This chapter is concerned with the interpolation of a function of one or more variables. When provided with the value of the function (and possibly one or more of its lowest-order derivatives) at each of a number of values of the variable(s), the NAG Library functions provide either an interpolating function or an interpolated value. For some of the interpolating functions, there are supporting NAG Library functions to evaluate, differentiate or integrate them.
2Background to the Problems
In motivation and in some of its numerical processes, this chapter has much in common with Chapter E02 (Curve and Surface Fitting). For this reason, we shall adopt the same terminology and refer to dependent variable and independent variable(s) instead of function and variable(s). Where there is only one independent variable, we shall denote it by and the dependent variable by . Thus, in the basic problem considered in this chapter, we are given a set of distinct values of and a corresponding set of values of , and we shall describe the problem as being one of interpolating the data points , rather than interpolating a function. In modern usage, however, interpolation can have either of two rather different meanings, both relevant to functions in this chapter. They are
(a)the determination of a function of which takes the value at , for (an interpolating function or interpolant),
(b)the determination of the value (interpolated value or interpolate) of an interpolating function at any given value, say , of within the range of the (so as to estimate the value at of the function underlying the data).
The latter is the older meaning, associated particularly with the use of mathematical tables. The term ‘function underlying the data’, like the other terminology described above, is used so as to cover situations additional to those in which the data points have been computed from a known function, as with a mathematical table. In some contexts, the function may be unknown, perhaps representing the dependency of one physical variable on another, say temperature upon time.
Whether the underlying function is known or unknown, the object of interpolation will usually be to approximate it to acceptable accuracy by a function which is easy to evaluate anywhere in some range of interest. Polynomials, rational functions (ratios of two polynomials) and piecewise polynomials, such as cubic splines (see Section 2.2 in the E02 Chapter Introduction for definitions of terms in the latter case), being easy to evaluate and also capable of approximating a wide variety of functions, are the types of function mostly used in this chapter as interpolating functions. An interpolating polynomial is taken to have degree when there are data points, and so it is unique. It is called the Lagrange interpolating polynomial. The rational function, in the special form used, is also unique. An interpolating spline, on the other hand, depends on the choice made for the knots.
One way of achieving the objective in (b) above is, of course, through (a), but there are also methods which do not involve the explicit computation of the interpolating function. Everett's formula and Aitken's successive linear interpolation (see Dahlquist and Björck (1974)) provide two such methods. Both are used in this chapter and determine a value of the Lagrange interpolating polynomial.
It is important to appreciate, however, that the Lagrange interpolating polynomial often exhibits unwanted fluctuations between the data points. These tend to occur particularly towards the ends of the data range, and to get larger with increasing number of data points. In severe cases, such as with or equally spaced values of , the polynomial can take on values several orders of magnitude larger than the data values. (Closer spacing near the ends of the range tends to improve the situation, and wider spacing tends to make it worse.) Clearly, therefore, the Lagrange polynomial often gives a very poor approximation to the function underlying the data. On the other hand, it can be perfectly satisfactory when its use is restricted to providing interpolated values away from the ends of the data range from a reasonably small number of data values.
In contrast, a cubic spline which interpolates a large number of data points can often be used satisfactorily over the whole of the data range. Unwanted fluctuations can still arise but much less frequently and much less severely than with polynomials. Rational functions, when appropriate, would also be used over the whole data range. The main danger with these functions is that their polynomial denominators may take zero values within that range. Unwanted fluctuations are avoided altogether by a function using piecewise cubic polynomials having only first derivative continuity. It is designed especially for monotonic data, but for other data still provides an interpolant which increases, or decreases, over the same intervals as the data.
The concept of interpolation can be generalized in a number of ways. Firstly, at each , the interpolating function may be required to take on not only a given value but also given values for all its derivatives up to some specified order (which can vary with ). This is the Hermite–Birkoff interpolation problem. Secondly, we may be required to estimate the value of the underlying function at a value outside the range of the data. This is the process of extrapolation. In general, it is a good deal less accurate than interpolation and is to be avoided whenever possible.
