Annotation of OpenXM/src/asir-doc/parts/builtin/array.texi, Revision 1.12
1.12 ! ohara 1: @comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/array.texi,v 1.11 2009/03/24 08:00:50 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::
20:
1.1 noro 21: * qsort::
22: @end menu
23:
1.11 ohara 24: \JP @node newvect vector vect,,, $BG[Ns(B
25: \EG @node newvect vector vect,,, Arrays
26: @subsection @code{newvect}, @code{vector}, @code{vect}
1.1 noro 27: @findex newvect
1.11 ohara 28: @findex vector
29: @findex vect
1.1 noro 30:
31: @table @t
32: @item newvect(@var{len}[,@var{list}])
1.11 ohara 33: @item vector(@var{len}[,@var{list}])
1.2 noro 34: \JP :: $BD9$5(B @var{len} $B$N%Y%/%H%k$r@8@.$9$k(B.
35: \EG :: Creates a new vector object with its length @var{len}.
1.11 ohara 36: @item vect([@var{elements}])
37: \JP :: @var{elements} $B$rMWAG$H$9$k%Y%/%H%k$r@8@.$9$k(B.
38: \EG :: Creates a new vector object by @var{elements}.
1.1 noro 39: @end table
40:
41: @table @var
42: @item return
1.2 noro 43: \JP $B%Y%/%H%k(B
44: \EG vector
1.1 noro 45: @item len
1.2 noro 46: \JP $B<+A3?t(B
47: \EG non-negative integer
1.1 noro 48: @item list
1.2 noro 49: \JP $B%j%9%H(B
50: \EG list
1.11 ohara 51: @item elements
52: \JP $BMWAG$NJB$S(B
53: \EG elements of the vector
1.1 noro 54: @end table
55:
56: @itemize @bullet
1.2 noro 57: \BJP
1.1 noro 58: @item
1.11 ohara 59: @code{vect} $B$OMWAG$NJB$S$+$i%Y%/%H%k$r@8@.$9$k(B.
60: @item
61: @code{vector} $B$O(B @code{newvect} $B$NJLL>$G$"$k(B.
62: @item
63: @code{newvect} $B$OD9$5(B @var{len} $B$N%Y%/%H%k$r@8@.$9$k(B. $BBh(B 2 $B0z?t$,$J$$>l9g(B,
1.1 noro 64: $B3F@.J,$O(B 0 $B$K=i4|2=$5$l$k(B. $BBh(B 2 $B0z?t$,$"$k>l9g(B,
65: $B%$%s%G%C%/%9$N>.$5$$@.J,$+$i(B, $B%j%9%H$N(B
66: $B3FMWAG$K$h$j=i4|2=$5$l$k(B. $B3FMWAG$O(B, $B@hF,$+$i=g$K(B
67: $B;H$o$l(B, $BB-$j$J$$J,$O(B 0 $B$,Kd$a$i$l$k(B.
68: @item
69: $B%Y%/%H%k$N@.J,$O(B, $BBh(B 0 $B@.J,$+$iBh(B @var{len}-1 $B@.J,$H$J$k(B.
70: ($BBh(B 1 $B@.J,$+$i$G$O$J$$;v$KCm0U(B. )
71: @item
72: $B%j%9%H$O3F@.J,$,(B, $B%]%$%s%?$rC)$k;v$K$h$C$F%7!<%1%s%7%c%k$K(B
73: $B8F$S=P$5$l$k$N$KBP$7(B, $B%Y%/%H%k$O3F@.J,$,(B
74: $BBh0l@.J,$+$i$N%a%b%j>e$N(B displacement ($BJQ0L(B)$B$K$h$C$F%i%s%@%`%"%/%;%9$G(B
75: $B8F$S=P$5$l(B, $B$=$N7k2L(B, $B@.J,$N%"%/%;%9;~4V$KBg$-$J:9$,=P$F$/$k(B.
76: $B@.J,%"%/%;%9$O(B, $B%j%9%H$G$O(B, $B@.J,$NNL$,A}$($k$K=>$C$F(B
77: $B;~4V$,$+$+$k$h$&$K$J$k$,(B, $B%Y%/%H%k$G$O(B, $B@.J,$NNL$K0MB8$;$:$[$\0lDj$G$"$k(B.
