math::interpolate(n)						 Tcl Math Library					      math::interpolate(n)

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NAME

       math::interpolate - Interpolation routines

SYNOPSIS

       package require Tcl  ?8.4?

       package require struct

       package require math::interpolate  ?1.0.2?

       ::math::interpolate::defineTable name colnames values

       ::math::interpolate::interp-1d-table name xval

       ::math::interpolate::interp-table name xval yval

       ::math::interpolate::interp-linear xyvalues xval

       ::math::interpolate::interp-lagrange xyvalues xval

       ::math::interpolate::prepare-cubic-splines xcoord ycoord

       ::math::interpolate::interp-cubic-splines coeffs x

       ::math::interpolate::interp-spatial xyvalues coord

       ::math::interpolate::interp-spatial-params max_search power

       ::math::interpolate::neville xlist ylist x

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DESCRIPTION

       This package implements several interpolation algorithms:

       o      Interpolation  into a table (one or two independent variables), this is useful for example, if the data are static, like with tables
	      of statistical functions.

       o      Linear interpolation into a given set of data (organised as (x,y) pairs).

       o      Lagrange interpolation. This is mainly of theoretical interest, because there is no guarantee about error bounds. One possible  use:
	      if you need a line or a parabola through given points (it will calculate the values, but not return the coefficients).

	      A variation is Neville's method which has better behaviour and error bounds.

       o      Spatial interpolation using a straightforward distance-weight method. This procedure allows any number of spatial dimensions and any
	      number of dependent variables.

       o      Interpolation in one dimension using cubic splines.

       This document describes the procedures and explains their usage.

PROCEDURES

       The interpolation package defines the following public procedures:

       ::math::interpolate::defineTable name colnames values
	      Define a table with one or two independent variables (the distinction is implicit in the data). The procedure returns  the  name	of
	      the  table  -  this  name  is used whenever you want to interpolate the values. Note: this procedure is a convenient wrapper for the
	      struct::matrix procedure. Therefore you can access the data at any location in your program.

	      string name (in)
		     Name of the table to be created

	      list colnames (in)
		     List of column names

	      list values (in)
		     List of values (the number of elements should be a multiple of the number of columns. See EXAMPLES for  more  information	on
		     the interpretation of the data.

		     The values must be sorted with respect to the independent variable(s).

       ::math::interpolate::interp-1d-table name xval
	      Interpolate into the one-dimensional table "name" and return a list of values, one for each dependent column.

	      string name (in)
		     Name of an existing table

	      float xval (in)
		     Value of the independent row variable

       ::math::interpolate::interp-table name xval yval
	      Interpolate into the two-dimensional table "name" and return the interpolated value.

	      string name (in)
		     Name of an existing table

	      float xval (in)
		     Value of the independent row variable

	      float yval (in)
		     Value of the independent column variable

       ::math::interpolate::interp-linear xyvalues xval
	      Interpolate linearly into the list of x,y pairs and return the interpolated value.

	      list xyvalues (in)
		     List of pairs of (x,y) values, sorted to increasing x.  They are used as the breakpoints of a piecewise linear function.

	      float xval (in)
		     Value of the independent variable for which the value of y must be computed.

       ::math::interpolate::interp-lagrange xyvalues xval
	      Use the list of x,y pairs to construct the unique polynomial of lowest degree that passes through all points and return the interpo-
	      lated value.

	      list xyvalues (in)
		     List of pairs of (x,y) values

	      float xval (in)
		     Value of the independent variable for which the value of y must be computed.

       ::math::interpolate::prepare-cubic-splines xcoord ycoord
	      Returns a list of coefficients for the second routine interp-cubic-splines to actually interpolate.

	      list xcoord
		     List of x-coordinates for the value of the function to be interpolated is known. The coordinates must be strictly	ascending.
		     At least three points are required.

	      list ycoord
		     List of y-coordinates (the values of the function at the given x-coordinates).

       ::math::interpolate::interp-cubic-splines coeffs x
	      Returns the interpolated value at coordinate x. The coefficients are computed by the procedure prepare-cubic-splines.

	      list coeffs
		     List of coefficients as returned by prepare-cubic-splines

	      float x
		     x-coordinate at which to estimate the function. Must be between the first and last x-coordinate for which values were given.

       ::math::interpolate::interp-spatial xyvalues coord
	      Use  a  straightforward interpolation method with weights as function of the inverse distance to interpolate in 2D and N-dimensional
	      space

	      The list xyvalues is a list of lists:

		  {   {x1 y1 z1 {v11 v12 v13 v14}}
		   {x2 y2 z2 {v21 v22 v23 v24}}
		   ...
		  }

	      The last element of each inner list is either a single number or a list in itself.  In the latter case the return value  is  a  list
	      with the same number of elements.

	      The method is influenced by the search radius and the power of the inverse distance

	      list xyvalues (in)
		     List of lists, each sublist being a list of coordinates and of dependent values.

	      list coord (in)
		     List of coordinates for which the values must be calculated

       ::math::interpolate::interp-spatial-params max_search power
	      Set the parameters for spatial interpolation

	      float max_search (in)
		     Search radius (data points further than this are ignored)

	      integer power (in)
		     Power for the distance (either 1 or 2; defaults to 2)

       ::math::interpolate::neville xlist ylist x
	      Interpolates  between the tabulated values of a function whose abscissae are xlist and whose ordinates are ylist to produce an esti-
	      mate for the value of the function at x.	The result is a two-element list; the first element is the function's estimated value, and
	      the second is an estimate of the absolute error of the result.  Neville's algorithm for polynomial interpolation is used.  Note that
	      a large table of values will use an interpolating polynomial of high degree, which is likely to result in  numerical  instabilities;
	      one is better off using only a few tabulated values near the desired abscissa.

EXAMPLES

       TODO Example of using the cubic splines:

       Suppose the following values are given:

	   x	   y
	 0.1	 1.0
	 0.3	 2.1
	 0.4	 2.2
	 0.8	 4.11
	 1.0	 4.12

       Then to estimate the values at 0.1, 0.2, 0.3, ... 1.0, you can use:

	  set coeffs [::math::interpolate::prepare-cubic-splines  {0.1 0.3 0.4 0.8  1.0}  {1.0 2.1 2.2 4.11 4.12}]
	  foreach x {0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0} {
	     puts "$x: [::math::interpolate::interp-cubic-splines $coeffs $x]"
	  }

       to get the following output:

       0.1: 1.0
       0.2: 1.68044117647
       0.3: 2.1
       0.4: 2.2
       0.5: 3.11221507353
       0.6: 4.25242647059
       0.7: 5.41804227941
       0.8: 4.11
       0.9: 3.95675857843
       1.0: 4.12

       As you can see, the values at the abscissae are reproduced perfectly.

BUGS, IDEAS, FEEDBACK
       This  document, and the package it describes, will undoubtedly contain bugs and other problems.	Please report such in the category math ::
       interpolate of the Tcllib SF Trackers [http://sourceforge.net/tracker/?group_id=12883].	Please also report any ideas for enhancements  you
       may have for either package and/or documentation.

KEYWORDS

       interpolation, math, spatial interpolation

CATEGORY

       Mathematics

COPYRIGHT

       Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>
       Copyright (c) 2004 Kevn B. Kenny <kennykb@users.sourceforge.net>

math								       1.0.2						      math::interpolate(n)