engleski

Stable Calculation by Splines in Tension

For given strictly increasing partition of the interval $[0, 1]$, $0=x_0 < x_1< ldots x_n=1$, spline in tension is a function that on each subinterval $(x_i, x_{;i+1};)$ satisfies differential equation egin{;displaymath}; u^{;(4)}; - frac{;p_i^2};{;h_i^2}; u^{;(2)}; = 0, end{;displaymath}; where $p_i$ are nonnegative real numbers, called tension parameters, and $h_i=x_{;i+1};-x_i$. Therefore, on subinterval $(x_i, x_{;i+1};)$, spline in tension is a linear combination of functions $1, x, sinh p_i x / h_i$ and $cosh p_i x / h_i$ when $p_i>0$, and it is a cubic polynomial when $p_i=0$. For large tension parameters $p_i$, spline in tension is close to a linear function. Schweikert cite{;sch}; introduced spline in tension in the problem of convex interpolation.He derived algorithm which can remove spurious inflection points resulted from interpolation by cubic spline by increasing of the tension parameters $p_i$. Another problem where splines in tension were applied is a singularly perturbed boundary value problem for ODE. While in interpolation algorithms there were no evidences of unstability, Flaherty and Mathon observed that in their collocation method by spline in tension collocation matrix was almost singular for large values of tension parameters $p_i$. In analogy to cubic spline, solution for this problem could be an algorithm based on B-spline representation of spline in tension. Here we construct B-spline in tension and develop an algorithm for their stable evaluation.

Spline in tension; B-spline

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