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Maximum Marks: 30
Due Date: 12 May, 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.
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muller method konsa ha
Muller's method is a generalization of the secant method. Instead of starting with two initial values and then joining them with a straight line in secant method, Mullers method starts with three initial approximations to the root and then join them with a second degree polynomial (a parabola), then the quadratic formula is used to find a root of the quadratic for the next approximation. That is if x0, x1 and x 2 are the initial approximations then x3 is obtained by solving the quadratic which is obtained by means of x0, x 1 and x2. Then two values among x0, x1 and x2 which are close to x3are chosen for the next iteration
x3 = x2 + z ( * )
|where z =||-2c|
a = D1/D2 , b = D 2 /D and c = f(x2)
D = h0h1(h0-h1) , D1 = (f0-c)h1-(f1-c)h0 , D2 = (f1 -c)h02 - (f0-c)h12
h0 = x0-x2 , h1 = x1-x2
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)
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, . . .
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koi es asgnmnt ka sol nhiiiiiiiiiiiiiiiiiiiiii day ga kya
Numerical methods for solving either algebraic or transcendental equation are classified
into two groups
Those methods which do not require any information about the initial approximation of
root to start the solution are known as direct methods.
(All these methods do not require any type of initial approximation.)
These methods require an initial approximation to start.
neton raphson method ka solution k ly b guid kr dain
What is root?
x + 7= 0 is a linear equation
x = -7 -7 is root of equation x+7= 0
x-3=2 is a linear equation
x=5 5 is root of equation x-3=2
Regula-Falsi method (Method of false position)
Here we choose two points xn and n 1 x − such that 1 ( ) ( ) n n f x and f x− have opposite signs.
Intermediate value property suggests that the graph of the y=f(x) crosses the x-axis
between these two points and therefore, a root lies between these two points.