78: @item
79: @b{Asir} $B$G$O(B, $B=D%Y%/%H%k(B, $B2#%Y%/%H%k$N6hJL$O$J$$(B.
80: $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
81: $B$_$J$5$l$k(B.
82: @item
83: $B%Y%/%H%k$ND9$5$O(B @code{size()} $B$K$h$C$FF@$i$l$k(B.
84: @item
85: $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
86: $B$r=q$-49$($k$3$H$,$G$-$k(B.
1.2 noro 87: \E
88: \BEG
1.11 ohara 89: @item
90: @code{vect} creates a new vector object by its elements.
91: @item
92: @code{vector} is an alias of @code{newvect}.
1.2 noro 93: @item
1.11 ohara 94: @code{newvect} creates a new vector object with its length @var{len} and its elements
1.2 noro 95: all cleared to value 0.
96: If the second argument, a list, is given, the vector is initialized by
97: the list elements.
98: Elements are used from the first through the last.
99: If the list is short for initializing the full vector,
100: 0's are filled in the remaining vector elements.
101: @item
102: Elements are indexed from 0 through @var{len}-1. Note that the first
103: element has not index 1.
104: @item
105: List and vector are different types in @b{Asir}.
106: Lists are conveniently used for representing many data objects whose
107: size varies dynamically as computation proceeds.
108: By its flexible expressive power, it is also conveniently used to
109: describe initial values for other structured objects as you see
110: for vectors.
111: Access for an element of a list is performed by following pointers to
112: next elements. By this, access costs for list elements differ for
113: each element.
114: In contrast to lists, vector elements can be accessed in a same time,
115: because they are accessed by computing displacements from the top memory
116: location of the vector object.
117:
118: Note also, in @b{Asir}, modification of an element of a vector causes
119: modification of the whole vector itself,
120: while modification of a list element does not cause the modification
121: of the whole list object.
122:
123: By this, in @b{Asir} language,
124: a vector element designator can be a left value of
125: assignment statement, but a list element designator can NOT be a left
126: value of assignment statement.
127:
128: @item
129: No distinction of column vectors and row vectors in @b{Asir}.
130: If a matrix is applied to a vector from left, the vector shall be taken
131: as a column vector, and if from right it shall be taken as a row vector.
132: @item
133: The length (or size or dimension) of a vector is given by function
134: @code{size()}.
135: @item
136: When a vector is passed to a function as its argument
137: (actual parameter), the vector element can be modified in that
138: function.
139:
140: @item
141: A vector is displayed in a similar format as for a list.
142: Note, however, there is a distinction: Elements of a vector are
143: separated simply by a `blank space', while those of a list by a `comma.'
144: \E
1.1 noro 145: @end itemize
146:
147: @example
148: [0] A=newvect(5);
149: [ 0 0 0 0 0 ]
150: [1] A=newvect(5,[1,2,3,4,[5,6]]);
151: [ 1 2 3 4 [5,6] ]
152: [2] A[0];
153: 1
154: [3] A[4];
155: [5,6]
156: [4] size(A);
157: [5]
1.11 ohara 158: [5] length(A);
159: 5
160: [6] vect(1,2,3,4,[5,6]);
161: [ 1 2 3 4 [5,6] ]
162: [7] def afo(V) @{ V[0] = x; @}
163: [8] afo(A)$
164: [9] A;
1.1 noro 165: [ x 2 3 4 [5,6] ]
166: @end example
167:
168: @table @t
1.2 noro 169: \JP @item $B;2>H(B
170: \EG @item References
1.12 ! ohara 171: @fref{newmat matrix}, @fref{size}, @fref{ltov}, @fref{vtol}.
1.9 ohara 172: @end table
173:
174: \JP @node ltov,,, $BG[Ns(B
175: \EG @node ltov,,, Arrays
176: @subsection @code{ltov}
177: @findex ltov
178:
179: @table @t
180: @item ltov(@var{list})
181: \JP :: $B%j%9%H$r%Y%/%H%k$KJQ49$9$k(B.
182: \EG :: Converts a list into a vector.
