lira

Lovasz Integer Relation Algorithm: find a basis for the lattice of integer vectors orthogonal to the one specified

The Lovasz Integer Relation Algorithm from Finding Integer Relations in Polynomial Time by Hastad, Just, Lagrias and Schnorr.

Find a basis for the collection of integer vectors orthogonal to v_x. Paraphrasing the paper:

On input v_x in Z^n, this algorithm finds a basis of the n-1-dimensional integer lattice L whose complement is the integer-span of v_x. The algorithm and output have nice properties by Theorem 6.1.

Input

v_x = nonzero integer n-vector

Output

mtx_C = n*(n-1) integer matrix, each column an integer vector orthogonal to v_x, with properties according to

Usage Example

>> v_x = [1;4;2;1];
>> mtx_C = lira(v_x)

ans =

    -1    -1     1
     1     0     0
    -1     1     0
    -1    -1    -1

>> v_x'*mtx_C 

ans = 

    0     0     0

enthdegree/lira

lira

Input

Output

Usage Example