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 ?1.1?
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 interpretation of the tables in the ::math::interpolate::interpo‐
67 late-1d-table command has been changed to be compatible with the inter‐
68 pretation for 2D interpolation in the ::math::interpolate::interpolate-
69 table command. As a consequence this version is incompatible with the
70 previous versions of the command (1.0.x).
71
73 The interpolation package defines the following public procedures:
74 ::math::interpolate::defineTable name colnames values
76 Define a table with one or two independent variables (the dis‐
77 tinction is implicit in the data). The procedure returns the
78 name of the table - this name is used whenever you want to in‐
79 terpolate the values. Note: this procedure is a convenient wrap‐
80 per for the struct::matrix procedure. Therefore you can access
81 the data at any location in your program.
82 string name (in)
84 Name of the table to be created
85 list colnames (in)
87 List of column names
88 list values (in)
90 List of values (the number of elements should be a multi‐
91 ple of the number of columns. See EXAMPLES for more in‐
92 formation on the interpretation of the data.
93 The values must be sorted with respect to the independent
95 variable(s).
96 ::math::interpolate::interp-1d-table name xval
99 Interpolate into the one-dimensional table "name" and return a
100 list of values, one for each dependent column.
101 string name (in)
103 Name of an existing table
104 float xval (in)
106 Value of the independent row variable
107 ::math::interpolate::interp-table name xval yval
110 Interpolate into the two-dimensional table "name" and return the
111 interpolated value.
112 string name (in)
114 Name of an existing table
115 float xval (in)
117 Value of the independent row variable
118 float yval (in)
120 Value of the independent column variable
121 ::math::interpolate::interp-linear xyvalues xval
124 Interpolate linearly into the list of x,y pairs and return the
125 interpolated value.
126 list xyvalues (in)
128 List of pairs of (x,y) values, sorted to increasing x.
129 They are used as the breakpoints of a piecewise linear
130 function.
131 float xval (in)
133 Value of the independent variable for which the value of
134 y must be computed.
135 ::math::interpolate::interp-lagrange xyvalues xval
138 Use the list of x,y pairs to construct the unique polynomial of
139 lowest degree that passes through all points and return the in‐
140 terpolated value.
141 list xyvalues (in)
143 List of pairs of (x,y) values
144 float xval (in)
146 Value of the independent variable for which the value of
147 y must be computed.
148 ::math::interpolate::prepare-cubic-splines xcoord ycoord
151 Returns a list of coefficients for the second routine interp-cu‐
152 bic-splines to actually interpolate.
153 list xcoord
155 List of x-coordinates for the value of the function to be
156 interpolated is known. The coordinates must be strictly
157 ascending. At least three points are required.
158 list ycoord
160 List of y-coordinates (the values of the function at the
161 given x-coordinates).
162 ::math::interpolate::interp-cubic-splines coeffs x
165 Returns the interpolated value at coordinate x. The coefficients
166 are computed by the procedure prepare-cubic-splines.
167 list coeffs
169 List of coefficients as returned by prepare-cubic-splines
170 float x
172 x-coordinate at which to estimate the function. Must be
173 between the first and last x-coordinate for which values
174 were given.
175 ::math::interpolate::interp-spatial xyvalues coord
178 Use a straightforward interpolation method with weights as func‐
179 tion of the inverse distance to interpolate in 2D and N-dimen‐
180 sional space
181 The list xyvalues is a list of lists:
183 { {x1 y1 z1 {v11 v12 v13 v14}}
186 {x2 y2 z2 {v21 v22 v23 v24}}
187 ...
188 }
189 The last element of each inner list is either a single number or
192 a list in itself. In the latter case the return value is a list
193 with the same number of elements.
194 The method is influenced by the search radius and the power of
196 the inverse distance
197 list xyvalues (in)
199 List of lists, each sublist being a list of coordinates
200 and of dependent values.
201 list coord (in)
203 List of coordinates for which the values must be calcu‐
204 lated
205 ::math::interpolate::interp-spatial-params max_search power
208 Set the parameters for spatial interpolation
209 float max_search (in)
211 Search radius (data points further than this are ignored)
212 integer power (in)
214 Power for the distance (either 1 or 2; defaults to 2)
215 ::math::interpolate::neville xlist ylist x
217 Interpolates between the tabulated values of a function whose
218 abscissae are xlist and whose ordinates are ylist to produce an
219 estimate for the value of the function at x. The result is a
220 two-element list; the first element is the function's estimated
221 value, and the second is an estimate of the absolute error of
222 the result. Neville's algorithm for polynomial interpolation is
223 used. Note that a large table of values will use an interpolat‐
224 ing polynomial of high degree, which is likely to result in nu‐
225 merical instabilities; one is better off using only a few tabu‐
226 lated values near the desired abscissa.
227
229 Example of using one-dimensional tables:
230 Suppose you have several tabulated functions of one variable:
232 x y1 y2
235 0.0 0.0 0.0
236 1.0 1.0 1.0
237 2.0 4.0 8.0
238 3.0 9.0 27.0
239 4.0 16.0 64.0
240 Then to estimate the values at 0.5, 1.5, 2.5 and 3.5, you can use:
242 set table [::math::interpolate::defineTable table1 {x y1 y2} { - 1 2
245 0.0 0.0 0.0
246 1.0 1.0 1.0
247 2.0 4.0 8.0
248 3.0 9.0 27.0
249 4.0 16.0 64.0}]
250 foreach x {0.5 1.5 2.5 3.5} {
251 puts "$x: [::math::interpolate::interp-1d-table $table $x]"
252 }
253 For one-dimensional tables the first row is not used. For two-dimen‐
255 sional tables, the first row represents the values for the second inde‐
256 pendent variable.
257 Example of using the cubic splines:
259 Suppose the following values are given:
261 x y
264 0.1 1.0
265 0.3 2.1
266 0.4 2.2
267 0.8 4.11
268 1.0 4.12
269 Then to estimate the values at 0.1, 0.2, 0.3, ... 1.0, you can use:
271 set coeffs [::math::interpolate::prepare-cubic-splines {0.1 0.3 0.4 0.8 1.0} {1.0 2.1 2.2 4.11 4.12}]
274 foreach x {0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0} {
275 puts "$x: [::math::interpolate::interp-cubic-splines $coeffs $x]"
276 }
277 to get the following output:
279 0.1: 1.0
282 0.2: 1.68044117647
283 0.3: 2.1
284 0.4: 2.2
285 0.5: 3.11221507353
286 0.6: 4.25242647059
287 0.7: 5.41804227941
288 0.8: 4.11
289 0.9: 3.95675857843
290 1.0: 4.12
291 As you can see, the values at the abscissae are reproduced perfectly.
293
295 This document, and the package it describes, will undoubtedly contain
296 bugs and other problems. Please report such in the category math ::
297 interpolate of the Tcllib Trackers [http://core.tcl.tk/tcllib/re‐
298 portlist]. Please also report any ideas for enhancements you may have
299 for either package and/or documentation.
300 When proposing code changes, please provide unified diffs, i.e the out‐
302 put of diff -u.
303 Note further that attachments are strongly preferred over inlined
305 patches. Attachments can be made by going to the Edit form of the
306 ticket immediately after its creation, and then using the left-most
307 button in the secondary navigation bar.
308
310 interpolation, math, spatial interpolation
311
313 Mathematics
314
316 Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>
317 Copyright (c) 2004 Kevn B. Kenny <kennykb@users.sourceforge.net>
318tcllib 1.1 math::interpolate(n)