MTH603-Numerical Analysis Assignments Fall 2014 Close Date 12 May, 2014

[education-2014]

MTH603-Numerical Analysis Assignments Discussion and Solution Fall 2014

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Question: 1 Marks: 10

The roots of the following equation are and. Using Muller method approximates the root upto four decimal places.


Question: 2 Marks: 10

Using Newton-Raphson method approximates the root of the following equation in the interval upto the three decimal places.

Note: All the calculations should be in radian.

Question: 3 Marks: 10

Using Regula Falsi method approximates the root of the following equation upto four decimal places.

Note: The following graph of above equation can be very helpful in finding the interval of the root. So, I am not mentioning the interval there as mentioned above. So, try to find the appropriate interval then apply the method. Otherwise, the convergence would be very slow.

[Vuhelper]

Numerical methods for solving either algebraic or transcendental equation are classified

into two groups
1. Direct Methods
2. Iterative Methods

Direct Methods:

Those methods which do not require any information about the initial approximation of

root to start the solution are known as direct methods.
●
1. Graefee root squaring method
2. Gauss elimination method
3. Gauss Jordan method

(All these methods do not require any type of initial approximation.)



Iteratative Methods:

These methods require an initial approximation to start.
1. Bisection method
2. Newton raphson method
3. Secant method
4. jacobie method
5. Muller’s Method

[Entboy]

Algorithm - Muller's Scheme
Given an equation f(x) = 0
Let the initial guesses be x0, x1 and x2
Let xi = x2, xi-1 = x1 and xi-2 = x0
Compute f(xi-2), f(xi-1) and f(xi)
Do
Compute
h = x - xi, hi = xi - xi-1 and hi-1 = xi-1 - xi-2
li = hi / hi-1, di = 1 + li
gi = li2 fi-2 - di2 fi-1 + ( li + di ) fi
ci = li (li fi-2 - di fi-1 + fi )
li+1 = min ( -2di fi / (gi ± Ö(gi2- 4di fici) ) )
(take minimum in absolute sense)
Then xi+1 = xi + ( xi - xi-1 ) li+1 , i = 2, 3, 4, . . .