Annotation of OpenXM/src/asir-doc/parts/builtin/array.texi, Revision 1.14
1.14 ! nisiyama 1: @comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/array.texi,v 1.13 2009/03/24 17:02:06 ohara Exp $
1.2 noro 2: \BJP
1.1 noro 3: @node $BG[Ns(B,,, $BAH$_9~$_H!?t(B
4: @section $BG[Ns(B
1.2 noro 5: \E
6: \BEG
7: @node Arrays,,, Built-in Function
8: @section Arrays
9: \E
1.1 noro 10:
11: @menu
1.11 ohara 12: * newvect vector vect::
1.9 ohara 13: * ltov::
14: * vtol::
1.4 noro 15: * newbytearray::
1.11 ohara 16: * newmat matrix::
1.12 ohara 17: * mat matr matc::
1.1 noro 18: * size::
1.10 noro 19: * det nd_det invmat::
1.13 ohara 20: * rowx rowm rowa colx colm cola::
1.10 noro 21:
1.1 noro 22: * qsort::
23: @end menu
24:
1.11 ohara 25: \JP @node newvect vector vect,,, $BG[Ns(B
26: \EG @node newvect vector vect,,, Arrays
27: @subsection @code{newvect}, @code{vector}, @code{vect}
1.1 noro 28: @findex newvect
1.11 ohara 29: @findex vector
30: @findex vect
1.1 noro 31:
32: @table @t
33: @item newvect(@var{len}[,@var{list}])
1.11 ohara 34: @item vector(@var{len}[,@var{list}])
1.2 noro 35: \JP :: $BD9$5(B @var{len} $B$N%Y%/%H%k$r@8@.$9$k(B.
36: \EG :: Creates a new vector object with its length @var{len}.
1.11 ohara 37: @item vect([@var{elements}])
38: \JP :: @var{elements} $B$rMWAG$H$9$k%Y%/%H%k$r@8@.$9$k(B.
39: \EG :: Creates a new vector object by @var{elements}.
1.1 noro 40: @end table
41:
42: @table @var
43: @item return
1.2 noro 44: \JP $B%Y%/%H%k(B
45: \EG vector
1.1 noro 46: @item len
1.2 noro 47: \JP $B<+A3?t(B
48: \EG non-negative integer
1.1 noro 49: @item list
1.2 noro 50: \JP $B%j%9%H(B
51: \EG list
1.11 ohara 52: @item elements
53: \JP $BMWAG$NJB$S(B
54: \EG elements of the vector
1.1 noro 55: @end table
56:
57: @itemize @bullet
1.2 noro 58: \BJP
1.1 noro 59: @item
1.11 ohara 60: @code{vect} $B$OMWAG$NJB$S$+$i%Y%/%H%k$r@8@.$9$k(B.
61: @item
62: @code{vector} $B$O(B @code{newvect} $B$NJLL>$G$"$k(B.
63: @item
64: @code{newvect} $B$OD9$5(B @var{len} $B$N%Y%/%H%k$r@8@.$9$k(B. $BBh(B 2 $B0z?t$,$J$$>l9g(B,
1.1 noro 65: $B3F@.J,$O(B 0 $B$K=i4|2=$5$l$k(B. $BBh(B 2 $B0z?t$,$"$k>l9g(B,
66: $B%$%s%G%C%/%9$N>.$5$$@.J,$+$i(B, $B%j%9%H$N(B
67: $B3FMWAG$K$h$j=i4|2=$5$l$k(B. $B3FMWAG$O(B, $B@hF,$+$i=g$K(B
68: $B;H$o$l(B, $BB-$j$J$$J,$O(B 0 $B$,Kd$a$i$l$k(B.
69: @item
70: $B%Y%/%H%k$N@.J,$O(B, $BBh(B 0 $B@.J,$+$iBh(B @var{len}-1 $B@.J,$H$J$k(B.
71: ($BBh(B 1 $B@.J,$+$i$G$O$J$$;v$KCm0U(B. )
72: @item
73: $B%j%9%H$O3F@.J,$,(B, $B%]%$%s%?$rC)$k;v$K$h$C$F%7!<%1%s%7%c%k$K(B
74: $B8F$S=P$5$l$k$N$KBP$7(B, $B%Y%/%H%k$O3F@.J,$,(B
75: $BBh0l@.J,$+$i$N%a%b%j>e$N(B displacement ($BJQ0L(B)$B$K$h$C$F%i%s%@%`%"%/%;%9$G(B
76: $B8F$S=P$5$l(B, $B$=$N7k2L(B, $B@.J,$N%"%/%;%9;~4V$KBg$-$J:9$,=P$F$/$k(B.
