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
              interpolated 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
              estimate 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)
