hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_linsys_real_posdef_band_solve (f04bf)

Purpose

nag_linsys_real_posdef_band_solve (f04bf) computes the solution to a real system of linear equations AX = BAX=B, where AA is an nn by nn symmetric positive definite band matrix of band width 2k + 12k+1, and XX and BB are nn by rr matrices. An estimate of the condition number of AA and an error bound for the computed solution are also returned.

Syntax

[ab, b, rcond, errbnd, ifail] = f04bf(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, b, rcond, errbnd, ifail] = nag_linsys_real_posdef_band_solve(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)

Description

The Cholesky factorization is used to factor AA as A = UTUA=UTU, if uplo = 'U'uplo='U', or A = LLTA=LLT, if uplo = 'L'uplo='L', where UU is an upper triangular band matrix with kk superdiagonals, and LL is a lower triangular band matrix with kk subdiagonals. The factored form of AA is then used to solve the system of equations AX = BAX=B.

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 http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
If uplo = 'U'uplo='U', the upper triangle of the matrix AA is stored.
If uplo = 'L'uplo='L', the lower triangle of the matrix AA is stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     kd – int64int32nag_int scalar
The number of superdiagonals kk (and the number of subdiagonals) of the band matrix AA.
Constraint: kd0kd0.
3:     ab(ldab, : :) – double array
The first dimension of the array ab must be at least kd + 1kd+1
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn symmetric band matrix AA. The upper or lower triangular part of the symmetric matrix is stored in the first kd + 1kd+1 rows of the array. The jjth column of AA is stored in the jjth column of the array ab as follows:
  • if uplo = 'U'uplo='U', ab ( k + 1 + ij ,j) = aij ab ( k+1+i-j ,j) =aij for max (1,jk)ijmax(1,j-k)ij;
  • if uplo = 'L'uplo='L', ab (1 + ij,j) = aij ab (1+i-j,j) =aij for jimin (n,j + k)jimin(n,j+k).
See Section [Further Comments] below for further details.
4:     b(ldb, : :) – double array
The first dimension of the array b must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,nrhs)max(1,nrhs)
The nn by rr matrix of right-hand sides BB.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b.
The number of linear equations nn, i.e., the order of the matrix AA.
Constraint: n0n0.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
The number of right-hand sides rr, i.e., the number of columns of the matrix BB.
Constraint: nrhs0nrhs0.

Input Parameters Omitted from the MATLAB Interface

ldab ldb

Output Parameters

1:     ab(ldab, : :) – double array
The first dimension of the array ab will be kd + 1kd+1
The second dimension of the array will be max (1,n)max(1,n)
ldabkd + 1ldabkd+1.
If ifail = 0ifail=0 or n + 1n+1, the factor UU or LL from the Cholesky factorization A = UTUA=UTU or A = LLTA=LLT, in the same storage format as AA.
2:     b(ldb, : :) – double array
The first dimension of the array b will be max (1,n)max(1,n)
The second dimension of the array will be max (1,nrhs)max(1,nrhs)
ldbmax (1,n)ldbmax(1,n).
If ifail = 0ifail=0 or n + 1n+1, the nn by rr solution matrix XX.
3:     rcond – double scalar
If ifail = 0ifail=0 or n + 1n+1, an estimate of the reciprocal of the condition number of the matrix AA, computed as rcond = 1 / (A1A11)rcond=1/(A1A-11).
4:     errbnd – double scalar
If ifail = 0ifail=0 or n + 1n+1, an estimate of the forward error bound for a computed solution x^, such that x1 / x1errbndx^-x1/x1errbnd, where x^ is a column of the computed solution returned in the array b and xx is the corresponding column of the exact solution XX. If rcond is less than machine precision, then errbnd is returned as unity.
5:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

  ifail < 0andifail999ifail<0andifail-999
If ifail = iifail=-i, the iith argument had an illegal value.
  ifail = 999ifail=-999
Allocation of memory failed. The integer allocatable memory required is n, and the double allocatable memory required is 3 × n3×n. Allocation failed before the solution could be computed.
  ifail > 0andifailNifail>0andifailN
If ifail = iifail=i, the leading minor of order ii of AA is not positive definite. The factorization could not be completed, and the solution has not been computed.
W ifail = N + 1ifail=N+1
rcond is less than machine precision, so that the matrix AA is numerically singular. A solution to the equations AX = BAX=B has nevertheless been computed.

