?herfsx

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 11/07/2023

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?herfsx

Uses extra precise iterative refinement to improve the solution to the system of linear equations with a symmetric indefinite coefficient matrix A and provides error bounds and backward error estimates.

Syntax

lapack_int LAPACKE_cherfsx( int matrix_layout, char uplo, char equed, lapack_int n, lapack_int nrhs, const lapack_complex_float* a, lapack_int lda, const lapack_complex_float* af, lapack_int ldaf, const lapack_int* ipiv, const float* s, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* rcond, float* berr, lapack_int n_err_bnds, float* err_bnds_norm, float* err_bnds_comp, lapack_int nparams, float* params );

lapack_int LAPACKE_zherfsx( int matrix_layout, char uplo, char equed, lapack_int n, lapack_int nrhs, const lapack_complex_double* a, lapack_int lda, const lapack_complex_double* af, lapack_int ldaf, const lapack_int* ipiv, const double* s, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* rcond, double* berr, lapack_int n_err_bnds, double* err_bnds_norm, double* err_bnds_comp, lapack_int nparams, double* params );

Include Files

mkl.h

Description

The routine improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite, and provides error bounds and backward error estimates for the solution. In addition to a normwise error bound, the code provides a maximum componentwise error bound, if possible. See comments for err_bnds_norm and err_bnds_comp for details of the error bounds.

The original system of linear equations may have been equilibrated before calling this routine, as described by the parameters equed and s below. In this case, the solution and error bounds returned are for the original unequilibrated system.

Input Parameters

matrix_layout

Specifies whether two-dimensional array storage is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

uplo

Must be 'U' or 'L'.

Indicates whether the upper or lower triangular part of A is stored:

If uplo = 'U', the upper triangle of A is stored.

If uplo = 'L', the lower triangle of A is stored.

equed

Must be 'N' or 'Y'.

Specifies the form of equilibration that was done to A before calling this routine.

If equed = 'N', no equilibration was done.

If equed = 'Y', both row and column equilibration was done, that is, A has been replaced by diag(s)*A*diag(s). The right-hand side B has been changed accordingly.

n

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

nrhs

The number of right-hand sides; the number of columns of the matrices B and X; nrhs≥ 0.

a, af, b

The array a of size max(1, lda*n) contains the Hermitian matrix A as specified by uplo. 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.

The array af of size max(1, ldaf*n) contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U^T or A = L*D*L^T as computed by ssytrf for cherfsx or dsytrf for zherfsx.

The array b of size max(1, ldb*nrhs) for row major layout and max(1, ldb*n) for column major layout 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).

ipiv

Array, size at least max(1, n). Contains details of the interchanges and the block structure of D as determined by ssytrf for real flavors or dsytrf for complex flavors.

s

Array, size (n). The array s contains the scale factors for A.

If equed = 'N', s is not accessed.

If equed = 'Y', each element of s must be positive.

Each element of s should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.

ldb

The leading dimension of the array b; ldb≥ max(1, n) for column major layout and ldb≥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.

The solution matrix X as computed by ?hetrs

ldx

The leading dimension of the output array x; ldx≥ max(1, n) for column major layout and ldx≥nrhs for row major layout.

n_err_bnds

Number of error bounds to return for each right hand side and each type (normwise or componentwise). See err_bnds_norm and err_bnds_comp descriptions in Output Arguments section below.

nparams

Specifies the number of parameters set in params. If ≤ 0, the params array is never referenced and default values are used.

params

Array, size nparams. Specifies algorithm parameters. If an entry is less than 0.0, that entry will be filled with the default value used for that parameter. Only positions up to nparams are accessed; defaults are used for higher-numbered parameters. If defaults are acceptable, you can pass nparams = 0, which prevents the source code from accessing the params argument.

params[0] : Whether to perform iterative refinement or not. Default: 1.0 (for cherfsx), 1.0D+0 (for zherfsx).

=0.0	No refinement is performed and no error bounds are computed.
=1.0	Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support double precision.

(Other values are reserved for future use.)

params[1] : Maximum number of residual computations allowed for refinement.

