math::interpolate -
Interpolation routines
package require Tcl ? 8.4 ?
package require struct
package require math::interpolate ? 1.0.1 ?
::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
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.
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.
| Type | Name | Mode |
| 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.
| Type | Name | Mode |
| 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.
| Type | Name | Mode |
| 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.
| Type | Name | Mode |
| 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.
| Type | Name | Mode |
| 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.
| Type | Name | Mode |
| 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.
| Type | Name | Mode |
| 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
| Type | Name | Mode |
| 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
| Type | Name | Mode |
| 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.
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.
math, interpolation, spatial interpolation