Uses extra precise iterative refinement to compute the solution to the system of linear equations with a banded coefficient matrix A and multiple right-hand sides

Syntax

lapack_int LAPACKE_sgbsvxx( int matrix_layout, char fact, char trans, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, float* ab, lapack_int ldab, float* afb, lapack_int ldafb, lapack_int* ipiv, char* equed, float* r, float* c, float* b, lapack_int ldb, float* x, lapack_int ldx, float* rcond, float* rpvgrw, float* berr, lapack_int n_err_bnds, float* err_bnds_norm, float* err_bnds_comp, lapack_int nparams, const float* params );

lapack_int LAPACKE_dgbsvxx( int matrix_layout, char fact, char trans, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, double* ab, lapack_int ldab, double* afb, lapack_int ldafb, lapack_int* ipiv, char* equed, double* r, double* c, double* b, lapack_int ldb, double* x, lapack_int ldx, double* rcond, double* rpvgrw, double* berr, lapack_int n_err_bnds, double* err_bnds_norm, double* err_bnds_comp, lapack_int nparams, const double* params );

lapack_int LAPACKE_cgbsvxx( int matrix_layout, char fact, char trans, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, lapack_complex_float* ab, lapack_int ldab, lapack_complex_float* afb, lapack_int ldafb, lapack_int* ipiv, char* equed, float* r, float* c, lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* rcond, float* rpvgrw, float* berr, lapack_int n_err_bnds, float* err_bnds_norm, float* err_bnds_comp, lapack_int nparams, const float* params );

lapack_int LAPACKE_zgbsvxx( int matrix_layout, char fact, char trans, lapack_int n, lapack_int kl, lapack_int ku, lapack_int nrhs, lapack_complex_double* ab, lapack_int ldab, lapack_complex_double* afb, lapack_int ldafb, lapack_int* ipiv, char* equed, double* r, double* c, lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* rcond, double* rpvgrw, double* berr, lapack_int n_err_bnds, double* err_bnds_norm, double* err_bnds_comp, lapack_int nparams, const double* params );

Include Files

  • mkl.h

Description

The routine uses the LU factorization to compute the solution to a real or complex system of linear equations A*X = B, AT*X = B, or AH*X = B, where A is an n-by-n banded matrix, the columns of the matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.

Both normwise and maximum componentwise error bounds are also provided on request. The routine returns a solution with a small guaranteed error (O(eps), where eps is the working machine precision) unless the matrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers are also calculated and returned.

The routine accepts user-provided factorizations and equilibration factors; see definitions of the fact and equed options. Solving with refinement and using a factorization from a previous call of the routine also produces a solution with O(eps) errors or warnings but that may not be true for general user-provided factorizations and equilibration factors if they differ from what the routine would itself produce.

The routine ?gbsvxx performs the following steps:

  1. If fact = 'E', scaling factors r and c are computed to equilibrate the system:

    trans = 'N': diag(r)*A*diag(c)*inv(diag(c))*X = diag(r)*B

    trans = 'T': (diag(r)*A*diag(c))T*inv(diag(r))*X = diag(c)*B

    trans = 'C': (diag(r)*A*diag(c))H*inv(diag(r))*X = diag(c)*B

    Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(r)*A*diag(c) and B by diag(r)*B (if trans='N') or diag(c)*B (if trans = 'T' or 'C').

  2. If fact = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if fact = 'E') as A = P*L*U, where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular.

  3. If some Ui,i= 0, so that U 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 (see the rcond parameter). If the reciprocal of the condition number is less than machine precision, the routine still goes on to solve for X and compute error bounds.

  4. The system of equations is solved for X using the factored form of A.

  5. By default, unless params[0] is set to zero, the routine applies iterative refinement to improve the computed solution matrix and calculate error bounds. Refinement calculates the residual to at least twice the working precision.

  6. If equilibration was used, the matrix X is premultiplied by diag(c) (if trans = 'N') or diag(r) (if trans = 'T' or 'C') so that it solves the original system before equilibration.

Input Parameters

matrix_layout

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

fact

Must be 'F', 'N', or 'E'.

Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.

If fact = 'F', on entry, afb and ipiv contain the factored form of A. If equed is not 'N', the matrix A has been equilibrated with scaling factors given by r and c. Parameters ab, afb, and ipiv are not modified.

