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1 @menu
2 * Introduction to interpol::
3 * Functions and Variables for interpol::
4 @end menu
6 @node Introduction to interpol, Functions and Variables for interpol, Package interpol, Package interpol
7 @section Introduction to interpol
9 Package @code{interpol} defines the Lagrangian, the linear and the cubic
10 splines methods for polynomial interpolation.
12 For comments, bugs or suggestions, please contact me at @var{'mario AT edu DOT xunta DOT es'}.
14 @opencatbox{Categories:}
15 @category{Numerical methods}
16 @category{Share packages}
17 @category{Package interpol}
18 @closecatbox
20 @node Functions and Variables for interpol, , Introduction to interpol, Package interpol
21 @section Functions and Variables for interpol
24 @anchor{lagrange}
25 @deffn {Function} lagrange @
26 @fname{lagrange} (@var{points}) @
27 @fname{lagrange} (@var{points}, @var{option})
29 Computes the polynomial interpolation by the Lagrangian method. Argument @var{points} must be either:
31 @itemize @bullet
32 @item
33 a two column matrix, @code{p:matrix([2,4],[5,6],[9,3])},
34 @item
35 a list of pairs, @code{p: [[2,4],[5,6],[9,3]]},
36 @item
37 a list of numbers, @code{p: [4,6,3]}, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
38 @end itemize
40 In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
42 With the @var{option} argument it is possible to select the name for the independent variable, which is @code{'x} by default; to define another one, write something like @code{varname='z}.
44 Note that when working with high degree polynomials, floating point evaluations are unstable.
46 See also @mrefcomma{linearinterpol} @mrefcomma{cspline} and @mrefdot{ratinterpol}
48 Examples:
50 @example
51 (%i1) load("interpol")$
52 (%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
53 (%i3) lagrange(p);
54 (x - 7) (x - 6) (x - 3) (x - 1)
55 (%o3) -------------------------------
56 35
57 (x - 8) (x - 6) (x - 3) (x - 1)
58 - -------------------------------
59 12
60 7 (x - 8) (x - 7) (x - 3) (x - 1)
61 + ---------------------------------
62 30
63 (x - 8) (x - 7) (x - 6) (x - 1)
64 - -------------------------------
65 60
66 (x - 8) (x - 7) (x - 6) (x - 3)
67 + -------------------------------
68 84
69 (%i4) f(x):=''%;
70 (x - 7) (x - 6) (x - 3) (x - 1)
71 (%o4) f(x) := -------------------------------
72 35
73 (x - 8) (x - 6) (x - 3) (x - 1)
74 - -------------------------------
75 12
76 7 (x - 8) (x - 7) (x - 3) (x - 1)
77 + ---------------------------------
78 30
79 (x - 8) (x - 7) (x - 6) (x - 1)
80 - -------------------------------
81 60
82 (x - 8) (x - 7) (x - 6) (x - 3)
83 + -------------------------------
84 84
85 (%i5) /* Evaluate the polynomial at some points */
86 expand(map(f,[2.3,5/7,%pi]));
87 4 3 2
88 919062 73 %pi 701 %pi 8957 %pi
89 (%o5) [- 1.567535, ------, ------- - -------- + ---------
90 84035 420 210 420
91 5288 %pi 186
92 - -------- + ---]
93 105 5
94 (%i6) %,numer;
95 (%o6) [- 1.567535, 10.9366573451538, 2.89319655125692]
96 (%i7) load("draw")$ /* load draw package */
97 (%i8) /* Plot the polynomial together with points */
98 draw2d(
99 color = red,
100 key = "Lagrange polynomial",
101 explicit(f(x),x,0,10),
102 point_size = 3,
103 color = blue,
104 key = "Sample points",
105 points(p))$
106 (%i9) /* Change variable name */
107 lagrange(p, varname=w);
108 (w - 7) (w - 6) (w - 3) (w - 1)
109 (%o9) -------------------------------
110 35
111 (w - 8) (w - 6) (w - 3) (w - 1)
112 - -------------------------------
113 12
114 7 (w - 8) (w - 7) (w - 3) (w - 1)
115 + ---------------------------------
116 30
117 (w - 8) (w - 7) (w - 6) (w - 1)
118 - -------------------------------
119 60
120 (w - 8) (w - 7) (w - 6) (w - 3)
121 + -------------------------------
122 84
123 @end example
125 @opencatbox{Categories:}
126 @category{Package interpol}
127 @closecatbox
129 @end deffn
132 @anchor{charfun2}
133 @deffn {Function} charfun2 (@var{x}, @var{a}, @var{b})
135 The characteristic or indicator function on the half-open interval @math{[@var{a}, @var{b})},
136 that is, including @var{a} and excluding @var{b}.
138 When @math{@var{x} >= @var{a}} and @math{@var{x} < @var{b}} evaluates to @code{true} or @code{false},
139 @code{charfun2} returns 1 or 0, respectively.
