NAG CL Interface
f07bec (dgbtrs)

1 Purpose

f07bec solves a real band system of linear equations with multiple right-hand sides,
AX=B   or   ATX=B ,  
where A has been factorized by f07bdc.

2 Specification

#include <nag.h>
void  f07bec (Nag_OrderType order, Nag_TransType trans, Integer n, Integer kl, Integer ku, Integer nrhs, const double ab[], Integer pdab, const Integer ipiv[], double b[], Integer pdb, NagError *fail)
The function may be called by the names: f07bec, nag_lapacklin_dgbtrs or nag_dgbtrs.

3 Description

f07bec is used to solve a real band system of linear equations AX=B or ATX=B, the function must be preceded by a call to f07bdc which computes the LU factorization of A as A=PLU. The solution is computed by forward and backward substitution.
If trans=Nag_NoTrans, the solution is computed by solving PLY=B and then UX=Y.
If trans=Nag_Trans or Nag_ConjTrans, the solution is computed by solving UTY=B and then LTPTX=Y.

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 equations.
AX=B is solved for X.
trans=Nag_Trans or Nag_ConjTrans
ATX=B is solved 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: 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 max1,pdab×n.
On entry: the LU factorization of A, as returned by f07bdc.
8: pdab Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array ab.
Constraint: pdab2×kl+ku+1.
9: ipiv[dim] const Integer Input
Note: the dimension, dim, of the array ipiv must be at least max1,n.
On entry: the pivot indices, as returned by f07bdc.
10: b[dim] double Input/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth 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 by r right-hand side matrix B.
On exit: the n by r solution matrix X.
11: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
12: 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

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
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, pdb=value.
Constraint: pdb>0.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
On entry, pdab=value, kl=value and ku=value.
Constraint: pdab2×kl+ku+1.
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.
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

For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where
EckεPLU ,  
ck is a modest linear function of k=kl+ku+1, and ε is the machine precision. This assumes kn.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x ckcondA,xε  
where condA,x=A-1Ax/xcondA=A-1AκA.
Note that condA,x can be much smaller than condA, and condAT can be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling f07bhc, and an estimate for κA can be obtained by calling f07bgc with norm=Nag_InfNorm.

8 Parallelism and Performance

f07bec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bec 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 is approximately 2n2kl+kur, assuming nkl and nku.
This function may be followed by a call to f07bhc to refine the solution and return an error estimate.
The complex analogue of this function is f07bsc.

10 Example

This example solves the system of equations AX=B, 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 (f07bece.c)

10.2 Program Data

Program Data (f07bece.d)

10.3 Program Results

Program Results (f07bece.r)