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 name string (in) Name of the table to be created
77 colnames list (in) List of column names
79 values list (in) List of values (the number of elements should
81 be a
82 multiple of the number of columns. See EXAMPLES for more
83 information on the interpretation of the data.
84 The values must be sorted with respect to the independent
86 variable(s).
87 ::math::interpolate::interp-1d-table name xval
90 Interpolate into the one-dimensional table "name" and return a
91 list of values, one for each dependent column.
92 name string (in) Name of an existing table
94 xval float (in) Value of the independent row variable
96 ::math::interpolate::interp-table name xval yval
99 Interpolate into the two-dimensional table "name" and return the
100 interpolated value.
101 name string (in) Name of an existing table
103 xval float (in) Value of the independent row variable
105 yval float (in) Value of the independent column variable
107 ::math::interpolate::interp-linear xyvalues xval
110 Interpolate linearly into the list of x,y pairs and return the
111 interpolated value.
112 xyvalues list (in) List of pairs of (x,y) values, sorted to
114 increasing x.
115 They are used as the breakpoints of a piecewise linear
116 function.
117 xval float (in) Value of the independent variable for which the
119 value of y
120 must be computed.
121 ::math::interpolate::interp-lagrange xyvalues xval
124 Use the list of x,y pairs to construct the unique polynomial of
125 lowest degree that passes through all points and return the
126 interpolated value.
127 xyvalues list (in) List of pairs of (x,y) values
129 xval float (in) Value of the independent variable for which the
131 value of y
132 must be computed.
133 ::math::interpolate::prepare_cubic_splines xcoord ycoord
136 Returns a list of coefficients for the second routine
137 interp_cubic_splines to actually interpolate.
138 xcoord list List of x-coordinates for the value of the
140 function to be interpolated is known. The coordinates
141 must be strictly ascending. At least three points are
142 required.
143 ycoord list List of y-coordinates (the values of the
145 function at the given x-coordinates).
146 ::math::interpolate::interp_cubic_splines coeffs x
149 Returns the interpolated value at coordinate x. The coefficients
150 are computed by the procedure prepare_cubic_splines.
151 coeffs list List of coefficients as returned by
153 prepare_cubic_splines
154 x float x-coordinate at which to estimate the function. Must
156 be between the first and last x-coordinate for which val‐
157 ues were given.
158 ::math::interpolate::interp-spatial xyvalues coord
161 Use a straightforward interpolation method with weights as func‐
162 tion of the inverse distance to interpolate in 2D and N-dimen‐
163 sional space
164 The list xyvalues is a list of lists:
166 { {x1 y1 z1 {v11 v12 v13 v14}}
168 {x2 y2 z2 {v21 v22 v23 v24}}
169 ...
170 }
171 The last element of each inner list is either a single number or
173 a list in itself. In the latter case the return value is a list
174 with the same number of elements.
175 The method is influenced by the search radius and the power of
177 the inverse distance
178 xyvalues list (in) List of lists, each sublist being a list of
180 coordinates and
181 of dependent values.
182 coord list (in) List of coordinates for which the values must be
184 calculated
185 ::math::interpolate::interp-spatial-params max_search power
188 Set the parameters for spatial interpolation
189 max_search float (in) Search radius (data points further than
191 this are ignored)
192 power integer (in) Power for the distance (either 1 or 2;
194 defaults to 2)
195 ::math::interpolate::neville xlist ylist x
197 Interpolates between the tabulated values of a function whose
198 abscissae are xlist and whose ordinates are ylist to produce an
199 estimate for the value of the function at x. The result is a
200 two-element list; the first element is the function's estimated
201 value, and the second is an estimate of the absolute error of
202 the result. Neville's algorithm for polynomial interpolation is
203 used. Note that a large table of values will use an interpolat‐
204 ing polynomial of high degree, which is likely to result in
205 numerical instabilities; one is better off using only a few tab‐
206 ulated values near the desired abscissa.
207
209 TODO Example of using the cubic splines:
210 Suppose the following values are given:
212 x y
214 0.1 1.0
215 0.3 2.1
216 0.4 2.2
217 0.8 4.11
218 1.0 4.12
219 Then to estimate the values at 0.1, 0.2, 0.3, ... 1.0, you can use:
221 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}]
223 foreach x {0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0} {
224 puts "$x: [::math::interpolate::interp_cubic_splines $coeffs $x]"
225 }
226 to get the following output:
228 0.1: 1.0
230 0.2: 1.68044117647
231 0.3: 2.1
232 0.4: 2.2
233 0.5: 3.11221507353
234 0.6: 4.25242647059
235 0.7: 5.41804227941
236 0.8: 4.11
237 0.9: 3.95675857843
238 1.0: 4.12
239 As you can see, the values at the abscissae are reproduced perfectly.
241
243 interpolation, math, spatial interpolation
244
246 Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>
247 Copyright (c) 2004 Kevn B. Kenny <kennykb@users.sourceforge.net>
248math 1.0.2 math::interpolate(n)