77: $B@.J,%"%/%;%9$O(B, $B%j%9%H$G$O(B, $B@.J,$NNL$,A}$($k$K=>$C$F(B
78: $B;~4V$,$+$+$k$h$&$K$J$k$,(B, $B%Y%/%H%k$G$O(B, $B@.J,$NNL$K0MB8$;$:$[$\0lDj$G$"$k(B.
79: @item
80: @b{Asir} $B$G$O(B, $B=D%Y%/%H%k(B, $B2#%Y%/%H%k$N6hJL$O$J$$(B.
81: $B9TNs$r:8$+$i3]$1$l$P=D%Y%/%H%k$H$_$J$5$l$k$7(B, $B1&$+$i3]$1$l$P2#%Y%/%H%k$H(B
82: $B$_$J$5$l$k(B.
83: @item
84: $B%Y%/%H%k$ND9$5$O(B @code{size()} $B$K$h$C$FF@$i$l$k(B.
85: @item
86: $BH!?t$N0z?t$H$7$F%Y%/%H%k$rEO$7$?>l9g(B, $BEO$5$l$?H!?t$O(B, $B$=$N%Y%/%H%k$N@.J,(B
87: $B$r=q$-49$($k$3$H$,$G$-$k(B.
1.2 noro 88: \E
89: \BEG
1.11 ohara 90: @item
91: @code{vect} creates a new vector object by its elements.
92: @item
93: @code{vector} is an alias of @code{newvect}.
1.2 noro 94: @item
1.11 ohara 95: @code{newvect} creates a new vector object with its length @var{len} and its elements
1.2 noro 96: all cleared to value 0.
97: If the second argument, a list, is given, the vector is initialized by
98: the list elements.
99: Elements are used from the first through the last.
100: If the list is short for initializing the full vector,
101: 0's are filled in the remaining vector elements.
102: @item
103: Elements are indexed from 0 through @var{len}-1. Note that the first
104: element has not index 1.
105: @item
106: List and vector are different types in @b{Asir}.
107: Lists are conveniently used for representing many data objects whose
108: size varies dynamically as computation proceeds.
109: By its flexible expressive power, it is also conveniently used to
110: describe initial values for other structured objects as you see
111: for vectors.
112: Access for an element of a list is performed by following pointers to
113: next elements. By this, access costs for list elements differ for
114: each element.
115: In contrast to lists, vector elements can be accessed in a same time,
116: because they are accessed by computing displacements from the top memory
117: location of the vector object.
118:
119: Note also, in @b{Asir}, modification of an element of a vector causes
120: modification of the whole vector itself,
121: while modification of a list element does not cause the modification
122: of the whole list object.
123:
124: By this, in @b{Asir} language,
125: a vector element designator can be a left value of
126: assignment statement, but a list element designator can NOT be a left
127: value of assignment statement.
128:
129: @item
130: No distinction of column vectors and row vectors in @b{Asir}.
131: If a matrix is applied to a vector from left, the vector shall be taken
132: as a column vector, and if from right it shall be taken as a row vector.
133: @item
134: The length (or size or dimension) of a vector is given by function
135: @code{size()}.
136: @item
137: When a vector is passed to a function as its argument
138: (actual parameter), the vector element can be modified in that
139: function.
140:
141: @item
142: A vector is displayed in a similar format as for a list.
143: Note, however, there is a distinction: Elements of a vector are
144: separated simply by a `blank space', while those of a list by a `comma.'
145: \E
1.1 noro 146: @end itemize
147:
148: @example
149: [0] A=newvect(5);
150: [ 0 0 0 0 0 ]
151: [1] A=newvect(5,[1,2,3,4,[5,6]]);
152: [ 1 2 3 4 [5,6] ]
153: [2] A[0];
154: 1
155: [3] A[4];
156: [5,6]
157: [4] size(A);
158: [5]
1.11 ohara 159: [5] length(A);
160: 5
161: [6] vect(1,2,3,4,[5,6]);
162: [ 1 2 3 4 [5,6] ]
163: [7] def afo(V) @{ V[0] = x; @}
164: [8] afo(A)$
165: [9] A;
1.1 noro 166: [ x 2 3 4 [5,6] ]
167: @end example
168:
169: @table @t
1.2 noro 170: \JP @item $B;2>H(B
171: \EG @item References
1.12 ohara 172: @fref{newmat matrix}, @fref{size}, @fref{ltov}, @fref{vtol}.
1.9 ohara 173: @end table
174:
175: \JP @node ltov,,, $BG[Ns(B
176: \EG @node ltov,,, Arrays
177: @subsection @code{ltov}
178: @findex ltov
179:
180: @table @t
181: @item ltov(@var{list})
182: \JP :: $B%j%9%H$r%Y%/%H%k$KJQ49$9$k(B.
183: \EG :: Converts a list into a vector.