141 Otherwise, @code{charfun2} returns a partially-evaluated result in terms of @mref{charfun}.
143 Package @code{interpol} loads this function.
145 See also @mrefdot{charfun}
147 Examples:
149 When @math{@var{x} >= @var{a}} and @math{@var{x} < @var{b}} evaluates to @code{true} or @code{false},
150 @code{charfun2} returns 1 or 0, respectively.
152 @c ===beg===
153 @c load ("interpol") $
154 @c charfun2 (5, 0, 100);
155 @c charfun2 (-5, 0, 100);
156 @c ===end===
157 @example
158 (%i1) load ("interpol") $
159 (%i2) charfun2 (5, 0, 100);
160 (%o2) 1
161 (%i3) charfun2 (-5, 0, 100);
162 (%o3) 0
163 @end example
165 Otherwise, @code{charfun2} returns a partially-evaluated result in terms of @mref{charfun}.
167 @c ===beg===
168 @c load ("interpol") $
169 @c charfun2 (t, 0, 100);
170 @c charfun2 (5, u, v);
171 @c assume (v > u, u > 5);
172 @c charfun2 (5, u, v);
173 @c ===end===
174 @example
175 (%i1) load ("interpol") $
176 (%i2) charfun2 (t, 0, 100);
177 (%o2) charfun((0 <= t) and (t < 100))
178 (%i3) charfun2 (5, u, v);
179 (%o3) charfun((u <= 5) and (5 < v))
180 (%i4) assume (v > u, u > 5);
181 (%o4) [v > u, u > 5]
182 (%i5) charfun2 (5, u, v);
183 (%o5) 0
184 @end example
186 @opencatbox{Categories:}
187 @category{Package interpol}
188 @closecatbox
190 @end deffn
193 @anchor{linearinterpol}
194 @deffn {Function} linearinterpol @
195 @fname{linearinterpol} (@var{points}) @
196 @fname{linearinterpol} (@var{points}, @var{option})
198 Computes the polynomial interpolation by the linear method. Argument @var{points} must be either:
200 @itemize @bullet
201 @item
202 a two column matrix, @code{p:matrix([2,4],[5,6],[9,3])},
203 @item
204 a list of pairs, @code{p: [[2,4],[5,6],[9,3]]},
205 @item
206 a list of numbers, @code{p: [4,6,3]}, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
207 @end itemize
209 In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
211 With the @var{option} argument it is possible to select the name for the independent variable, which is @code{'x} by default; to define another one, write something like @code{varname='z}.
213 See also @mrefcomma{lagrange} @mrefcomma{cspline} and @mrefdot{ratinterpol}
215 Examples:
217 @example
218 (%i1) load("interpol")$
219 (%i2) p: matrix([7,2],[8,3],[1,5],[3,2],[6,7])$
220 (%i3) linearinterpol(p);
221 13 3 x
222 (%o3) (-- - ---) charfun2(x, minf, 3)
223 2 2
224 + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
225 5 x
226 + (--- - 3) charfun2(x, 3, 6)
227 3
229 (%i4) f(x):=''%;
230 13 3 x
231 (%o4) f(x) := (-- - ---) charfun2(x, minf, 3)
232 2 2
233 + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
234 5 x
235 + (--- - 3) charfun2(x, 3, 6)
236 3
237 (%i5) /* Evaluate the polynomial at some points */
238 map(f,[7.3,25/7,%pi]);
239 62 5 %pi
240 (%o5) [2.3, --, ----- - 3]
241 21 3
242 (%i6) %,numer;
243 (%o6) [2.3, 2.952380952380953, 2.235987755982989]
244 (%i7) load("draw")$ /* load draw package */
245 (%i8) /* Plot the polynomial together with points */
246 draw2d(
247 color = red,
248 key = "Linear interpolator",
249 explicit(f(x),x,-5,20),
250 point_size = 3,
251 color = blue,
252 key = "Sample points",
253 points(args(p)))$
254 (%i9) /* Change variable name */
255 linearinterpol(p, varname='s);
256 13 3 s
257 (%o9) (-- - ---) charfun2(s, minf, 3)
258 2 2
259 + (s - 5) charfun2(s, 7, inf) + (37 - 5 s) charfun2(s, 6, 7)
260 5 s
261 + (--- - 3) charfun2(s, 3, 6)
262 3
263 @end example
265 @opencatbox{Categories:}
266 @category{Package interpol}
267 @closecatbox
269 @end deffn
273 @anchor{cspline}
274 @deffn {Function} cspline @
275 @fname{cspline} (@var{points}) @
276 @fname{cspline} (@var{points}, @var{option1}, @var{option2}, ...)
278 Computes the polynomial interpolation by the cubic splines method. Argument @var{points} must be either:
280 @itemize @bullet
281 @item
282 a two column matrix, @code{p:matrix([2,4],[5,6],[9,3])},
283 @item
284 a list of pairs, @code{p: [[2,4],[5,6],[9,3]]},
285 @item
286 a list of numbers, @code{p: [4,6,3]}, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
287 @end itemize
289 In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
291 There are three options to fit specific needs:
292 @itemize @bullet
293 @item
294 @code{'d1}, default @code{'unknown}, is the first derivative at @math{x_1}; if it is @code{'unknown}, the second derivative at @math{x_1} is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
296 @item
297 @code{'dn}, default @code{'unknown}, is the first derivative at @math{x_n}; if it is @code{'unknown}, the second derivative at @math{x_n} is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
299 @item
300 @code{'varname}, default @code{'x}, is the name of the independent variable.
