STEP 18

FINITE DIFFERENCES 1

Tables

Historically speaking, numerical analysts have always been concerned with tables of numbers, and many techniques have been developed for dealing with mathematical functions, represented in this way. For example, the value of the function at an untabulated point may be required, so that a interpolation is necessary. It is also possible to estimate the derivative or the definite integral of a tabulated function, using some finite processes to approximate the corresponding (infinitesimal) limiting procedures of calculus. In each case, it has been traditional to use finite differences. Another application of finite differences, which is outside the scope of this book, is the numerical solution of partial differential equations.

  1. Tables of values

    Many books contain tables of mathematical functions. One of the most comprehensive is the Handbook of Mathematical Functions, edited by Abramowitz and Stegun (see the Bibliography for publication details), which also contains useful information about numerical methods.

    Although most tables use constant argument intervals, some functions do change rapidly in value in particular regions of their argument, and hence may best be tabulated using intervals varying according to the local behaviour of the function. Tables with varying argument intervals are more difficult to work with, however, and it is common to adopt uniform argument intervals wherever possible. As a simple example, consider the 6S table of the exponential function over 0.10 (0.01 ) 0.18 (a notation which specifies the domain 0.10

    It is extremely important that the interval between successive values is small enough to display the variation of the tabulated function, because usually the value of the function will be needed at some argument value between values specified (for example, from the above table). If the table is constructed in this manner, we can obtain such intermediate values to a reasonable accuracy by using a polynomial representation (hopefully, of low degree) of the function f.

  2. Finite differences

    Since Newton, finite differences have been used extensively. The construction of a table of finite differences for a tabulated function is simple: One obtains first differences by subtracting each value from the succeeding value in the table, second differences by repeating this operation on the first differences, and so on for higher order differences. From the above table of one has the (note the standard layout, with decimal points and leading zeros omitted from the differences):

    (In this case, the differences must be multiplied by 10-5 for comparison with the function values.)

  3. Influence of round-off errors

    Consider the difference table given below for to 6S, constructed as in Section 2. As before, differences of increasing order decrease rapidly in magnitude, but the third differences are irregular. This is largely a consequence of round-off errors, as tabulation of the function to 7S and differencing to fourth order illustrates (compare Exercise 3 ).

    Although the round-off errors in f should be less than 1/2 in the last significant place, they may accumulate; the greatest error that can be obtained corresponds to:

    A rough working criterion for the expected fluctuations (noise level) due to round-off error is shown in the table:

    Checkpoint

    1. What factors determine the intervals of tabulation of a function?
    2. What is the name of the procedure to determine a value of a tabulated function at an intermediate point?
    3. What may be the cause of irregularity in the highest order differences in a difference table?

    EXERCISES

    1. Construct the difference table for the function f (x) = x3 for x = 0(1) 6.

    2. Construct difference tables for each of the polynomials:

    1. 2x - l for x = 0(1)3.
    2. 3x2 + 2x - 4 for x = 0(1)4.
    3. 2x3 + 3x - 3 for x = 0(1)5.

    Study your resulting tables carefully; note what happens in the final few columns of each table. Suggest a general result for polynomials of degree n and compare your answer with the theorem in STEP 20.

    3. Construct a difference table for the function f (x) = ex, given to 7D for x = 0.1(0.05) 0.5

Answers

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