math::interpolate - Interpolation routines
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
This package implements several interpolation algorithms:
This document describes the procedures and explains their
usage.
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.
- ::math::interpolate::interp-table name xval
yval
- Interpolate into the two-dimensional table "name" and return the
interpolated value.
- ::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.
- ::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
- ::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.
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.
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.
interpolation, math, spatial interpolation
Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>
Copyright (c) 2004 Kevn B. Kenny <kennykb@users.sourceforge.net>