301 @end itemize
303 See also @mrefcomma{lagrange} @mrefcomma{linearinterpol} and @mrefdot{ratinterpol}
305 Examples:
306 @example
307 (%i1) load("interpol")$
308 (%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
309 (%i3) /* Unknown first derivatives at the extremes
310 is equivalent to natural cubic splines */
311 cspline(p);
312 3 2
313 1159 x 1159 x 6091 x 8283
314 (%o3) (------- - ------- - ------ + ----) charfun2(x, minf, 3)
315 3288 1096 3288 1096
316 3 2
317 2587 x 5174 x 494117 x 108928
318 + (- ------- + ------- - -------- + ------) charfun2(x, 7, inf)
319 1644 137 1644 137
320 3 2
321 4715 x 15209 x 579277 x 199575
322 + (------- - -------- + -------- - ------) charfun2(x, 6, 7)
323 1644 274 1644 274
324 3 2
325 3287 x 2223 x 48275 x 9609
326 + (- ------- + ------- - ------- + ----) charfun2(x, 3, 6)
327 4932 274 1644 274
329 (%i4) f(x):=''%$
330 (%i5) /* Some evaluations */
331 map(f,[2.3,5/7,%pi]), numer;
332 (%o5) [1.991460766423356, 5.823200187269903, 2.227405312429507]
333 (%i6) load("draw")$ /* load draw package */
334 (%i7) /* Plotting interpolating function */
335 draw2d(
336 color = red,
337 key = "Cubic splines",
338 explicit(f(x),x,0,10),
339 point_size = 3,
340 color = blue,
341 key = "Sample points",
342 points(p))$
343 (%i8) /* New call, but giving values at the derivatives */
344 cspline(p,d1=0,dn=0);
345 3 2
346 1949 x 11437 x 17027 x 1247
347 (%o8) (------- - -------- + ------- + ----) charfun2(x, minf, 3)
348 2256 2256 2256 752
349 3 2
350 1547 x 35581 x 68068 x 173546
351 + (- ------- + -------- - ------- + ------) charfun2(x, 7, inf)
352 564 564 141 141
353 3 2
354 607 x 35147 x 55706 x 38420
355 + (------ - -------- + ------- - -----) charfun2(x, 6, 7)
356 188 564 141 47
357 3 2
358 3895 x 1807 x 5146 x 2148
359 + (- ------- + ------- - ------ + ----) charfun2(x, 3, 6)
360 5076 188 141 47
361 (%i8) /* Defining new interpolating function */
362 g(x):=''%$
363 (%i9) /* Plotting both functions together */
364 draw2d(
365 color = black,
366 key = "Cubic splines (default)",
367 explicit(f(x),x,0,10),
368 color = red,
369 key = "Cubic splines (d1=0,dn=0)",
370 explicit(g(x),x,0,10),
371 point_size = 3,
372 color = blue,
373 key = "Sample points",
374 points(p))$
375 @end example
377 @opencatbox{Categories:}
378 @category{Package interpol}
379 @closecatbox
381 @end deffn
383 @anchor{ratinterpol}
384 @deffn {Function} ratinterpol @
385 @fname{ratinterpol} (@var{points}, @var{numdeg}) @
386 @fname{ratinterpol} (@var{points}, @var{numdeg}, @var{option1})
388 Generates a rational interpolator for data given by @var{points} and the degree of the numerator
389 being equal to @var{numdeg}; the degree of the denominator is calculated
390 automatically. Argument @var{points} must be either:
392 @itemize @bullet
393 @item
394 a two column matrix, @code{p:matrix([2,4],[5,6],[9,3])},
395 @item
396 a list of pairs, @code{p: [[2,4],[5,6],[9,3]]},
397 @item
398 a list of numbers, @code{p: [4,6,3]}, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
399 @end itemize
401 In the first two cases the pairs are ordered with respect to the first coordinate before making computations.
403 There is one option to fit specific needs:
404 @itemize @bullet
405 @item
406 @code{'varname}, default @code{'x}, is the name of the independent variable.
407 @end itemize
409 See also @mrefcomma{lagrange} @mrefcomma{linearinterpol} @mrefcomma{cspline} @mrefcomma{minpack_lsquares} and @ref{Package lbfgs}
411 Examples:
413 @example
414 (%i1) load("interpol")$
415 (%i2) load("draw")$
416 (%i3) p:[[7.2,2.5],[8.5,2.1],[1.6,5.1],[3.4,2.4],[6.7,7.9]]$
417 (%i4) for k:0 thru length(p)-1 do
418 draw2d(
419 explicit(ratinterpol(p,k),x,0,9),
420 point_size = 3,
421 points(p),
422 title = concat("Degree of numerator = ",k),
423 yrange=[0,10])$
424 @end example
426 @opencatbox{Categories:}
427 @category{Package interpol}
428 @closecatbox
430 @end deffn