap, afp, b |
Arrays: ap (size max(1,n*(n+1)/2), afp (size max(1,n*(n+1)/2), bof size max(1, ldb*nrhs) for column major layout and max(1, ldb*n) for row major layout. The array ap contains the upper or lower triangle of the Hermitian matrix A in packed storage (see Matrix Storage Schemes). The array afp is an input argument if fact = 'F'. It contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*UH or A = L*D*LH as computed by ?hptrf, in the same storage format as A. The array b contains the matrix B whose columns are the right-hand sides for the systems of equations. |
ldb |
The leading dimension of b; ldb≥ max(1, n) for column major layout and ldb≥nrhs for row major layout. |
ipiv |
Array, size at least max(1, n). The array ipiv is an input argument if fact = 'F'. It contains details of the interchanges and the block structure of D, as determined by ?hptrf. If ipiv[i-1] = k > 0, then dii is a 1-by-1 block, and the i-th row and column of A was interchanged with the k-th row and column. If uplo = 'U'and ipiv[i]=ipiv[i-1] = -m < 0, then D has a 2-by-2 block in rows/columns i and i+1, and i-th row and column of A was interchanged with the m-th row and column. If uplo = 'L'and ipiv[i-1] =ipiv[i] = -m < 0, then D has a 2-by-2 block in rows/columns i and i+1, and (i+1)-th row and column of A was interchanged with the m-th row and column. |
ldx |
The leading dimension of the output array x; ldx≥ max(1, n) for column major layout and ldx≥nrhs for row major layout. |
x |
Array, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n) for row major layout. If info = 0 or info = n+1, the array x contains the solution matrix X to the system of equations. |
afp, ipiv |
These arrays are output arguments if fact = 'N'. See the description of afp, ipiv in Input Arguments section. |
rcond |
An estimate of the reciprocal condition number of the matrix A. If rcond is less than the machine precision (in particular, if rcond = 0), the matrix is singular to working precision. This condition is indicated by a return code of info > 0. |
ferr |
Array, size at least max(1, nrhs). Contains the estimated forward error bound for each solution vector xj (the j-th column of the solution matrix X). If xtrue is the true solution corresponding to xj, ferr[j-1] is an estimated upper bound for the magnitude of the largest element in (xj - xtrue) divided by the magnitude of the largest element in xj. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error. |
berr |
Array, size at least max(1, nrhs). Contains the component-wise relative backward error for each solution vector xj, that is, the smallest relative change in any element of A or B that makes xj an exact solution. |
This function returns a value info.
If info = 0, the execution is successful.
If info = -i, parameter i had an illegal value.
If info = i, and i≤n, then dii is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned.
If info = i, and i = n + 1, then D is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.