Interpolation can also be extended to the case of two or more independent variables. If the data values are given at the intersections of a regular two-dimensional mesh bicubic splines (see Section 2.3.2 in the E02 Chapter Introduction) are very suitable and usually very effective for the problem. For other cases, perhaps where the data values are quite arbitrarily scattered, polynomials and splines are not at all appropriate and special forms of interpolating function have to be employed. Many such forms have been devised and two of the most successful are in functions in this chapter. They both have continuity in first, but not higher, derivatives.
3Recommendations on Choice and Use of Available Functions
Before undertaking interpolation, in other than the simplest cases, you should seriously consider the alternative of using a function from Chapter E02 to approximate the data by a polynomial or spline containing significantly fewer coefficients than the corresponding interpolating function. This approach is much less liable to produce unwanted fluctuations and so can often provide a better approximation to the function underlying the data.
When interpolation is employed to approximate either an underlying function or its values, you will need to be satisfied that the accuracy of approximation achieved is adequate. There may be a means for doing this which is particular to the application, or the function used may itself provide a means. In other cases, one possibility is to repeat the interpolation using one or more extra data points, if they are available, or otherwise one or more fewer, and to compare the results. Other possibilities, if it is an interpolating function which is determined, are to examine the function graphically, if that gives sufficient accuracy, or to observe the behaviour of the differences in a finite difference table, formed from evaluations of the interpolating function at equally-spaced values of over the range of interest. The spacing should be small enough to cause the typical size of the differences to decrease as the order of difference increases.
3.2One Independent Variable
3.2.1Interpolated values: data without derivatives
When the underlying function is well represented by data points on both sides of the value, , at which an interpolated value is required, e01abc should be tried first if the data points are equally spaced, e01aac if they are not. Both compute a value of the Lagrange interpolating polynomial, the first using Everett's formula, the second Aitken's successive linear interpolation. The first function requires an equal (or nearly equal) number of data points on each side of ; such a distribution of points is preferable also for the second function. If there are many data points, this will be achieved simply by using only an appropriate subset for each value of . Ten to twelve data points are the most that would be required for many problems. Both functions provide a means of assessing the accuracy of an interpolated value, with e01abc by examination of the size of the finite differences supplied, with e01aac by intercomparison of the set of interpolated values obtained from polynomials of increasing degree.
In other cases, or when the above functions fail to produce a satisfactory result, one of the functions discussed in the next section should be used. The spline and other piecewise polynomial functions are the most generally applicable. They are particularly appropriate when interpolated values towards the ends of the range are required. They are also likely to be preferable, for reasons of economy, when many interpolated values are required.
e01aac above, and three of the functions discussed in the next section, can be used to compute extrapolated values. These three are e01aec,e01becande01rac based on polynomials, piecewise polynomials and rational functions respectively. Extrapolation is not recommended in general, but can sometimes give acceptable results if it is to a point not far outside the data range, and only the few nearest data points are used in the process. e01rac is most likely to be successful.
3.2.2Interpolating function: data without derivatives
e01aec computes the Lagrange interpolating polynomial by a method (based on Newton's formula with divided differences (see Fröberg (1970)) which has proved numerically very stable.
However, the likelihood of the polynomial having unwanted fluctuations, particularly near the ends of the data range when a moderate or large number of data points are used, should be remembered.
Such fluctuations of the polynomial can be avoided if you are at liberty to choose the values at which to provide data points. In this case, a function from Chapter E02, namely e02afc, should be used in the manner and with the values discussed in Section 3.2.2 in the E02 Chapter Introduction.
Usually however, when the whole of the data range is of interest, it is preferable to use a cubic spline as the interpolating function. e01bac computes an interpolating cubic spline, using a particular choice for the set of knots which has proved generally satisfactory in practice. If you wish to choose a different set, a cubic spline function from Chapter E02, namely e02bac, may be used in its interpolating mode, setting
and all elements of the argument weights to unity.
The cubic spline does not always avoid unwanted fluctuations, especially when the data shows a steep slope close to a region of small slope, or when the data inadequately represents the underlying curve. In such cases, e01bec can be very useful. It derives a piecewise cubic polynomial (with first derivative continuity) which, between any adjacent pair of data points, either increases all the way, or decreases all the way (or stays constant). It is especially suited to data which is monotonic over the whole range. If it is important to preserve both monotonicity and convexity, for example when constructing yield curves in Financial Mathematics, then e01cec should be used.
