1. Manuals
  2. Tumbleweed
  3. tcllib
  4. interpolate(n)

Scroll to navigation

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 ?1.1?


::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.






INCOMPATIBILITY WITH VERSION 1.0.3


The interpretation of the tables in the
    ::math::interpolate::interpolate-1d-table command has been changed to
    be compatible with the interpretation for 2D interpolation in the
    ::math::interpolate::interpolate-table command. As a consequence this
    version is incompatible with the previous versions of the command
  (1.0.x).






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.







  

  
Name of the table to be created

  

  
List of column names

  

  
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.







  

  
Name of an existing table

  

  
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.







  

  
Name of an existing table

  

  
Value of the independent row variable

  

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

  

  
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 of pairs of (x,y) values

  

  
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 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 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 of coefficients as returned by prepare-cubic-splines

  

  
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 of lists, each sublist being a list of coordinates and of dependent
      values.

  

  
List of coordinates for which the values must be calculated









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

  
Set the parameters for spatial interpolation







  

  
Search radius (data points further than this are ignored)

  

  
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


Example of using one-dimensional tables:


Suppose you have several tabulated functions of one variable:






    x     y1     y2


  0.0    0.0    0.0


  1.0    1.0    1.0


  2.0    4.0    8.0


  3.0    9.0   27.0


  4.0   16.0   64.0




Then to estimate the values at 0.5, 1.5, 2.5 and 3.5, you can use:





   set table [::math::interpolate::defineTable table1  {x y1 y2} {   -      1      2


                   0.0    0.0    0.0


                   1.0    1.0    1.0


                   2.0    4.0    8.0


                   3.0    9.0   27.0


                   4.0   16.0   64.0}]


   foreach x {0.5 1.5 2.5 3.5} {


       puts "$x: [::math::interpolate::interp-1d-table $table $x]"


   }




For one-dimensional tables the first row is not used. For two-dimensional
  tables, the first row represents the values for the second independent
  variable.

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 Trackers
    [http://core.tcl.tk/tcllib/reportlist]. Please also report any ideas for
    enhancements you may have for either package and/or documentation.


When proposing code changes, please provide unified diffs,
    i.e the output of diff -u.


Note further that attachments are strongly preferred over
    inlined patches. Attachments can be made by going to the Edit form of
    the ticket immediately after its creation, and then using the left-most
    button in the secondary navigation bar.






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>








  

    1.1
    tcllib