In mathematics I like to think when some mathematical ideas, I think only mastered the formula derivation idea can really achieve mastery of mathematics, rather than mechanically rigid back out derivation.
Value in a graduate course in analysis, interpolation is a most important chapter two interpolation method of Newton and Lagrange polynomial interpolation. Both methods produce a polynomial in the same order after simplification are the same, the remainder is the same, but generally are written, so just a different form of its most typical form of writing. When data changes for its own unique written form to suit their usage scenarios brought distinction. Newton interpolation method is suitable for the case to be inserted in the growing point, but the point is the same interpolation function itself is located (can not change the function value has slipped to the point), with the insertion point is increased, we fit a function of high accuracy, this method can reuse the previous calculation result. The Lagrange polynomial interpolation point applies to which have been identified (can not continue to add new interpolation points), but the values of these functions have been interpolated point where they can change, in this case Lagrange you can take advantage of previous results.
Derivation Newton interpolation method is easy to understand, the first node can be calculated using each interpolation step difference of each supplier, the respective difference quotient is the first derivative of the approximation discrete case. So long as the two interpolation points, you can get a point and the initial first-order difference quotient, we can estimate the simplest any point in the interpolation function. But still more than items, as long as there is a sufficient number of interpolation points, remainder can continue to be broken down into higher-order difference quotient and the remainder of the remainder. In the Lagrange polynomial textbooks principle and not the derivation process, and therefore can be understood as derived by Newton interpolation format conversion and then becomes the Lagrange polynomial. In this format conversion is calculated to achieve the purpose to reduce the amount of dynamic data specific application scenario.