In this function, the interpolating function is represented simply by its value and first derivative at the data points. Supporting functions compute its value and first derivative elsewhere, as well as its definite integral over an arbitrary interval. The other functions mentioned, namely e01aecande01bac, provide the interpolating function either in Chebyshev series form or in B-spline form (see Sections 2.2.1 and 2.2.2 in the E02 Chapter Introduction). Functions for evaluating, differentiating and integrating these forms are discussed in Section 3.7 in the E02 Chapter Introduction. The splines and other piecewise cubics will normally provide better estimates of the derivatives of the underlying function than will interpolating polynomials, at any rate away from the central part of the data range.
e01rac computes an interpolating rational function. It is intended mainly for those cases where you know that this form of function is appropriate. However, it is also worth trying in cases where the other functions have proved unsatisfactory. e01rbc is available to compute values of the function provided by e01rac.
3.2.3Data containing derivatives
e01aec (see Section 3.2.2) can also compute the polynomial which, at each , has not only a specified value but also a specified value of each derivative up to order .
3.3Two Independent Variables
3.3.1Data on a rectangular mesh
Given the value of the dependent variable at the point in the plane of the independent variables and , for each and (so that the points lie at the intersections of a rectangular mesh), e01dac computes an interpolating bicubic spline, using a particular choice for each of the spline's knot-set. This choice, the same as in e01bac, has proved generally satisfactory in practice.
If, instead, you wish to specify your own knots, a function from Chapter E02, namely e02dac, may be used (it is more cumbersome for the purpose, however, and much slower for larger problems). Using and in the above sense, the argument
m must be set to ,
spline.nx and spline.ny
must be set to and respectively and all elements of
w should be set to unity. The recommended value for
eps is zero.
As remarked at the end of Section 2, specific methods of interpolating are required for this problem, which can often be difficult to solve satisfactorily. Two of the most successful are employed in
the two functions which (with their respective evaluation functions
are provided for the problem. Definitions can be found in the function documents. Both interpolants have first derivative continuity and are ‘local’, in that their value at any point depends only on data in the immediate neighbourhood of the point. This latter feature is necessary for large sets of data to avoid prohibitive computing time. e01shc allows evaluation of the interpolant and its first partial derivatives.
The relative merits of the two methods vary with the data and it is not possible to predict which will be the better in any particular case.
e01sgcande01sjc each perform
a triangulation of the scattered data points and then
a bicubic interpolant based on this triangulation and on the function values at the scattered points (which can be evaluated by
Where derivative continuity is not essential and where bilinear interpolated values are sufficient, e01eac (which performs the same triangulation as
and e01ebc (which performs barycentric interpolation using the set of function values) may be used. It should be noted that the triangulation computed by e01eac can be used to obtain the ordered list of connected data nodes that comprise the triangulated boundary, i.e., a convex hull for the given points; this is demonstrated in Section 10 in e01eac.
3.4Three Independent Variables
The function e01tgc and its evaluation function e01thc are provided for interpolation of three-dimensional scattered data. As in the case of two independent variables, the method is local, and produces an interpolant with first derivative continuity. e01thc allows evaluation of the interpolant and its first partial derivatives.
3.5Four and Five Independent Variables
The function e01tkc and its evaluation function e01tlc allow interpolation of four-dimensional scattered data, while the function e01tmc and its evaluation function e01tnc allow interpolation of five-dimensional scattered data. e01tkcande01tmc are higher dimensional analogues to the functions e01sgcande01tgc, while e01tlcande01tnc are analogous to e01shcande01thc.
Interpolation of scattered data in -dimensions, where , is provided by function e01zmc. This extends the local method of e01tgcande01tkc to higher dimensions. Evaluation of the interpolant, which has continuous first derivatives, is carried out by function e01znc.
Interpolation of gridded data in -dimensions is provided by function e01zac. This allows three methods of interpolation: a modified Shepard method, as used in function e01zmc; linear interpolation; and cubic interpolation, which is based on cubic convolution.