We employ a range of approached include cubic
splines, polynomial and rational interpolation for functions with 1 variable
and bicubic splines for functions of 2 variables.
Continuously Differentiable Curves and Surfaces
For any tabulated function in 1 or 2 dimensions all of the these interpolation
procedures will (for suitable choices of the parameters) produce continuously
differential functions in a neighborhood around any point at which they exist.
The differentiable nature of these curves and surfaces allows the application
of methods from differential geometry to be applied to the study of these curves
and surfaces.
Introduction to Interpolation/Extrapolation
functions from discrete data points. We consider here the problem of
constructing a curve form a
Detailed Overview of the Functionality Offered within this XL Service
Within this XL Service we consider the problem of approximating a function
in 1 or 2 variables which is known on a discrete data set with a function which is
continuously differentiable in a neighborhood of all points at which it exists.
Below we list in sections corresponding to each of the generic types of interpolation
function the business methods which are contained within this XL Service.
Cubic Interpolation of Variables (i.e. cubic splines)
CubicSpline2ndDifferential - Calculates the
second derivatives of the cubic spline interpolation polynomial at the tabulation points.
CubicSplinePointwisePreEvaluation - Evaluates
the cubic spline interpolation of a function at a point when the 2nd differentials at the
tabulation points are known.
CubicSplinePointwise - Calculates the value of the
cubic spline interpolation when the end point derivatives are known.
CoefficientsInterpolatingPolynomialStable
- A stable (and slower) means by which to evaluate the coefficients of the interpolating polynomial of
1 variable when the tabulation points are known.
CoefficientsInterpolatingPolynomial
- A fast (and less stable) means by which to evaluate the coefficients of the interpolating polynomial of
1 variable when the tabulation points are known.
Rational Function Interpolation of one variable
RationalInterpolationExtrapolation - Evaluates
the value and the corresponding error estimate at a given point of the rational interpolation
function.
Evaluation of the Bicubic interpolation function of two variables
BicubicCoefficients - Calculates
the coefficients for the bicubic interpolation polynomial.
BicubicInterpolation(double[], ...)
- Evaluates the value of the bicubic interpolation polynomial at a point and the derivatives in the directions
of the grid.
BicubicInterpolation(double[][], ...)
- Evaluates the values of the bicubic interpolation polynomial at a point and the derivatives in the directions
of the grid when its coefficients are known.