Challenge: Implement and Compare Root-Finding Methods
You will implement two classic numerical methods for finding roots of equations of the form:
f(x)=0Because many equations cannot be solved analytically, numerical root-finding methods are widely used in scientific computing and engineering to approximate solutions.
Methods to Implement
Bisection Method
- Requires an interval ([a,b]) where the function changes sign.
- Repeatedly halves the interval to narrow down the root.
- Guaranteed to converge, but relatively slow compared to other methods.
Newton-Raphson Method
- Uses the derivative of the function.
- Starts from an initial guess and iteratively refines the solution.
- Converges faster, but may fail if the derivative is zero or the initial guess is poor.
Swipe to start coding
You must:
- Implement both root-finding methods.
- Stop the iteration when: the approximation error is less than or equal to
tolor the maximum number of iterationsmax_iteris reached. - Return:
- The estimated root
- The number of iterations used to reach the result.
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Can you explain how the bisection method works step by step?
What are the advantages and disadvantages of the Newton-Raphson method?
Can you provide example functions where these methods are commonly used?
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You will implement two classic numerical methods for finding roots of equations of the form:
f(x)=0Because many equations cannot be solved analytically, numerical root-finding methods are widely used in scientific computing and engineering to approximate solutions.
Methods to Implement
Bisection Method
- Requires an interval ([a,b]) where the function changes sign.
- Repeatedly halves the interval to narrow down the root.
- Guaranteed to converge, but relatively slow compared to other methods.
Newton-Raphson Method
- Uses the derivative of the function.
- Starts from an initial guess and iteratively refines the solution.
- Converges faster, but may fail if the derivative is zero or the initial guess is poor.
Swipe to start coding
You must:
- Implement both root-finding methods.
- Stop the iteration when: the approximation error is less than or equal to
tolor the maximum number of iterationsmax_iteris reached. - Return:
- The estimated root
- The number of iterations used to reach the result.
Рішення
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