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1 @c -----------------------------------------------------------------------------
2 @c File : diag.de.texi
3 @c License : GNU General Public License (GPL)
4 @c Language : German
5 @c Original : diag.texi revision 1.6
6 @c Date : 08.11.2010
7 @c Revision : 08.04.2011
8 @c
9 @c This file is part of Maxima -- GPL CAS based on DOE-MACSYMA
10 @c -----------------------------------------------------------------------------
12 @menu
13 * Functions and Variables for diag::
14 @end menu
16 @c -----------------------------------------------------------------------------
17 @node Functions and Variables for diag, , diag, diag
18 @section Functions and Variables for diag
19 @c -----------------------------------------------------------------------------
21 @c -----------------------------------------------------------------------------
22 @deffn {Function} diag (@var{lm})
24 Constructs a square matrix with the matrices of @var{lm} in the diagonal.
25 @var{lm} is a list of matrices or scalars.
27 Example:
29 @example
30 (%i1) load("diag")$
31 (%i2) a1:matrix([1,2,3],[0,4,5],[0,0,6])$
32 (%i3) a2:matrix([1,1],[1,0])$
33 (%i4) diag([a1,x,a2]);
34 @end example
35 @example
36 @group
37 [ 1 2 3 0 0 0 ]
38 [ ]
39 [ 0 4 5 0 0 0 ]
40 [ ]
41 [ 0 0 6 0 0 0 ]
42 (%o4) [ ]
43 [ 0 0 0 x 0 0 ]
44 [ ]
45 [ 0 0 0 0 1 1 ]
46 [ ]
47 [ 0 0 0 0 1 0 ]
48 @end group
49 @end example
51 To use this function write first @code{load("diag")}.
52 @end deffn
54 @c -----------------------------------------------------------------------------
55 @deffn {Function} JF (@var{lambda}, @var{n})
57 Returns the Jordan cell of order @var{n} with eigenvalue @var{lambda}.
59 Example:
61 @example
62 (%i1) load("diag")$
63 (%i2) JF(2,5);
64 @end example
65 @example
66 @group
67 [ 2 1 0 0 0 ]
68 [ ]
69 [ 0 2 1 0 0 ]
70 [ ]
71 (%o2) [ 0 0 2 1 0 ]
72 [ ]
73 [ 0 0 0 2 1 ]
74 [ ]
75 [ 0 0 0 0 2 ]
76 @end group
77 @end example
78 @example
79 (%i3) JF(3,2);
80 @group
81 [ 3 1 ]
82 (%o3) [ ]
83 [ 0 3 ]
84 @end group
85 @end example
87 To use this function write first @code{load("diag")}.
88 @end deffn
90 @c -----------------------------------------------------------------------------
91 @deffn {Function} jordan (@var{mat})
93 Returns the Jordan form of matrix @var{mat}, but codified in a Maxima list.
94 To get the corresponding matrix, call function @code{dispJordan} using as
95 argument the output of @code{jordan}.
97 Example:
99 @example
100 (%i1) load("diag")$
101 (%i3) a:matrix([2,0,0,0,0,0,0,0],
102 [1,2,0,0,0,0,0,0],
103 [-4,1,2,0,0,0,0,0],
104 [2,0,0,2,0,0,0,0],
105 [-7,2,0,0,2,0,0,0],
106 [9,0,-2,0,1,2,0,0],
107 [-34,7,1,-2,-1,1,2,0],
108 [145,-17,-16,3,9,-2,0,3])$
109 (%i34) jordan(a);
110 (%o4) [[2, 3, 3, 1], [3, 1]]
111 (%i5) dispJordan(%);
112 [ 2 1 0 0 0 0 0 0 ]
113 [ ]
114 [ 0 2 1 0 0 0 0 0 ]
115 [ ]
116 [ 0 0 2 0 0 0 0 0 ]
117 [ ]
118 [ 0 0 0 2 1 0 0 0 ]
119 (%o5) [ ]
120 [ 0 0 0 0 2 1 0 0 ]
121 [ ]
122 [ 0 0 0 0 0 2 0 0 ]
123 [ ]
124 [ 0 0 0 0 0 0 2 0 ]
125 [ ]
126 [ 0 0 0 0 0 0 0 3 ]
127 @end example
129 To use this function write first @code{load("diag")}. See also @code{dispJordan}
130 and @code{minimalPoly}.
131 @end deffn
133 @c -----------------------------------------------------------------------------
134 @deffn {Function} dispJordan (@var{l})
136 Returns the Jordan matrix associated to the codification given by the Maxima
137 list @var{l}, which is the output given by function @code{jordan}.
139 Example:
141 @example
142 (%i1) load("diag")$
143 (%i2) b1:matrix([0,0,1,1,1],
144 [0,0,0,1,1],
145 [0,0,0,0,1],
146 [0,0,0,0,0],
147 [0,0,0,0,0])$
148 (%i3) jordan(b1);
149 (%o3) [[0, 3, 2]]
150 (%i4) dispJordan(%);
151 [ 0 1 0 0 0 ]
152 [ ]
153 [ 0 0 1 0 0 ]
154 [ ]
155 (%o4) [ 0 0 0 0 0 ]
156 [ ]
157 [ 0 0 0 0 1 ]
158 [ ]
159 [ 0 0 0 0 0 ]
160 @end example
162 To use this function write first @code{load("diag")}. See also @code{jordan}
163 and @code{minimalPoly}.
164 @end deffn
166 @c -----------------------------------------------------------------------------
167 @deffn {Function} minimalPoly (@var{l})
169 Returns the minimal polynomial associated to the codification given by the
170 Maxima list @var{l}, which is the output given by function @code{jordan}.
172 Example:
174 @example
175 (%i1) load("diag")$
176 (%i2) a:matrix([2,1,2,0],
177 [-2,2,1,2],
178 [-2,-1,-1,1],
179 [3,1,2,-1])$
180 (%i3) jordan(a);
181 (%o3) [[- 1, 1], [1, 3]]
182 (%i4) minimalPoly(%);
183 3
184 (%o4) (x - 1) (x + 1)
185 @end example
187 To use this function write first @code{load("diag")}. See also @code{jordan}
188 and @code{dispJordan}.
189 @end deffn
191 @c -----------------------------------------------------------------------------
192 @deffn {Function} ModeMatrix (@var{A},@var{l})
194 Returns the matrix @var{M} such that @math{(M^^-1).A.M=J}, where @var{J} is
195 the Jordan form of @var{A}. The Maxima list @var{l} is the codified form of
196 the Jordan form as returned by function @code{jordan}.
198 Example:
200 @example
201 (%i1) load("diag")$
202 (%i2) a:matrix([2,1,2,0],
203 [-2,2,1,2],
204 [-2,-1,-1,1],
205 [3,1,2,-1])$
207 (%i3) jordan(a);
208 (%o3) [[- 1, 1], [1, 3]]
209 (%i4) M: ModeMatrix(a,%);
210 @end example
211 @example
212 @group
213 [ 1 - 1 1 1 ]
214 [ ]
215 [ 1 ]
216 [ - - - 1 0 0 ]
217 [ 9 ]
218 [ ]
219 (%o4) [ 13 ]
220 [ - -- 1 - 1 0 ]
221 [ 9 ]
222 [ ]
223 [ 17 ]
224 [ -- - 1 1 1 ]
225 [ 9 ]
226 @end group
227 @end example
228 @example
229 (%i5) is( (M^^-1).a.M = dispJordan(%o3) );
230 (%o5) true
231 @end example
233 Note that @code{dispJordan(%o3)} is the Jordan form of matrix @code{a}.
235 To use this function write first @code{load("diag")}. See also @code{jordan}
236 and @code{dispJordan}.
