is a symmetric positive definite variable-bandwidth matrix, which has previously been factorized by nag_matop_real_vband_posdef_fac (f01mc). Related systems may also be solved.

Syntax

[x, ifail] = f04mc(al, d, nrow, b, iselct, 'n', n, 'lal', lal, 'ir', ir)

[x, ifail] = nag_linsys_real_posdef_vband_solve(al, d, nrow, b, iselct, 'n', n, 'lal', lal, 'ir', ir)

Description

The normal use of this function is the solution of the systems

A X = B

, following a call of nag_matop_real_vband_posdef_fac (f01mc) to determine the Cholesky factorization

A = L D L^{T}

of the symmetric positive definite variable-bandwidth matrix

A

However, the function may be used to solve any one of the following systems of linear algebraic equations:

1.	$L D L^{T} X = B$ (usual system),
2.	$L D X = B$ (lower triangular system),
3.	$D L^{T} X = B$ (upper triangular system),
4.	$L L^{T} X = B$
5.	$L X = B$ (unit lower triangular system),
6.	$L^{T} X = B$ (unit upper triangular system).

L

denotes a unit lower triangular variable-bandwidth matrix of order

n

D

a diagonal matrix of order

n

, and

B

a set of right-hand sides.

The matrix

L

is represented by the elements lying within its envelope, i.e., between the first nonzero of each row and the diagonal (see Example for an example). The width

nrow (i)

of the

i

th row is the number of elements between the first nonzero element and the element on the diagonal inclusive.

References

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

Parameters

Compulsory Input Parameters

1: $al (lal)$ – double array

The elements within the envelope of the lower triangular matrix

L

, taken in row by row order, as returned by nag_matop_real_vband_posdef_fac (f01mc). The unit diagonal elements of

L

must be stored explicitly.

2: $d (:)$ – double array

The dimension of the array d must be at least

1

iselct \geq 4

, and at least

n

otherwise

The diagonal elements of the diagonal matrix

D

. d is not referenced if

iselct \geq 4

3: $nrow (n)$ – int64int32nag_int array

nrow (i)

must contain the width of row

i

L

, i.e., the number of elements between the first (leftmost) nonzero element and the element on the diagonal, inclusive.

Constraint:

1 \leq nrow (i) \leq i

4: $b (ldb, ir)$ – double array

ldb, the first dimension of the array, must satisfy the constraint

ldb \geq n

The

n

r

right-hand side matrix

B

. See also Further Comments.

5: $iselct$ – int64int32nag_int scalar

Must specify the type of system to be solved, as follows:

$iselct = 1$: Solve $L D L^{T} X = B$ .
$iselct = 2$: Solve $L D X = B$ .
$iselct = 3$: Solve $D L^{T} X = B$ .
$iselct = 4$: Solve $L L^{T} X = B$ .
$iselct = 5$: Solve $L X = B$ .
$iselct = 6$: Solve $L^{T} X = B$ .

Constraint:

iselct = 1

2

3

4

5

6

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array nrow and the first dimension of the array b. (An error is raised if these dimensions are not equal.)
$n$ , the order of the matrix $L$ .

Constraint: $n \geq 1$ .
2: $lal$ – int64int32nag_int scalar: Default: the dimension of the array al.
The dimension of the array al.

Constraint: $lal \geq nrow (1) + nrow (2) + \dots + nrow (n)$ .
3: $ir$ – int64int32nag_int scalar: Default: the second dimension of the array b.
$r$ , the number of right-hand sides.

Constraint: $ir \geq 1$ .

Output Parameters

1: $x (ldx, ir)$ – double array: The $n$ by $r$ solution matrix $X$ . See also Further Comments.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$n < 1$ ,
or	for some $i$ , $nrow (i) < 1$ or $nrow (i) > i$ ,
or	$lal < nrow (1) + nrow (2) + \dots + nrow (n)$ .

$ifail = 2$

On entry,	$ir < 1$ ,
or	$ldb < n$ ,
or	$ldx < n$ .

$ifail = 3$

On entry,	$iselct < 1$ ,
or	$iselct > 6$ .

$ifail = 4$: The diagonal matrix $D$ is singular, i.e., at least one of the elements of d is zero. This can only occur if $iselct \leq 3$ .

$ifail = 5$: At least one of the diagonal elements of $L$ is not equal to unity.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The usual backward error analysis of the solution of triangular system applies: each computed solution vector is exact for slightly perturbed matrices

L

and

D

, as appropriate (see pages 25–27 and 54–55 of Wilkinson and Reinsch (1971)).

Further Comments

The time taken by nag_linsys_real_posdef_vband_solve (f04mc) is approximately proportional to

p r

, where

p = nrow (1) + nrow (2) + \dots + nrow (n)

Example

This example solves the system of equations

A X = B

, where

A = (\begin{array}{r} 1 & 2 & 0 & 0 & 5 & 0 \\ 2 & 5 & 3 & 0 & 14 & 0 \\ 0 & 3 & 13 & 0 & 18 & 0 \\ 0 & 0 & 0 & 16 & 8 & 24 \\ 5 & 14 & 18 & 8 & 55 & 17 \\ 0 & 0 & 0 & 24 & 17 & 77 \end{array}) and B = (\begin{array}{r} 6 & - 10 \\ 15 & - 21 \\ 11 & - 3 \\ 0 & 24 \\ 51 & - 39 \\ 46 & 67 \end{array})

Here

A

is symmetric and positive definite and must first be factorized by nag_matop_real_vband_posdef_fac (f01mc).

Open in the MATLAB editor: f04mc_example

function f04mc_example


fprintf('f04mc example results\n\n');

a = [1;
     2;     5;
            3;     13;
                           16;
     5;    14;     18;      8;     55;
                   24;     17;     77];
nrow = [int64(1);  2; 2; 1; 5; 3];

% Factorize
[L, D, ifail] = f01mc(a, nrow);

% Solve Ax = b
b = [  6.0 -10.0;
      15.0 -21.0;
      11.0  -3.0;
       0.0  24.0;
      51.0 -39.0;
      46.0  67.0];
iselct = int64(1);
[x, ifail] = f04mc(L, D, nrow, b, iselct);

disp('Solution');
disp(x);

f04mc example results

Solution
    -3     4
     2    -2
    -1     3
    -2     1
     1    -2
     1     1

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox