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.
- 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¶
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.
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>