184: @end table
185:
186: @table @var
187: @item return
188: \JP $B%Y%/%H%k(B
189: \EG vector
190: @item list
191: \JP $B%j%9%H(B
192: \EG list
193: @end table
194:
195: @itemize @bullet
196: \BJP
197: @item
198: $B%j%9%H(B @var{list} $B$rF1$8D9$5$N%Y%/%H%k$KJQ49$9$k(B.
199: @item
200: $B$3$N4X?t$O(B @code{newvect(length(@var{list}), @var{list})} $B$KEy$7$$(B.
201: \E
202: \BEG
203: @item
204: Converts a list @var{list} into a vector of same length.
205: See also @code{newvect()}.
206: \E
207: @end itemize
208:
209: @example
210: [3] A=[1,2,3];
211: [4] ltov(A);
212: [ 1 2 3 ]
213: @end example
214:
215: @table @t
216: \JP @item $B;2>H(B
217: \EG @item References
1.12 ohara 218: @fref{newvect vector vect}, @fref{vtol}.
1.1 noro 219: @end table
220:
1.2 noro 221: \JP @node vtol,,, $BG[Ns(B
222: \EG @node vtol,,, Arrays
1.1 noro 223: @subsection @code{vtol}
224: @findex vtol
225:
226: @table @t
227: @item vtol(@var{vect})
1.2 noro 228: \JP :: $B%Y%/%H%k$r%j%9%H$KJQ49$9$k(B.
229: \EG :: Converts a vector into a list.
1.1 noro 230: @end table
231:
232: @table @var
233: @item return
1.2 noro 234: \JP $B%j%9%H(B
235: \EG list
1.1 noro 236: @item vect
1.2 noro 237: \JP $B%Y%/%H%k(B
238: \EG vector
1.1 noro 239: @end table
240:
241: @itemize @bullet
1.2 noro 242: \BJP
1.1 noro 243: @item
244: $BD9$5(B @var{n} $B$N%Y%/%H%k(B @var{vect} $B$r(B
245: @code{[@var{vect}[0],...,@var{vect}[@var{n}-1]]} $B$J$k%j%9%H$KJQ49$9$k(B.
246: @item
247: $B%j%9%H$+$i%Y%/%H%k$X$NJQ49$O(B @code{newvect()} $B$G9T$&(B.
1.2 noro 248: \E
249: \BEG
250: @item
251: Converts a vector @var{vect} of length @var{n} into
252: a list @code{[@var{vect}[0],...,@var{vect}[@var{n}-1]]}.
253: @item
254: A conversion from a list to a vector is done by @code{newvect()}.
255: \E
1.1 noro 256: @end itemize
257:
258: @example
259: [3] A=newvect(3,[1,2,3]);
260: [ 1 2 3 ]
261: [4] vtol(A);
262: [1,2,3]
1.4 noro 263: @end example
264:
265: @table @t
266: \JP @item $B;2>H(B
267: \EG @item References
1.12 ohara 268: @fref{newvect vector vect}, @fref{ltov}.
1.4 noro 269: @end table
270:
271: \JP @node newbytearray,,, $BG[Ns(B
272: \EG @node newbytearray,,, Arrays
273: @subsection @code{newbytearray}
274: @findex newbytearray
275:
276: @table @t
277: @item newbytearray(@var{len},[@var{listorstring}])
278: \JP :: $BD9$5(B @var{len} $B$N(B byte array $B$r@8@.$9$k(B.
279: \EG :: Creates a new byte array.
280: @end table
281:
282: @table @var
283: @item return
284: byte array
285: @item len
286: \JP $B<+A3?t(B
287: \EG non-negative integer
288: @item listorstring
289: \JP $B%j%9%H$^$?$OJ8;zNs(B
290: \EG list or string
291: @end table
292:
293: @itemize @bullet
294: @item
295: \JP @code{newvect} $B$HF1MM$K$7$F(B byte array $B$r@8@.$9$k(B.
296: \EG This function generates a byte array. The specification is
297: similar to that of @code{newvect}.
298: @item
299: \JP $BJ8;zNs$G=i4|CM$r;XDj$9$k$3$H$b2DG=$G$"$k(B.
300: \EG The initial value can be specified by a character string.
301: @item
302: \JP byte array $B$NMWAG$N%"%/%;%9$OG[Ns$HF1MM$G$"$k(B.
303: \EG One can access elements of a byte array just as an array.
304: @end itemize
305:
306: @example
307: [182] A=newbytearray(3);
308: |00 00 00|
309: [183] A=newbytearray(3,[1,2,3]);
310: |01 02 03|
311: [184] A=newbytearray(3,"abc");
312: |61 62 63|
313: [185] A[0];
314: 97
315: [186] A[1]=123;
316: 123
317: [187] A;
318: |61 7b 63|
1.1 noro 319: @end example
320:
321: @table @t
1.2 noro 322: \JP @item $B;2>H(B
323: \EG @item References
1.12 ohara 324: @fref{newvect vector vect}.
