W. D. Hoskins

A faster method of computing the square root of a matrix

By considering the square root of a matrix as being a special case of a matrix Riccati-type equation, a fast economical algorithm is developed as a stable generalization of the process given in [1].

The numerical solution of the matrix equationXA+AY=F

An iterative method for solving the matrix equationXA+AY=F is discussed Algorithms and techniques for accelerating convergence are outlined. The method compares favourably with existing techniques.

Linear dependence relations for polynomial splines at midknots

Linear relations between midknot values of a smooth polynomial spline and its derivatives are derived for the case of a uniformly spaced set of knots. The leading term of the truncation error for midknot interpolating splines is determined for these relations. Comparisons are made with the corresponding relations and truncation errors associated with knot-interpolating splines.

Some approximation properties of periodic parametric cubic splines

The parametric cubic splines interpolating to such closed curves as the circle and ellipse are derived in a form where their parameters are given by simple algebraic expressions.The structure of these expressions enables the error in approximation of the given curves to be precisely determined and some additional features of the spline deduced.

A rapidly convergent method for the solution of of the matrix equation XA+AY=F

An iterative method for solving the matrix equation XA + AY = F is outlined. An algorithm with acceleration parameters is provided.

Solution of bi-linear systems arising from high order discretizations of poisson-type equations

Bi-linear systems of the formAV+WA=G are obtained by approximating Poisson-type equations using higher-order finite difference formulae whereV,W andG are known matrices. Solution of the bi-linear system requiresO(n3) operations for ann×n mesh. However, due to the increased accuracy obtained when using a high-order discretization formula,n can be made much smaller than in the conventional methods and indicates that faster Poisson-solvers which are numerically stable can be obtained by considering the bilinear system rather than the composite matrix form.

Successive polynomial spline function approximation

The use of successive polynomial spline approximation is established as a method of improving the accuracy of estimates of derivatives of periodic functions approximated by interpolating odd order splines defined on a uniformly spaced set of data points. For the various configurations possible with this multiple-approximation method, bounds for the leading error terms are explicitly given. In particular, for the quintic spline, the variety of approximation sequences is described in detail.

Explicit calculation of interpolating cubic splines on equi-distant knots

The coefficients of the cubic splines(x) interpolating to the functionf(x) on the equi-distant knots,xi=ih(i=0(1)n andh=1/n) in the interval [0, 1], are determined explicitly in the cases whenf(x) is either periodic or has linear combinations of the first and second derivatives specified as boundary conditions.The effects of perturbations in the boundary conditions are analysed in closed form and exact results given for the ensuing changes in the spline fit. As illustration of the techniques a numerical example is given.