Math Problem Statement
- Find the root of x2-2=0 by using
a. Newton Raphson method( ) where Xo=1.5
b. Secant formula () Xo=1.45 X1=1.5
Solution
Let's solve the problem step by step for each method.
(a) Newton-Raphson Method
The Newton-Raphson formula is:
Here, and .
Iteration Steps:
-
Start with :
-
Next iteration ():
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Next iteration ():
The root converges to after sufficient iterations.
(b) Secant Method
The Secant formula is:
Given and :
Iteration Steps:
-
Compute and :
-
Find :
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Compute :
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Find :
The root converges to after sufficient iterations.
Final Results:
- Newton-Raphson root:
- Secant method root:
Would you like a detailed explanation of any part or need further assistance?
Here are 5 related questions to explore:
- Why does the Newton-Raphson method converge faster than the Secant method?
- What are the conditions for convergence of these iterative methods?
- How can the initial guesses influence the convergence of the Secant method?
- What happens if the derivative in the Newton-Raphson method is zero at some point?
- Can you apply these methods to find the root of a cubic polynomial?
Tip: Always check if the function and its derivative behave well around the initial guesses to avoid divergence in iterative methods.
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Math Problem Analysis
Mathematical Concepts
Numerical Methods
Root Finding
Iteration Methods
Formulas
Newton-Raphson formula: x_(n+1) = x_n - f(x_n)/f'(x_n)
Secant formula: x_(n+1) = x_n - f(x_n) * (x_n - x_(n-1)) / (f(x_n) - f(x_(n-1)))
Theorems
Convergence of Newton-Raphson method
Convergence of Secant method
Suitable Grade Level
Undergraduate - First Year