183: @end table
184:
185: @table @var
186: @item return
187: \JP $B%Y%/%H%k(B
188: \EG vector
189: @item list
190: \JP $B%j%9%H(B
191: \EG list
192: @end table
193:
194: @itemize @bullet
195: \BJP
196: @item
197: $B%j%9%H(B @var{list} $B$rF1$8D9$5$N%Y%/%H%k$KJQ49$9$k(B.
198: @item
199: $B$3$N4X?t$O(B @code{newvect(length(@var{list}), @var{list})} $B$KEy$7$$(B.
200: \E
201: \BEG
202: @item
203: Converts a list @var{list} into a vector of same length.
204: See also @code{newvect()}.
205: \E
206: @end itemize
207:
208: @example
209: [3] A=[1,2,3];
210: [4] ltov(A);
211: [ 1 2 3 ]
212: @end example
213:
214: @table @t
215: \JP @item $B;2>H(B
216: \EG @item References
1.12 ! ohara 217: @fref{newvect vector vect}, @fref{vtol}.
1.1 noro 218: @end table
219:
1.2 noro 220: \JP @node vtol,,, $BG[Ns(B
221: \EG @node vtol,,, Arrays
1.1 noro 222: @subsection @code{vtol}
223: @findex vtol
224:
225: @table @t
226: @item vtol(@var{vect})
1.2 noro 227: \JP :: $B%Y%/%H%k$r%j%9%H$KJQ49$9$k(B.
228: \EG :: Converts a vector into a list.
1.1 noro 229: @end table
230:
231: @table @var
232: @item return
1.2 noro 233: \JP $B%j%9%H(B
234: \EG list
1.1 noro 235: @item vect
1.2 noro 236: \JP $B%Y%/%H%k(B
237: \EG vector
1.1 noro 238: @end table
239:
240: @itemize @bullet
1.2 noro 241: \BJP
1.1 noro 242: @item
243: $BD9$5(B @var{n} $B$N%Y%/%H%k(B @var{vect} $B$r(B
244: @code{[@var{vect}[0],...,@var{vect}[@var{n}-1]]} $B$J$k%j%9%H$KJQ49$9$k(B.
245: @item
246: $B%j%9%H$+$i%Y%/%H%k$X$NJQ49$O(B @code{newvect()} $B$G9T$&(B.
1.2 noro 247: \E
248: \BEG
249: @item
250: Converts a vector @var{vect} of length @var{n} into
251: a list @code{[@var{vect}[0],...,@var{vect}[@var{n}-1]]}.
252: @item
253: A conversion from a list to a vector is done by @code{newvect()}.
254: \E
1.1 noro 255: @end itemize
256:
257: @example
258: [3] A=newvect(3,[1,2,3]);
259: [ 1 2 3 ]
260: [4] vtol(A);
261: [1,2,3]
1.4 noro 262: @end example
263:
264: @table @t
265: \JP @item $B;2>H(B
266: \EG @item References
1.12 ! ohara 267: @fref{newvect vector vect}, @fref{ltov}.
1.4 noro 268: @end table
269:
270: \JP @node newbytearray,,, $BG[Ns(B
271: \EG @node newbytearray,,, Arrays
272: @subsection @code{newbytearray}
273: @findex newbytearray
274:
275: @table @t
276: @item newbytearray(@var{len},[@var{listorstring}])
277: \JP :: $BD9$5(B @var{len} $B$N(B byte array $B$r@8@.$9$k(B.
278: \EG :: Creates a new byte array.
279: @end table
280:
281: @table @var
282: @item return
283: byte array
284: @item len
285: \JP $B<+A3?t(B
286: \EG non-negative integer
287: @item listorstring
288: \JP $B%j%9%H$^$?$OJ8;zNs(B
289: \EG list or string
290: @end table
291:
292: @itemize @bullet
293: @item
294: \JP @code{newvect} $B$HF1MM$K$7$F(B byte array $B$r@8@.$9$k(B.
295: \EG This function generates a byte array. The specification is
296: similar to that of @code{newvect}.
297: @item
298: \JP $BJ8;zNs$G=i4|CM$r;XDj$9$k$3$H$b2DG=$G$"$k(B.
299: \EG The initial value can be specified by a character string.
