japanese translation updated for linear-algebra.
[scilab.git] / scilab / modules / linear_algebra / help / ja_JP / linear / qr.xml
1 <?xml version="1.0" encoding="UTF-8"?>
2
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14
15 <refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:lang="ja" xml:id="qr">
16     
17     <refnamediv>
18         
19         <refname>qr</refname>
20         
21         <refpurpose>QR 分解</refpurpose>
22         
23     </refnamediv>
24     
25     <refsynopsisdiv>
26         
27         <title>呼び出し手順</title>
28         
29         <synopsis>[Q,R]=qr(X [,"e"])
30             
31             [Q,R,E]=qr(X [,"e"])
32             
33             [Q,R,rk,E]=qr(X [,tol])
34             
35         </synopsis>
36         
37     </refsynopsisdiv>
38     
39     <refsection>
40         
41         <title>引数</title>
42         
43         <variablelist>
44             
45             <varlistentry>
46                 
47                 <term>X</term>
48                 
49                 <listitem>
50                     
51                     <para>実数または複素数の行列</para>
52                     
53                 </listitem>
54                 
55             </varlistentry>
56             
57             <varlistentry>
58                 
59                 <term>tol</term>
60                 
61                 <listitem>
62                     
63                     <para>非負の実数</para>
64                     
65                 </listitem>
66                 
67             </varlistentry>
68             
69             <varlistentry>
70                 
71                 <term>Q</term>
72                 
73                 <listitem>
74                     
75                     <para>正方直交またはユニタリ行列</para>
76                     
77                 </listitem>
78                 
79             </varlistentry>
80             
81             <varlistentry>
82                 
83                 <term>R</term>
84                 
85                 <listitem>
86                     
87                     <para>
88                         
89                         <literal>X</literal>と同じ次元の行列
90                         
91                     </para>
92                     
93                 </listitem>
94                 
95             </varlistentry>
96             
97             <varlistentry>
98                 
99                 <term>E</term>
100                 
101                 <listitem>
102                     
103                     <para>置換行列</para>
104                     
105                 </listitem>
106                 
107             </varlistentry>
108             
109             <varlistentry>
110                 
111                 <term>rk</term>
112                 
113                 <listitem>
114                     
115                     <para>
116                         
117                         整数 (<literal>X</literal>のQRランク)
118                         
119                     </para>
120                     
121                 </listitem>
122                 
123             </varlistentry>
124             
125         </variablelist>
126         
127     </refsection>
128     
129     <refsection>
130         
131         <title>説明</title>
132         
133         <variablelist>
134             
135             <varlistentry>
136                 
137                 <term>[Q,R] = qr(X)</term>
138                 
139                 <listitem>
140                     
141                     <para>
142                         
143                         <literal>X= Q*R</literal>となるような
144                         
145                         <literal>X</literal>と同じ次元の
146                         
147                         上三角行列<literal>R</literal>および直交(複素数の場合はユニタリ)行列
148                         
149                         <literal>Q</literal>を出力します.
150                         
151                         <literal>[Q,R] = qr(X,"e")</literal>は次にように
152                         
153                         "エコノミーサイズ"で出力します:
154                         
155                         <literal>X</literal> が m行n列 (m &gt; n)の場合,
156                         
157                         <literal>Q</literal>の最初のn列のみが
158                         
159                         <literal>R</literal>の最初のn行と同時に計算されます.
160                         
161                     </para>
162                     
163                     <para>
164                         
165                         <literal>Q*R = X</literal> から,
166                         
167                         行列 <literal>X</literal>のk番目の列は,
168                         
169                         (係数 <literal> R(1,k), ..., R(k,k) </literal>を用いて)
170                         
171                         <literal>Q</literal>の最初のk列の線形結合で表されます. 
172                         
173                         <literal>Q</literal>の最初のk列は,<literal>X</literal>の最初のk列
174                         
175                         に広がる部分空間の直交基底を作成します.
176                         
177                         <literal>X</literal>の列<literal>k</literal>(すなわち, <literal>X(:,k)</literal> )
178                         
179                         が<literal>X</literal>の最初の<literal>p</literal>列の線形結合の場合,
180                         
181                         エントリ<literal>R(p+1,k), ..., R(k,k)</literal>は 0 となります.
182                         
183                         この場合,<literal>R</literal>は上台形となります.
184                         
185                         <literal>X</literal> がランク<literal>rk</literal>を有する場合,
186                         
187                         行 <literal>R(rk+1,:), R(rk+2,:), ...</literal> は 0 となります.
188                         
189                     </para>
190                     
191                 </listitem>
192                 
193             </varlistentry>
194             
195             <varlistentry>
196                 
197                 <term>[Q,R,E] = qr(X)</term>
198                 
199                 <listitem>
200                     
201                     <para>
202                         
203                         <literal>X*E =    Q*R</literal>となるような
204                         
205                         (列)置換行列<literal>E</literal>,
206                         
207                         降順の対角要素を有する上三角行列 <literal>R</literal>,
208                         
209                         直交(またはユニタリ)<literal>Q</literal>
210                         
211                         を出力します.
