NAG FL Interface
f04bbf (real_​band_​solve)

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1 Purpose

f04bbf computes the solution to a real system of linear equations AX=B, where A is an n×n band matrix, with kl subdiagonals and ku superdiagonals, and X and B are n×r matrices. An estimate of the condition number of A and an error bound for the computed solution are also returned.

2 Specification

Fortran Interface
Subroutine f04bbf ( n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, rcond, errbnd, ifail)
Integer, Intent (In) :: n, kl, ku, nrhs, ldab, ldb
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: ipiv(n)
Real (Kind=nag_wp), Intent (Inout) :: ab(ldab,*), b(ldb,*)
Real (Kind=nag_wp), Intent (Out) :: rcond, errbnd
C Header Interface
#include <nag.h>
void  f04bbf_ (const Integer *n, const Integer *kl, const Integer *ku, const Integer *nrhs, double ab[], const Integer *ldab, Integer ipiv[], double b[], const Integer *ldb, double *rcond, double *errbnd, Integer *ifail)
The routine may be called by the names f04bbf or nagf_linsys_real_band_solve.

3 Description

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.

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

5 Arguments

1: n Integer Input
On entry: the number of linear equations n, i.e., the order of the matrix A.
Constraint: n0.
2: kl Integer Input
On entry: the number of subdiagonals kl, within the band of A.
Constraint: kl0.
3: ku Integer Input
On entry: the number of superdiagonals ku, within the band of A.
Constraint: ku0.
4: nrhs Integer Input
On entry: the number of right-hand sides r, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
5: ab(ldab,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array ab must be at least max(1,n).
On entry: the n×n matrix A.
The matrix is stored in rows kl+1 to 2kl+ku+1; the first kl rows need not be set, more precisely, the element Aij must be stored in
ab(kl+ku+1+i-j,j)=Aij  for ​max(1,j-ku)imin(n,j+kl). 
See Section 9 for further details.
On exit: if ifail0, ab is overwritten by details of the factorization.
The upper triangular band matrix U, with kl+ku superdiagonals, is stored in rows 1 to kl+ku+1 of the array, and the multipliers used to form the matrix L are stored in rows kl+ku+2 to 2kl+ku+1.
6: ldab Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f04bbf is called.
Constraint: ldab2×kl+ku+1.
7: ipiv(n) Integer array Output
On exit: if ifail0, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row ipiv(i). ipiv(i)=i indicates a row interchange was not required.
8: b(ldb,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max(1,nrhs).
On entry: the n×r matrix of right-hand sides B.
On exit: if ifail=0 or n+1, the n×r solution matrix X.
9: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f04bbf is called.
Constraint: ldbmax(1,n).
10: rcond Real (Kind=nag_wp) 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).
11: errbnd Real (Kind=nag_wp) Output
On exit: if ifail=0 or n+1, 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.
12: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of -1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value -1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail>0andifailn
Diagonal element value of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.
ifail=n+1
A solution has been computed, but rcond is less than machine precision so that the matrix A is numerically singular.
ifail=-1
On entry, n=value.
Constraint: n0.
ifail=-2
On entry, kl=value.
Constraint: kl0.
ifail=-3
On entry, ku=value.
Constraint: ku0.
ifail=-4
On entry, nrhs=value.
Constraint: nrhs0.
ifail=-6
On entry, ldab =value, kl =value and ku=value.
Constraint: ldab2×kl+ku+1.
ifail=-9
On entry, ldb=value and n=value.
Constraint: ldbmax(1,n).
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
The integer allocatable memory required is n, and the real 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 9 in the Introduction to the NAG Library FL 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
E1 = O(ε) A1  
and ε is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κ(A) E1 A1 ,  
where κ(A) = A-11 A1 , the condition number of A with respect to the solution of the linear equations. f04bbf uses the approximation E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

f04bbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f04bbf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The band storage scheme for the array ab is illustrated by the following example, when n=6, kl=1, and ku=2. Storage of the band matrix A in the array ab:
* * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 *  
Array elements marked * need not be set and are not referenced by the routine. Array elements marked + need not be set, but are defined on exit from the routine and contain the elements u14, u25 and u36.
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(nkl(kl+ku)) for the factorization and O(n(2kl+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 f04bbf is f04cbf.

10 Example

This example solves the equations
AX=B,  
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.

10.1 Program Text

Program Text (f04bbfe.f90)

10.2 Program Data

Program Data (f04bbfe.d)

10.3 Program Results

Program Results (f04bbfe.r)