For a homogeneous linear-recurrence f(n) with integer coefficients and integer starting points, we derive a deterministic algorithm that finds the upper bound of the last non-periodic position n where f(n)=0, for a large family of special cases. First, when theta is a given irrational constant, then we show that, an eventual lower bound of minimum(absolute(cos(m PI theta)), over positive integers m less than n), for large positive integers n, is (2 theta / (sqrt(5) n)). Our deterministic algorithm is based on the key concept that this lower bound decreases at a lower rate than the nth power of the ratio of root-moduli since the ratio is lesser than 1. Our deterministic algorithm is developed for the special cases where G(x), the characteristic polynomial of f(n), has either equal absolute values of arguments or commensurable arguments of those complex roots whose moduli are equal. In an attempt to extend this algorithm as a general solution to Skolems problem, we obtain the lower bound of the distance between a zero and the next (2^(m+1))th zero, in the weighted sum of m continuous cosine functions, where the weights are given real-algebraic constants.