Question

Type in the following code and run it: eps = 1.0 while 1.0 != 1.0 + eps: print ’...............’, eps eps = eps/2.0 print ’final eps:’, eps Explain with words what the code is doing, line by line. Then examine the output. How can it be that the "equation" \(1 \neq 1+\) eps is not true? Or in other words, that a number of approximately size \(10^{-16}\) (the final eps value when the loop terminates) gives the same result as if eps \(^{9}\) were zero? Name of program file: machine_zero.py. If somebody shows you this interactive session >>> 0.5 + 1.45E-22 0.5 and claims that Python cannot add numbers correctly, what is your answer? \(\diamond\)

Short answer

The loop demonstrates that when `eps` becomes too small, it does not affect `1.0` due to floating-point precision limits. Hence, `0.5 + 1.45E-22` equals `0.5` in Python.

Step-by-step solution

Initialize epsilon

The `eps` variable is initialized to the value of 1.0. This variable will be used in the comparison condition of the while loop.

Begin while loop

The program enters a while loop with the condition `1.0 != 1.0 + eps`. This loop will continue as long as 1.0 is not equal to 1.0 + eps.

Print current epsilon

Inside the loop, the code prints the current value of `eps` each time the loop iterates. The print statement outputs '...............' followed by the value of `eps`.

Halve epsilon

After printing, the value of `eps` is halved by the assignment `eps = eps/2.0`. This reduces the value of `eps` for the next iteration of the loop.

Loop termination and final print

The loop continues halving `eps` until it is so small that `1.0 + eps` is indistinguishable from `1.0` in floating-point representation, causing the while loop condition to eventually evaluate to false and the loop to terminate. The program then prints 'final eps:' followed by the value of `eps`.

Understanding floating-point precision

In floating-point arithmetic, very small numbers can become effectively zero when added to larger numbers due to limited precision. `eps` becomes so small that adding it to 1.0 does not change the number in the computer's memory.

Addressing the interactive session claim

The result of `0.5 + 1.45E-22` being 0.5 is due to the small size of `1.45E-22`, which falls below the precision threshold of floating-point arithmetic. This does not indicate an error in addition but is a characteristic of how floating-point arithmetic handles very small numbers.

Key concepts

Epsilon

Epsilon is a crucial concept in understanding floating-point arithmetic. It represents the smallest possible difference between 1 and a number greater than 1 in a computer's floating-point system. This value is used to determine the precision of calculations.

In Python, this concept is explored using a loop where `eps` starts at 1.0 and gets halved repeatedly. The loop stops when `1.0 + eps` becomes indistinguishable from `1.0`, showing us the limit of epsilon.

• This limit helps detect the machine precision limit where two distinct floats become equal due to limited precision.
• Epsilon, often around \(10^{-16}\) for double-precision floats, highlights the smallest discernible change possible.

It emphasizes that calculations involving numbers smaller than epsilon will not affect the result, crucial for accurate numeric computations.

Machine Precision

Machine precision is the smallest gap each machine can notice between numbers, defining its limits in distinguishing adjacent values.

This precision is foundational in scientific computing and floating-point representation.

• It is primarily about how finely a floating-point number can be adjusted, affecting numerical computations significantly.
• This is why small numbers, when added to much larger ones, might seem to "disappear."

In the code example, machine precision becomes evident when `eps`, though still a non-zero number, no longer impacts the sum `1.0 + eps`.
The machine can't represent the extremely small changes beyond a certain point, reinforcing the concept's importance in mathematical and programming contexts.

Python Programming

Python programming offers an intuitive way to experiment with concepts like epsilon and machine precision through its simple syntax and comprehensive libraries.

The code in the exercise uses a while-loop to demonstrate these concepts. Python empowers users to interactively explore floating-point arithmetic issues.

• Using `print` statements and loops, as shown, help visualize the point at which further precision doesn't affect results.
• Python's consistency in handling floating-point precision is crucial for developers to ensure accuracy in computations.

Python also has built-in functions and libraries, like `sys.float_info.epsilon`, which allow precise extraction and manipulation of float characteristics.
This makes Python a preferred choice for scientific computing and numerical simulations.

Scientific Computing

Scientific computing leverages mathematical models and computer simulations to solve complex scientific problems. Floating-point arithmetic plays a pivotal role in these simulations.

Understanding concepts like epsilon and machine precision is vital for scientists to ensure meaningful and reliable results.

• Floating-point limitations can lead to errors in calculations, especially in iterative processes or simulations involving many computations.
• Programs must account for this to maintain accuracy, especially when dealing with very small or very large numbers.

The interaction session example with `0.5 + 1.45E-22` serves as a practical illustration of potential pitfalls in scientific computations.
It underscores how scientific computing depends on proper management of precision to prevent incorrect conclusions based on minute numerical discrepancies.