1.1 noro 325: @end table
326:
1.11 ohara 327: \JP @node newmat matrix,,, $BG[Ns(B
328: \EG @node newmat matrix,,, Arrays
329: @subsection @code{newmat}, @code{matrix}
1.1 noro 330: @findex newmat
1.11 ohara 331: @findex matrix
1.1 noro 332:
333: @table @t
1.6 noro 334: @item newmat(@var{row},@var{col} [,[[@var{a},@var{b},...],[@var{c},@var{d},...],...]])
1.11 ohara 335: @item matrix(@var{row},@var{col} [,[[@var{a},@var{b},...],[@var{c},@var{d},...],...]])
1.2 noro 336: \JP :: @var{row} $B9T(B @var{col} $BNs$N9TNs$r@8@.$9$k(B.
337: \EG :: Creates a new matrix with @var{row} rows and @var{col} columns.
1.1 noro 338: @end table
339:
340: @table @var
341: @item return
1.2 noro 342: \JP $B9TNs(B
343: \EG matrix
1.6 noro 344: @item row col
1.2 noro 345: \JP $B<+A3?t(B
346: \EG non-negative integer
1.6 noro 347: @item a b c d
1.2 noro 348: \JP $BG$0U(B
349: \EG arbitrary
1.1 noro 350: @end table
351:
352: @itemize @bullet
1.2 noro 353: \BJP
1.1 noro 354: @item
1.11 ohara 355: @code{matrix} $B$O(B @code{newmat} $B$NJLL>$G$"$k(B.
356: @item
1.1 noro 357: @var{row} $B9T(B @var{col} $BNs$N9TNs$r@8@.$9$k(B. $BBh(B 3 $B0z?t$,$J$$>l9g(B,
358: $B3F@.J,$O(B 0 $B$K=i4|2=$5$l$k(B. $BBh(B 3 $B0z?t$,$"$k>l9g(B,
359: $B%$%s%G%C%/%9$N>.$5$$@.J,$+$i(B, $B3F9T$,(B, $B%j%9%H$N(B
360: $B3FMWAG(B ($B$3$l$O$^$?%j%9%H$G$"$k(B) $B$K$h$j=i4|2=$5$l$k(B. $B3FMWAG$O(B, $B@hF,$+$i=g$K(B
361: $B;H$o$l(B, $BB-$j$J$$J,$O(B 0 $B$,Kd$a$i$l$k(B.
362: @item
363: $B9TNs$N%5%$%:$O(B @code{size()} $B$GF@$i$l$k(B.
364: @item
365: @code{M} $B$,9TNs$N$H$-(B, @code{M[I]} $B$K$h$jBh(B @code{I} $B9T$r%Y%/%H%k$H$7$F(B
366: $B<h$j=P$9$3$H$,$G$-$k(B. $B$3$N%Y%/%H%k$O(B, $B$b$H$N9TNs$H@.J,$r6&M-$7$F$*$j(B,
367: $B$$$:$l$+$N@.J,$r=q$-49$($l$P(B, $BB>$NBP1~$9$k@.J,$b=q$-49$o$k$3$H$K$J$k(B.
368: @item
369: $BH!?t$N0z?t$H$7$F9TNs$rEO$7$?>l9g(B, $BEO$5$l$?H!?t$O(B, $B$=$N9TNs$N@.J,(B
370: $B$r=q$-49$($k$3$H$,$G$-$k(B.
1.2 noro 371: \E
372: \BEG
1.11 ohara 373: @item
374: @code{matrix} is an alias of @code{newmat}.
1.2 noro 375: @item
376: If the third argument, a list, is given, the newly created matrix
377: is initialized so that each element of the list (again a list)
378: initializes each of the rows of the matrix.
379: Elements are used from the first through the last.
380: If the list is short, 0's are filled in the remaining matrix elements.
381: If no third argument is given all the elements are cleared to 0.
382: @item
383: The size of a matrix is given by function @code{size()}.
384: @item
385: Let @code{M} be a program variable assigned to a matrix.
386: Then, @code{M[I]} denotes a (row) vector which corresponds with
387: the @code{I}-th row of the matrix.
388: Note that the vector shares its element with the original matrix.
389: Subsequently, if an element of the vector is modified, then the
390: corresponding matrix element is also modified.
391: @item
392: When a matrix is passed to a function as its argument
393: (actual parameter), the matrix element can be modified within that
394: function.
