1math::interpolate(n) Tcl Math Library math::interpolate(n)
2______________________________________________________________________________
6
8 math::interpolate - Interpolation routines
9
11 package require Tcl ?8.4?
12 package require struct
14 package require math::interpolate ?1.0.2?
16 ::math::interpolate::defineTable name colnames values
18 ::math::interpolate::interp-1d-table name xval
20 ::math::interpolate::interp-table name xval yval
22 ::math::interpolate::interp-linear xyvalues xval
24 ::math::interpolate::interp-lagrange xyvalues xval
26 ::math::interpolate::prepare-cubic-splines xcoord ycoord
28 ::math::interpolate::interp-cubic-splines coeffs x
30 ::math::interpolate::interp-spatial xyvalues coord
32 ::math::interpolate::interp-spatial-params max_search power
34 ::math::interpolate::neville xlist ylist x
36_________________________________________________________________
38
40 This package implements several interpolation algorithms:
41 · Interpolation into a table (one or two independent variables),
43 this is useful for example, if the data are static, like with
44 tables of statistical functions.
45 · Linear interpolation into a given set of data (organised as
47 (x,y) pairs).
48 · Lagrange interpolation. This is mainly of theoretical interest,
50 because there is no guarantee about error bounds. One possible
51 use: if you need a line or a parabola through given points (it
52 will calculate the values, but not return the coefficients).
53 A variation is Neville's method which has better behaviour and
55 error bounds.
56 · Spatial interpolation using a straightforward distance-weight
58 method. This procedure allows any number of spatial dimensions
59 and any number of dependent variables.
60 · Interpolation in one dimension using cubic splines.
62 This document describes the procedures and explains their usage.
64
66 The interpolation package defines the following public procedures:
67 ::math::interpolate::defineTable name colnames values
69 Define a table with one or two independent variables (the dis‐
70 tinction is implicit in the data). The procedure returns the
71 name of the table - this name is used whenever you want to
72 interpolate the values. Note: this procedure is a convenient
73 wrapper for the struct::matrix procedure. Therefore you can
74 access the data at any location in your program.
75 string name (in)
77 Name of the table to be created
78 list colnames (in)
80 List of column names
81 list values (in)
83 List of values (the number of elements should be a multi‐
84 ple of the number of columns. See EXAMPLES for more
85 information on the interpretation of the data.
86 The values must be sorted with respect to the independent
88 variable(s).
89 ::math::interpolate::interp-1d-table name xval
92 Interpolate into the one-dimensional table "name" and return a
93 list of values, one for each dependent column.
94 string name (in)
96 Name of an existing table
97 float xval (in)
99 Value of the independent row variable
100 ::math::interpolate::interp-table name xval yval
103 Interpolate into the two-dimensional table "name" and return the
104 interpolated value.
105 string name (in)
107 Name of an existing table
108 float xval (in)
110 Value of the independent row variable
111 float yval (in)
113 Value of the independent column variable
114 ::math::interpolate::interp-linear xyvalues xval
117 Interpolate linearly into the list of x,y pairs and return the
118 interpolated value.
119 list xyvalues (in)
121 List of pairs of (x,y) values, sorted to increasing x.
122 They are used as the breakpoints of a piecewise linear
123 function.
124 float xval (in)
126 Value of the independent variable for which the value of
127 y must be computed.
128 ::math::interpolate::interp-lagrange xyvalues xval
131 Use the list of x,y pairs to construct the unique polynomial of
132 lowest degree that passes through all points and return the
133 interpolated value.
134 list xyvalues (in)
136 List of pairs of (x,y) values
137 float xval (in)
139 Value of the independent variable for which the value of
140 y must be computed.
141 ::math::interpolate::prepare-cubic-splines xcoord ycoord
144 Returns a list of coefficients for the second routine interp-
145 cubic-splines to actually interpolate.
146 list xcoord
148 List of x-coordinates for the value of the function to be
149 interpolated is known. The coordinates must be strictly
150 ascending. At least three points are required.
151 list ycoord
153 List of y-coordinates (the values of the function at the
154 given x-coordinates).
155 ::math::interpolate::interp-cubic-splines coeffs x
158 Returns the interpolated value at coordinate x. The coefficients
159 are computed by the procedure prepare-cubic-splines.
160 list coeffs
162 List of coefficients as returned by prepare-cubic-splines
163 float x
165 x-coordinate at which to estimate the function. Must be
166 between the first and last x-coordinate for which values
167 were given.
168 ::math::interpolate::interp-spatial xyvalues coord
171 Use a straightforward interpolation method with weights as func‐
172 tion of the inverse distance to interpolate in 2D and N-dimen‐
173 sional space
174 The list xyvalues is a list of lists:
176 { {x1 y1 z1 {v11 v12 v13 v14}}
178 {x2 y2 z2 {v21 v22 v23 v24}}
179 ...
180 }
181 The last element of each inner list is either a single number or
183 a list in itself. In the latter case the return value is a list
184 with the same number of elements.
185 The method is influenced by the search radius and the power of
187 the inverse distance
188 list xyvalues (in)
190 List of lists, each sublist being a list of coordinates
191 and of dependent values.
192 list coord (in)
194 List of coordinates for which the values must be calcu‐
195 lated
196 ::math::interpolate::interp-spatial-params max_search power
199 Set the parameters for spatial interpolation
200 float max_search (in)
202 Search radius (data points further than this are ignored)
203 integer power (in)
205 Power for the distance (either 1 or 2; defaults to 2)
206 ::math::interpolate::neville xlist ylist x
208 Interpolates between the tabulated values of a function whose
209 abscissae are xlist and whose ordinates are ylist to produce an
210 estimate for the value of the function at x. The result is a
211 two-element list; the first element is the function's estimated
212 value, and the second is an estimate of the absolute error of
213 the result. Neville's algorithm for polynomial interpolation is
214 used. Note that a large table of values will use an interpolat‐
215 ing polynomial of high degree, which is likely to result in
216 numerical instabilities; one is better off using only a few tab‐
217 ulated values near the desired abscissa.
218
220 TODO Example of using the cubic splines:
221 Suppose the following values are given:
223 x y
225 0.1 1.0
226 0.3 2.1
227 0.4 2.2
228 0.8 4.11
229 1.0 4.12
230 Then to estimate the values at 0.1, 0.2, 0.3, ... 1.0, you can use:
232 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}]
234 foreach x {0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0} {
235 puts "$x: [::math::interpolate::interp-cubic-splines $coeffs $x]"
236 }
237 to get the following output:
239 0.1: 1.0
241 0.2: 1.68044117647
242 0.3: 2.1
243 0.4: 2.2
244 0.5: 3.11221507353
245 0.6: 4.25242647059
246 0.7: 5.41804227941
247 0.8: 4.11
248 0.9: 3.95675857843
249 1.0: 4.12
250 As you can see, the values at the abscissae are reproduced perfectly.
252
254 This document, and the package it describes, will undoubtedly contain
255 bugs and other problems. Please report such in the category math ::
256 interpolate of the Tcllib SF Trackers [http://source‐
257 forge.net/tracker/?group_id=12883]. Please also report any ideas for
258 enhancements you may have for either package and/or documentation.
259
261 interpolation, math, spatial interpolation
262
264 Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>
265 Copyright (c) 2004 Kevn B. Kenny <kennykb@users.sourceforge.net>
266math 1.0.2 math::interpolate(n)