NAG C Library Function Document

nag_real_band_lin_solve (f04bbc)


nag_real_band_lin_solve (f04bbc) computes the solution to a real system of linear equations AX=B, where A is an n by n band matrix, with kl subdiagonals and ku superdiagonals, and X and B are n by r matrices. An estimate of the condition number of A and an error bound for the computed solution are also returned.


#include <nag.h>
#include <nagf04.h>
void  nag_real_band_lin_solve (Nag_OrderType order, Integer n, Integer kl, Integer ku, Integer nrhs, double ab[], Integer pdab, Integer ipiv[], double b[], Integer pdb, double *rcond, double *errbnd, NagError *fail)


The LU decomposition with partial pivoting and row interchanges is used to factor A as A=PLU, where P is a permutation matrix, L is the product of permutation matrices and unit lower triangular matrices with kl subdiagonals, and U is upper triangular with kl+ku superdiagonals. The factored form of A is then used to solve the system of equations AX=B.


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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia


1:     order Nag_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 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     n IntegerInput
On entry: the number of linear equations n, i.e., the order of the matrix A.
Constraint: n0.
3:     kl IntegerInput
On entry: the number of subdiagonals kl, within the band of A.
Constraint: kl0.
4:     ku IntegerInput
On entry: the number of superdiagonals ku, within the band of A.
Constraint: ku0.
5:     nrhs IntegerInput
On entry: the number of right-hand sides r, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
6:     ab[dim] doubleInput/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the n by n matrix A.
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=max1,i-kl,,minn,i+ku, depends on the order argument as follows:
  • if order=Nag_ColMajor, Aij is stored as ab[j-1×pdab+kl+ku+i-j];
  • if order=Nag_RowMajor, Aij is stored as ab[i-1×pdab+kl+j-i].
See Section 9 for further details.
On exit: ab is overwritten by details of the factorization.
The elements, uij, of the upper triangular band factor U with kl+ku super-diagonals, and the multipliers, lij, used to form the lower triangular factor L are stored. The elements uij, for i=1,,n and j=i,,minn,i+kl+ku, and lij, for i=1,,n and j=max1,i-kl,,i, are stored where Aij is stored on entry.
7:     pdab IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdab2×kl+ku+1.
8:     ipiv[n] IntegerOutput
On exit: if fail.code= NE_NOERROR, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row ipiv[i-1]. ipiv[i-1]=i indicates a row interchange was not required.
9:     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 matrix of right-hand sides B.
On exit: if fail.code= NE_NOERROR or NE_RCOND, the n by r solution matrix X.
10:   pdb IntegerInput
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.
11:   rcond double *Output
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix A, computed as rcond=1/A1A-11.
12:   errbnd double *Output
On exit: if fail.code= NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution x^, such that x^-x1/x1errbnd, where x^ is a column of the computed solution returned in the array b and x is the corresponding column of the exact solution X. If rcond is less than machine precision, errbnd is returned as unity.
13:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
The Integer allocatable memory required is n, and the double allocatable memory required is 3×n. In this case the factorization and the solution X have been computed, but rcond and errbnd have not been computed.
See Section in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
A solution has been computed, but rcond is less than machine precision so that the matrix A is numerically singular.
Diagonal element value of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.


The computed solution for a single right-hand side, x^, satisfies an equation of the form
A+E x^=b,  
E1 = Oε A1  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κA E1 A1 ,  
where κA = A-11 A1 , the condition number of A with respect to the solution of the linear equations. nag_real_band_lin_solve (f04bbc) uses the approximation E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Parallelism and Performance

nag_real_band_lin_solve (f04bbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_real_band_lin_solve (f04bbc) 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.

Further Comments

The band storage scheme for the array ab stored in Nag_ColMajor is illustrated by the following example, when n=5, kl=2, and ku=1. Storage of the band matrix A in the array ab:
Band matrix A Band storage in array ab
order=Nag_ColMajor order=Nag_RowMajor
a11 a12 a21 a22 a23 a31 a32 a33 a34 a42 a43 a44 a45 a53 a54 a55 * * * + + * * + + + * a12 a23 a34 a45 a11 a22 a33 a44 a55 a21 a32 a43 a54 * a31 a42 a53 * * * * a11 a12 + + * a21 a22 a23 + + a31 a32 a33 a34 + * a42 a43 a44 a45 * * a53 a54 a55 * * *
Array elements marked * need not be set and are not referenced by the function. Array elements marked + need not be set, but are defined on exit from the function and contain the elements u13, u14, u24, u25 and u35. In this example when order=Nag_ColMajor the first referenced element of ab is ab[3]=a11; while for order=Nag_RowMajor the first referenced element is ab[2]=a11.
In general, elements aij are stored as follows: where max1,i-kl j minn,i+ku .
The total number of floating-point operations required to solve the equations AX=B depends upon the pivoting required, but if nkl+ku then it is approximately bounded by O n kl kl + ku  for the factorization and O n 2 kl + ku r  for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The complex analogue of nag_real_band_lin_solve (f04bbc) is nag_complex_band_lin_solve (f04cbc).


This example solves the equations
where A is the band matrix
A= -0.23 2.54 -3.66 0 -6.98 2.46 -2.73 -2.13 0 2.56 2.46 4.07 0 0 -4.78 -3.82   and   B= 4.42 -36.01 27.13 -31.67 -6.14 -1.16 10.50 -25.82 .  
An estimate of the condition number of A and an approximate error bound for the computed solutions are also printed.

Program Text

Program Text (f04bbce.c)

Program Data

Program Data (f04bbce.d)

Program Results

Program Results (f04bbce.r)