A univariate quasi-multiquadric interpolation with better smoothness

In this paper, we propose a multilevel univariate quasi-interpolation scheme using multiquadric basis. It is practical as it does not require derivative values of the function being interpolated. It has a higher degree of smoothness than the original level-0 formula as it allows a shape parameter c = O(h). Our level-1 quasi-interpolation costs O(n log n) flops to set up. It preserves strict convexity and monotonicity. When c = O(h), we prove the proposed scheme converges with a rate of O(h(2.5) log h). Furthermore, if both \f(a)\ and \f(b)\ are relatively small compared with parallel tofparallel to(infinity), the convergence rate will increase. We verify numerically that c = h is a good shape parameter to use for our method, hence we need not find the optimal parameter. For all test functions, both convergence speed and error are optimized for c between 0.5h and 1.5h. Our method can be generalized to a multilevel scheme; we include the numerical results for the level-2 scheme. The shape parameter of the level-2 scheme can be chosen between 2h to 3h. (C) 2004 Elsevier Ltd. All rights reserved.