300: @item
301: \JP byte array $B$NMWAG$N%"%/%;%9$OG[Ns$HF1MM$G$"$k(B.
302: \EG One can access elements of a byte array just as an array.
303: @end itemize
304:
305: @example
306: [182] A=newbytearray(3);
307: |00 00 00|
308: [183] A=newbytearray(3,[1,2,3]);
309: |01 02 03|
310: [184] A=newbytearray(3,"abc");
311: |61 62 63|
312: [185] A[0];
313: 97
314: [186] A[1]=123;
315: 123
316: [187] A;
317: |61 7b 63|
1.1 noro 318: @end example
319:
320: @table @t
1.2 noro 321: \JP @item $B;2>H(B
322: \EG @item References
1.12 ! ohara 323: @fref{newvect vector vect}.
1.1 noro 324: @end table
325:
1.11 ohara 326: \JP @node newmat matrix,,, $BG[Ns(B
327: \EG @node newmat matrix,,, Arrays
328: @subsection @code{newmat}, @code{matrix}
1.1 noro 329: @findex newmat
1.11 ohara 330: @findex matrix
1.1 noro 331:
332: @table @t
1.6 noro 333: @item newmat(@var{row},@var{col} [,[[@var{a},@var{b},...],[@var{c},@var{d},...],...]])
1.11 ohara 334: @item matrix(@var{row},@var{col} [,[[@var{a},@var{b},...],[@var{c},@var{d},...],...]])
1.2 noro 335: \JP :: @var{row} $B9T(B @var{col} $BNs$N9TNs$r@8@.$9$k(B.
336: \EG :: Creates a new matrix with @var{row} rows and @var{col} columns.
1.1 noro 337: @end table
338:
339: @table @var
340: @item return
1.2 noro 341: \JP $B9TNs(B
342: \EG matrix
1.6 noro 343: @item row col
1.2 noro 344: \JP $B<+A3?t(B
345: \EG non-negative integer
1.6 noro 346: @item a b c d
1.2 noro 347: \JP $BG$0U(B
348: \EG arbitrary
1.1 noro 349: @end table
350:
351: @itemize @bullet
1.2 noro 352: \BJP
1.1 noro 353: @item
1.11 ohara 354: @code{matrix} $B$O(B @code{newmat} $B$NJLL>$G$"$k(B.
355: @item
1.1 noro 356: @var{row} $B9T(B @var{col} $BNs$N9TNs$r@8@.$9$k(B. $BBh(B 3 $B0z?t$,$J$$>l9g(B,
357: $B3F@.J,$O(B 0 $B$K=i4|2=$5$l$k(B. $BBh(B 3 $B0z?t$,$"$k>l9g(B,
358: $B%$%s%G%C%/%9$N>.$5$$@.J,$+$i(B, $B3F9T$,(B, $B%j%9%H$N(B
359: $B3FMWAG(B ($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
360: $B;H$o$l(B, $BB-$j$J$$J,$O(B 0 $B$,Kd$a$i$l$k(B.
361: @item
362: $B9TNs$N%5%$%:$O(B @code{size()} $B$GF@$i$l$k(B.
363: @item
364: @code{M} $B$,9TNs$N$H$-(B, @code{M[I]} $B$K$h$jBh(B @code{I} $B9T$r%Y%/%H%k$H$7$F(B
365: $B<h$j=P$9$3$H$,$G$-$k(B. $B$3$N%Y%/%H%k$O(B, $B$b$H$N9TNs$H@.J,$r6&M-$7$F$*$j(B,
366: $B$$$:$l$+$N@.J,$r=q$-49$($l$P(B, $BB>$NBP1~$9$k@.J,$b=q$-49$o$k$3$H$K$J$k(B.
367: @item
368: $BH!?t$N0z?t$H$7$F9TNs$rEO$7$?>l9g(B, $BEO$5$l$?H!?t$O(B, $B$=$N9TNs$N@.J,(B
369: $B$r=q$-49$($k$3$H$,$G$-$k(B.
1.2 noro 370: \E
371: \BEG
1.11 ohara 372: @item
373: @code{matrix} is an alias of @code{newmat}.
1.2 noro 374: @item
375: If the third argument, a list, is given, the newly created matrix
376: is initialized so that each element of the list (again a list)
377: initializes each of the rows of the matrix.
