NAG CL Interface
f07bhc (dgbrfs)

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1 Purpose

f07bhc returns error bounds for the solution of a real band system of linear equations with multiple right-hand sides, AX=B or ATX=B. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

2 Specification

#include <nag.h>
void  f07bhc (Nag_OrderType order, Nag_TransType trans, Integer n, Integer kl, Integer ku, Integer nrhs, const double ab[], Integer pdab, const double afb[], Integer pdafb, const Integer ipiv[], const double b[], Integer pdb, double x[], Integer pdx, double ferr[], double berr[], NagError *fail)
The function may be called by the names: f07bhc, nag_lapacklin_dgbrfs or nag_dgbrfs.

3 Description

f07bhc returns the backward errors and estimated bounds on the forward errors for the solution of a real band system of linear equations with multiple right-hand sides AX=B or ATX=B. The function handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of f07bhc in terms of a single right-hand side b and solution x.
Given a computed solution x, the function computes the component-wise backward error β. This is the size of the smallest relative perturbation in each element of A and b such that x is the exact solution of a perturbed system
(A+δA)x=b+δb |δaij|β|aij|   and   |δbi|β|bi| .  
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
maxi |xi-x^i| / maxi |xi|  
where x^ is the true solution.
For details of the method, see the F07 Chapter Introduction.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: trans Nag_TransType Input
On entry: indicates the form of the linear equations for which X is the computed solution.
trans=Nag_NoTrans
The linear equations are of the form AX=B.
trans=Nag_Trans or Nag_ConjTrans
The linear equations are of the form ATX=B.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: kl Integer Input
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
5: ku Integer Input
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
6: nrhs Integer Input
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
7: ab[dim] const double Input
Note: the dimension, dim, of the array ab must be at least max(1,pdab×n).
On entry: the original n×n band matrix A as supplied to f07bdc but with reduced requirements since the matrix is not factorized.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements Aij, for row i=1,,n and column j=max(1,i-kl),,min(n,i+ku), depends on the order argument as follows:
  • if order=Nag_ColMajor, Aij is stored as ab[(j-1)×pdab+ku+i-j];
  • if order=Nag_RowMajor, Aij is stored as ab[(i-1)×pdab+kl+j-i].
See Section 9 in f07bac for further details.
8: pdab Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabkl+ku+1.
9: afb[dim] const double Input
Note: the dimension, dim, of the array afb must be at least max(1,pdafb×n).
On entry: the LU factorization of A, as returned by f07bdc.
10: pdafb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array afb.
Constraint: pdafb2×kl+ku+1.
11: ipiv[dim] const Integer Input
Note: the dimension, dim, of the array ipiv must be at least max(1,n).
On entry: the pivot indices, as returned by f07bdc.
12: b[dim] const double Input
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×nrhs) when order=Nag_ColMajor;
  • max(1,n×pdb) when order=Nag_RowMajor.
The (i,j)th element of the matrix B is stored in
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On entry: the n×r right-hand side matrix B.
13: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax(1,n);
  • if order=Nag_RowMajor, pdbmax(1,nrhs).
14: x[dim] double Input/Output
Note: the dimension, dim, of the array x must be at least
  • max(1,pdx×nrhs) when order=Nag_ColMajor;
  • max(1,n×pdx) when order=Nag_RowMajor.
The (i,j)th element of the matrix X is stored in
  • x[(j-1)×pdx+i-1] when order=Nag_ColMajor;
  • x[(i-1)×pdx+j-1] when order=Nag_RowMajor.
On entry: the n×r solution matrix X, as returned by f07bec.
On exit: the improved solution matrix X.
15: pdx Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxmax(1,n);
  • if order=Nag_RowMajor, pdxmax(1,nrhs).
16: ferr[nrhs] double Output
On exit: ferr[j-1] contains an estimated error bound for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
17: berr[nrhs] double Output
On exit: berr[j-1] contains the component-wise backward error bound β for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
18: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, kl=value.
Constraint: kl0.
On entry, ku=value.
Constraint: ku0.
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdafb=value.
Constraint: pdafb>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and nrhs=value.
Constraint: pdbmax(1,nrhs).
On entry, pdx=value and n=value.
Constraint: pdxmax(1,n).
On entry, pdx=value and nrhs=value.
Constraint: pdxmax(1,nrhs).
NE_INT_3
On entry, pdab=value, kl=value and ku=value.
Constraint: pdabkl+ku+1.
On entry, pdafb=value, kl=value and ku=value.
Constraint: pdafb2×kl+ku+1.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07bhc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bhc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

For each right-hand side, computation of the backward error involves a minimum of 4n(kl+ku) floating-point operations. Each step of iterative refinement involves an additional 2n(4kl+3ku) operations. This assumes nkl and nku. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form Ax=b or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 2n(2kl+ku) operations.
The complex analogue of this function is f07bvc.

10 Example

This example solves the system of equations AX=B using iterative refinement and to compute the forward and backward error bounds, where
A= ( -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 )   and   B= ( 4.42 -36.01 27.13 -31.67 -6.14 -1.16 10.50 -25.82 ) .  
Here A is nonsymmetric and is treated as a band matrix, which must first be factorized by f07bdc.

10.1 Program Text

Program Text (f07bhce.c)

10.2 Program Data

Program Data (f07bhce.d)

10.3 Program Results

Program Results (f07bhce.r)