212                         
213                         <literal>rk</literal>が<literal>X</literal>のランクの場合,
214                         
215                         <literal>R</literal>の主対角項に沿った
216                         
217                         最初の<literal>rk</literal>個のエントリ,
218                         
219                         すなわち,<literal>R(1,1), R(2,2), ..., R(rk,rk)</literal>は
220                         
221                         全て0以外となります.
222                         
223                         <literal>[Q,R,E] =  qr(X,"e")</literal> は
224                         
225                         "エコノミーサイズ"で出力します:
226                         
227                         <literal>X</literal> が m行n列 (m &gt; n)の場合,
228                         
229                         <literal>Q</literal>の最初のn列のみが
230                         
231                         <literal>R</literal>の最初のn行と同時に計算されます.
232                         
233                     </para>
234                     
235                 </listitem>
236                 
237             </varlistentry>
238             
239             <varlistentry>
240                 
241                 <term>[Q,R,rk,E] = qr(X ,tol)</term>
242                 
243                 <listitem>
244                     
245                     <para>
246                         
247                         <literal>rk</literal> = <literal>X</literal>のランクの推定値
248                         
249                         を返します.
250                         
251                         すなわち, <literal>rk</literal>は,
252                         
253                         指定した閾値<literal>tol</literal>より大きな
254                         
255                         <literal>R</literal>の対角要素の数となります.
256                         
257                     </para>
258                     
259                 </listitem>
260                 
261             </varlistentry>
262             
263             <varlistentry>
264                 
265                 <term>[Q,R,rk,E] = qr(X) </term>
266                 
267                 <listitem>
268                     
269                     <para>
270                         
271                         <literal>rk</literal> = <literal>X</literal>のランクの推定値
272                         
273                         を返します.
274                         
275                         すなわち,<literal>rk</literal> は
276                         
277                         <literal>tol=R(1,1)*%eps*max(size(R))</literal>より大きな
278                         
279                         <literal>R</literal>の対角要素の数となります.
280                         
281                         <literal>R</literal>の条件数を用いる
282                         
283                         ランク計算型のQR分解については,<literal>rankqr</literal>を
284                         
285                         参照してください.
286                         
287                     </para>
288                     
289                 </listitem>
290                 
291             </varlistentry>
292             
293         </variablelist>
294         
295     </refsection>
296     
297     <refsection>
298         
299         <title>例</title>
300         
301         <programlisting role="example"><![CDATA[ 
302 // QR factorization, generic case
303 // X is tall (full rank)
304 X=rand(5,2);[Q,R]=qr(X); [Q'*X R]
305 //X is fat (full rank)
306 X=rand(2,3);[Q,R]=qr(X); [Q'*X R]
307 //Column 4 of X is a linear combination of columns 1 and 2:
308 X=rand(8,5);X(:,4)=X(:,1)+X(:,2); [Q,R]=qr(X); R, R(:,4)
309 //X has rank 2, rows 3 to $ of R are zero:
310 X=rand(8,2)*rand(2,5);[Q,R]=qr(X); R
311 //Evaluating the rank rk: column pivoting ==> rk first
312 //diagonal entries of R are non zero :
313 A=rand(5,2)*rand(2,5);
314 [Q,R,rk,E] = qr(A,1.d-10);
315 norm(Q'*A-R)
316 svd([A,Q(:,1:rk)])    //span(A) =span(Q(:,1:rk))
317  ]]></programlisting>
318         
319     </refsection>
320     
321     <refsection role="see also">
322         
323         <title>参照</title>
324         
325         <simplelist type="inline">
326             
327             <member>
328                 
329                 <link linkend="rankqr">rankqr</link>
330                 
331             </member>
332             
333             <member>
334                 
335                 <link linkend="rank">rank</link>
336                 
337             </member>
338             
339             <member>
340                 
341                 <link linkend="svd">svd</link>
342                 
343             </member>
344             
345             <member>
346                 
347                 <link linkend="rowcomp">rowcomp</link>
348                 
349             </member>
350             
351             <member>
352                 
353                 <link linkend="colcomp">colcomp</link>
354                 
355             </member>
356             
357         </simplelist>
358         
359     </refsection>
360     
361     <refsection>
362         
363         <title>使用する関数</title>
364         
365         <para>
366             
367             qr 分解はLapack ルーチン DGEQRF, DGEQPF,
368             
369             DORGQR (実数行列)および  ZGEQRF, ZGEQPF, ZORGQR (複素数の場合)
370             
371             に基づいています.
372             
373         </para>
374         
375     </refsection>
376     
377 </refentry>
378