Question

7-find minimum point of ( ) 2 2 ( ) 0.1 0.1 0.1 x f x x x = + + + using i. bisection method ii. newton raphson method iii. golden search method iv. fibonnacci method

Answer 1

To find the minimum point of a given function, you can use different methods such as the bisection method, Newton-Raphson method, golden search method, or Fibonacci method.

Step-by-step explanation:

To find the minimum point of the function f(x) = 2x^2 + 0.1x + 0.1 using different methods, we can follow these steps:

i. Bisection Method:

1. Choose two initial points 'a' and 'b' such that f(a) and f(b) have opposite signs.
2. Calculate the midpoint 'c' between 'a' and 'b' using the formula c = (a + b) / 2.
3. Evaluate f(c).
4. If f(c) is close enough to 0 or reaches a desired level of accuracy, stop and consider c as the minimum point.
5. Otherwise, update 'a' or 'b' based on the sign of f(c) and repeat steps 2-5.

ii. Newton-Raphson Method:

1. Choose an initial guess 'x0'.
2. Calculate the derivative f'(x) of the function f(x).
3. Iterate using the formula xn+1 = xn - f(xn) / f'(xn).
4. Repeat the iteration until the value of f(x) is close enough to 0 or reaches the desired accuracy.

iii. Golden Search Method:

1. Choose an interval [a, b] that contains the minimum point.
2. Calculate the values of two additional points inside the interval using the Golden Section Ratio.
3. Compare the values of the function at these two points and update the interval accordingly.
4. Repeat step 2 and 3 until the interval becomes small enough or the desired accuracy is achieved.

iv. Fibonacci Method:

1. Choose an interval [a, b] that contains the minimum point.
2. Calculate the number of iterations 'n' based on the desired accuracy.
3. Find the values of the Fibonacci sequence up to the 'n'th term.
4. Calculate two new points using the Fibonacci numbers and update the interval accordingly.
5. Repeat step 3 and 4 until the desired accuracy is achieved.