?gbrfsx

Developer Reference for Intel® oneAPI Math Kernel Library for C

Download PDF

ID 766684

Date 11/07/2023

Version

Public

A newer version of this document is available. Customers should click here to go to the newest version.

Visible to Intel only — GUID: GUID-A7AB402C-4E8E-4B0D-A670-DF9D4E0FC2C1

View Details

?gbrfsx

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

Syntax

lapack_int LAPACKE_sgbrfsx( int matrix_layout, char trans, char equed, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, const float* ab, lapack_int ldab, const float* afb, lapack_int ldafb, const lapack_int* ipiv, const float* r, const float* c, const float* b, lapack_int ldb, 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_dgbrfsx( int matrix_layout, char trans, char equed, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, const double* ab, lapack_int ldab, const double* afb, lapack_int ldafb, const lapack_int* ipiv, const double* r, const double* c, const double* b, lapack_int ldb, 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 );

lapack_int LAPACKE_cgbrfsx( int matrix_layout, char trans, char equed, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, const lapack_complex_float* ab, lapack_int ldab, const lapack_complex_float* afb, lapack_int ldafb, const lapack_int* ipiv, const float* r, const float* c, 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_zgbrfsx( int matrix_layout, char trans, char equed, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, const lapack_complex_double* ab, lapack_int ldab, const lapack_complex_double* afb, lapack_int ldafb, const lapack_int* ipiv, const double* r, const double* c, 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 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, r, and c 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).

trans

Must be 'N', 'T', or 'C'.

Specifies the form of the system of equations:

If trans = 'N', the system has the form A*X = B (No transpose).

If trans = 'T', the system has the form A^T*X = B (Transpose).

If trans = 'C', the system has the form A^H*X = B (Conjugate transpose for complex flavors, Transpose for real flavors).

equed

Must be 'N', 'R', 'C', or 'B'.

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

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

If equed = 'R', row equilibration was done, that is, A has been premultiplied by diag(r).

If equed = 'C', column equilibration was done, that is, A has been postmultiplied by diag(c).

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

n

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

kl

The number of subdiagonals within the band of A; kl≥ 0.

ku

The number of superdiagonals within the band of A; ku≥ 0.

nrhs

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

ab, afb, b

The array abof size max(1, ldab*n) contains the original matrix A in band storage, in rows from 1 to kl+ku + 1 for column major layout, and in columns from 1 to kl+ku + 1 for row major layout.

The array afbof size max(1, ldafb*n) contains details of the LU factorization of the banded matrix A as computed by ?gbtrf.

The array bof size max(1, ldb*nrhs) for column major layout and max(1, ldb*n) for row major layout contains the matrix B whose columns are the right-hand sides for the systems of equations.

ldab

The leading dimension of the array ab; ldab≥kl+ku+1.

ldafb

The leading dimension of the array afb; ldafb≥ 2*kl+ku+1.

ipiv

Array, size at least max(1, n). Contains the pivot indices as computed by ?gbtrf; for row 1 ≤i≤n, row i of the matrix was interchanged with row ipiv[i-1].

r, c

Arrays: r(n), c(n). The array r contains the row scale factors for A, and the array c contains the column scale factors for A.

If equed = 'R' or 'B', A is multiplied on the left by diag(r); if equed = 'N' or 'C', r is not accessed.

If equed = 'R' or 'B', each element of r must be positive.

If equed = 'C' or 'B', A is multiplied on the right by diag(c); if equed = 'N' or 'R', c is not accessed.

If equed = 'C' or 'B', each element of c must be positive.

Each element of r or c 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 sgbtrs/dgbtrs for real flavors or cgbtrs/zgbtrs for complex flavors.

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 single precision flavors), 1.0D+0 (for double precision flavors).

=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	10.0
Aggressive	Set to 100.0 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 single precision flavors and sqrt(n)*dlamch(ε) for double precision flavors.

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 single precision flavors and sqrt(n)*dlamch(ε) for double precision flavors. 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

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 single precision flavors and sqrt(n)*dlamch(ε) for double precision flavors. 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], and 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 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[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

Select Your Language

Using Intel.com Search

Quick Links

Recent Searches

Advanced Search

Only search in

Developer Reference for Intel® oneAPI Math Kernel Library for C

?gbrfsx

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

See Also