Approximation

The Problem: given data (x,f(x)) for x in a set X

a function g(x,P) , and a norm ||.|| on X

find parameters P such that

||f(.)-g(.,P)||is minimal with respect to P

For approximations to special functions search the netlib-libraries a, amos, cephes, elefunt, specfunc and the file netlib/toms/757 .
Here a list of some representative approximation algorithms is given. There are much more possibilities, e.g. using interpolation by splines, by splines under tension, by radial basis functions .... For more information use the search facility of netlib

NETLIB

or the HOTGAMS service

HOTGAMS

	APPROX/EXCH	best polynomial approximation to a discrete one-dimensional data set in the Chebyshev (minimax) sense
	L1GNR	generating test problems for discrete linear l1-approximation
	CDFCOR	rational approximation to a finite set of data f90 version
	COCA	COmplex linear Chebyshev Approximation (Matlab)
	SPLPAK	Cubic spline interpolation and fitting package (Matlab)
	l2appr	construct least squares approximation of B-splines to given data
	l2main	driver for l2appr.f
	titan	sample data for l2appr.f
	cubgcv	cubic spline data smoother using cross validation (used to produce the picture above)
	APPROX	discrete linear l1 approximation
	STL2	approximation by a piecewise linear function
	CL1(B&R)	linearly constrained approximation in l1-sense by linear programming
	CL1(B&C)	overdetermined system, linear constraints, discrete approximation in l1 sense
	RQN	unconstrained l1 approximation, interior point (several nonstandard languages)
	infSolve	Solves sparse minimax problems, with linear and nonlinear constraints (Matlab).
	DIFCOR	rational approximation to finite set of data (differential correction algorithm)
	DIFCORDOC	documentation for difcor
	REMEZ	continuous minimax rational approximation (in C)
	EVAL	needed by REMEZ (in C)
	CSHEP2D	Cubic {Shepard} Method for Bivariate Interpolation of Scattered Data
	TSHEP2D	Cosine Series {Shepard} Method for Bivariate Interpolation of Scattered Data
	CS2TST	accuracy tests for interpolation of scattered planar data
	PCB2NURB	smooth free form surfacing with linearly trimmed bicubic B-splines
	SSRFPACK	Interpolation of Scattered Data on the Surface of a Sphere with a Splines under Tension
	SDBI3P	Scattered-data surface fitting that has the accuracy of a cubic polynomial
	TSPACK	tension spline curve-fitting package
	QSHEP2D	quadratic Shepard method for bivariate interpolation of scattered data
	CUBGCV	O(n) computation of a cubic smoothing spline fitted to n noisy data points
	CAPROX	Chebyshev solution of systems of complex linear equations with linear inequality constraints and simple bound constraints
	CONST&EVAL	fitting multinomials in the least squares sense