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 (x0, x1) and f(x0), f(x1) are of opposite sign (say f(x0) < 0, f(x1) > 0), then
x2 = x0 – $$\frac{\left(x_{1}-x_{0}\right) f\left(x_{0}\right)}{f\left(x_{1}\right)-f\left(x_{0}\right)}$$
Working rule

• Calculate f(x0) and f(x1), if these are of opposite sign then the root lies between x0 and x1.
• Calculate x2 by the above formula.
• Now if f(x2) = 0, then x2 is the required root.
• If f(x2) is negative, then the root lies in (x2, x1)
• If f(x2) is positive, then the root lies in (x0, x2).

(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 = x0 otherwise b = x0
• Find x1 = x0 – $$\frac{f\left(x_{0}\right)}{f^{\prime}\left(x_{0}\right)}$$
• If f(x1) = 0, then x1 is the required root, otherwise by taking x1 as starting point find x2

⇒ x2 = x1 – $$\frac{f\left(x_{1}\right)}{f^{\prime}\left(x_{1}\right)}$$
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 = $$\frac{h}{2}$$[(y0 + yn) + 2(y1 + y2 + ……. + yn-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 = $$\frac{h}{3}$$[(y0 + yn) + 4(y1 + y3 + ……. + yn-1) + 2(y2 + y4 + ……. + yn-2)]