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



____________________________________________________________________________________________________________

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

____________________________________________________________________________________________________________

DESCRIPTION
       This package implements several interpolation algorithms:

       •      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.

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

       •      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.

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

       •      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  break-points breakpoints
                     points 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  inter-polate. interpolate.
              polate.

              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 xcoordinate
                     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 interpola-tion interpolation
              tion 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://source -
       forge.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)