Piecewise cubic spline

EnHT20.7B Piecewise Cubic Spline Swap Zero Curve by Currency (continuously compounded)... 

The linear algebraic system consists of 3n - 3 equations and 3n - 3 unknowns that can be solved to produce the optimal piecewise cubic spline. Press, Teukolsky, Vetterling, and Flannery describe a routine for cubic spline interpolation. " ... 

This section provides a discussion of piecewise cubic spline interpolation methodology and its application to the term stmcture. Our intent is to provide an accessible approach to cubic spline interpolation for... 

The answer to this difficulty lies in the use of piecewise approximants, such as cubic splines, which are in general use in the mathematics literature (11). Carey and Finlayson (12) have introduced a finite-element collocation method along these lines, which uses polynomial approximants on sub-intervals of the domain, and apply continuity conditions at the break-points to smooth the solution. It would seem more straight-forward, however, to use piecewise polynomials which do not require explicit continuity... 

Fig. 4. (A) Parametrization of a shaped pulse envelope with the help of a cubic spline interpolation between a small number of anchor points (circles). (B) Approximation of the smooth pulse envelope by rectangular pulses with piecewise constant rf amplitude. (Adapted from Ewing et al., 1990, p. 123, with kind permission from Elsevier Science—NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)...

Simple cubic splines, 5 i(x) and 2(x), Eqs. (20.9), are piecewise continuous third-order polynomials governed by various constraints (depending on the type of cubic spline—e.g., free or clamped) that link three or more contiguous (x, y) points together. Spline segments are sequentially added until all n (x, y) data points along the interval, Min(x) < x < Max(x), have been included to form S(x). 

Another structure for expressing a nonlinear relationship between X and Y is splines [333] or smoothing functions [75]. Splines are piecewise polynomials joined at knots (denoted by Zj) with continuity constraints on the function and all its derivatives except the highest. Splines have good approximation power, high flexibility and smooth appearance as a result of continuity constraints. For example, if cubic splines are used for representing the inner relation ... 

One solution is to use a piecewise cubic polynomial, where d l) is associated with a different cubic polynomial, with different coefficients. A special case of this approach, the cubic spline, is discussed in the next section. 

In mathematics a spline is a piecewise polynomial function, made up of individual polynomial sections or segments that are joined together at (user-selected) points known as knot points. Splines used in term structure modeling are generally made up of cubic polynomials. The reason they are often cubic polynomials, as opposed to polynomials of order, say, two or five, is explained in straightforward fashion by de la Grandville (2001). A cubic spline is a function of order three and a piecewise cubic polynomial that is twice differentiable at each knot point. At each knot point the slope and curvature of the curve on either side must match. The cubic spline approach is employed to fit a smooth curve to bond prices (yields) given by the term discount factors. 

MATLAB has several functions for interpolation. The function = interpl(x, y, x) takes the values of the independent variable x and the dependent variable y (base points) and does the one-dimensional interpolation based on x, to find yj. The default method of interpolation is linear. However, the user can choose the method of interpolation in the fourth input argument from nearest (nearest neighbor interpolation), linear (linear interpolation), spline (cubic spline interpolation), and cubic (cubic inteipolation). If the vector of independent variable is not equally spaced, the function interplq may be used instead. It is faster than interpl because it does not check the input arguments. MATLAB also has the function sp/ine to perform one-dimensional interpolation by cubic. splines, using nat-a- not method, ft can also return coefficients of piecewise poiynomiais, if required. The functions interp2, inte.rp3, and interpn perform two-, three-, and n-dimensional interpolation, respectively. 

A piecewise polynomial will typically have / = 1 at the places where the pieces meet, so that the cubic B-spline is C2+1 at its knots. It is, of course C°° over the open intervals between the knots. 

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