Question

Because of round-off errors, it could happen that a mathematical rule like \((a b)^{3}=a^{3} b^{3}\) does not hold (exactly) on a computer. The idea of this exercise is to check such identities for a large number of random numbers. We can make random numbers using the random module in Python: import random \(\mathrm{a}=\) random. uniform \((A, B)\) \(b=\) random. uniform \((A, B)\) Here, a and b will be random numbers which are always larger than or equal to A and smaller than \(B\). Make a program that reads the number of tests to be performed from the command line. Set \(A\) and \(B\) to fixed values (say - 100 and 100\()\). Perform the test in a loop. Inside the loop, draw random numbers a and b and test if the two mathematical expressions (a*b)**3 and a**3*b**3 are equivalent. Count the number of failures of equivalence and write out the percentage of failures at the end of the program. Duplicate the code segment outlined above to also compare the expressions \(a / b\) and \(1 /(b / a)\). Name of program file: math_identities_failures.py.

Short answer

The percentage of failures indicates how often round-off errors affect the mathematical identity equivalences in this test.

Step-by-step solution

Import Necessary Modules

Begin by importing the random module, which will be used to generate random numbers within a specified range.

Set the Range for Random Numbers

Define fixed values for \(A\) and \(B\). In this example, use \(A = -100\) and \(B = 100\) to ensure a wide range of random numbers.

Define the Main Function

Create a function called `main` that will encapsulate the logic of the program, including reading input for the number of tests to perform.

Read Number of Tests from Input

Use `input` to prompt the user for the number of tests. Convert this input to an integer, which represents how many times the equivalence tests should be performed.

Initialize Counters for Failures

Prepare variables to count the number of failures for each identity test, starting them at zero.

Set Up the Loop for Random Tests

Use a `for` loop to iterate over the specified number of tests. Each iteration will generate new random numbers \(a\) and \(b\).

Test the First Identity

Inside the loop, compute both \((a*b)^3\) and \(a^3*b^3\). Use Python's `math.isclose` function to check their equivalence within a small tolerance, indicative of round-off error.

Count Failures for the First Identity

If the two expressions from Step 7 are not close enough, increment the counter for equivalence failures.

Test the Second Identity

Also within the loop, verify if \(a/b\) is approximately equal to \(1/(b/a)\) by evaluating these expressions and checking their equivalence with `math.isclose`.

Count Failures for the Second Identity

Increment a separate counter if these expressions are found not to be equivalent, based on the check from Step 9.

Calculate Failure Percentages

At the end of the loop, compute the percentage of failures for each identity by dividing the number of failures by the total number of tests and multiplying by 100.

Output Results

Print the results, including the percentage of failure for each mathematical identity, to provide an overview of how often the identities were not equivalent due to round-off errors.

Call the Main Function

Ensure that `main` is called if the script is executed. This is often done with an `if __name__ == '__main__':` block at the end of the program.

Key concepts

Round-off Errors

Round-off errors occur when finite computer precision is used to represent real numbers. Computers use binary floating-point arithmetic, which cannot represent every decimal number exactly. This leads to discrepancies in calculations. In the exercise, round-off errors are critical because they can prevent a mathematical identity that holds true in theory from holding true on a computer. For example,

• The expression \( (a*b)^3 = a^3*b^3 \) might seem straightforward, but due to round-off errors, both sides can yield slightly different results when computed separately.

Round-off errors are especially significant when dealing with operations like division and exponentiation, where accumulated errors can be more pronounced. Therefore, when programming, it's important to account for these errors, for instance using functions that test for equivalence within a certain tolerance, rather than checking for exact equality.

Python Programming

Python is a versatile and widely-used programming language ideal for tasks that involve mathematical computations and data analysis. In the exercise, Python's flexibility allows us to automate the testing of mathematical identities across a range of randomly generated numbers. Using Python, you can easily set up loops, perform arithmetic operations, and use libraries for greater functionality.

• The `random` module is used to generate random numbers within a defined range using the `random.uniform(A, B)` function.
• The `math.isclose` function is a convenient way to compare floating point numbers, considering a tolerance for round-off errors.

Python's simplicity and readability make it a perfect choice for quickly implementing and testing mathematical problems, especially when dealing with randomness and approximate comparisons.

Random Number Generation

Random number generation is a key concept in this exercise as it introduces natural variability into the tests. Python's `random` module is a simple tool that provides functions to generate random numbers, which is crucial in simulations or tests like those in our exercise.

When you use:

• `random.uniform(A, B)`, it returns a float number that's >= A and < B, introducing unpredictability to the values of \(a\) and \(b\) used in the mathematical identity tests.

This unpredictability is important because it tests the mathematical expressions over a vast, unpredictable range of numbers, increasing the robustness of the examination by simulating a variety of real-world scenarios where numbers can vary widely.

Mathematical Identities

Mathematical identities are equations that hold true for all numbers the variables can logically take. However, in computing, these identities may not always hold perfectly due to the precision limitations of floating point arithmetic.

The exercise requires testing:

• \((a*b)^3 = a^3*b^3\)
• \(a/b = 1/(b/a)\)

These tests highlight how even universally true identities can fail under finite precision calculations, emphasizing the importance of understanding mathematical rigor and computational limitations. By evaluating these identities using random values, we explore how often they deviate due to inherent limitations when implemented on a computer.

Floating Point Arithmetic

Floating point arithmetic refers to how real numbers are represented and manipulated in a computer. Unlike integer arithmetic, floating point numbers allow for a much wider range of values but can introduce rounding errors.

In floating point arithmetic:

• Numbers are represented in a form akin to scientific notation, with a base and an exponent, allowing for very large or very small numbers to be represented.
• This representation is not exact for many numbers, especially those with non-terminating decimal representations.

This imprecision means that operations like addition, subtraction, multiplication, and division can accumulate errors, leading to inaccuracies in complex expressions or iterative processes. Recognizing these inaccuracies is crucial, particularly in scientific computations where precision is paramount. In this exercise, observing floating point arithmetic helps explain the failures in mathematical identities, prompting an understanding of how to manage such errors effectively in computational tasks.