378: Elements are used from the first through the last.
379: If the list is short, 0's are filled in the remaining matrix elements.
380: If no third argument is given all the elements are cleared to 0.
381: @item
382: The size of a matrix is given by function @code{size()}.
383: @item
384: Let @code{M} be a program variable assigned to a matrix.
385: Then, @code{M[I]} denotes a (row) vector which corresponds with
386: the @code{I}-th row of the matrix.
387: Note that the vector shares its element with the original matrix.
388: Subsequently, if an element of the vector is modified, then the
389: corresponding matrix element is also modified.
390: @item
391: When a matrix is passed to a function as its argument
392: (actual parameter), the matrix element can be modified within that
393: function.
394: \E
1.1 noro 395: @end itemize
396:
397: @example
398: [0] A = newmat(3,3,[[1,1,1],[x,y],[x^2]]);
399: [ 1 1 1 ]
400: [ x y 0 ]
401: [ x^2 0 0 ]
402: [1] det(A);
403: -y*x^2
404: [2] size(A);
405: [3,3]
406: [3] A[1];
407: [ x y 0 ]
408: [4] A[1][3];
409: getarray : Out of range
410: return to toplevel
411: @end example
412:
413: @table @t
1.2 noro 414: \JP @item $B;2>H(B
415: \EG @item References
1.12 ! ohara 416: @fref{newvect vector vect}, @fref{size}, @fref{det nd_det invmat}.
! 417: @end table
! 418:
! 419: \JP @node mat matr matc,,, $BG[Ns(B
! 420: \EG @node mat matr matc,,, Arrays
! 421: @subsection @code{mat}, @code{matr}, @code{matc}
! 422: @findex mat
! 423: @findex matr
! 424: @findex matc
! 425:
! 426: @table @t
! 427: @item mat(@var{vector}[,...])
! 428: @item matr(@var{vector}[,...])
! 429: \JP :: $B9T%Y%/%H%k$NJB$S$+$i9TNs$r@8@.$9$k(B.
! 430: \EG :: Creates a new matrix by list of row vectors.
! 431: @item matc(@var{vector}[,...])
! 432: \JP :: $BNs%Y%/%H%k$NJB$S$+$i9TNs$r@8@.$9$k(B.
! 433: \EG :: Creates a new matrix by list of column vectors.
! 434: @end table
! 435:
! 436: @table @var
! 437: @item return
! 438: \JP $B9TNs(B
! 439: \EG matrix
! 440: @item @var{vector}
! 441: \JP $BG[Ns$^$?$O%j%9%H(B
! 442: \EG array or list
! 443: @end table
! 444:
! 445: @itemize @bullet
! 446: \BJP
! 447: @item
! 448: @code{mat} $B$O(B @code{matr} $B$NJLL>$G$"$k(B.
! 449: @item
! 450: $B0z?t$N3F%Y%/%H%k$OF1$8D9$5$r$b$D(B.
! 451: $B3FMWAG$O(B, $B@hF,$+$i=g$K;H$o$l(B, $BB-$j$J$$J,$O(B 0 $B$,Kd$a$i$l$k(B.
! 452: \E
! 453: \BEG
! 454: @item
! 455: @code{mat} is an alias of @code{matr}.
! 456: @item
! 457: Each vector has same length.
! 458: Elements are used from the first through the last.
! 459: If the list is short, 0's are filled in the remaining matrix elements.
! 460: \E
! 461: @end itemize
! 462:
! 463: @example
! 464: [0] matr([1,2,3],[4,5,6],[7,8]);
! 465: [ 1 2 3 ]
! 466: [ 4 5 6 ]
! 467: [ 7 8 0 ]
! 468: [1] matc([1,2,3],[4,5,6],[7,8]);
! 469: [ 1 4 7 ]
! 470: [ 2 5 8 ]
! 471: [ 3 6 0 ]
! 472: @end example
! 473:
! 474: @table @t
! 475: \JP @item $B;2>H(B
! 476: \EG @item References
! 477: @fref{newmat matrix}
1.1 noro 478: @end table
479:
1.2 noro 480: \JP @node size,,, $BG[Ns(B
481: \EG @node size,,, Arrays
1.1 noro 482: @subsection @code{size}
483: @findex size
484:
485: @table @t
486: @item size(@var{vect|mat})
1.2 noro 487: \JP :: @code{[@var{vect} $B$ND9$5(B]} $B$^$?$O(B @code{[@var{mat} $B$N9T?t(B,@var{mat} $B$NNs?t(B]}.
