(d) Explain with suitable examples the concepts of
(i) singular system and
(ii) ill-conditional system of linear equations. (2 Marks)

d.i

A set of equations which is linear in all its unknowns has the special property, like a single linear equation, that it has what is effectively an analytical solution.

It is not usual to attempt to write out such a solution, for a system of more than two equations it becomes very cumbersome, but the systematic procedure which enables an analytical solution to be found can more conveniently be used to compute the numerical values of the unknowns.

These properties formally exist only for a set of equations which is linear in all its unknowns. A set of equations is linear if and only if all its unknowns appear only as first powers.

The solution of sets of linear equations is a standard problem and many computer programs are available to perform it

Two things may go wrong in the course of attempts to solve a set of linear equations.

1. The set of equations may not have a solution. For example, consider the following, all of which can be written as sets of equations, and indeed have coefficients and constants entered into one of the solvers, but none of which have valid solutions.
x = 3
2x = 4
The two equations are inconsistent.

2. x1 = 3
2x1 = 6
The equations are consistent, but if intended for solution in two unknowns, i.e. x1 and x2, the second equation provides no information additional to the first, and so there is no means of solving for x2.

Depending on the values on the RHS the two equations are either inconsistent or one is redundant. In neither case can a solution be found.

In all the above situations the equations are said to be linearly dependent. This occurs whenever one LHS is a multiple of that of another equation. More generally this occurs if any LHS can be constructed from any linear combination of another, i.e. by adding together multiples of right hand sides. The solution procedure will fail for such a set of equations regardless of the values on the RHS.
The problem is not always obvious:
x1 + x2 + x3 = 1
3 x1 -2 x2 + x3 = 2
5 x1 + 3 x3 = 2
The 3rd LHS is equal to twice the first plus the second.

The above are all examples of equations sets which are said to be singular. This depends only on the structure or numerical values of the coefficient matrix, which is also said to be singular.

d.ii

considering a graphic interpretation of the equations
3x1 + 6x2 = 12 ………. (1)
5x1 – 3x2 = 7 ………..(2)

In a practical computation, arithmetic cannot be performed exactly, so round-off errors will affect the answers. Their effect can be seen by considering a graphic interpretation of an equation: an equation of the form (1) represents a straight line on a graph relating x1 and x2, as in Figure 6. Any point on the line L1 shown has coordinates x1 and x2 that satisfy equation (1). Similarly, equation (2) represents another straight line shown as L2 in Figure 1d1. Any point on that line satisfies equation (2). Hence, the values of x1 and x2 that satisfy both (1) and (2) must lie on both lines, so that these values are the coordinates of the intersection of the two lines. If the lines are not parallel, they have a unique intersection, so that the equations have a unique solution.

Round-off errors in the computation have an effect similar to that of changing the coefficients of the equations. Hence, in the presence of round-off errors, a computer will work with a narrow region that can be thought of as a "thick line," as shown in Figure 1d1, rather than an ideal line. The coordinates of any point inside the shaded region satisfy the equation of the line within the accuracy of the errors introduced into the coefficients. A pair of equations represents two thick lines within the accuracy of their coefficients, and the solution determined by the pair of equations lies somewhere within the intersection of the two thick lines.

considering a graphic interpretation of the equations
3x + 2y =42
x + y =20

The equations represent a pair of lines that are nearly parallel, therefore the numerical problem represented by the pair of thick lines in Figure 1d3 does not have an accurately determined answer because the region of intersection is large. The problem is said to be ill-conditioned. Any slight changes in the coefficients can cause large changes in the answer, so any round-off errors in the numerical procedure may cause large changes in the answer.

The algorithmic solution to this is to reorder the equations during the elimination process to ensure that the Aii ( coefficients ) elements are large. This is called partial pivoting. Even with partial pivoting, solution errors for ill-conditioned problems can be large.