The number of ways to shuffle a standard deck of cards is 52 factorial

1. The number of ways to shuffle a standard deck of cards is 52 factorial, written as 52! and equal to 52 x 51 x… x 1 or 80658175170943878571660636856403766975289505440883277824000000000000.
   In scientific notation, its value is about 8.066 x 1067. It cannot be represented exactly in IEEE double precision, but about how accurate (in absolute and relative terms) can you expect the best floating point approximation to be?
2. Determine what output is produced by the MATLAB code
   t = 0.1
   n = 1:10
   e = n/10 – n*t.
   What does this demonstrate about floating point arithmetic (in particular, as- sociativity)? Explain.

The Correct Answer and Explanation is :

1. Floating Point Approximation of 52!

To estimate how accurate the best floating point approximation of ( 52! ) can be, we need to consider the limitations of floating-point arithmetic, specifically in IEEE double precision format.

IEEE double precision uses 64 bits, and it provides a significant number of digits (about 15–17 decimal digits of precision). Since ( 52! ) has 68 digits, the approximation to this number in double precision can only be accurate up to the first 15–17 digits. This means that the approximation to ( 52! ) in IEEE double precision will have some loss of precision in the least significant digits.

For ( 52! \approx 8.066 \times 10^{67} ), the absolute error will depend on the exact rounding used by the floating-point system. Given that IEEE double precision can represent approximately 15–17 digits accurately, the relative error is expected to be quite small but non-zero, especially as the size of the number increases. The relative error can be expressed as:

[
\text{Relative error} \approx \frac{\text{Absolute error}}{\text{Value of the number}}.
]

Since the value of ( 52! ) is on the order of ( 10^{67} ), the relative error will be on the order of ( 10^{-17} ), which is still very small, but not negligible.

Thus, the best floating point approximation of ( 52! ) will have an error on the order of ( 10^{-17} ) in relative terms, and the absolute error will depend on the implementation details of the floating-point system.

2. MATLAB Code and Floating Point Arithmetic

The MATLAB code:

t = 0.1;
n = 1:10;
e = n/10 - n*t;

The output of this code is calculated as follows:

• t = 0.1 is a scalar value.
• n = 1:10 creates a vector of values from 1 to 10.
• n/10 produces a vector of values ([0.1, 0.2, 0.3, …, 1.0]).
• n*t computes ([0.1, 0.2, 0.3, …, 1.0]), as ( n \times 0.1 ).
• Subtracting n*t from n/10 results in a vector of zeros.

However, due to the limitations of floating-point arithmetic, the result will not be exactly zero. The output might look like:

e = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

However, because floating point arithmetic is not always perfectly associative due to rounding errors, there may be small deviations from zero for each element in e.

Explanation of Associativity Issue

Floating-point numbers have finite precision, meaning that operations like addition and multiplication are subject to rounding errors. Associativity means that the result of an operation should not depend on the order of operations. However, in floating point arithmetic, this is not always the case. When you perform operations like subtraction with numbers that are close in value, the result can suffer from “catastrophic cancellation,” which results in a loss of precision.

In this code, ( n/10 – n \times t ) is expected to produce zero because both terms are mathematically equal. But, due to rounding errors inherent in floating point representation, small errors accumulate and prevent the result from being exactly zero. This demonstrates that floating point arithmetic does not always satisfy exact mathematical properties such as associativity and distributivity, especially when dealing with numbers that are close together, leading to precision loss.

In conclusion, this example highlights how floating-point arithmetic can lead to small errors, which can be amplified in more complex computations. It also demonstrates the sensitivity of floating-point results to the order in which operations are performed.