488: \BEG
489: :: A list containing the number of elements of the given vector,
490: @code{[size of @var{vect}]},
491: or a list containing row size and column size of the given matrix,
492: @code{[row size of @var{mat}, column size of @var{mat}]}.
493: \E
1.1 noro 494: @end table
495:
496: @table @var
497: @item return
1.2 noro 498: \JP $B%j%9%H(B
499: \EG list
1.1 noro 500: @item vect
1.2 noro 501: \JP $B%Y%/%H%k(B
502: \EG vector
1.1 noro 503: @item mat
1.2 noro 504: \JP $B9TNs(B
505: \EG matrix
1.1 noro 506: @end table
507:
508: @itemize @bullet
1.2 noro 509: \BJP
1.1 noro 510: @item
1.9 ohara 511: @var{vect} $B$ND9$5(B, $B$^$?$O(B @var{mat} $B$NBg$-$5$r%j%9%H$G=PNO$9$k(B.
512: @item
513: @var{vect} $B$ND9$5$O(B @code{length()} $B$G5a$a$k$3$H$b$G$-$k(B.
1.1 noro 514: @item
1.9 ohara 515: @var{list} $B$ND9$5$O(B @code{length()}$B$r(B, $BM-M}<0$K8=$l$kC19`<0$N?t$O(B @code{nmono()} $B$rMQ$$$k(B.
1.2 noro 516: \E
517: \BEG
518: @item
519: Return a list consisting of the dimension of the vector @var{vect},
520: or a list consisting of the row size and column size of the matrix
521: @var{matrix}.
522: @item
523: Use @code{length()} for the size of @var{list}, and
524: @code{nmono()} for the number of monomials with non-zero coefficients
525: in a rational expression.
526: \E
1.1 noro 527: @end itemize
528:
529: @example
530: [0] A = newvect(4);
531: [ 0 0 0 0 ]
532: [1] size(A);
533: [4]
1.9 ohara 534: [2] length(A);
535: 4
536: [3] B = newmat(2,3,[[1,2,3],[4,5,6]]);
1.1 noro 537: [ 1 2 3 ]
538: [ 4 5 6 ]
1.9 ohara 539: [4] size(B);
1.1 noro 540: [2,3]
541: @end example
542:
543: @table @t
1.2 noro 544: \JP @item $B;2>H(B
545: \EG @item References
1.1 noro 546: @fref{car cdr cons append reverse length}, @fref{nmono}.
547: @end table
548:
1.10 noro 549: \JP @node det nd_det invmat,,, $BG[Ns(B
550: \EG @node det nd_det invmat,,, Arrays
1.11 ohara 551: @subsection @code{det}, @code{nd_det}, @code{invmat}
1.1 noro 552: @findex det
1.11 ohara 553: @findex nd_det
1.5 noro 554: @findex invmat
1.1 noro 555:
556: @table @t
557: @item det(@var{mat}[,@var{mod}])
1.10 noro 558: @itemx nd_det(@var{mat}[,@var{mod}])
1.2 noro 559: \JP :: @var{mat} $B$N9TNs<0$r5a$a$k(B.
560: \EG :: Determinant of @var{mat}.
1.5 noro 561: @item invmat(@var{mat})
1.8 takayama 562: \JP :: @var{mat} $B$N5U9TNs$r5a$a$k(B.
1.5 noro 563: \EG :: Inverse matrix of @var{mat}.
1.1 noro 564: @end table
565:
566: @table @var
567: @item return
1.5 noro 568: \JP @code{det}: $B<0(B, @code{invmat}: $B%j%9%H(B
569: \EG @code{det}: expression, @code{invmat}: list
1.1 noro 570: @item mat
1.2 noro 571: \JP $B9TNs(B
572: \EG matrix
1.1 noro 573: @item mod
1.2 noro 574: \JP $BAG?t(B
575: \EG prime
1.1 noro 576: @end table
577:
578: @itemize @bullet
1.2 noro 579: \BJP
1.1 noro 580: @item
1.10 noro 581: @code{det} $B$*$h$S(B @code{nd_det} $B$O9TNs(B @var{mat} $B$N9TNs<0$r5a$a$k(B.
