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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dpttrs (f07je)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_dpttrs (f07je) computes the solution to a real system of linear equations AX=B , where A  is an n  by n  symmetric positive definite tridiagonal matrix and X  and B  are n  by r  matrices, using the LDLT  factorization returned by nag_lapack_dpttrf (f07jd).


[b, info] = f07je(d, e, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dpttrs(d, e, b, 'n', n, 'nrhs_p', nrhs_p)


nag_lapack_dpttrs (f07je) should be preceded by a call to nag_lapack_dpttrf (f07jd), which computes a modified Cholesky factorization of the matrix A  as
A=LDLT ,  
where L  is a unit lower bidiagonal matrix and D  is a diagonal matrix, with positive diagonal elements. nag_lapack_dpttrs (f07je) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form UTDU , where U  is a unit upper bidiagonal matrix.


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


Compulsory Input Parameters

1:     d: – double array
The dimension of the array d must be at least max1,n
Must contain the n diagonal elements of the diagonal matrix D from the LDLT factorization of A.
2:     e: – double array
The dimension of the array e must be at least max1,n-1
Must contain the n-1 subdiagonal elements of the unit lower bidiagonal matrix L. (e can also be regarded as the superdiagonal of the unit upper bidiagonal matrix U from the UTDU factorization of A.)
3:     bldb: – double array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,nrhs_p.
The n by r matrix of right-hand sides B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array b and the dimension of the array d.
n, the order of the matrix A.
Constraint: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs_p0.

Output Parameters

1:     bldb: – double array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,nrhs_p.
The n by r solution matrix X.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.


The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,  
E1 =OεA1  
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^ - x 1 x1 κA E1 A1 ,  
where κA = A-11 A1 , the condition number of A  with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Following the use of this function nag_lapack_dptcon (f07jg) can be used to estimate the condition number of A  and nag_lapack_dptrfs (f07jh) can be used to obtain approximate error bounds.

Further Comments

The total number of floating-point operations required to solve the equations AX=B  is proportional to nr .
The complex analogue of this function is nag_lapack_zpttrs (f07js).


This example solves the equations
AX=B ,  
where A  is the symmetric positive definite tridiagonal matrix
A = 4.0 -2.0 0.0 0.0 0.0 -2.0 10.0 -6.0 0.0 0.0 0.0 -6.0 29.0 15.0 0.0 0.0 0.0 15.0 25.0 8.0 0.0 0.0 0.0 8.0 5.0   and   B = 6.0 10.0 9.0 4.0 2.0 9.0 14.0 65.0 7.0 23.0 .  
function f07je_example

fprintf('f07je example results\n\n');

% Symmetric tridiagonal A stored as two diagonals
d = [ 4     10     29     25     5];
e = [-2     -6     15     8       ];

b = [ 6, 10;
      9,  4;
      2,  9;
     14, 65;
      7, 23];

% Factorize
[df, ef, info] = f07jd( ...
                        d, e);

[x, info] = f07je( ...
                   df, ef, b);


f07je example results

    2.5000    2.0000
    2.0000   -1.0000
    1.0000   -3.0000
   -1.0000    6.0000
    3.0000   -5.0000

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

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