Accuracy

The computed solution for a single right-hand side, x^, satisfies an equation of the form
(A + E) = b,
(A+E) x^=b,
where
E1 = O(ε) A1
E1=O(ε) A1
and εε is the machine precision. An approximate error bound for the computed solution is given by
(x1)/(x1) κ(A) (E1)/(A1) ,
x^-x1 x1 κ(A) E1 A1 ,
where κ(A) = A11A1κ(A)=A-11A1, the condition number of AA with respect to the solution of the linear equations. nag_linsys_real_posdef_band_solve (f04bf) uses the approximation E1 = εA1E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The band storage scheme for the array ab is illustrated by the following example, when n = 6n=6, k = 2k=2, and uplo = 'U'uplo='U':
On entry:
* * a13 a24 a35 a46
* a12 a23 a34 a45 a56
a11 a22 a33 a44 a55 a66
* * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66
On exit:
* * u13 u24 u35 u46
* u12 u23 u34 u45 u56
u11 u22 u33 u44 u55 u66
* * u13 u24 u35 u46 * u12 u23 u34 u45 u56 u11 u22 u33 u44 u55 u66
Similarly, if uplo = 'L'uplo='L' the format of ab is as follows:
On entry:
a11 a22 a33 a44 a55 a66
a21 a32 a43 a54 a65 *
a31 a42 a53 a64 * *
a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 * a31 a42 a53 a64 * *
On exit:
l11 l22 l33 l44 l55 l66
l21 l32 l43 l54 l65 *
l31 l42 l53 l64 * *
l11 l22 l33 l44 l55 l66 l21 l32 l43 l54 l65 * l31 l42 l53 l64 * *
Array elements marked * * need not be set and are not referenced by the function.
Assuming that nknk, the total number of floating point operations required to solve the equations AX = BAX=B is approximately n(k + 1)2n(k+1)2 for the factorization and 4nkr4nkr 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_linsys_real_posdef_band_solve (f04bf) is nag_linsys_complex_posdef_band_solve (f04cf).

Example

function nag_linsys_real_posdef_band_solve_example
uplo = 'U';
kd = int64(1);
ab = [5.495816452771857e+222, 2.68, -2.39, -2.22;
     5.49, 5.63, 2.6, 5.17];
b = [22.09, 5.1;
     9.31, 30.81;
     -5.24, -25.82;
     11.83, 22.9];
[abOut, bOut, rcond, errbnd, ifail] = nag_linsys_real_posdef_band_solve(uplo, kd, ab, b)
 

abOut =

  1.0e+222 *

    5.4958    0.0000   -0.0000   -0.0000
    0.0000    0.0000    0.0000    0.0000


bOut =

    5.0000   -2.0000
   -2.0000    6.0000
   -3.0000   -1.0000
    1.0000    4.0000


rcond =

    0.0135


errbnd =

   8.2325e-15


ifail =

                    0


function f04bf_example
uplo = 'U';
kd = int64(1);
ab = [5.495816452771857e+222, 2.68, -2.39, -2.22;
     5.49, 5.63, 2.6, 5.17];
b = [22.09, 5.1;
     9.31, 30.81;
     -5.24, -25.82;
     11.83, 22.9];
[abOut, bOut, rcond, errbnd, ifail] = f04bf(uplo, kd, ab, b)
 

abOut =

  1.0e+222 *

    5.4958    0.0000   -0.0000   -0.0000
    0.0000    0.0000    0.0000    0.0000


bOut =

    5.0000   -2.0000
   -2.0000    6.0000
   -3.0000   -1.0000
    1.0000    4.0000


rcond =

    0.0135


errbnd =

   8.2325e-15


ifail =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013