395: \E
1.1 noro 396: @end itemize
397:
398: @example
399: [0] A = newmat(3,3,[[1,1,1],[x,y],[x^2]]);
400: [ 1 1 1 ]
401: [ x y 0 ]
402: [ x^2 0 0 ]
403: [1] det(A);
404: -y*x^2
405: [2] size(A);
406: [3,3]
407: [3] A[1];
408: [ x y 0 ]
409: [4] A[1][3];
410: getarray : Out of range
411: return to toplevel
412: @end example
413:
414: @table @t
1.2 noro 415: \JP @item $B;2>H(B
416: \EG @item References
1.12 ohara 417: @fref{newvect vector vect}, @fref{size}, @fref{det nd_det invmat}.
418: @end table
419:
420: \JP @node mat matr matc,,, $BG[Ns(B
421: \EG @node mat matr matc,,, Arrays
422: @subsection @code{mat}, @code{matr}, @code{matc}
423: @findex mat
424: @findex matr
425: @findex matc
426:
427: @table @t
428: @item mat(@var{vector}[,...])
429: @item matr(@var{vector}[,...])
430: \JP :: $B9T%Y%/%H%k$NJB$S$+$i9TNs$r@8@.$9$k(B.
431: \EG :: Creates a new matrix by list of row vectors.
432: @item matc(@var{vector}[,...])
433: \JP :: $BNs%Y%/%H%k$NJB$S$+$i9TNs$r@8@.$9$k(B.
434: \EG :: Creates a new matrix by list of column vectors.
435: @end table
436:
437: @table @var
438: @item return
439: \JP $B9TNs(B
440: \EG matrix
441: @item @var{vector}
442: \JP $BG[Ns$^$?$O%j%9%H(B
443: \EG array or list
444: @end table
445:
446: @itemize @bullet
447: \BJP
448: @item
449: @code{mat} $B$O(B @code{matr} $B$NJLL>$G$"$k(B.
450: @item
451: $B0z?t$N3F%Y%/%H%k$OF1$8D9$5$r$b$D(B.
452: $B3FMWAG$O(B, $B@hF,$+$i=g$K;H$o$l(B, $BB-$j$J$$J,$O(B 0 $B$,Kd$a$i$l$k(B.
453: \E
454: \BEG
455: @item
456: @code{mat} is an alias of @code{matr}.
457: @item
458: Each vector has same length.
459: Elements are used from the first through the last.
460: If the list is short, 0's are filled in the remaining matrix elements.
461: \E
462: @end itemize
463:
464: @example
465: [0] matr([1,2,3],[4,5,6],[7,8]);
466: [ 1 2 3 ]
467: [ 4 5 6 ]
468: [ 7 8 0 ]
469: [1] matc([1,2,3],[4,5,6],[7,8]);
470: [ 1 4 7 ]
471: [ 2 5 8 ]
472: [ 3 6 0 ]
473: @end example
474:
475: @table @t
476: \JP @item $B;2>H(B
477: \EG @item References
478: @fref{newmat matrix}
1.1 noro 479: @end table
480:
1.2 noro 481: \JP @node size,,, $BG[Ns(B
482: \EG @node size,,, Arrays
1.1 noro 483: @subsection @code{size}
484: @findex size
485:
486: @table @t
487: @item size(@var{vect|mat})
1.2 noro 488: \JP :: @code{[@var{vect} $B$ND9$5(B]} $B$^$?$O(B @code{[@var{mat} $B$N9T?t(B,@var{mat} $B$NNs?t(B]}.
489: \BEG
490: :: A list containing the number of elements of the given vector,
491: @code{[size of @var{vect}]},
492: or a list containing row size and column size of the given matrix,
493: @code{[row size of @var{mat}, column size of @var{mat}]}.
494: \E
1.1 noro 495: @end table
496:
497: @table @var
498: @item return
1.2 noro 499: \JP $B%j%9%H(B
500: \EG list
1.1 noro 501: @item vect
1.2 noro 502: \JP $B%Y%/%H%k(B
503: \EG vector
1.1 noro 504: @item mat
1.2 noro 505: \JP $B9TNs(B
506: \EG matrix
1.1 noro 507: @end table
508:
509: @itemize @bullet
1.2 noro 510: \BJP
1.1 noro 511: @item
1.9 ohara 512: @var{vect} $B$ND9$5(B, $B$^$?$O(B @var{mat} $B$NBg$-$5$r%j%9%H$G=PNO$9$k(B.
513: @item
514: @var{vect} $B$ND9$5$O(B @code{length()} $B$G5a$a$k$3$H$b$G$-$k(B.
1.1 noro 515: @item
1.9 ohara 516: @var{list} $B$ND9$5$O(B @code{length()}$B$r(B, $BM-M}<0$K8=$l$kC19`<0$N?t$O(B @code{nmono()} $B$rMQ$$$k(B.
1.2 noro 517: \E
518: \BEG
519: @item
520: Return a list consisting of the dimension of the vector @var{vect},
521: or a list consisting of the row size and column size of the matrix
522: @var{matrix}.
