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.
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![The text describes a task to create a function in a programming context related to numerical analysis. Here's the transcription as it might appear on an educational website:
---
**Programming Task: Implementing Newton's Method**
Create a function named `newt(f, df, x0)` to implement Newton's method for finding a root of the function `f`. The first derivative of this function is provided by `df`. The algorithm should begin with an initial guess given by `x0`.
**Details:**
- **Function Name:** `newt`
- **Parameters:**
- `f`: The function for which we want to find the root.
- `df`: The derivative of the function `f`.
- `x0`: The starting value (initial guess) for the root-finding process.
**Objective:**
Use Newton's method to iteratively approximate a root of the function `f` using its derivative `df`. Newton's method is a powerful technique for solving equations numerically, particularly when combined with a suitable starting point `x0`.
**Concept Overview:**
Newton’s method is a root-finding algorithm that uses a sequence of function evaluations to approximate a solution. The method applies the formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{df(x_n)} \]
Repeat this process until the change in root estimates is within a desired tolerance. The effectiveness of the method depends on the choice of starting point `x0`.
---
This representation provides a clear, educational explanation suitable for users learning about numerical methods and programming.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad10dbe2-69d5-44c1-bbde-1a4f79a6621c%2Fd9638fab-7bac-42fb-99d9-88e0ac22db46%2Fbw72r7f_processed.png&w=3840&q=75)
Transcribed Image Text:The text describes a task to create a function in a programming context related to numerical analysis. Here's the transcription as it might appear on an educational website:
---
**Programming Task: Implementing Newton's Method**
Create a function named `newt(f, df, x0)` to implement Newton's method for finding a root of the function `f`. The first derivative of this function is provided by `df`. The algorithm should begin with an initial guess given by `x0`.
**Details:**
- **Function Name:** `newt`
- **Parameters:**
- `f`: The function for which we want to find the root.
- `df`: The derivative of the function `f`.
- `x0`: The starting value (initial guess) for the root-finding process.
**Objective:**
Use Newton's method to iteratively approximate a root of the function `f` using its derivative `df`. Newton's method is a powerful technique for solving equations numerically, particularly when combined with a suitable starting point `x0`.
**Concept Overview:**
Newton’s method is a root-finding algorithm that uses a sequence of function evaluations to approximate a solution. The method applies the formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{df(x_n)} \]
Repeat this process until the change in root estimates is within a desired tolerance. The effectiveness of the method depends on the choice of starting point `x0`.
---
This representation provides a clear, educational explanation suitable for users learning about numerical methods and programming.
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