EXPLANATION FILE OF PROGRAM NEWTON ================================== Newton Divided-Differences Interpolation and Error Estimates ------------------------------------------------------------ Another approach to obtaining the Nth-degree polynomial interpolation curve required to piece-wise fit a given table is to consider the Taylor series ex- pansion about a particular nearby table value: 2 2 3 3 df| (x-xi) d f | (x-xi) d f | f(x) = f(xi) + (x-xi)--| + ------ ---2| + ------ ---3| dx|xi 2! d x |xi 3! d x |xi 4 4 n n (x-xi) d f | (x-xi) d f | + ------ ---4| + ... + ------ ---n| + ... (5.3.1) 4! d x |xi n! d x |xi n By truncating the series at the (x - xi) term, the function f(x) is approximated with an Nth-degree polynomial. The resulting error is contained largely in the first term neglected. It can be shown that the error is n+1 n+1 (x - xi) d f | -------- ------| (n+1)! dx^n+1|x' where x' is sorne value in the interval between x and xi. To generate an algorithm that implements this expansion, table values neighboring the position of interest are employed to find estimates for the (1/ N!) dnf / dxn terms in a form called dioided differences. These factors can be derived very methodically as follows: (1) fi+1 - fi First divided difference = Fi = --------- xi+l - xi (1) (1) (2) Fi+1 - Fi Second divided difference = Fi = --------- xi+1 - xi (5.3.2) (2) (2) (3) Fi+1 - Fi Third divided difference = Fi = --------- xi+1 - xi (N-1) (N-1) (N) Fi+1 - Fi Nth divided difference = Fi = --------- xi+1 - xi The polynomial fitted between xi and xi+1 is then (1) 2 (2) 3 (3) f(x) = p(x) = fi + (x - xi)Fi + (x - xi) Fi + (x - xi) Fi + ... (5.3.3) Observe that the divided differences automatically include the factorials appearing in equation (5.3.1). Note also that the divided difference corresponding to dnf / dxn is calculated by employing fi values somewhat remote from the region of interest: Fi(l) uses fi and fi+1, Fi(2) uses fi, fi+1 and fi+2, etc. ln fact, all but one of the points invol- ved is forward of the interpolation position. Therefore, the required derivatives are not being estimated at the exact position desired, but rather a little after it. This same bias is built into the Lagrange formulation. The effect can be reduced by applying a for- ward/backward, or central divided-differences hybrid formula instead. However, this alter- ation to improve the accuracy is not worth the effort when the increased complexity (and slower execution speed) of the resulting program is considered along with the round-off error, which appears to dominate the accuracy anyway. There are two sources of error in the Newton divided-differences interpolation scheme. The first and most obvious source is the intrinsic error due to truncating the Taylor series. If Nth-order interpolation is employed, then the truncation error is asiated with the first term not included: N+1 (N+1) (x - xi) d f Erroe = ----------- --------- (5.3.4) (N+1) (N+1)! dx (The error for the actual numerical calculation has this sa me form, but it is a little different. It will be discussed shortly). Truncation error is, in principle, amenable to estimation. However, in the specific case of Newton interpolation, it is usually not the dominant source of inaccucy. The Nth divided difference involves N calculations of the differences between nearly equal numbers. This leads to significant round-off error. Unfortunately, this round-off error cannot easily be estimated. We will see its very large effect in a later example. A subroutine for third-order (cubic) Newton interpolation appears in program Newton. The associated input/output variables are the same as those for the Lagrange routine, with the exception of N. Because the interpolation is cubic, the order input, N, is super- fluous. The NEWTON subroutine can easily be extended to higher orders of interpolation, but this is not wise. Such an extension leads to an increase in the round-off error which, as we will soon see, is bad even at the cubic level. A priori, the only obvious advantages to the Lagrange interpolation routine are that it is shorter in length, and that it does not require several additional large-diensioned arrays. Mathematically, Lagrange and Newton interpolation of the same type (forward, backward, etc.) and order are equivalent--only the evaluation forms are different. Therefore, they have the same intrinsic accuracy. ln fact, the truncation error for the cubic case in each is 4 (x - xi)(x - xi+1)(x - xi+2)(x - xi+3) d f | Error = -------------------------------------- ---4| (5.3.5) 4! dx |x' where x' is an evaluation point somewhere in the interval between xi and xi+3. Equations appear to be different, with the latter method giving larger values for the error estimate. The difference between the two equations can be rationalized as follows. Equation (5.3.4) assumes implicitly that we know the value of all the required derivatives at the position xi. However, such is not the case in the numerical calculation. Rather, a group of table values is used to estimate the required derivaes, thereby increasing the error. For example, the divided- difference third derivative calculation involves using the points xi, xi+l, xi+2, and xi+3. The value found is the third derivative at some point in that interval (by the mean value theorem). This introduces into the algorithm an intrinsic inaccuracy beyond that expected from equation (5.3.4). For a further discussion of this issue, see Applied Numerical Methods, Ref.10. From the form of the error estimate, it is apparent that a cubic polynomial is fitted exactly because the fourth derivative of a cubic is necessarily zero throughout the interval. This error estimate is also calculated in the Newton program. The Newton method is two orders of magnitude less accurate than the Lagrange method in this particular example. It is not sufficient to simply reduce the truncation error by using more closely spaced table values or higher-order inter- polation; round-off error must also be guarded against. It is obvious that Lagrange interpolation is superior to Newton's divided-differences technique in terms of final accuracy. The precautions associated with using NEWTON are identical to those en- countered with LAGRANGE (see file Lagrange.txt). From [BIBLI 01] -------------------------------------------------- End of file Newton.txt