523: @item
524: Use @code{length()} for the size of @var{list}, and
525: @code{nmono()} for the number of monomials with non-zero coefficients
526: in a rational expression.
527: \E
1.1 noro 528: @end itemize
529:
530: @example
531: [0] A = newvect(4);
532: [ 0 0 0 0 ]
533: [1] size(A);
534: [4]
1.9 ohara 535: [2] length(A);
536: 4
537: [3] B = newmat(2,3,[[1,2,3],[4,5,6]]);
1.1 noro 538: [ 1 2 3 ]
539: [ 4 5 6 ]
1.9 ohara 540: [4] size(B);
1.1 noro 541: [2,3]
542: @end example
543:
544: @table @t
1.2 noro 545: \JP @item $B;2>H(B
546: \EG @item References
1.1 noro 547: @fref{car cdr cons append reverse length}, @fref{nmono}.
548: @end table
549:
1.10 noro 550: \JP @node det nd_det invmat,,, $BG[Ns(B
551: \EG @node det nd_det invmat,,, Arrays
1.11 ohara 552: @subsection @code{det}, @code{nd_det}, @code{invmat}
1.1 noro 553: @findex det
1.11 ohara 554: @findex nd_det
1.5 noro 555: @findex invmat
1.1 noro 556:
557: @table @t
558: @item det(@var{mat}[,@var{mod}])
1.10 noro 559: @itemx nd_det(@var{mat}[,@var{mod}])
1.2 noro 560: \JP :: @var{mat} $B$N9TNs<0$r5a$a$k(B.
561: \EG :: Determinant of @var{mat}.
1.5 noro 562: @item invmat(@var{mat})
1.8 takayama 563: \JP :: @var{mat} $B$N5U9TNs$r5a$a$k(B.
1.5 noro 564: \EG :: Inverse matrix of @var{mat}.
1.1 noro 565: @end table
566:
567: @table @var
568: @item return
1.5 noro 569: \JP @code{det}: $B<0(B, @code{invmat}: $B%j%9%H(B
570: \EG @code{det}: expression, @code{invmat}: list
1.1 noro 571: @item mat
1.2 noro 572: \JP $B9TNs(B
573: \EG matrix
1.1 noro 574: @item mod
1.2 noro 575: \JP $BAG?t(B
576: \EG prime
1.1 noro 577: @end table
578:
579: @itemize @bullet
1.2 noro 580: \BJP
1.1 noro 581: @item
1.10 noro 582: @code{det} $B$*$h$S(B @code{nd_det} $B$O9TNs(B @var{mat} $B$N9TNs<0$r5a$a$k(B.
1.5 noro 583: @code{invmat} $B$O9TNs(B @var{mat} $B$N5U9TNs$r5a$a$k(B. $B5U9TNs$O(B @code{[$BJ,Jl(B, $BJ,;R(B]}
1.14 ! nisiyama 584: $B$N7A$GJV$5$l(B, @code{$BJ,Jl(B}$B$,9TNs(B, @code{$BJ,;R(B/$BJ,Jl(B} $B$,5U9TNs$H$J$k(B.
1.1 noro 585: @item
586: $B0z?t(B @var{mod} $B$,$"$k;~(B, GF(@var{mod}) $B>e$G$N9TNs<0$r5a$a$k(B.
587: @item
588: $BJ,?t$J$7$N%,%&%9>C5nK!$K$h$C$F$$$k$?$a(B, $BB?JQ?tB?9`<0$r@.J,$H$9$k(B
589: $B9TNs$KBP$7$F$O>.9TNs<0E83+$K$h$kJ}K!$N$[$&$,8zN($,$h$$>l9g$b$"$k(B.
1.10 noro 590: @item
591: @code{nd_det} $B$OM-M}?t$^$?$OM-8BBN>e$NB?9`<09TNs$N9TNs<0(B
592: $B7W;;@lMQ$G$"$k(B. $B%"%k%4%j%:%`$O$d$O$jJ,?t$J$7$N%,%&%9>C5nK!$@$,(B,
593: $B%G!<%?9=B$$*$h$S>h=|;;$N9)IW$K$h$j(B, $B0lHL$K(B @code{det} $B$h$j9bB.$K(B
594: $B7W;;$G$-$k(B.
1.2 noro 595: \E
596: \BEG
597: @item
1.10 noro 598: @code{det} and @code{nd_det} compute the determinant of matrix @var{mat}.
1.5 noro 599: @code{invmat} computes the inverse matrix of matrix @var{mat}.
600: @code{invmat} returns a list @code{[num,den]}, where @code{num}
601: is a matrix and @code{num/den} represents the inverse matrix.
1.2 noro 602: @item
603: The computation is done over GF(@var{mod}) if @var{mod} is specitied.
604: @item
605: The fraction free Gaussian algorithm is employed. For matrices with
606: multi-variate polynomial entries, minor expansion algorithm sometimes
607: is more efficient than the fraction free Gaussian algorithm.
