We offer refined numerical procedures to either construct a function of one or two variables from a set of points (i.e. interpolate), or solve an equation of one variable. The interpolation procedures provided include Newton polynomials, Lagrange's formula, Burlisch-Stoer algorithm, Cubic splines (natural and free), Bicubic interpolation and procedures for find the interpolation functions coefficients. In order to solve an equation we provide the Van Wijngaarden-Dekker-Brent algorithm, interval bisection method, secant and false position, Newton-Raphson method and Ridders' method.
This suite includes the following features:
Polynomial Interpolation and extrapolation
- Lagrange's formula - for interpolating a function known at N points with a polynomial of degree N-1
- Burlisch-Stoer algorithm - interpolates functions using rational functions, this method gives error estimates
- Cubic Splines - we give algorithms for natural and clamped cubic splines
- Sorting - efficient techniques are used for finding tabulated values
Coefficients of an Interpolating Polynomial
- Matrix method - this method relies upon diagonalizing a matrix (or solving a system of equations), and is of the order N squared
- Zero method - by evaluating the interpolating polynomial at particular values we deduce the coefficients, this method is of the order N cubed
Interpolation and extrapolation in two or more dimensions
- Grid - functions can be interpolated on an n-dimensional grid
- Bilinear interpolation - we consider a multidimensional interpolation by breaking the problem into successive one dimensional interpolations
- Accuracy - the use of higher order polynomials to obtain increased accuracy
- Smoothness - the use of higher order polynomials to enforce smoothness on some of the derivatives
- Bicubic interpolation - finds an interpolating function with a specified derivatives and cross derivatives which vary smoothly at the grid points
- Bicubic spline - a special case of Bicubic interpolation involving the use of successive one-dimensional splines
Equation Solver Module
- Interval Bisection Method - A robust method that always finds a solution or a singularity inside a bracketed interval.
- Secant Method - Generally this procedure converges and is much faster than the interval bisection method.
- Brent's Algorithm - The method of choice to find a bracketed root of a one dimensional equation when you cannot easily compute the function's derivative.
- Ridders' Method - Concise and almost as reliable as Brent's Algorithm for finding a bracketed root of an equation.
- Method of Regula Falsi - This procedure uses a slight alteration on the secant method to ensure convergence. The procedure is generally faster than the interval bisection method and slightly slower than the secant method.
- Newton-Raphson Method - Given a first approximation to a root and the differential of the function this procedure will always produce a solution. We implement this procedure for polynomial functions of one variable.
- Fail-Safe Newton-Raphson Method - This method combines the Newton-Raphson method and the Interval Bisection Method in order to produce very stable and fast convergence. Given a first approximation to a root and the differential of the function this procedure will always produce a solution.