1.5 noro 582: @code{invmat} $B$O9TNs(B @var{mat} $B$N5U9TNs$r5a$a$k(B. $B5U9TNs$O(B @code{[$BJ,Jl(B, $BJ,;R(B]}
583: $B$N7A$GJV$5$l(B, @code{$BJ,Jl(B}$B$,9TNs(B, @code{$BJ,Jl(B/$BJ,;R(B} $B$,5U9TNs$H$J$k(B.
1.1 noro 584: @item
585: $B0z?t(B @var{mod} $B$,$"$k;~(B, GF(@var{mod}) $B>e$G$N9TNs<0$r5a$a$k(B.
586: @item
587: $BJ,?t$J$7$N%,%&%9>C5nK!$K$h$C$F$$$k$?$a(B, $BB?JQ?tB?9`<0$r@.J,$H$9$k(B
588: $B9TNs$KBP$7$F$O>.9TNs<0E83+$K$h$kJ}K!$N$[$&$,8zN($,$h$$>l9g$b$"$k(B.
1.10 noro 589: @item
590: @code{nd_det} $B$OM-M}?t$^$?$OM-8BBN>e$NB?9`<09TNs$N9TNs<0(B
591: $B7W;;@lMQ$G$"$k(B. $B%"%k%4%j%:%`$O$d$O$jJ,?t$J$7$N%,%&%9>C5nK!$@$,(B,
592: $B%G!<%?9=B$$*$h$S>h=|;;$N9)IW$K$h$j(B, $B0lHL$K(B @code{det} $B$h$j9bB.$K(B
593: $B7W;;$G$-$k(B.
1.2 noro 594: \E
595: \BEG
596: @item
1.10 noro 597: @code{det} and @code{nd_det} compute the determinant of matrix @var{mat}.
1.5 noro 598: @code{invmat} computes the inverse matrix of matrix @var{mat}.
599: @code{invmat} returns a list @code{[num,den]}, where @code{num}
600: is a matrix and @code{num/den} represents the inverse matrix.
1.2 noro 601: @item
602: The computation is done over GF(@var{mod}) if @var{mod} is specitied.
603: @item
604: The fraction free Gaussian algorithm is employed. For matrices with
605: multi-variate polynomial entries, minor expansion algorithm sometimes
606: is more efficient than the fraction free Gaussian algorithm.
1.10 noro 607: @item
608: @code{nd_det} can be used for computing the determinant of a matrix with
609: polynomial entries over the rationals or finite fields. The algorithm
610: is an improved vesion of the fraction free Gaussian algorithm
611: and it computes the determinant faster than @code{det}.
1.2 noro 612: \E
1.1 noro 613: @end itemize
614:
615: @example
616: [91] A=newmat(5,5)$
617: [92] V=[x,y,z,u,v];
618: [x,y,z,u,v]
619: [93] for(I=0;I<5;I++)for(J=0,B=A[I],W=V[I];J<5;J++)B[J]=W^J;
620: [94] A;
621: [ 1 x x^2 x^3 x^4 ]
622: [ 1 y y^2 y^3 y^4 ]
623: [ 1 z z^2 z^3 z^4 ]
624: [ 1 u u^2 u^3 u^4 ]
625: [ 1 v v^2 v^3 v^4 ]
626: [95] fctr(det(A));
1.7 noro 627: [[1,1],[u-v,1],[-z+v,1],[-z+u,1],[-y+u,1],[y-v,1],[-y+z,1],[-x+u,1],
628: [-x+z,1],[-x+v,1],[-x+y,1]]
1.5 noro 629: [96] A = newmat(3,3)$
630: [97] for(I=0;I<3;I++)for(J=0,B=A[I],W=V[I];J<3;J++)B[J]=W^J;
631: [98] A;
632: [ 1 x x^2 ]
633: [ 1 y y^2 ]
634: [ 1 z z^2 ]
635: [99] invmat(A);
636: [[ -z*y^2+z^2*y z*x^2-z^2*x -y*x^2+y^2*x ]
637: [ y^2-z^2 -x^2+z^2 x^2-y^2 ]
638: [ -y+z x-z -x+y ],(-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y]
639: [100] A*B[0];
640: [ (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y 0 0 ]
641: [ 0 (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y 0 ]
642: [ 0 0 (-y+z)*x^2+(y^2-z^2)*x-z*y^2+z^2*y ]
643: [101] map(red,A*B[0]/B[1]);
644: [ 1 0 0 ]
645: [ 0 1 0 ]
646: [ 0 0 1 ]
1.1 noro 647: @end example
648:
649: @table @t
1.2 noro 650: \JP @item $B;2>H(B
651: \EG @item References
1.12 ! ohara 652: @fref{newmat matrix}.
