NumericConstantApproximator

Approximate numeric constants such as Pi (3.1415926...) and Phi (1.6180339...) to arbitrary precision using Java's BigDecimal class (and analyze the results).

License

This project is licensed under the MIT license.

Compile

Note: this project requires a minimum of Java 14 to run. Java 15 is recommended as it is the most recent version as of the writing of this documentation.

javac ComputePhi.java ComputePi.java PiAlgorithmExaminer.java

1. ComputePhi

ComputePhi demonstrates algorithmically that no matter which two terms you start with, if you generate a series of numbers by adding the two previous terms, the ratio between consecutive terms converges to Phi (the Golden Ratio).

Usage:

java ComputePhi <first-term> <second-term> <iterations> <precision> [options]

Options:

--print_steps (-p)      # Print approximation for each iteration of algorithm.
--compare_values (-c)   # Compare sequential iterations' results, and stop iterating early if they are equal.

Examples:

# The Fibonacci sequence
java ComputePhi 1 1 10 10

# The Lucas numbers (printing each step)
java ComputePhi 2 1 10 10 -p

# Go wild if you want to!
java ComputePhi 23 9001 200 1000 --print_steps
java ComputePhi 314 159 200 20000 --compare_values

2. ComputePi

ComputePi demonstrates 7 different algorithms for approximating Pi. The result is then compared to a precomputed file of one million digits of Pi (sourced from here) to determine accuracy.

Usage:

java ComputePi <algorithm> <iterations> <precision> [options]

Notes:

• <algorithm> can be a number, or an algorithm name (case-insensitive):

Number	Algorithm Name	Reference
1	Gregory-Leibniz	[1]
2	Nilakantha	[2]
3	Newton	[3]
4	Viete	[4]
5	Wallis	[5]
6	Chudnovsky	[6]
7	Brent-Salamin (or Gauss-Legendre)	[7]

Options:

--all_digits (-a)            # Print all digits (default only prints accurate digits).
--print_steps (-p)           # Print approximation for each iteration of algorithm.
--compare_values (-c)        # Compare sequential iterations' results, and stop iterating early if they are equal.
--estimate_memory_usage (-e) # Print an estimate of memory usage.

Examples:

java ComputePi 1          1000  10
java ComputePi 2          1000  10 --print_steps
java ComputePi CHUDNOVSKY   10 100 --compare_values --estimate_memory_usage

3. PiAlgorithmExaminer

PiAlgorithmExaminer compares the approximations, runtimes, and memory usage of the algorithms implemented in ComputePi iteration-by-iteration, prints the results to a file output.txt, and then prints an analysis of the data to a table in standard output.

Usage:

java PiAlgorithmExaminer <algorithms> <iterations> <precision> [options]

Notes:

• <algorithms> is a comma-separated list of algorithm names and numbers (see above for values).

Options:

--skip_tests (-s)  # Skip running the tests (only use to re-analyze already generated output data).
--print_table (-p) # Print the analysis as a table (defaults to printing results in a "key:value" format, which is more useful for large datasets).

Examples:

java PiAlgorithmExaminer CHUDNOVSKY,BRENT_SALAMIN 1000 2000
java PiAlgorithmExaminer 1,2,3,4,5,6,7 10 2000 --print_table # <----- outputs results shown below

Note that some algorithms are far slower to compute than others. For example, Viete's formula computes a square root for every iteration, and is thus comparatively slower than other algorithms.

Example Results:

Here's an example output of the program showing a comparison of the first 10 iterations of all 7 implemented algorithms. The first section shows the arguments passed to ComputePi (with the runtime and memory usage info), and the second section shows a table with the number of accurate leading digits of pi per iteration.

GREGORY_LEIBNIZ 10 2000 --print_steps --compare_values --estimate_memory_usage     0.103 seconds      1.640 KB
NILAKANTHA      10 2000 --print_steps --compare_values --estimate_memory_usage     0.027 seconds      1.652 KB
NEWTON          10 2000 --print_steps --compare_values --estimate_memory_usage     0.096 seconds      3.263 KB
VIETE           10 2000 --print_steps --compare_values --estimate_memory_usage     0.395 seconds     10.548 KB
WALLIS          10 2000 --print_steps --compare_values --estimate_memory_usage     0.017 seconds      1.647 KB
CHUDNOVSKY      10 2000 --print_steps --compare_values --estimate_memory_usage     0.133 seconds     13.305 KB
BRENT_SALAMIN   10 2000 --print_steps --compare_values --estimate_memory_usage     0.331 seconds     11.368 KB

ITERATIONS	GREGORY_LEIBNIZ	NILAKANTHA	NEWTON	VIETE	WALLIS	CHUDNOVSKY	BRENT_SALAMIN
1	0	2	1	0	0	14	0
2	1	2	3	1	0	28	3
3	0	3	3	2	0	42	8
4	1	2	4	2	0	56	19
5	0	3	4	3	1	70	41
6	1	3	6	4	1	85	84
7	1	3	6	5	1	99	171
8	1	4	7	5	1	113	345
9	1	4	8	5	1	127	694
10	1	4	9	6	1	142	1392