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On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grass...

Abstract We present a procedure to generate smooth interpolating curves on submanifolds, which are given in closed form in terms of the coordinates of the embedding space. In contrast to other existing methods, this approach makes the corresponding algorithm easy to implement. The idea is to project the prescribed data on the manifold onto the affine tangent space at a particular point, solve the interpolation...

Padé and Gregory error estimates for the logarithm of block triangular matrices

In this paper we give bounds for the error arising in the approximation of the logarithm of a block triangular matrix T by Padé approximants of the function f(x)=log[(1+x)/(1-x)] and partial sums of Gregory's series. These bounds show that if the norm of all diagonal blocks of the Cayley-transform B=(T-I)(T+I)-1 is sufficiently close to zero, then both approximation methods are accurate. This will contribute fo...

A multi-input/multi-output system representation of generalized splines in R^n

Theoretical and numerical considerations about Padé approximants for the matrix...

We show that for a vast class of matrix Lie groups, which includes the orthogonal and the symplectic, diagonal Padé approximants of log((1+x)/(1-x)) are structure preserving. The conditioning of these approximants is analyzed. We also present a new algorithm for the Briggs-Padé method, based on a strategy for reducing the number of square roots in the inverse scaling and squaring procedure. ; http://www.scie...