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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'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 x_{0}, x_{1} and x_{ 2} are the initial approximations then x_{3} is obtained by solving the quadratic which is obtained by means of x_{0}, x_{ 1} and x_{2}. Then two values among x_{0}, x_{1} and x_{2} which are close to x_{3}are chosen for the next iteration
x_{3} = x_{2} + z ( * )
where z = | -2c |
b±Ö( b^{2}-4ac) |
a = D_{1}/D_{2} , b = D_{ 2 }/D and c = f(x_{2})
D = h_{0}h_{1}(h_{0}-h_{1}) , D_{1} = (f_{0}-c)h_{1}-(f_{1}-c)h_{0 }, D_{2} = (f_{1} -c)h_{0}^{2} - (f_{0}-c)h_{1}^{2}
h_{0} = x_{0}-x_{2 }, h_{1 }= x_{1}-x_{2}
Given an equation f(x) = 0
Let the initial guesses be x_{0}, x_{1 }and x_{2}
Let x_{i} = x_{2}, x_{i-1} = x_{1 }and_{ }x_{i-2} = x_{0}
Compute f(x_{i-2}), f(x_{i-1}) and f(x_{i})
_{Do}
Compute
h = x - x_{i}, h_{i} = x_{i} - x_{i-1} and h_{i-1} = x_{i-1} - x_{i-2}
l_{i} = h_{i} / h_{i-1}, d_{i} = 1 + l_{i}
g_{i} = l_{i}^{2} f_{i-2} - d_{i}^{2} f_{i-1} + ( l_{i }+ d_{i} ) f_{i}
c_{i} = l_{i }(l_{i} f_{i-2} - d_{i} f_{i-1} + f_{i} )
l_{i+1} = min ( -2d_{i} f_{i} / (g_{i }± Ö(g_{i}^{2}- 4d_{i} f_{i}c_{i}) ) )
(take minimum in absolute sense)
Then x_{i+1} = x_{i} + ( x_{i } - _{ }x_{i-1 }) l_{i+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
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.
(All these methods do not require any type of initial approximation.)
Iteratative Methods:
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.
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