# 2.3 Apparent Errors

We now turn to the problem of estimating the mathematical error of the approximation. In practice, this estimate must be based on likely hypothe­ses, and estimating these apparent errors is currently an art as well as a science.

Let us now turn our attention to functions which have a dominant singularity of the form, see  and let us suppose that we have estimated this singularity, probably by using the methods of Section 2.2, to be Ihe three parameters A, fi, and y are in principle determined by three equations, which originate from accuracv-throueh-order equations. Let these and  then we know that equations be accurate to order where r/7 are small percentage errors. Obviously, if r/7 =0, the parameters are identical, and so the r/7 are to be regarded as the source of the error in the approximation scheme. We consider percentage errors, because fi determines the scale of the z-plane; if ju. is very different from unity, the magnitude of the terms in (3.3) can vary rapidly with j. If the approximation is a good one, we may use first-order expansions

A first-order analysis and logarithmic differentiation of (3.3) gives

The determination of the apparent absolute size of the errors is more difficult. We will be explicit about our procedure when it is used in conjunction with a method such as (i) or (iv) of the previous section. In those cases the parameter jit is just a zero of the reciprocal of the function, and so is proportional to the error of estimation of (e.g.)

Thus we can relate the error fyt/jw to the errors in a table of values, which are in practice relatively easy to compute.

First let us look at the structure of the difference between two adjacent Padé approximants. By the (* *) identity (3.4.5), the r/'s. The hypothesis involved here is that there are no unusual cancella­tions, so that a reasonable estimate for the magnitude of 17 is obtained.

Since B[L/M](0)= 1, one can estimate the coefficient of zL+M, and at the same time to some extent take into account higher-order terms, by forming a table of the left-hand side of (3.8) over the range 0<z<ju~' and fitting it to the monomial zI+M at the point in that range which gives the largest coefficient.Now, in the treatment of method (i) of Section 2.2 (i7,), a different reduction of (3.5) is convenient, namely plays the role of 8y, is of order JSp/n, in harmony with (3.7). Thus by this analysis we have, by (3.8), that -q is given by where we have used

Exercise Why is the variation in value of a diagonal Padé approximant with respect to small variations in c0 usually a poor estimate of the accuracy of the formation of the approximant?