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NAG Toolbox

NAG Toolbox: nag_linsys_real_tridiag_solve (f04bc)

Purpose

nag_linsys_real_tridiag_solve (f04bc) computes the solution to a real system of linear equations AX = BAX=B, where AA is an nn by nn tridiagonal matrix 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

[dl, d, du, du2, ipiv, b, rcond, errbnd, ifail] = f04bc(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)
[dl, d, du, du2, ipiv, b, rcond, errbnd, ifail] = nag_linsys_real_tridiag_solve(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)

Description

The LULU decomposition with partial pivoting and row interchanges is used to factor AA as A = PLUA=PLU, where PP is a permutation matrix, LL is unit lower triangular with at most one nonzero subdiagonal element, and UU is an upper triangular band matrix with two superdiagonals. The factored form of AA is then used to solve the system of equations AX = BAX=B.
Note that the equations ATX = BATX=B may be solved by interchanging the order of the arguments du and dl.

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:     dl( : :) – double array
Note: the dimension of the array dl must be at least max (1,n1)max(1,n-1).
Must contain the (n1)(n-1) subdiagonal elements of the matrix AA.
2:     d( : :) – double array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
Must contain the nn diagonal elements of the matrix AA.
3:     du( : :) – double array
Note: the dimension of the array du must be at least max (1,n1)max(1,n-1).
Must contain the (n1)(n-1) superdiagonal elements of the matrix AA
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 dimension of the array d.
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

ldb

Output Parameters

1:     dl( : :) – double array
Note: the dimension of the array dl must be at least max (1,n1)max(1,n-1).
If ifail0ifail0, dl stores the (n1)(n-1) multipliers that define the matrix LL from the LULU factorization of AA.
2:     d( : :) – double array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
If ifail0ifail0, d stores the nn diagonal elements of the upper triangular matrix UU from the LULU factorization of AA.
3:     du( : :) – double array
Note: the dimension of the array du must be at least max (1,n1)max(1,n-1).
If ifail0ifail0, du stores the (n1)(n-1) elements of the first superdiagonal of UU.
4:     du2(n2n-2) – double array
If ifail0ifail0, du2 returns the (n2)(n-2) elements of the second superdiagonal of UU.
5:     ipiv(n) – int64int32nag_int array
If ifail0ifail0, the pivot indices that define the permutation matrix PP; at the iith step row ii of the matrix was interchanged with row ipiv(i)ipivi. ipiv(i)ipivi will always be either ii or (i + 1)(i+1); ipiv(i) = iipivi=i indicates a row interchange was not required.
6:     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.
7:     rcond – double scalar
If no constraints are violated, an estimate of the reciprocal of the condition number of the matrix AA, computed as rcond = 1 / (A1A11)rcond=1/(A1A-11).
8:     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.
9:     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 2 × n2×n. In this case the factorization and the solution XX have been computed, but rcond and errbnd have not been computed.
W ifail > 0andifailNifail>0andifailN
If ifail = iifail=i, uiiuii is exactly zero. The factorization has been completed, but the factor UU is exactly singular, so the solution could not be 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) = A11 A1 κ(A) = A-11 A1 , the condition number of AA with respect to the solution of the linear equations. nag_linsys_real_tridiag_solve (f04bc) uses the approximation E1 = εA1E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

Further Comments

The total number of floating point operations required to solve the equations AX = BAX=B is proportional to nrnr. 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_tridiag_solve (f04bc) is nag_linsys_complex_tridiag_solve (f04cc).

Example

function nag_linsys_real_tridiag_solve_example
dl = [3.4;
     3.6;
     7;
     -6];
d = [3;
     2.3;
     -5;
     -0.9;
     7.1];
du = [2.1;
     -1;
     1.9;
     8];
b = [2.7, 6.6;
     -0.5, 10.8;
     2.6, -3.2;
     0.6, -11.2;
     2.7, 19.1];
[dlOut, dOut, duOut, du2, ipiv, bOut, rcond, errbnd, ifail] = ...
    nag_linsys_real_tridiag_solve(dl, d, du, b)
 

dlOut =

    0.8824
    0.0196
    0.1401
   -0.0148


dOut =

    3.4000
    3.6000
    7.0000
   -6.0000
   -1.0154


duOut =

    2.3000
   -5.0000
   -0.9000
    7.1000


du2 =

   -1.0000
    1.9000
    8.0000


ipiv =

                    2
                    3
                    4
                    5
                    5


bOut =

   -4.0000    5.0000
    7.0000   -4.0000
    3.0000   -3.0000
   -4.0000   -2.0000
   -3.0000    1.0000


rcond =

    0.0108


errbnd =

   1.0297e-14


ifail =

                    0


function f04bc_example
dl = [3.4;
     3.6;
     7;
     -6];
d = [3;
     2.3;
     -5;
     -0.9;
     7.1];
du = [2.1;
     -1;
     1.9;
     8];
b = [2.7, 6.6;
     -0.5, 10.8;
     2.6, -3.2;
     0.6, -11.2;
     2.7, 19.1];
[dlOut, dOut, duOut, du2, ipiv, bOut, rcond, errbnd, ifail] = f04bc(dl, d, du, b)
 

dlOut =

    0.8824
    0.0196
    0.1401
   -0.0148


dOut =

    3.4000
    3.6000
    7.0000
   -6.0000
   -1.0154


duOut =

    2.3000
   -5.0000
   -0.9000
    7.1000


du2 =

   -1.0000
    1.9000
    8.0000


ipiv =

                    2
                    3
                    4
                    5
                    5


bOut =

   -4.0000    5.0000
    7.0000   -4.0000
    3.0000   -3.0000
   -4.0000   -2.0000
   -3.0000    1.0000


rcond =

    0.0108


errbnd =

   1.0297e-14


ifail =

                    0



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