1.1 noro 653: @end table
654:
1.2 noro 655: \JP @node qsort,,, $BG[Ns(B
656: \EG @node qsort,,, Arrays
1.1 noro 657: @subsection @code{qsort}
658: @findex qsort
659:
660: @table @t
661: @item qsort(@var{array}[,@var{func}])
1.2 noro 662: \JP :: $B0l<!85G[Ns(B @var{array} $B$r%=!<%H$9$k(B.
663: \EG :: Sorts an array @var{array}.
1.1 noro 664: @end table
665:
666: @table @var
667: @item return
1.2 noro 668: \JP @var{array} ($BF~NO$HF1$8(B; $BMWAG$N$_F~$lBX$o$k(B)
669: \EG @var{array} (The same as the input; Only the elements are exchanged.)
1.1 noro 670: @item array
1.2 noro 671: \JP $B0l<!85G[Ns(B
672: \EG array
1.1 noro 673: @item func
1.2 noro 674: \JP $BHf3SMQ4X?t(B
675: \EG function for comparison
1.1 noro 676: @end table
677:
678: @itemize @bullet
1.2 noro 679: \BJP
1.1 noro 680: @item
681: $B0l<!85G[Ns$r(B quick sort $B$G%=!<%H$9$k(B.
682: @item
683: $BHf3SMQ4X?t$,;XDj$5$l$F$$$J$$>l9g(B, $B%*%V%8%'%/%H$I$&$7$NHf3S7k2L$G(B
684: $B=g=x$,2<$N$b$N$+$i=g$KJB$Y49$($i$l$k(B.
685: @item
686: 0, 1, -1 $B$rJV$9(B 2 $B0z?t4X?t$,(B @var{func} $B$H$7$FM?$($i$l$?>l9g(B,
687: @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
688: $B$b$N$+$i=g$KJB$Y49$($i$l$k(B.
689: @item
690: $BG[Ns$O?7$?$K@8@.$5$l$:(B, $B0z?t$NG[Ns$NMWAG$N$_F~$lBX$o$k(B.
1.2 noro 691: \E
692: \BEG
693: @item
694: This function sorts an array by @var{quick sort}.
695: @item
696: If @var{func} is not specified, the built-in comparison function
697: is used and the array is sorted in increasing order.
698: @item
699: If a function of two arguments @var{func} which returns 0, 1, or -1
700: is provided, then an ordering is detemined so that
701: @code{A<B} if @code{@var{func}(A,B)=1} holds, and
702: the array is sorted in increasing order with respect to the ordering.
703: @item
704: The returned array is the same as the input. Only the elements
705: are exchanged.
706: \E
1.1 noro 707: @end itemize
708:
709: @example
710: [0] qsort(newvect(10,[1,4,6,7,3,2,9,6,0,-1]));
711: [ -1 0 1 2 3 4 6 6 7 9 ]
712: [1] def rev(A,B) @{ return A>B?-1:(A<B?1:0); @}
713: [2] qsort(newvect(10,[1,4,6,7,3,2,9,6,0,-1]),rev);
714: [ 9 7 6 6 4 3 2 1 0 -1 ]
715: @end example
716:
717: @table @t
1.2 noro 718: \JP @item $B;2>H(B
719: \EG @item References
1.1 noro 720: @fref{ord}, @fref{vars}.
721: @end table
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