## Regula Falsi Method

Regula Falsi Method In Numerical Method is the oldest method for finding the real root of an equation, and it is likely the same as the bisection method. In this method, we choose two points x_{0} and x_{1} such that f(x_{0}) and f(x_{1}) are of opposite signs. Since the graph of y = f(x) crosses the x-axis between these two points, a root must lie in between these points. Now, the equation of the chord joining the two points, [x_{0}. f(x_{0})] and [x_{1}, f(x_{1})] is:

The method consists in replacing the part of the curve between the points [x_{0}. f(x_{0})] and [x_{1}, f(x_{1})] by means of the chord joining these points, and taking the point of intersection of the chord with the x-axis as an approximation to the root. The point of intersection in the present case is given by putting y = 0 in ** (equation 1)**. Thus, we obtain

Hence the second approximation to the root of f(x) = 0 is given by

If now f(x_{2}) and f(x_{0}) are of opposite signs, then the root lies between x_{0} and x_{2}, and we replace x_{1}, with x_{2}, and obtain the next approximation. Otherwise, we replace x_{0} with x_{2} and generate the next approximation.

The procedure is repeated till the root is obtained to the desired accuracy. The Figure gives a graphical representation of the method.

### The Method Of Regula Falsi in Numerical Method

- This is the oldest method for finding the real root of an equation
- we choose two points x
_{0}and x_{1}such that f(x_{0}) and f(x_{1}) are of opposite signs - Since the graph of y = f(x) crosses the x-axis between these two points, a root must lie in between these points
- if the f(x) value is a positive value then the (x
_{1}) value is replaced by the (x_{2}) value and if the f(x) value is negative then the (x_{0}) value is replaced by (x_{1}). - if the f(x) value is a positive value then the f(x
_{1}) value is replaced by the f(x_{2}) value and if the f(x) value is negative then the f(x_{0}) value is replaced by f(x_{2}).

### Example:

**Finding the root of a polynomial f(x) = x ^{6} – x^{4} – x^{3} – 1=0**, correct up to 4 decimal places

**by using Regula falsi method.**

Let the function is continuous

F(x) = x^{6} – x^{4} – x^{3} – 1= 0

Let put x=0 in f(x)

Then f(x)=-1

Let put x=1 in f(x)

Then f(x)=-2

Let put x=2 in f(x)

Then f(x)=39

Let put x=1.4 in f(x)

Then f(x)=-0.5606

Let put x=1.41 in f(x)

Then f(x)=0.10228

Then we take the value x_{0}=(1.4) and x_{1}= (1.41)

Then we calculate the value of x_{2} using the regula false method. i.e:

X_{0} | X_{1} | F(x_{0}) | F(x_{1}) | X_{2} | F(x_{2}) |

1.4 | 1.41 | -0.05608 | 0.10228 | 1.40354 | -0.0009 |

1.40354 | 1.41 | -0.00097 | 0.10228 | 1.40360 | -0.00003 |

1.40360 | 1.41 | -0.00003 | 0.10228 | 1.40360 | -3.33932 |

From the above calculation, we come to a conclusion the approximation root of the function f(x) x^{6} – x^{4} – x^{3} – 1=0 is **1.4036** correct up to 4 decimal places by using Regula False method.

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