nag_dgbtrs (f07bec) (PDF version)
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f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgbtrs (f07bec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgbtrs (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 nag_dgbtrf (f07bdc).

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dgbtrs (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)

3  Description

nag_dgbtrs (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 nag_dgbtrf (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:     orderNag_OrderTypeInput
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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     transNag_TransTypeInput
On entry: indicates the form of the equations.
trans=Nag_NoTrans
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:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     klIntegerInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
5:     kuIntegerInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
6:     nrhsIntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
7:     ab[dim]const doubleInput
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 nag_dgbtrf (f07bdc).
8:     pdabIntegerInput
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 IntegerInput
Note: the dimension, dim, of the array ipiv must be at least max1,n.
On entry: the pivot indices, as returned by nag_dgbtrf (f07bdc).
10:   b[dim]doubleInput/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:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
NE_INT_3
On entry, pdab=value, kl=value and ku=value.
Constraint: pdab2×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.

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 nag_dgbrfs (f07bhc), and an estimate for κA can be obtained by calling nag_dgbcon (f07bgc) with norm=Nag_InfNorm.

8  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 nag_dgbrfs (f07bhc) to refine the solution and return an error estimate.
The complex analogue of this function is nag_zgbtrs (f07bsc).

9  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 nag_dgbtrf (f07bdc).

9.1  Program Text

Program Text (f07bece.c)

9.2  Program Data

Program Data (f07bece.d)

9.3  Program Results

Program Results (f07bece.r)


nag_dgbtrs (f07bec) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012