Default	10
Aggressive	Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy.

params[2] : Flag determining if the code will attempt to find a solution with a small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt componentwise convergence).

Output Parameters

x

The improved solution matrix X.

rcond

Reciprocal scaled condition number. An estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If rcond is less than the machine precision, in particular, if rcond = 0, the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned.

berr

Array, size at least max(1, nrhs). Contains the componentwise 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.

err_bnds_norm

Array of size nrhs*n_err_bnds. For each right-hand side, contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

Normwise relative error in the i-th solution vector

The array is indexed by the type of error information as described below. There are currently up to three pieces of information returned.

err=1

"Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n)*slamch(ε) for cherfsx and sqrt(n)*dlamch(ε) for zherfsx.

err=2

"Guaranteed" error bound. The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n)*slamch(ε) for cherfsx and sqrt(n)*dlamch(ε) for zherfsx. This error bound should only be trusted if the previous boolean is true.

err=3

Reciprocal condition number. Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n)*slamch(ε) for single precision flavors and sqrt(n)*dlamch(ε) for double precision flavors to determine if the error estimate is "guaranteed". These reciprocal condition numbers for some appropriately scaled matrix Z are:

Let z=s*a, where s scales each row by a power of the radix so all absolute row sums of z are approximately 1.

The information for right-hand side i, where 1 ≤i≤nrhs, and type of error err is stored in:

Column major layout: err_bnds_norm[(err - 1)*nrhs + i - 1].
Row major layout: err_bnds_norm[err - 1 + (i - 1)*n_err_bnds]

err_bnds_comp

Array of size nrhs*n_err_bnds. For each right-hand side, contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

Componentwise relative error in the i-th solution vector:

The array is indexed by the type of error information as described below. There are currently up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (params[2] = 0.0), then err_bnds_comp is not accessed.

err=1

"Trust/don't trust" boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n)*slamch(ε) for cherfsx and sqrt(n)*dlamch(ε) for zherfsx.

err=2

"Guaranteed" error bpound. The estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n)*slamch(ε) for cherfsx and sqrt(n)*dlamch(ε) for zherfsx. This error bound should only be trusted if the previous boolean is true.

err=3

Reciprocal condition number. Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n)*slamch(ε) for single precision flavors and sqrt(n)*dlamch(ε) for double precision flavors to determine if the error estimate is "guaranteed". These reciprocal condition numbers for some appropriately scaled matrix Z are:

Let z=s*(a*diag(x)), where x is the solution for the current right-hand side and s scales each row of a*diag(x) by a power of the radix so all absolute row sums of z are approximately 1.

The information for right-hand side i, where 1 ≤i≤nrhs, and type of error err is stored in:

Column major layout: err_bnds_comp[(err - 1)*nrhs + i - 1].
Row major layout: err_bnds_comp[err - 1 + (i - 1)*n_err_bnds]

params

Output parameter only if the input contains erroneous values, namely, in params[0], params[1], params[2]. In such a case, the corresponding elements of params are filled with default values on output.

Return Values

This function returns a value info.

If info = 0, the execution is successful. The solution to every right-hand side is guaranteed.

If info = -i, parameter i had an illegal value.

If 0 < info≤n: U_info,info is exactly zero. The factorization has been completed, but the factor D is exactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned.

If info = n+j: The solution corresponding to the j-th right-hand side is not guaranteed. The solutions corresponding to other right-hand sides k with k > j may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested params[2] = 0.0, then the j-th right-hand side is the first with a normwise error bound that is not guaranteed (the smallest j such that for column major layout err_bnds_norm[j - 1] = 0.0 or err_bnds_comp[j - 1] = 0.0; or for row major layout err_bnds_norm[(j - 1)*n_err_bnds] = 0.0 or err_bnds_comp[(j - 1)*n_err_bnds] = 0.0). See the definition of err_bnds_norm and err_bnds_comp for err = 1. To get information about all of the right-hand sides, check err_bnds_norm or err_bnds_comp.

Parent topic: Refining the Solution and Estimating Its Error: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?herfsx

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

See Also