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:
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 dis-
tinction 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 multi-
ple 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 func-
tion of the inverse distance to interpolate in 2D and N-dimen-
sional 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 calcu-
lated
::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 interpolat-
ing polynomial of high degree, which is likely to result in
numerical instabilities; one is better off using only a few tab-
ulated 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)
Mac OS X 10.8 - Generated Fri Sep 7 19:46:16 CDT 2012