1.10 noro 608: @item
609: @code{nd_det} can be used for computing the determinant of a matrix with
610: polynomial entries over the rationals or finite fields. The algorithm
611: is an improved vesion of the fraction free Gaussian algorithm
612: and it computes the determinant faster than @code{det}.
1.2 noro 613: \E
1.1 noro 614: @end itemize
615:
616: @example
617: [91] A=newmat(5,5)$
618: [92] V=[x,y,z,u,v];
619: [x,y,z,u,v]
620: [93] for(I=0;I<5;I++)for(J=0,B=A[I],W=V[I];J<5;J++)B[J]=W^J;
621: [94] A;
622: [ 1 x x^2 x^3 x^4 ]
623: [ 1 y y^2 y^3 y^4 ]
624: [ 1 z z^2 z^3 z^4 ]
625: [ 1 u u^2 u^3 u^4 ]
626: [ 1 v v^2 v^3 v^4 ]
627: [95] fctr(det(A));
1.7 noro 628: [[1,1],[u-v,1],[-z+v,1],[-z+u,1],[-y+u,1],[y-v,1],[-y+z,1],[-x+u,1],
629: [-x+z,1],[-x+v,1],[-x+y,1]]
1.5 noro 630: [96] A = newmat(3,3)$
631: [97] for(I=0;I<3;I++)for(J=0,B=A[I],W=V[I];J<3;J++)B[J]=W^J;
632: [98] A;
633: [ 1 x x^2 ]
634: [ 1 y y^2 ]
635: [ 1 z z^2 ]
636: [99] invmat(A);
637: [[ -z*y^2+z^2*y z*x^2-z^2*x -y*x^2+y^2*x ]
638: [ y^2-z^2 -x^2+z^2 x^2-y^2 ]
639: [ -y+z x-z -x+y ],(-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y]
640: [100] A*B[0];
641: [ (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y 0 0 ]
642: [ 0 (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y 0 ]
643: [ 0 0 (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y ]
644: [101] map(red,A*B[0]/B[1]);
645: [ 1 0 0 ]
646: [ 0 1 0 ]
647: [ 0 0 1 ]
1.1 noro 648: @end example
649:
650: @table @t
1.2 noro 651: \JP @item $B;2>H(B
652: \EG @item References
1.12 ohara 653: @fref{newmat matrix}.
1.1 noro 654: @end table
655:
1.2 noro 656: \JP @node qsort,,, $BG[Ns(B
657: \EG @node qsort,,, Arrays
1.1 noro 658: @subsection @code{qsort}
659: @findex qsort
660:
661: @table @t
662: @item qsort(@var{array}[,@var{func}])
1.2 noro 663: \JP :: $B0l<!85G[Ns(B @var{array} $B$r%=!<%H$9$k(B.
664: \EG :: Sorts an array @var{array}.
1.1 noro 665: @end table
666:
667: @table @var
668: @item return
1.2 noro 669: \JP @var{array} ($BF~NO$HF1$8(B; $BMWAG$N$_F~$lBX$o$k(B)
670: \EG @var{array} (The same as the input; Only the elements are exchanged.)
1.1 noro 671: @item array
1.2 noro 672: \JP $B0l<!85G[Ns(B
673: \EG array
1.1 noro 674: @item func
1.2 noro 675: \JP $BHf3SMQ4X?t(B
676: \EG function for comparison
1.1 noro 677: @end table
678:
679: @itemize @bullet
1.2 noro 680: \BJP
1.1 noro 681: @item
682: $B0l<!85G[Ns$r(B quick sort $B$G%=!<%H$9$k(B.
683: @item
684: $BHf3SMQ4X?t$,;XDj$5$l$F$$$J$$>l9g(B, $B%*%V%8%'%/%H$I$&$7$NHf3S7k2L$G(B
685: $B=g=x$,2<$N$b$N$+$i=g$KJB$Y49$($i$l$k(B.
686: @item
687: 0, 1, -1 $B$rJV$9(B 2 $B0z?t4X?t$,(B @var{func} $B$H$7$FM?$($i$l$?>l9g(B,
688: @code{@var{func}(A,B)=1} $B$N>l9g$K(B @code{A<B} $B$H$7$F(B, $B=g=x$,2<$N(B
689: $B$b$N$+$i=g$KJB$Y49$($i$l$k(B.
690: @item
691: $BG[Ns$O?7$?$K@8@.$5$l$:(B, $B0z?t$NG[Ns$NMWAG$N$_F~$lBX$o$k(B.
1.2 noro 692: \E
693: \BEG
694: @item
695: This function sorts an array by @var{quick sort}.
696: @item
697: If @var{func} is not specified, the built-in comparison function
698: is used and the array is sorted in increasing order.
