27-07-2012, 01:42 PM
Constrained Cubic Spline Interpolation
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Introduction
Interpolation is used to estimate the value of a function between known data points without knowing the
actual function. Interpolation methods can be divided into two main categories [1,2]:
• Global interpolation. These methods rely on a constructing single equation that fits all the data points.
This equation is usually a high degree polynomial equation. Although these methods result in smooth
curves, they are usually not well suited for engineering applications, as they are prone to severe
oscillation and overshoot at intermediate points.
• Piecewise interpolation. These methods rely on constructing a polynomial of low degree between each
pair of known data points. If a first degree polynomial is used, it is called linear interpolation. For
second and third degree polynomials, it is called quadratic and cubic splines respectively. The higher the
degree of the spline, the smoother the curve. Splines of degree m, will have continuous derivatives up to
degree m-1 at the data points.
Linear interpolation result in straight line between each pair of points and all derivatives are
discontinuous at the data points. As it never overshoots or oscillates, it is frequently used in chemical
engineering despite the fact that the curves are not smooth.
To obtain a smoother curve, cubic splines are frequently recommended. They are generally well behaved
and continuous up to the second order derivative at the data points. Even though cubic splines are less
prone to oscillation or overshoot than global polynomial equations, they do not prevent it. Thus, the use
of cubic splines in chemical engineering is limited to applications where oscillation and overshoot are
acceptable or desirable.
Traditional Cubic Splines
Consider a collection of known points (x0, y0), (x1, y1), ... (xi-1, yi-1), (xi, yi), (xi+1, yi+1), ... (xn, yn). To
interpolate between these data points using traditional cubic splines, a third degree polynomial is constructed
between each point. The equation to the left of point (xi, yi) is indicated as fi with a y value of fi(xi) at point
xi. Similarly, the equation to the right of point (xi, yi) is indicated as fi+1 with a y value of fi+1(xi) at point xi.
Proposed Constrained Cubic Splines
The principle behind the proposed constrained cubic spline is to prevent overshooting by sacrificing
smoothness. This is achieved by eliminating the requirement for equal second order derivatives at every
point (equation 4) and replacing it with specified first order derivatives.
Conclusions
A modified cubic spline interpolation method has been developed for chemical engineering application. The
main benefits of the proposed constrained cubic spline are:
• It is a relatively smooth curve;
• It never overshoots intermediate values;
• Interpolated values can be calculated directly without solving a system of equations;
• The actual parameters (ai, bi, ci and di) for each of the cubic spline equations can still be calculated. This
permits analytical integration of the data.