差商展开系数的递推公式和算法

A Recurrent Formula and Algorithm for the Divided Difference Expanded Coefficients

推导出了差商展开系数的一个递推公式 ,基于该公式给出了计算差商展开系数的一个新算法 .本算法比已有的算法更易于理解和实现 ,而且可同时计算一个节点向量上多个相邻的 k阶差商的展开系数 .当计算一个节点向量上的所有 k阶差商的展开系数时 ,本算法效率较高 ,时间复杂性为 O( k2 max( k,n +1 ) ) ,其中 k为差商的阶 ,n +k

A recurrent formula for the divided difference expanded coefficients is derived. Based on this recurrent formula, a new algorithm for the calculation of the divided difference expanded coefficients is given. This algorithm is more understandable than the existing algorithm and is more suitable for the calculation of the expanded coefficients of several contiguous divided differences of order k on a knot vector simultaneously. When being used to calculate the expanded coefficients of all divided differences of order k on a knot vector, this algorithm is more efficient. In this case, the time complexity of the algorithm is O(k 2 max (k,n+1)), where k is the order of divided difference, n+k+1 is the number of knots in the given knot vector.