237 @end deffn
239 @c -----------------------------------------------------------------------------
240 @deffn {Function} mat_function (@var{f},@var{mat})
242 Returns @math{f(mat)}, where @var{f} is an analytic function and @var{mat}
243 a matrix. This computation is based on Cauchy's integral formula, which states
244 that if @code{f(x)} is analytic and
245 @code{mat = diag([JF(m1,n1), ..., JF(mk,nk)])}, then @code{f(mat) =
246 ModeMatrix * diag([f(JF(m1,n1)), ..., f(JF(mk,nk))]) * ModeMatrix^^(-1)}.
247 Note that there are about 6 or 8 other methods for this calculation.
248 Some examples follow.
250 To use this function write first @code{load("diag")}.
252 Example 1:
254 @example
255 (%i1) load("diag")$
256 (%i2) b2:matrix([0,1,0], [0,0,1], [-1,-3,-3])$
257 (%i3) mat_function(exp,t*b2);
258 2 - t
259 t %e - t - t
260 (%o3) matrix([-------- + t %e + %e ,
261 2
262 - t - t - t
263 2 %e %e - t - t %e
264 t (- ----- - ----- + %e ) + t (2 %e - -----)
265 t 2 t
266 t
267 - t - t - t
268 - t - t %e 2 %e %e
269 + 2 %e , t (%e - -----) + t (----- - -----)
270 t 2 t
271 2 - t - t - t
272 - t t %e 2 %e %e - t
273 + %e ], [- --------, - t (- ----- - ----- + %e ),
274 2 t 2
275 t
276 - t - t 2 - t
277 2 %e %e t %e - t
278 - t (----- - -----)], [-------- - t %e ,
279 2 t 2
280 - t - t - t
281 2 %e %e - t - t %e
282 t (- ----- - ----- + %e ) - t (2 %e - -----),
283 t 2 t
284 t
285 - t - t - t
286 2 %e %e - t %e
287 t (----- - -----) - t (%e - -----)])
288 2 t t
289 @end example
290 @example
291 (%i4) ratsimp(%);
292 @group
293 [ 2 - t ]
294 [ (t + 2 t + 2) %e ]
295 [ -------------------- ]
296 [ 2 ]
297 [ ]
298 [ 2 - t ]
299 (%o4) Col 1 = [ t %e ]
300 [ - -------- ]
301 [ 2 ]
302 [ ]
303 [ 2 - t ]
304 [ (t - 2 t) %e ]
305 [ ---------------- ]
306 [ 2 ]
307 @end group
308 @end example
309 @example
310 @group
311 [ 2 - t ]
312 [ (t + t) %e ]
313 [ ]
314 Col 2 = [ 2 - t ]
315 [ - (t - t - 1) %e ]
316 [ ]
317 [ 2 - t ]
318 [ (t - 3 t) %e ]
319 @end group
320 @end example
321 @example
322 @group
323 [ 2 - t ]
324 [ t %e ]
325 [ -------- ]
326 [ 2 ]
327 [ ]
328 [ 2 - t ]
329 Col 3 = [ (t - 2 t) %e ]
330 [ - ---------------- ]
331 [ 2 ]
332 [ ]
333 [ 2 - t ]
334 [ (t - 4 t + 2) %e ]
335 [ -------------------- ]
336 [ 2 ]
337 @end group
338 @end example
340 Example 2:
342 @example
343 (%i5) b1:matrix([0,0,1,1,1],
344 [0,0,0,1,1],
345 [0,0,0,0,1],
346 [0,0,0,0,0],
347 [0,0,0,0,0])$
348 @end example
349 @example
350 (%i6) mat_function(exp,t*b1);
351 @group
352 [ 2 ]
