?herfsx

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/22/2024

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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

call cherfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv, s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info )

call zherfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv, s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info )

Include Files

mkl.fi, lapack.f90

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

uplo

CHARACTER*1. 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

CHARACTER*1. 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If equed = 'Y', both row and column equilibration was done, that is, A has been replaced by diag(s)*A*diag(s).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

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

nrhs

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

a, af, b, work

COMPLEX for cherfsx

DOUBLE COMPLEX for zherfsx.

Arrays: a(lda,*), af(ldaf,*), b(ldb,*), work(*).

The array a 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 second dimension of a must be at least max(1,n).

The array af 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 contains the matrix B whose columns are the right-hand sides for the systems of equations. The second dimension of b must be at least max(1,nrhs).

work(*) is a workspace array. The dimension of work must be at least max(1,2*n).

lda

INTEGER. The leading dimension of a; lda≥ max(1, n).

ldaf

INTEGER. The leading dimension of af; ldaf≥ max(1, n).

ipiv

INTEGER.

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

REAL for single precision flavors

DOUBLE PRECISION for double precision flavors.

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

INTEGER. The leading dimension of the array b; ldb≥ max(1, n).

x

COMPLEX for cherfsx

DOUBLE COMPLEX for zherfsx.

Array, size (ldx,*).

The solution matrix X as computed by ?hetrs

ldx

INTEGER. The leading dimension of the output array x; ldx≥ max(1, n).

n_err_bnds

INTEGER. 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

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

params

REAL for single precision flavors

DOUBLE PRECISION for double precision flavors.

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(1) : 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(2) : 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(3) : 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).

rwork

REAL for cherfsx

DOUBLE PRECISION for zherfsx.

Workspace array, size at least max(1, 3*n).

Output Parameters

x

COMPLEX for cherfsx

DOUBLE COMPLEX for zherfsx.

The improved solution matrix X.

rcond

REAL for cherfsx

DOUBLE PRECISION for zherfsx.

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

REAL for cherfsx

DOUBLE PRECISION for zherfsx.

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

REAL for cherfsx

DOUBLE PRECISION for zherfsx.

Array of size nrhs by 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.

The first index in err_bnds_norm(i,:) corresponds to the i-th right-hand side.

The second index in err_bnds_norm(:,err) contains the following three fields:

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.

err_bnds_comp

REAL for cherfsx

DOUBLE PRECISION for zherfsx.

Array of size nrhs by 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 right-hand side i, on which the componentwise relative error depends, and 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(3) = 0.0), then err_bnds_comp is not accessed. If n_err_bnds < 3, then at most the first (:,n_err_bnds) entries are returned.

The first index in err_bnds_comp(i,:) corresponds to the i-th right-hand side.

The second index in err_bnds_comp(:,err) contains the following three fields:

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.

params

REAL for single precision flavors

DOUBLE PRECISION for double precision flavors.

Output parameter only if the input contains erroneous values, namely, in params(1), params(2), params(3). In such a case, the corresponding elements of params are filled with default values on output.

info

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

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

If 0 < info≤n: U_info,info is exactly zero. The factorization has been completed, but the factor U 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(3) = 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 err_bnds_norm(j,1) = 0.0 or err_bnds_comp(j,1) = 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 Fortran

?herfsx

Syntax

Include Files

Description

Input Parameters

Output Parameters

See Also