naginterfaces.library.sparse.direct_real_gen_refine¶
- naginterfaces.library.sparse.direct_real_gen_refine(trans, icolzp, irowix, a, iprm, il, lval, iu, uval, nrhs, b, x)[source]¶
direct_real_gen_refine
returns error bounds for the solution of a real sparse system of linear equations with multiple right-hand sides, or . It improves the solution by iterative refinement in standard precision, in order to reduce the backward error as much as possible.For full information please refer to the NAG Library document for f11mh
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f11/f11mhf.html
- Parameters
- transstr, length 1
Specifies whether or is solved.
is solved.
is solved.
- icolzpint, array-like, shape
The new column index array of sparse matrix . See the F11 Introduction.
- irowixint, array-like, shape
The row index array of sparse matrix . See the F11 Introduction.
- afloat, array-like, shape
The array of nonzero values in the sparse matrix .
- iprmint, array-like, shape
The column permutation which defines , the row permutation which defines , plus associated data structures as computed by
direct_real_gen_lu()
.- ilint, array-like, shape
Records the sparsity pattern of matrix as computed by
direct_real_gen_lu()
.- lvalfloat, array-like, shape
Records the nonzero values of matrix and some nonzero values of matrix as computed by
direct_real_gen_lu()
.- iuint, array-like, shape
Records the sparsity pattern of matrix as computed by
direct_real_gen_lu()
.- uvalfloat, array-like, shape
Records some nonzero values of matrix as computed by
direct_real_gen_lu()
.- nrhsint
, the number of right-hand sides in .
- bfloat, array-like, shape
The right-hand side matrix .
- xfloat, array-like, shape
The solution matrix , as returned by
direct_real_gen_solve()
.
- Returns
- xfloat, ndarray, shape
The improved solution matrix .
- ferrfloat, ndarray, shape
contains an estimated error bound for the th solution vector, that is, the th column of , for .
- berrfloat, ndarray, shape
contains the component-wise backward error bound for the th solution vector, that is, the th column of , for .
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: or .
- (errno )
Incorrect row permutations in array .
- (errno )
Incorrect column permutations in array .
- Notes
direct_real_gen_refine
returns the backward errors and estimated bounds on the forward errors for the solution of a real system of linear equations with multiple right-hand sides or . The function handles each right-hand side vector (stored as a column of the matrix ) independently, so we describe the function ofdirect_real_gen_refine
in terms of a single right-hand side and solution .Given a computed solution , the function computes the component-wise backward error . This is the size of the smallest relative perturbation in each element of and such that if is the exact solution of a perturbed system:
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
where is the true solution.
The function uses the factorization computed by
direct_real_gen_lu()
and the solution computed bydirect_real_gen_solve()
.
- References
Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore