# Enumeration of all existing nonnegative integer solutions of a linear Diophantine equation This function enumerates nonnegative integer solutions of a linear Diophantine equation (NLDE): [REMOVE_ME]$$ a_1s_1 +a_2s_2 +...+ a_ls_l =n, [REMOVE_ME_2]$$ where $a_1 <= a_2 <= ... <= a_l,$ $a_i > 0,$ $n > 0,$ $s_i >= 0,$ $i=1,2,...,l,$ and all variables involved are integers. The algorithm is based on a generating function of Hardy and Littlewood used by Voinov and Nikulin (1997). ## Description This function enumerates nonnegative integer solutions of a linear Diophantine equation (NLDE): $$ a_1s_1 +a_2s_2 +...+ a_ls_l =n, $$ where $a_1 <= a_2 <= ... <= a_l,$ $a_i > 0,$ $n > 0,$ $s_i >= 0,$ $i=1,2,...,l,$ and all variables involved are integers. The algorithm is based on a generating function of Hardy and Littlewood used by Voinov and Nikulin (1997). ```r nlde(a, n, M=NULL, at.most=TRUE, option=0) ``` ## Arguments - `a`: An `l`-vector of positive integers (coefficients of the left-hand-side of NLDE) with `l>= 2`. - `n`: A positive integer which is to be partitioned. - `M`: A positive integer, the number of parts of `n`, `M <= n`. - `at.most`: If `TRUE` partitioning of `n` into at most `M` parts, if `FALSE` partitioning on exactly `M` parts. - `option`: When set to `1` (or any positive number) finds only 0-1 solutions of the linear Diophantine equation. When set to `2` (or any positive number > 1) finds 0-1 solutions of the linear Diophantine inequality. ## Returns - **p.n**: total number of partitions obtained. - **solutions**: a matrix with each column forming a partition of `n`. ## Author(s) Vassilly Voinov, Natalya Pya Arnqvist, Yevgeniy Voinov ## References Voinov, V. and Nikulin, M. (1997) On a subset sum algorithm and its probabilistic and other applications. In: Advances in combinatorial methods and applications to probability and statistics, Ed. N. Balakrishnan, , Boston, 153-163. Hardy, G.H. and Littlewood, J.E. (1966) Collected Papers of G.H. Hardy, Including Joint Papers with J.E. Littlewood and Others. Clarendon Press, Oxford. ## See Also `nilde-package`, `get.partitions`, `get.subsetsum`, `get.knapsack` ## Examples ```r ## some examples... ## example 1... nlde(a=c(3,2,5,16),n=18,at.most=TRUE) b1 <- nlde(a=c(3,2,5,16),n=18,M=6,at.most=FALSE) b1 ## checking M, the number of parts that n=18 has been partitioned into... colSums(b1$solutions) ## checking the value of n... colSums(b1$solutions*c(3,2,5,16)) ## example 2: solving 0-1 nlde ... b2 <- nlde(a=c(3,2,5,16),n=18,M=6,option=1) b2 colSums(b2$solutions*c(3,2,5,16)) ## example 3... b3 <- nlde(c(15,21),261) b3 ## checking M, the number of parts that n has been partitioned into... colSums(b3$solutions) ## checking the value of n... colSums(b3$solutions*c(15,21)) ## example 4... nlde(c(5,6),19) ## no solutions ## example 5: solving 0-1 inequality... b4 <- nlde(a=c(70,60,50,33,33,33,11,7,3),n=100,at.most=TRUE,option=2) ```

nlde function