If fact = 'N', the matrix A will be copied to afb and factored.

If fact = 'E', the matrix A will be equilibrated, if necessary, copied to afb and factored.

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 AT*X = B (Transpose).

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

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

Arrays: ab (max(ldab*n)), afb (max(ldafb*n)), b(max(1, ldb*nrhs) for column major layout and max(1, ldb*n) for row major layout).

The array ab contains the matrix A in band storage.

If fact = 'F' and equed is not 'N', then AB must have been equilibrated by the scaling factors in r and/or c.

The array afb is an input argument if fact = 'F'. It contains the factored form of the banded matrix A, that is, the factors L and U from the factorization A = P*L*U as computed by ?gbtrf. U is stored as an upper triangular banded matrix with kl + ku superdiagonals. L is stored as lower triangular band matrix with kl subdiagonals. If equed is not 'N', then afb is the factored form of the equilibrated matrix A.

The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.

ldab

The leading dimension of the array ab; ldabkl+ku+1.

ldafb

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

ipiv

Array, size at least max(1, n). The array ipiv is an input argument if fact = 'F'. It contains the pivot indices from the factorization A = P*L*U as computed by ?gbtrf; row i of the matrix was interchanged with row ipiv[i-1].

equed

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

equed is an input argument if fact = 'F'. It specifies the form of equilibration that was done:

If equed = 'N', no equilibration was done (always true if fact = 'N').

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).

r, c

Arrays: r (size n), c (size n). The array r contains the row scale factors for A, and the array c contains the column scale factors for A. These arrays are input arguments if fact = 'F' only; otherwise they are output arguments.

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

If fact = 'F' and 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 fact = 'F' and 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 ldbnrhs for row major layout.

ldx

The leading dimension of the output array x; ldx max(1, n) for column major layout and ldxnrhs 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 max(1, nparams). Specifies algorithm parameters. If an entry is less than 0.0, that entry is 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 extra-precise refinement algorithm.

(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

Array, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n) for row major layout.

If info = 0, the array x contains the solution n-by-nrhs matrix X to the original system of equations. Note that A and B are modified on exit if equed'N', and the solution to the equilibrated system is:

inv(diag(c))*X, if trans = 'N' and equed = 'C' or 'B'; or inv(diag(r))*X, if trans = 'T' or 'C' and equed = 'R' or 'B'.

ab

Array ab is not modified on exit if fact = 'F' or 'N', or if fact = 'E' and equed = 'N'.

If equed'N', A is scaled on exit as follows:

equed = 'R': A = diag(r)*A

equed = 'C': A = A*diag(c)

equed = 'B': A = diag(r)*A*diag(c).

afb

If fact = 'N' or 'E', then afb is an output argument and on exit returns the factors L and U from the factorization A = PLU of the original matrix A (if fact = 'N') or of the equilibrated matrix A (if fact = 'E').

b

Overwritten by diag(r)*B if trans = 'N' and equed = 'R' or 'B';

overwritten by trans = 'T' or 'C' and equed = 'C' or 'B';

not changed if equed = 'N'.

r, c

These arrays are output arguments if fact'F'. Each element of these arrays is a power of the radix. See the description of r, c in Input Arguments section.

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.

rpvgrw

Contains the reciprocal pivot growth factor:

If this is much less than 1, the stability of the LU factorization of the (equlibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable. If factorization fails with 0 < infon, this parameter contains the reciprocal pivot growth factor for the leading info columns of A. In ?gbsvx, this quantity is returned in rpivot.

berr

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

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 inrhs, and type of error err is stored in err_bnds_norm[(err-1)*nrhs + i - 1].

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

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 inrhs, and type of error err is stored in err_bnds_comp[(err-1)*nrhs + i - 1].

ipiv

If fact = 'N' or 'E', then ipiv is an output argument and on exit contains the pivot indices from the factorization A = P*L*U of the original matrix A (if fact = 'N') or of the equilibrated matrix A (if fact = 'E').

equed

If fact'F', then equed is an output argument. It specifies the form of equilibration that was done (see the description of equed in Input Arguments section).

params

If an entry is less than 0.0, that entry is filled with the default value used for that parameter, otherwise the entry is not modified.

info

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 < infon: Uinfo,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 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.

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 < infon: Uinfo,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.

For more complete information about compiler optimizations, see our Optimization Notice.
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