Can somebody help me out with this? I have a matrix with unknown entry values that are either zero or positive. I know the exact sum of each matrix row and approximate sum of each matrix column. The matrix is large (>50000 columns, >1000 rows) but sparse (about 30000 nonzero entries total that are all known).

Is there any known algorithm or software that could be practically applied to find such entry values that sum of each row would be exactly the known value and the sum of each column would be as close as possible (with some criterion such as squared error) to target value?