699: @item
700: If a function of two arguments @var{func} which returns 0, 1, or -1
701: is provided, then an ordering is detemined so that
702: @code{A<B} if @code{@var{func}(A,B)=1} holds, and
703: the array is sorted in increasing order with respect to the ordering.
704: @item
705: The returned array is the same as the input. Only the elements
706: are exchanged.
707: \E
1.1 noro 708: @end itemize
709:
710: @example
711: [0] qsort(newvect(10,[1,4,6,7,3,2,9,6,0,-1]));
712: [ -1 0 1 2 3 4 6 6 7 9 ]
713: [1] def rev(A,B) @{ return A>B?-1:(A<B?1:0); @}
714: [2] qsort(newvect(10,[1,4,6,7,3,2,9,6,0,-1]),rev);
715: [ 9 7 6 6 4 3 2 1 0 -1 ]
716: @end example
717:
718: @table @t
1.2 noro 719: \JP @item $B;2>H(B
720: \EG @item References
1.1 noro 721: @fref{ord}, @fref{vars}.
722: @end table
1.13 ohara 723:
724: \JP @node rowx rowm rowa colx colm cola,,, $BG[Ns(B
725: \EG @node rowx rowm rowa colx colm cola,,, Arrays
726: @subsection @code{rowx}, @code{rowm}, @code{rowa}, @code{colx}, @code{colm}, @code{cola}
727: @findex rowx
728: @findex rowm
729: @findex rowa
730: @findex colx
731: @findex colm
732: @findex cola
733:
734: @table @t
735: @item rowx(@var{matrix},@var{i},@var{j})
736: \JP :: $BBh(B @var{i} $B9T$HBh(B @var{j} $B9T$r8r49$9$k(B.
737: \EG :: Exchanges the @var{i}-th and @var{j}-th rows.
738: @item rowm(@var{matrix},@var{i},@var{c})
739: \JP :: $BBh(B @var{i} $B9T$r(B @var{c} $BG\$9$k(B.
740: \EG :: Multiplies the @var{i}-th row by @var{c}.
741: @item rowa(@var{matrix},@var{i},@var{c})
742: \JP :: $BBh(B @var{i} $B9T$KBh(B @var{i} $B9T$N(B @var{c} $BG\$r2C$($k(B.
743: \EG :: Appends @var{c} times the @var{j}-th row to the @var{j}-th row.
744: @item colx(@var{matrix},@var{i},@var{j})
745: \JP :: $BBh(B @var{i} $B9T$HBh(B @var{j} $B9T$r8r49$9$k(B.
746: \EG :: Exchanges the @var{i}-th and @var{j}-th columns.
747: @item colm(@var{matrix},@var{i},@var{c})
748: \JP :: $BBh(B @var{i} $B9T$r(B @var{c} $BG\$9$k(B.
749: \EG :: Multiplies the @var{i}-th column by @var{c}.
750: @item cola(@var{matrix},@var{i},@var{c})
751: \JP :: $BBh(B @var{i} $B9T$KBh(B @var{i} $B9T$N(B @var{c} $BG\$r2C$($k(B.
752: \EG :: Appends @var{c} times the @var{j}-th column to the @var{j}-th column.
753: @end table
754:
755: @table @var
756: @item return
757: \JP $B9TNs(B
758: \EG matrix
759: @item @var{i}, @var{j}
760: \JP $B@0?t(B
761: \EG integers
762: @item @var{c}
763: \JP $B78?t(B
764: \EG coefficient
765: @end table
766:
767: @itemize @bullet
768: \BJP
769: @item
770: $B9TNs$N4pK\JQ7A$r9T$&$?$a$N4X?t$G$"$k(B.
771: @item
772: $B9TNs$,GK2u$5$l$k$3$H$KCm0U$9$k(B.
773: \E
774: \BEG
775: @item
776: These operations are destructive for the matrix.
777: \E
778: @end itemize
779:
780: @example
781: [0] A=newmat(3,3,[[1,2,3],[4,5,6],[7,8,9]]);
782: [ 1 2 3 ]
783: [ 4 5 6 ]
784: [ 7 8 9 ]
785: [1] rowx(A,1,2)$
786: [2] A;
787: [ 1 2 3 ]
788: [ 7 8 9 ]
789: [ 4 5 6 ]
790: [3] rowm(A,2,x);
791: [ 1 2 3 ]
792: [ 7 8 9 ]
793: [ 4*x 5*x 6*x ]
794: [4] rowa(A,0,1,z);
795: [ 7*z+1 8*z+2 9*z+3 ]
796: [ 7 8 9 ]
797: [ 4*x 5*x 6*x ]
798: @end example
799:
800: @table @t
801: \JP @item $B;2>H(B
802: \EG @item References
803: @fref{newmat matrix}
804: @end table
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