Interpolation with spline functions

Writing that the first cubic goes through the points (X i,y,--i) an we 

For n data points, (n — 1) equations, such as the one above, can be written with (n +1) unknowns y, (i = 0.n). Two additional equations are needed, which most often are end conditions at i=0 and i=n. The two conditions specifying the slopes 

Mis the (n +1)-vector of the unknowns y , while D is the (n +1)-vector with current 

Once the linear system is solved for the unknowns y, , the value interpolated at any arbitrary x can be calculated from either formula 

Let us build first the matrix A of coefficients au and the right-hand side vector D with the - admittedly questionable - assumption that the derivatives at the end-points are zero. 

---

R = 0 leads back to the problem of interpolation by spline functions. It should be noted at this point that the condition stated by eq. (4) is not sufficient for the construction of calibration curves and additional considerations have to take effect. A reformulation of the problem stated in Equations (2) and (3) gives us with 8y. = 1 for all i calibration points another look at the problem that clarifies the role of the integral in Equation (2) as balanced against a value of R. Find S (x) to... 

E. B. Smith, The Virial Coefficients of Pure Gases and Mixtures, Clarendon Press, Oxford, 1980. By interpolation (e.g., with cubic spline functions), virial coefficients can be determined for any temperature. 

The utility of spline functions to molecular dynamic studies has been tested by Sathyamurthy and Raff by carrying out quassiclassical trajectory and quantum mechanical calculations for various surfaces. However, the accuracy of spline interpolation deteriorated with an increase in dimension from 1 to 2 to 3. Various other numerical interpolation methods, such as Akima s interpolation in filling ab initio PES for reactive systems, have been used. 

There are a number of ways to model calibration data by regression. Host researchers have attempted to describe data with a linear function. Others ( 4,5 ) have chosen a higher order or a polynomial method. One report ( 6 ) compared the error in the interpolation using linear segments over a curved region verses using a curvilinear regression. Still others ( 7,8 ) chose empirical or spline functions. Mixed model descriptions have also been used ( 4,7 ). 

With the end condition flag EC = 0 on the input, the module determines the natural cubic spline function interpolating the function values stored in vector F. Otherwise, D1 and DN are additional input parameters specifying the first derivatives at the first and last points, respectively. Results are returned in the array S such that S(J,1), J = 0, 1, 2, 3 contain the 4 coefficients of the cubic defined on the I-th segment between Z(I) and ZII+l). Note that the i-th cubic is given in a coordinate system centered at Z(I). The module also calculates the area under the curve from the first point Z(l) to each grid point Z(I), and returns it in S(4,I). The entries in the array S can be directly used in applications, but we provide a further module to facilitate this step. 

A sculptured surface is obtained by interpolation with two-dimensional B-spline-functions. A B-spline surface is considered as a collection of surface patches and the whole surface is a mosaic of these patches linked together with proper continuity (Figure 9). Due to its computational efficiency a uniform bicubic B-spline surface has been implemented. The bicubic B-spline patch can be written in matrix form as... 

A cubic spline interpolation with periodic end conditions (IMSL-ICSPLN) was used to represent the load versus time function and a cubic spline evaluation routine (IMSL-ICSEVU) made load values available at any instant in time. 

We now turn our attention to the dual formulation of kriging which leads us to linear equations of the same type as those that arise from interpolation with radial basic functions such as thin plate splines. Some of the basic ideas can be found in [4]. The first studies go back to [8],... 

Local cubic interpolation results in a function whose derivative is not necessarily continuous at the grid points. With a non-local adjustment of the coefficients we can, however, achieve global differentiability up to the second derivatives. Such functions, still being cubic polynomials between each pair of grid points, are called cubic splines and offer a "stiffer" interpolation than the strictly local approach. 

The additional problem we face is determining the optimal value for p. It is important to note that the squared distance F2(p) increases with the value of p. Therefore, the algorithm can be viewed as starting with an interpolating spline obtained at p = 0, and then "streching" this function by gradually increasing the value of p until (4.27) holds. To find this particular p we solve the nonlinear equation /2... 

Spline Fitting in Two Variables. The methods described in the previous section may be extended to functions of two variables (9). The problem now is to find a surface S(x,y) which interpolates data values in a rectangular region a xsb, c ysd. By analogy with the previous section, we specify h "interior" knots in the x direction and l in the y direction. These knots, k, k2 ", kh and nl n2 ni divi< e... 

---