?hesvx

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?hesvx

Uses the diagonal pivoting factorization to compute the solution to the complex system of linear equations with a Hermitian coefficient matrix A, and provides error bounds on the solution.

Syntax

lapack_int LAPACKE_chesvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_float* a, lapack_int lda, lapack_complex_float* af, lapack_int ldaf, 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_zhesvx( int matrix_layout, char fact, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_double* a, lapack_int lda, lapack_complex_double* af, lapack_int ldaf, lapack_int* ipiv, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );

Include Files

mkl.h

Description

The routine uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations A*X = B, where A is an n-by-n Hermitian matrix, 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 ?hesvx 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*U^H or A = L*D*L^H, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
If some d_i,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.

Input Parameters

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, `af` and `ipiv` contain the factored form of `A`. Arrays `a`, `af`, and `ipiv` are not modified. If `fact = 'N'`, the matrix `A` is copied to `af` 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 `a` stores the upper triangular part of the Hermitian matrix `A`, and `A` is factored as `UDU^H`. If `uplo = 'L'`, the array `a` stores the lower triangular part of the Hermitian matrix `A`; `A` is factored as `LDL^H`.
n	The order of matrix `A`; `n`≥ 0.
nrhs	The number of right-hand sides, the number of columns in `B; nrhs≥ 0`.
a, af, b	Arrays: `a`(size max(1, ldan)), `af`(size max(1, ldafn)), `b`of size max(1, ldbnrhs) for column major layout and max(1, ldbn) for row major layout. The array `a` contains the upper or the lower triangular part of the Hermitian matrix `A` (see `uplo`). The array `af` is an input argument if `fact = 'F'`. It contains he block diagonal matrix `D` and the multipliers used to obtain the factor `U` or `L` from the factorization `A = UDU^H` or `A = LDL^H` as computed by ?hetrf. The array `b` contains the matrix `B` whose columns are the right-hand sides for the systems of equations.
lda	The leading dimension of `a`; `lda≥ max(1, n)`.
ldaf	The leading dimension of `af`; `ldaf≥ max(1, n)`.
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 ?hetrf. If `ipiv[i-1] = k > 0`, then `d_ii` is a 1-by-1 diagonal 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`] =`ipiv`[`i`-1] = -`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.

Output Parameters

x	Array, size max(1, ldxnrhs) for column major layout and max(1, ldxn) for row major layout. If `info = 0` or `info = n+1`, the array `x` contains the solution matrix `X` to the system of equations.
af, ipiv	These arrays are output arguments if `fact = 'N'`. See the description of `af`, `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 `x`_j (the `j`-th column of the solution matrix `X`). If `xtrue` is the true solution corresponding to `x`_j, `ferr[j-1]` is an estimated upper bound for the magnitude of the largest element in (`x`_j) - `xtrue`) divided by the magnitude of the largest element in `x`_j. The estimate is as reliable as the estimate for rcon, 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 `x`_j, that is, the smallest relative change in any element of `A` or `B` that makes `x`_j an exact solution.

Return Values

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 d_ii 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.

Parent topic: LAPACK Linear Equation Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?hesvx

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