javaalgorithmmathpolynomial-math

# Solving a Linear Diophantine Equation(see description for examples)

Let me start off by clarifying that(before you guys dismiss me), this is not a homework problem and I'm not a university student. :)

EDIT Thanks to @Klas and others, my question now boils down to a mathematical equation which needs to be solved programmatically.

I'm looking for an algorithm/code which solves `Linear Diophantine Equation`. For lesser mortals like me, here's how such an equation looks like:

Example 1: `3x + 4y + 5z = 25` (find all possible values of x,y,z)

Example 2: `10p + 5q + 6r + 11s = 224` (find all possible values of p,q,r,s)

Example 3: `8p + 9q + 10r + 11s + 12t = 1012` (find all possible values of p,q,r,s,t)

I tried googling to no avail. I would have thought that some code would already be written to solve this. Do let me know if you guys have come across some kind of library which has already implemented this. And if the solution is in Java, nothing can be cooler!. Algorithm/pseudo code will also do. Thanks much.

Solution

• Brute-force recursion is an option, depending on how large you will allow the value or number of values to become.

Assumptions: The user inputs (the multiplicands) are always distinct positive integers. The coefficients to be found must be non-negative integers.

Algorithm:

``````Of the multiplicands, let M be the largest.
Calculate C=floor(F/M).
If F=M*C, output solution of the form (0,0,...,C) and decrement C
If M is the only multiplicand, terminate processing
Loop from C down to 0 (call that value i)
Let F' = F - i*M
Recursively invoke this algorithm:
The multiplicands will be the current set minus M
The goal value will be F'
For each solution output by the recursive call:
append i to the coefficient list and output the solution
``````