# 'Change of Sign Method'

Mathematics Coursework

Numerical Methods can be used to solve those equations which cannot be solved analytically.

One such method is the ‘Change of Sign Method’, focusing on Decimal Search.

Decimal search is based on the principal that f(x) changes sign as a curve passes through the x axis at a root.

Above the x axis, f(x) is positive and below the x axis f(x) is negative.

Using this principal, values of f(x) are initially found and consecutive values are found in increments of 0.1 until a change in sign is found. If further accuracy is required, consecutives values could then be found in increments of 0.01 and 0.001, until a change of accuracy is found, therefore giving one root of the equation.

Evidence

Equation: y=0.6938x3 – 0.9157x2 – 1.421x + 1.671

Graph:

Table of results

First Interval

The table shows that the sign changes within the interval 1.6 to 1.7. This tell us that the roots lie somewhere within this interval;

Therefore the root is 1.65 ± 0.05

Second Interval

The table of values shows that the root lies in the interval [1.68, 1.69]. Therefore the root of the equation is 1.685 ± 0.05

Third Interval

Therefore, we can say that the one root of the equation;

y=0.6938x3 – 0.9157x2 – 1.421x + 1.671

is 1.6885 ± 0.0005

However, decimal search cannot be used in all circumstances. This can be demonstrated using the equation:

y=0.6918x3 – 0.2159x2 – 3.019x + 2.77

Graph

If we zoom in, the roots of the graph can be seen more clearly:

Table of Values

The graph does not display any change of sign, implying that there are no roots. However, it is evident from the graph that there are two roots (see magnified version of graph). However, as the two roots are so close together, the decimal search method does not detect this and so in this instance, this method cannot be used to help solve the equation y=0.6918x3 – 0.2159x2 – 3.019x + 2.77

Newton Raphson ...