Question

A widely used formula for numerical differentiation of a function \(f(x)\) takes the form $$ f^{\prime}(x) \approx \frac{f(x+h)-f(x-h)}{2 h} $$ This formula usually gives more accurate derivatives than ( \(7.1)\) because it applies a centered, rather than a one-sided, difference. The goal of this exercise is to use the formula (7.7) to automatically differentiate a mathematical function \(f(x)\) implemented as a Python function \(f(x)\). More precisely, the following code should work: def f(x): return 0.25*x**4 df = Central(f) # make function-like object df # df(x) computes the derivative of f(x) approximately: for x in (1, 5, 10): df_value = df(x) # approx value of derivative of f at point x exact = x**3 # exact value of derivative print "f’(%d)=%g (error=%.2E)" % (x, df_value, exact-df_value) Implement class Central and test that the code above works. Include an optional argument \(\mathrm{h}\) to the constructor in class Central so that one can specify the value of \(h\) in the approximation (7.7). Apply class Central to produce a table of the derivatives and the associated approximation errors for \(f(x)=\ln x, x=10\), and \(h=\) \(0.5,0.1,10^{-3}, 10^{-5}, 10^{-7}, 10^{-9}, 10^{-11} .\) Collect class Central and the two applications of the class in the same file, but organize the file as a module so that class Central can be imported in other files. Name of program file: Central.py.

Short answer

Implement a class `Central` for numerical differentiation, test it with `f(x) = 0.25*x^4`, and apply it to `f(x) = ln(x)` for various `h` values. Ensure the code is modular in `Central.py`. Compute and print the errors.

Step-by-step solution

Define the Central Class

Create a Python class named `Central` with an initializer `__init__` that accepts a function `f` and an optional parameter `h`. Store these as instance attributes for future use.

Initialize with Function and Parameter

In the `__init__` method, initialize the instance attributes with the function `f` and the small interval `h`. For flexibility, set `h` to a default value (e.g., `0.01`) if not provided.

Implement the Derivative Method

Define a `__call__` method in the `Central` class. This method accepts a value `x` and applies the numerical differentiation formula to compute the approximate derivative. Calculate:\[f^{\prime}(x) \approx \frac{f(x+h)-f(x-h)}{2 h}\]

Test the Central Class with Polynomial Function

Implement the function `f(x)` as specified, i.e. `f(x) = 0.25*x**4`. Instantiate the `Central` class with this function. Test it by calling the instance for values `x = 1`, `x = 5`, and `x = 10`.

Calculate Exact and Approximate Derivatives

For each point, compute the approximate derivative using the Central class instance and the exact derivative using `f'(x) = x^3`. Print the results and compute the error as `exact - df_value`.

Apply to Logarithmic Function

Implement the function `f(x) = ln(x)`. Instantiate the `Central` class with this function and test it with `x = 10` for various `h` values: `0.5, 0.1, 10^{-3}, 10^{-5}, 10^{-7}, 10^{-9}, 10^{-11}`.

Print the Derivative and Error Table

For each specified `h`, compute the approximate derivative and compare it to the exact derivative, `f'(x) = 1/x`, at `x = 10`. Print a table of the results, showing the derivative and error for each `h`.

Organize the Code as a Module

Ensure the `Central` class and its applications are contained within a single file named `Central.py`. Structuring it this way allows for easy import and reuse in other Python scripts.

Key concepts

Python Programming

Python is an incredibly versatile language that excels in mathematics and scientific computing. One of its strengths is the ability to create custom classes and functions for specific calculations or processes. In this educational exercise, we leveraged Python's object-oriented programming features to implement a `Central` class for numerical differentiation.

The Python programming language provides robust built-in functionalities such as the power operator `**`, which is crucial for performing mathematical operations like raising a number to a given power. In our `Central` class, we used this operator to define the function `f(x) = 0.25*x**4`. Furthermore, Python’s flexibility allows for customization, enabling us to use default parameters like setting `h` to `0.01` by default if another value isn't provided.

• Object-oriented design helps organize code.
• Python supports mathematical operations with ease.
• Custom classes can be structured as modules for reusability.

By encapsulating the differentiation logic within the `Central` class, it demonstrates an excellent way to maintain clean, reusable, and extendable code in Python programming.

Educational Exercise

This exercise serves as an excellent educational tool for understanding numerical differentiation. By transitioning from theoretical mathematical concepts to practical coding implementations, students develop a deeper connection to the underlying mathematics.

The primary goal is a practical understanding of calculating derivatives using numerical methods in Python. Students are tasked with building a `Central` class capable of applying the central difference method to approximate derivatives of given mathematical functions. The exercise guides learners through the process of implementing the class, using it on specific functions, and verifying its accuracy by comparing the approximate results with exact derivatives.

• The exercise involves both coding and theoretical mathematics.
• Encourages integration of classroom teaching and practical application.
• Importance is placed on testing and validating results.

This blend of programming and mathematics provides a compelling learning experience, encouraging students to think critically about both the implementation and the mathematics behind numerical solutions.

Derivative Calculations

Derivative calculations are central to this exercise. The goal is to approximate the derivative of a function using the central difference method. The formula used is:\[f^{\prime}(x) \approx \frac{f(x+h)-f(x-h)}{2 h}\]This approach offers a more accurate estimation than one-sided differences, making it a preferred choice in many applications.

In practice, different values of `h` affect the accuracy of approximations. A smaller `h` generally leads to more accurate results, but too small `h` can introduce numerical instability due to floating-point arithmetic limitations. Thus, choosing an appropriate `h` is crucial:

• Improve accuracy by finding the optimal `h` value.
• Understand advantages of central over one-sided differences.
• Balance between numerical precision and stability.

By applying these calculations programmatically, students can experiment with different `h` values and observe the immediate impact on approximation accuracy.

Mathematics Functions

Mathematics functions underpin the whole concept of numerical differentiation. In this exercise, we test the implementation using two distinct functions: a polynomial function, \(0.25x^4\), and a logarithmic function, \(\ln(x)\).

Understanding these functions' derivatives is key to evaluating the numerical approximation's accuracy. For the polynomial, the exact derivative is \(x^3\). For the logarithmic function, the derivative at the point of interest (say, \(x = 10\)) is simpler, \(1/x\), thus equating to \(0.1\) at \(x = 10\).

• Different types of functions require specific derivative formulas.
• Direct comparison between exact and approximate derivatives shows precision.
• Analyzing errors helps improve the approximation process.

The ability to approximate derivatives across various function types empowers students to deal with more complex real-world mathematical problems.