A univariate quasi-multiquadric interpolationwith better smoothness

In this paper, we propose a multilevel univariate quasi-interpolation scheme usingmultiquadric 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 . Our level-1 quasi-interpolation costs flops to set up. It preserves strict convexity and monotonicity. When , we prove the proposed scheme converges with a rate of .Furthermore, if both ƒ″(a) and ƒ″ are relatively small compared with ƒ″ ∞, 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.