Integer Solutions of Linear Inhomogeneous Equations

A linear equation with integer coefficients is represented as a pair of lists of non-zero integers, the coefficients and the constants. If there are no constants, the linear equation represented by (c, []) is the homogeneous equation:

c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1] = 0

where n is the length of c. Otherwise, (c, d) represents the inhomogeneous equation:

c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1] = g

where g = gcd(d[0], d[1], ..., d[m-1]), and m is the length of d. Thus g is the greatest common denominator of the elements of d.

A solution is a partial map from variables to terms, and a term is a pair of lists of integers, the variable part of the term followed by the constant part. The variable part may specify variables not in the input. In other words, the length of the coefficents in the answer may exceed the length of the coefficients in the input. For example, the solution of

64x - 41y = 1

is x = -41z - 16 and y = -64z - 25. The computed solution is read off the list returned as an answer.

intLinEq [64,-41] [1] = [(0,([0,0,0,0,0,0,-41],[-16])), (1,([0,0,0,0,0,0,-64],[-25]))]

The algorithm used to find solutions is described in Vol. 2 of The Art of Computer Programming / Seminumerical Alorithms, 2nd Ed., 1981, by Donald E. Knuth, pg. 327. To show sums, we write

sum[i] c[i]*x[i] for c[0]*x[0] + c[1]*x[1] + ... + c[n-1]*x[n-1].

The algorithm's initial values are the linear equation (c,d) and an empty substitution s.

- Let c[i] be the smallest non-zero coefficient in absolute value.
- If c[i] < 0, multiply c and d by -1 and goto step 1.
- If c[i] = 1, a general solution of the following form has been found:

x[i] = sum[j] -c'[j]*x[j] + d[k] for all k

where c' is c with c'[i] = 0. Use the equation to eliminate x[i] from the range of the current substitution s. If variable x[i] is in the original equation, add the mapping to substitution s.

- If c[i] divides every coefficient in c,

- if c[i] divides every constant in d, divide c and d by c[i] and goto step 3,
- otherwise fail because there is no solution.

- Otherwise, eliminate x[i] as above in favor of freshly created variable x[n], where n is the length of c.

x[n] = sum[j] (c[j] div c[i] * x[j])

Goto step 1 and solve the equation:

c[i]*x[n] + sum[j] (c[j] mod c[i])*x[j] = d[k] for all k

# Documentation

type LinEq = ([Int], [Int])Source

A linear equation with integer coefficients is represented as a pair of lists of non-zero integers, the coefficients and the constants.

type Subst = [(Int, LinEq)]Source

A solution to a linear equation is a partial map from variables to terms, and a term is a pair of lists of integers, the variable part of the term followed by the constant part. The variable part may specify variables not in the input. In other words, the length of the coefficents in the answer may exceed the length of the coefficients in the input.