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NAG Toolbox: nag_linsys_real_posdef_solve_1rhs (f04as)

Purpose

nag_linsys_real_posdef_solve_1rhs (f04as) calculates the accurate solution of a set of real symmetric positive definite linear equations with a single right-hand side, Ax = bAx=b, using a Cholesky factorization and iterative refinement.

Syntax

[a, c, ifail] = f04as(a, b, 'n', n)
[a, c, ifail] = nag_linsys_real_posdef_solve_1rhs(a, b, 'n', n)

Description

Given a set of real linear equations Ax = bAx=b, where AA is a symmetric positive definite matrix, nag_linsys_real_posdef_solve_1rhs (f04as) first computes a Cholesky factorization of AA as A = LLTA=LLT where LL is lower triangular. An approximation to xx is found by forward and backward substitution. The residual vector r = bAxr=b-Ax is then calculated using additional precision and a correction dd to xx is found by solving LLTd = rLLTd=r. xx is then replaced by x + dx+d, and this iterative refinement of the solution is repeated until machine accuracy is obtained.

References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

Parameters

Compulsory Input Parameters

1:     a(lda, : :) – double array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The upper triangle of the nn by nn positive definite symmetric matrix AA. The elements of the array below the diagonal need not be set.
2:     b(max (1,n)max(1,n)) – double array
Note: the dimension of the array b must be at least max (1,n)max(1,n).
The right-hand side vector bb.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the arrays a, b.
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda wk1 wk2

Output Parameters

1:     a(lda, : :) – double array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,n)ldamax(1,n).
The elements of the array below the diagonal are overwritten; the upper triangle of aa is unchanged.
2:     c(max (1,n)max(1,n)) – double array
The solution vector xx.
3:     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:
  ifail = 1ifail=1
The matrix AA is not positive definite, possibly due to rounding errors.
  ifail = 2ifail=2
Iterative refinement fails to improve the solution, i.e., the matrix AA is too ill-conditioned.
  ifail = 3ifail=3
On entry,n < 0n<0,
orlda < max (1,n)lda<max(1,n).

Accuracy

The computed solutions should be correct to full machine accuracy. For a detailed error analysis see page 39 of Wilkinson and Reinsch (1971).

Further Comments

The time taken by nag_linsys_real_posdef_solve_1rhs (f04as) is approximately proportional to n3n3.
The function must not be called with the same name for parameters b and c.

Example

function nag_linsys_real_posdef_solve_1rhs_example
a = [5, 7, 6, 5;
     7, 10, 8, 7;
     6, 8, 10, 9;
     5, 7, 9, 10];
b = [23;
     32;
     33;
     31];
[aOut, c, ifail] = nag_linsys_real_posdef_solve_1rhs(a, b)
 

aOut =

    5.0000    7.0000    6.0000    5.0000
    3.1305   10.0000    8.0000    7.0000
    2.6833   -0.8944   10.0000    9.0000
    2.2361         0    2.1213   10.0000


c =

     1
     1
     1
     1


ifail =

                    0


function f04as_example
a = [5, 7, 6, 5;
     7, 10, 8, 7;
     6, 8, 10, 9;
     5, 7, 9, 10];
b = [23;
     32;
     33;
     31];
[aOut, c, ifail] = f04as(a, b)
 

aOut =

    5.0000    7.0000    6.0000    5.0000
    3.1305   10.0000    8.0000    7.0000
    2.6833   -0.8944   10.0000    9.0000
    2.2361         0    2.1213   10.0000


c =

     1
     1
     1
     1


ifail =

                    0



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Chapter Introduction
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