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Properties of matrices in methods of constructing an interpolating spline via the coordinates of its derivatives in \(B\)-spline basis. (English) Zbl 0979.65014

The problem of constructing an interpolating spline of minimal defect is reduced to solving some system of linear equations. In [Vychisl. Sist. 147, 3-10 (1992; Zbl 0816.41009)] the author established that the condition number of the matrix of a widespread method for determining the \(B\)-spline decomposition coefficients (collocation matrix) can be arbitrary large on essentially nonuniform meshes. In [Vychisl. Sist. 159, 3-18 (1997)] the author proposed a new approach to the problem of constructing an interpolating spline and derived systems of equations for finding the derivatives of splines as coordinates with respect to a \(B\)-spline basis.
In the article under review, the properties of the matrices \(A_k\) (\(k\) is the order of derivatives) of such systems are considered. In the special case of a uniform mesh, the matrices \(A_k\) of all systems (for all \(k\)) are identical and coincide with the \(B\)-spline collocation matrix. The connection with convergence of an interpolation process in \(C^k\) is studied. Two systems of equations for \(k=n-1,n\) (the degree of splines is \(2n-1\)), probably, are well conditioned for arbitrary nonuniform meshes. Numerical data are presented of the condition values of the matrices of the methods proposed for splines of odd degrees from 3 to 15.

MSC:

65D07 Numerical computation using splines
65F35 Numerical computation of matrix norms, conditioning, scaling
15A12 Conditioning of matrices
41A15 Spline approximation
65D05 Numerical interpolation

Citations:

Zbl 0816.41009
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