2.3 Apparent Errors
With any approximation scheme, one must be able to estimate the approximation error. Using Padé approximants, there are three principal sources of error: (i) the given coefficients are known only to limited accuracy, (ii) accuracy is lost in forming the coefficients of the Padé approximant and in forming its value, and (iii) the Padé approximant is not the function itself, leading to the fundamental approximation error. There is little to be done about (i) except to note that accuracy in the given coefficients is essential. The Padé approximant is necessarily misguided by errors in the given series, no matter what their source. The variation of the coefficients of the given series within their accuracy limits provides a very useful and instant error estimate of the accuracy of Padé approximation. For (ii), loss of accuracy in the formation of Padé approximants is, in practice, usually inexcusable. Double precision on a modern computer gives 20 or 30 decimal places, and this precision should be used if necessary. Of course, the approximation problem is often ill conditioned, and the accuracy of the input coefficients is usually better than the accuracy of the output coefficients even with exact arithmetic. A rough and ready working guide is that one extra decimal place of working accuracy should be kept for every decimal place of accuracy required in the answer (cf. Section 2.4). As a purely empirical anthropological observation, most inexperienced Padé approximators use sufficiently high order approximants—often too high—and insufficient working accuracy to justify them.
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.
In the previous section, we encountered defects in the approximants. They are nearby pole and zero combinations which are significant when they occur within the region of interest. The region is frequently most conveniently taken to be the largest circle containing the origin and whatever singularities are of interest, but excluding, with a margin to spare, all other nonpolar singularities. A defect would then be any pole in that region with a residue less than some preassigned value, say 0.003 times the expected residue. We are inclined to exclude these defective Padé approximants from the set of Padé approximants expected to be useful approximations. By so doing, we expect to obtain a set which is uniformly bounded on a bounded region which excludes singularities of the given function. Provided that there are sufficiently many Padé approximants in this set, the theorems of Section 6.4 assure convergence. In the assessment of apparent errors, it should be noted that, as a practical matter, the occurrence of defects seems be approximately equal, and one can be misled about the rate of convergence if the defects are not detected either directly or indirectly. The occurrence of defects can be thought of as a near miss at the existence of a block in the Padé table, where the determinant does not vanish but is anomalously small. In the case where there is a block, certain consecutive Padé approximants are, of course, exactly equal (Section 1.4). The existence of defects shows that it is important to analyze as much as possible of the Padé table to decide which poles and zeros are significant, and which are defects indicating that the Padé approximant in question is unreliable. Furthermore, it is plain that a blind calculation of Padé approximants at any one value (e.g. by using the e-algorithm) ignores much of the information provided by the Padé approximants themselves about their convergence in the z-plane.
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?