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Write a function newt(f, df, x0) which implements Newton's method to identify a root of the function f whose first derivative is given by the function df, with the starting value being given by x0.

Write a function newt(f, df, x0) which implements Newton's method to identify a root of the function f whose first derivative is given by the function df, with the starting value being given by x0.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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### Newton's Method Implementation **Task:** Write a function `newt(f, df, x0)` which implements Newton’s method to identify a root of the function `f` whose first derivative is given by the function `df`, with the starting value being given by `x0`. --- **Explanation:** Newton's method, also known as the Newton-Raphson method, is a powerful technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. Here's how you can approach writing the function: - **Function Parameters:** - `f`: The function for which we are trying to find the root. - `df`: The derivative of the function `f`. - `x0`: The initial guess for the root. - **Concept:** - Start with an initial guess `x0`. - Apply the iterative formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{df(x_n)} \] - Repeat the iteration until the difference between successive iterations is smaller than a pre-defined threshold (indicating convergence), or until a maximum number of iterations is reached. - **Applications:** - Suitable for solving equations where an analytical solution may not exist or is difficult to obtain. - Common in numerical analysis and applied mathematics. Ensure that the derivative function `df` does not evaluate to zero at any point in the domain, as this would cause a division by zero error in the iteration formula.
Transcribed Image Text:### Newton's Method Implementation **Task:** Write a function `newt(f, df, x0)` which implements Newton’s method to identify a root of the function `f` whose first derivative is given by the function `df`, with the starting value being given by `x0`. --- **Explanation:** Newton's method, also known as the Newton-Raphson method, is a powerful technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. Here's how you can approach writing the function: - **Function Parameters:** - `f`: The function for which we are trying to find the root. - `df`: The derivative of the function `f`. - `x0`: The initial guess for the root. - **Concept:** - Start with an initial guess `x0`. - Apply the iterative formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{df(x_n)} \] - Repeat the iteration until the difference between successive iterations is smaller than a pre-defined threshold (indicating convergence), or until a maximum number of iterations is reached. - **Applications:** - Suitable for solving equations where an analytical solution may not exist or is difficult to obtain. - Common in numerical analysis and applied mathematics. Ensure that the derivative function `df` does not evaluate to zero at any point in the domain, as this would cause a division by zero error in the iteration formula.
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