An numerical algorithm for Laplace's integral by B-spline

Algorithm for Laplace's integral is given when the inverse image function has high order discontinuity. The multi-node technique of B-spline is used to describe the interruption point, cusp and non-smooth point of the inverse image function. The difference quotient and de Boor algorithm are used to derive the image function of the Laplace's integral under non-uniform partition. And a set of practical formula is got when the partition is quasi-uniform. The scheme enables the image function to be approximated within any prescribed tolerance. Experiments also show that good result is achieved. It is much faster than that of Simpsons rule, and much simpler than that of Berge method, the traditional efficient method. It is no longer to find the zero points and coefficients of Gauss-Laguerre or Gauss-Legendre polynomials. The image function of Laplace's integral can also be computed while the inverse image function is hyper-function with high order discontinuity.