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NAG Toolbox

NAG Toolbox: nag_lapack_dpbsv (f07ha)

Purpose

nag_lapack_dpbsv (f07ha) computes the solution to a real system of linear equations
AX = B ,
AX=B ,
where AA is an nn by nn symmetric positive definite band matrix of bandwidth (2kd + 1) ( 2 kd + 1 )  and XX and BB are nn by rr matrices.

Syntax

[ab, b, info] = f07ha(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, b, info] = nag_lapack_dpbsv(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dpbsv (f07ha) uses the Cholesky decomposition 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, and LL is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as AA. 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
If uplo = 'U'uplo='U', the upper triangle of AA is stored.
If uplo = 'L'uplo='L', the lower triangle of AA is stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     kd – int64int32nag_int scalar
kdkd, the number of superdiagonals of the matrix AA if uplo = 'U'uplo='U', or the number of subdiagonals if uplo = 'L'uplo='L'.
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 upper or lower triangle of the symmetric band matrix AA.
The matrix is stored in rows 11 to kd + 1kd+1, more precisely,
  • if uplo = 'U'uplo='U', the elements of the upper triangle of AA within the band must be stored with element AijAij in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ijabkd+1+i-jj​ for ​max(1,j-kd)ij;
  • if uplo = 'L'uplo='L', the elements of the lower triangle of AA within the band must be stored with element AijAij in ab(1 + ij,j)​ for ​jimin (n,j + kd).ab1+i-jj​ for ​jimin(n,j+kd).
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 right-hand side matrix BB.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b The second dimension of the array ab.
nn, the number of linear equations, i.e., the order of the matrix AA.
Constraint: n0n0.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
rr, the number of right-hand sides, 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 INFO = 0INFO=0, the triangular factor UU or LL from the Cholesky factorization A = UTUA=UTU or A = LLTA=LLT of the band matrix AA, 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 INFO = 0INFO=0, the nn by rr solution matrix XX.
3:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: kd, 4: nrhs_p, 5: ab, 6: ldab, 7: b, 8: ldb, 9: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
  INFO > 0INFO>0
If info = iinfo=i, the leading minor of order ii of AA is not positive definite, so the factorization could not be completed, and the solution has not 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 ε ε  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) = A11 A1 κ(A) = A-11 A1 , the condition number of A A  with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
nag_lapack_dpbsvx (f07hb) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_real_posdef_band_solve (f04bf) solves Ax = b Ax=b  and returns a forward error bound and condition estimate. nag_linsys_real_posdef_band_solve (f04bf) calls nag_lapack_dpbsv (f07ha) to solve the equations.

Further Comments

When nk nk , the total number of floating point operations is approximately n(k + 1)2 + 4nkr n(k+1)2+4nkr , where k k  is the number of superdiagonals and r r  is the number of right-hand sides.
The complex analogue of this function is nag_lapack_zpbsv (f07hn).

Example

function nag_lapack_dpbsv_example
uplo = 'U';
kd = int64(1);
ab = [0, 2.68, -2.39, -2.22;
     5.49, 5.63, 2.6, 5.17];
b = [22.09;
     9.31;
     -5.24;
     11.83];
[abOut, bOut, info] = nag_lapack_dpbsv(uplo, kd, ab, b)
 

abOut =

         0    1.1438   -1.1497   -1.9635
    2.3431    2.0789    1.1306    1.1465


bOut =

    5.0000
   -2.0000
   -3.0000
    1.0000


info =

                    0


function f07ha_example
uplo = 'U';
kd = int64(1);
ab = [0, 2.68, -2.39, -2.22;
     5.49, 5.63, 2.6, 5.17];
b = [22.09;
     9.31;
     -5.24;
     11.83];
[abOut, bOut, info] = f07ha(uplo, kd, ab, b)
 

abOut =

         0    1.1438   -1.1497   -1.9635
    2.3431    2.0789    1.1306    1.1465


bOut =

    5.0000
   -2.0000
   -3.0000
    1.0000


info =

                    0



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Chapter Introduction
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