# Numerical Methods Formulas

Numerical Methods is an important topic and is considered as difficult one by most of the people. To help all such people we have jotted the Numerical Methods Formulas Complete List here. Avail them during your work and simplify the calculation part. Formula Collection on Numerical Methods present here will definitely be of great help during your homework or exam preparation.

## Numerical Methods Formulae Collection

If you are looking for ways on simplifying problems related to Numerical Methods have a look at the Numerical Methods Formulae prevailing. The Numerical Methods Formula Sheet existing will help you overcome the hassle of lengthy calculations. Understand the concept of Numerical Methods easily taking the help of the Formula Cheat Sheet & Tables given.

**1. Iterative method of Solving Equations**

(i) Successive Bisection method

If f(x) is a continuous function in the interval [a, b] and f(a), f(b) are of opposite sign, then there exist at least one value of x say a ∈ (a, b) such that f(α) = 0 and a < α < b.

Working Rule

- Find f(a) and f(b).
- Let f(a) be negative and f(b) be positive then take \(\alpha=\frac{a+b}{2}\).
- If f(α) = 0, then a is a required root. If f(α) > 0 then roots lie between a and α. If f(α) < 0 then roots lies between α and b.
- Repeat the process until we get the root correct up to desired level of accuracy.

(ii) False Position method (Regula- falsi method)

If the root of f(x) = 0 belongs to the interval (x_{0}, x_{1}) and f(x_{0}), f(x_{1}) are of opposite sign (say f(x_{0}) < 0, f(x_{1}) > 0), then

x_{2} = x_{0} – \(\frac{\left(x_{1}-x_{0}\right) f\left(x_{0}\right)}{f\left(x_{1}\right)-f\left(x_{0}\right)}\)

Working rule

- Calculate f(x
_{0}) and f(x_{1}), if these are of opposite sign then the root lies between x_{0}and x_{1}. - Calculate x
_{2}by the above formula. - Now if f(x
_{2}) = 0, then x_{2}is the required root. - If f(x
_{2}) is negative, then the root lies in (x_{2}, x_{1}) - If f(x
_{2}) is positive, then the root lies in (x_{0}, x_{2}).

(iii) Newton- Raphson method

If f(x) = 0 and f(a) and f(b) are of opposite sign, then to find the root in interval (a, b).

- Find |f (a)|, |f (b)|. If |f(a)| < |f (b)| then assume a = x
_{0}otherwise b = x_{0 } - Find x
_{1}= x_{0}– \(\frac{f\left(x_{0}\right)}{f^{\prime}\left(x_{0}\right)}\) - If f(x
_{1}) = 0, then x_{1}is the required root, otherwise by taking x_{1}as starting point find x_{2}

⇒ x_{2} = x_{1} – \(\frac{f\left(x_{1}\right)}{f^{\prime}\left(x_{1}\right)}\)

This process is continued till we get a value for the root up to the desired level of accuracy.

**2. Trapezoidal rule**

Let y = f(x) be a function defined on [a, b] which is subdivided into n equal sub intervals each of width h so that b – a = nh.

\(\int_{a}^{b}\)f(x)dx = \(\frac{h}{2}\)[(y_{0} + y_{n}) + 2(y_{1} + y_{2} + ……. + y_{n-1})]

where n is any positive integer and yr is the value of f(x) for x = a + rh.

**3. Simpson’s one third rule**

Let y = f(x) be a function defined in the interval [a, b] which is divided into n (an even number) equal parts of width h so that b – a = nh and yr, is the value of f(x) for x = a + rh, Then

\(\int_{a}^{b}\)f(x)dx = \(\frac{h}{3}\)[(y_{0} + y_{n}) + 4(y_{1} + y_{3} + ……. + y_{n-1}) + 2(y_{2} + y_{4} + ……. + y_{n-2})]