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NAG Toolbox

NAG Toolbox: nag_lapack_dptsv (f07ja)

Purpose

nag_lapack_dptsv (f07ja) computes the solution to a real system of linear equations
AX = B ,
AX=B ,
where AA is an nn by nn symmetric positive definite tridiagonal matrix, and XX and BB are nn by rr matrices.

Syntax

[d, e, b, info] = f07ja(d, e, b, 'n', n, 'nrhs_p', nrhs_p)
[d, e, b, info] = nag_lapack_dptsv(d, e, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dptsv (f07ja) factors AA as A = LDLTA=LDLT. The factored form of AA is then used to solve the system of equations.

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:     d( : :) – double array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
The nn diagonal elements of the tridiagonal matrix AA.
2:     e( : :) – double array
Note: the dimension of the array e must be at least max (1,n1)max(1,n-1).
The (n1)(n-1) subdiagonal elements of the tridiagonal matrix AA.
3:     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 dimension of the array d.
nn, 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

ldb

Output Parameters

1:     d( : :) – double array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
The nn diagonal elements of the diagonal matrix DD from the factorization A = LDLTA=LDLT.
2:     e( : :) – double array
Note: the dimension of the array e must be at least max (1,n1)max(1,n-1).
The (n1)(n-1) subdiagonal elements of the unit bidiagonal factor LL from the LDLTLDLT factorization of AA. (e can also be regarded as the superdiagonal of the unit bidiagonal factor UU from the UTDUUTDU factorization of AA.)
3:     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.
4:     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: n, 2: nrhs_p, 3: d, 4: e, 5: b, 6: ldb, 7: 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 is not positive definite, and the solution has not been computed. The factorization has not been completed unless i = ni=n.

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_dptsvx (f07jb) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_real_posdef_tridiag_solve (f04bg) solves Ax = b Ax=b  and returns a forward error bound and condition estimate. nag_linsys_real_posdef_tridiag_solve (f04bg) calls nag_lapack_dptsv (f07ja) to solve the equations.

Further Comments

The number of floating point operations required for the factorization of A A  is proportional to n n , and the number of floating point operations required for the solution of the equations is proportional to nr nr , where r r  is the number of right-hand sides.
The complex analogue of this function is nag_lapack_zptsv (f07jn).

Example

function nag_lapack_dptsv_example
d = [4;
     10;
     29;
     25;
     5];
e = [-2;
     -6;
     15;
     8];
b = [6;
     9;
     2;
     14;
     7];
[dOut, eOut, bOut, info] = nag_lapack_dptsv(d, e, b)
 

dOut =

     4
     9
    25
    16
     1


eOut =

   -0.5000
   -0.6667
    0.6000
    0.5000


bOut =

    2.5000
    2.0000
    1.0000
   -1.0000
    3.0000


info =

                    0


function f07ja_example
d = [4;
     10;
     29;
     25;
     5];
e = [-2;
     -6;
     15;
     8];
b = [6;
     9;
     2;
     14;
     7];
[dOut, eOut, bOut, info] = f07ja(d, e, b)
 

dOut =

     4
     9
    25
    16
     1


eOut =

   -0.5000
   -0.6667
    0.6000
    0.5000


bOut =

    2.5000
    2.0000
    1.0000
   -1.0000
    3.0000


info =

                    0



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Chapter Introduction
NAG Toolbox

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