NAG CL Interface
f07chc (dgtrfs)

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1 Purpose

f07chc computes error bounds and refines the solution to a real system of linear equations AX=B or ATX=B , where A is an n × n tridiagonal matrix and X and B are n × r matrices, using the LU factorization returned by f07cdc and an initial solution returned by f07cec. Iterative refinement is used to reduce the backward error as much as possible.

2 Specification

#include <nag.h>
void  f07chc (Nag_OrderType order, Nag_TransType trans, Integer n, Integer nrhs, const double dl[], const double d[], const double du[], const double dlf[], const double df[], const double duf[], const double du2[], 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: f07chc, nag_lapacklin_dgtrfs or nag_dgtrfs.

3 Description

f07chc should normally be preceded by calls to f07cdc and f07cec. f07cdc uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix A as
A=PLU ,  
where P is a permutation matrix, L is unit lower triangular with at most one nonzero subdiagonal element in each column, and U is an upper triangular band matrix, with two superdiagonals. f07cec then utilizes the factorization to compute a solution, X^ , to the required equations. Letting x^ denote a column of X^ , f07chc computes a component-wise backward error, β , the smallest relative perturbation in each element of A and b such that x^ is the exact solution of a perturbed system
(A+E) x^=b+f , with  |eij| β |aij| , and  |fj| β |bj| .  
The function also estimates a bound for the component-wise forward error in the computed solution defined by max |xi-xi^| / max |xi^| , where x is the corresponding column of the exact solution, X .

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug

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: specifies the equations to be solved as follows:
trans=Nag_NoTrans
Solve AX=B for X.
trans=Nag_Trans or Nag_ConjTrans
Solve ATX=B for X.
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: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5: dl[dim] const double Input
Note: the dimension, dim, of the array dl must be at least max(1,n-1).
On entry: must contain the (n-1) subdiagonal elements of the matrix A.
6: d[dim] const double Input
Note: the dimension, dim, of the array d must be at least max(1,n).
On entry: must contain the n diagonal elements of the matrix A.
7: du[dim] const double Input
Note: the dimension, dim, of the array du must be at least max(1,n-1).
On entry: must contain the (n-1) superdiagonal elements of the matrix A.
8: dlf[dim] const double Input
Note: the dimension, dim, of the array dlf must be at least max(1,n-1).
On entry: must contain the (n-1) multipliers that define the matrix L of the LU factorization of A.
9: df[dim] const double Input
Note: the dimension, dim, of the array df must be at least max(1,n).
On entry: must contain the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
10: duf[dim] const double Input
Note: the dimension, dim, of the array duf must be at least max(1,n-1).
On entry: must contain the (n-1) elements of the first superdiagonal of U.
11: du2[dim] const double Input
Note: the dimension, dim, of the array du2 must be at least max(1,n-2).
On entry: must contain the (n-2) elements of the second superdiagonal of U.
12: ipiv[dim] const Integer Input
Note: the dimension, dim, of the array ipiv must be at least max(1,n).
On entry: must contain the n pivot indices that define the permutation matrix P. At the ith step, row i of the matrix was interchanged with row ipiv[i-1], and ipiv[i-1] must always be either i or (i+1), ipiv[i-1]=i indicating that a row interchange was not performed.
13: 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 matrix of right-hand sides B.
14: 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).
15: 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 initial solution matrix X.
On exit: the n×r refined solution matrix X.
16: 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).
17: ferr[nrhs] double Output
On exit: estimate of the forward error bound for each computed solution vector, such that x^j-xj/x^jferr[j-1], where x^j is the jth column of the computed solution returned in the array x and xj is the corresponding column of the exact solution X. The estimate is almost always a slight overestimate of the true error.
18: berr[nrhs] double Output
On exit: estimate of the component-wise relative backward error of each computed solution vector x^j (i.e., the smallest relative change in any element of A or B that makes x^j an exact solution).
19: 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, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
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_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 computed solution for a single right-hand side, x^ , satisfies an equation of the form
(A+E) x^=b ,  
where
E=O(ε)A  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x x κ(A) E A ,  
where κ(A)=A-1 A , the condition number of A with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Function f07cgc can be used to estimate the condition number of A .

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07chc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07chc 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

The total number of floating-point operations required to solve the equations AX=B or ATX=B is proportional to nr . At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The complex analogue of this function is f07cvc.

10 Example

This example solves the equations
AX=B ,  
where A is the tridiagonal matrix
A = ( 3.0 2.1 0.0 0.0 0.0 3.4 2.3 -1.0 0.0 0.0 0.0 3.6 -5.0 1.9 0.0 0.0 0.0 7.0 -0.9 8.0 0.0 0.0 0.0 -6.0 7.1 )   and   B = ( 2.7 6.6 -0.5 10.8 2.6 -3.2 0.6 -11.2 2.7 19.1 ) .  
Estimates for the backward errors and forward errors are also output.

10.1 Program Text

Program Text (f07chce.c)

10.2 Program Data

Program Data (f07chce.d)

10.3 Program Results

Program Results (f07chce.r)