Numerical Solutions of Algebraic Equations - CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

NUMERICAL SOLUTIONS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS

Aim: To find a root of f(x) = 0

Algebraic Equation: An Equation which contains algebraic terms is called as an algebraic Equation.

Example: x² + x + 1 = 0, Here Highest power of x is finite. So it is an algebraic Equation.

Transcendental Equation: An equation which contains trigonometric ratios, exponential function and logarithmic functions is called as a Transcendental Equation.

Example: e^x + 2 = 0, sinx + 1 = 0, log(l + x) = 0 etc.

• Directive Methods: The methods which are used to find solutions of given equations in the direct process is called as directive methods.
  
  Example: Synthetic division, remainder theorem, Factorization method etc
  
  Note: By using Directive Methods, it is possible to find exact solutions of the given equation.
   
• Iterative Methods (Indirect Methods): The methods which are used to find solutions of the given equation in some indirect process is called as Iterative Methods
  
  Note: By using Iterative methods, it is possible to find approximate solution of the given equation and also it is possible to find single solution of the given equation at the same time.

To find a root of the given equation, we have following methods

• Bisection Method
• The Method of false position (Or) Regular folsi Method
• Iteration Method (Successive approximation Method)
• Newton Raphson Method.

Bisection Method

Consider f(x) = 0 be the given equation.

Let us choose the constants and in such a way that 

 are of opposite signs) i.e. 

(or)

Then the curve y =  f(x) crosses - in between a and b, so that there exists a root of the given equation in between a and b.

Let us define initial approximation 

Now, If  then x₀ is root of the given equation.

If f(x₀) = ≠ 0 then either f(x₀) < 0 (or) f(x₀) > 0

Here, the logic is that, we have to select first negative or first positive from bottom approximations only, but not from top. i.e. the approximation which we have recently found should be selected.

If f(x₁) = 0, then x₁ is root of the given equation. Otherwise repeat above process until we obtain solution of the given equation.

Regular Folsi Method (Or) Method of False position

Let us consider that y = f(x) be the given equation.

Let us choose two points a and b in such a way that f(a) and f(b) are of opposite signs. i.e. f(x₀).f(x₁) < 0

Consider f(a) > 0,f(b) < 0, so that the graph y = f(x) crosses .Y-axis in between a and b. then, there exists a root of the given equation in between a and b.

Let us define a straight line joining the points A(a,f(a)), B(b,f(b)) , then the equation of straight line is given by 

This method consists of replacing the part of the curve by means of a straight line and then takes the point of intersection of the straight line with -axis, which give an approximation to the required root. 

In the present case, it can be obtained by substituting in equation I, and it is denoted by x₀.

Now, any point on .Y-axis ⇒ y = 0 and let initial approximation be x₀ i.e. x = x₀

If f(x₀) = 0 , then x₀ is root of the given equation.

If fix₀) ≠ 0 , then either f(x₀) < 0 (or) f(x₀) > 0

Case (i): Let us consider that f(x₀) < 0

We know that f(a) > 0 and f(x₀) < 0

Hence root of the given equation lies between a and x₀.

In order to obtain next approximation x₁, replace b with x₀ in equation (II)

Case (ii): Let us consider that f(x₀) > 0

We know that f(b) < 0 and f(x₀) > 0

Hence root of the given equation lies between b and x₀.

In order to obtain next approximation x₁, replace a with x₀ in equation (II)

If f(x₁) = 0, then x₁ is root of the given equation. Otherwise repeat above process until we obtain a root of the given equation to the desired accuracy.

Newton Raphson Method

Let us consider that f(x) = 0 be the given equation.

Let us choose initial approximation to be x₀.

Let us assume that x₁ = x₀ + h be exact root of f(x) = 0 where h << 0, so that f(x₀ + h) = 0

Expanding above relation by means of Taylor's expansion method, we get

Since is h very small, h and higher powers of h are neglected. Then the above relation becomes 

If f(x₁)= 0, then is x₁ root of the given equation. Otherwise repeat above process until we obtain a root of the given equation to the desired accuracy.

Successive approximations are given by

Iteration Method

Let us consider that f(x) = 0 be the given equation.

Let us choose initial approximation to be x₀.

Rewrite f(x) = 0 as x = φ(x) such that |φ'(x₀)| < 0.

Then successive approximations are given by

Repeat the above process until we get successive approximation equal, which will gives the required root of the given equation.