353 [ t ]
354 [ 1 0 t t -- + t ]
355 [ 2 ]
356 [ ]
357 (%o6) [ 0 1 0 t t ]
358 [ ]
359 [ 0 0 1 0 t ]
360 [ ]
361 [ 0 0 0 1 0 ]
362 [ ]
363 [ 0 0 0 0 1 ]
364 @end group
365 @end example
366 (%i7) minimalPoly(jordan(b1));
367 3
368 (%o7) x
369 @example
370 (%i8) ident(5)+t*b1+1/2*(t^2)*b1^^2;
371 @group
372 [ 2 ]
373 [ t ]
374 [ 1 0 t t -- + t ]
375 [ 2 ]
376 [ ]
377 (%o8) [ 0 1 0 t t ]
378 [ ]
379 [ 0 0 1 0 t ]
380 [ ]
381 [ 0 0 0 1 0 ]
382 [ ]
383 [ 0 0 0 0 1 ]
384 @end group
385 @end example
386 @example
387 (%i9) mat_function(exp,%i*t*b1);
388 @group
389 [ 2 ]
390 [ t ]
391 [ 1 0 %i t %i t %i t - -- ]
392 [ 2 ]
393 [ ]
394 (%o9) [ 0 1 0 %i t %i t ]
395 [ ]
396 [ 0 0 1 0 %i t ]
397 [ ]
398 [ 0 0 0 1 0 ]
399 [ ]
400 [ 0 0 0 0 1 ]
401 @end group
402 @end example
403 (%i10) mat_function(cos,t*b1)+%i*mat_function(sin,t*b1);
404 @example
405 @group
406 [ 2 ]
407 [ t ]
408 [ 1 0 %i t %i t %i t - -- ]
409 [ 2 ]
410 [ ]
411 (%o10) [ 0 1 0 %i t %i t ]
412 [ ]
413 [ 0 0 1 0 %i t ]
414 [ ]
415 [ 0 0 0 1 0 ]
416 [ ]
417 [ 0 0 0 0 1 ]
418 @end group
419 @end example
421 Example 3:
423 @example
424 (%i11) a1:matrix([2,1,0,0,0,0],
425 [-1,4,0,0,0,0],
426 [-1,1,2,1,0,0],
427 [-1,1,-1,4,0,0],
428 [-1,1,-1,1,3,0],
429 [-1,1,-1,1,1,2])$
430 (%i12) fpow(x):=block([k],declare(k,integer),x^k)$
431 (%i13) mat_function(fpow,a1);
432 @end example
433 @example
434 @group
435 [ k k - 1 ] [ k - 1 ]
436 [ 3 - k 3 ] [ k 3 ]
437 [ ] [ ]
438 [ k - 1 ] [ k k - 1 ]
439 [ - k 3 ] [ 3 + k 3 ]
440 [ ] [ ]
441 [ k - 1 ] [ k - 1 ]
442 [ - k 3 ] [ k 3 ]
443 (%o13) Col 1 = [ ] Col 2 = [ ]
444 [ k - 1 ] [ k - 1 ]
445 [ - k 3 ] [ k 3 ]
446 [ ] [ ]
447 [ k - 1 ] [ k - 1 ]
448 [ - k 3 ] [ k 3 ]
449 [ ] [ ]
450 [ k - 1 ] [ k - 1 ]
451 [ - k 3 ] [ k 3 ]
452 @end group
453 @end example
454 @example
455 @group
456 [ 0 ] [ 0 ]
457 [ ] [ ]
458 [ 0 ] [ 0 ]
459 [ ] [ ]
460 [ k k - 1 ] [ k - 1 ]
461 [ 3 - k 3 ] [ k 3 ]
462 [ ] [ ]
463 Col 3 = [ k - 1 ] Col 4 = [ k k - 1 ]
464 [ - k 3 ] [ 3 + k 3 ]
465 [ ] [ ]
466 [ k - 1 ] [ k - 1 ]
467 [ - k 3 ] [ k 3 ]
468 [ ] [ ]
469 [ k - 1 ] [ k - 1 ]
470 [ - k 3 ] [ k 3 ]
471 @end group
472 @end example
473 @example
474 @group
475 [ 0 ]
476 [ ] [ 0 ]
477 [ 0 ] [ ]
478 [ ] [ 0 ]
479 [ 0 ] [ ]
480 [ ] [ 0 ]
481 Col 5 = [ 0 ] Col 6 = [ ]
482 [ ] [ 0 ]
483 [ k ] [ ]
484 [ 3 ] [ 0 ]
485 [ ] [ ]
486 [ k k ] [ k ]
487 [ 3 - 2 ] [ 2 ]
488 @end group
489 @end example
490 @end deffn
492 @c --- End of file diag.de.texi ------------------------------------------------