scilab/modules/differential_equations/help/ru_RU/bvode.xml

   1 <?xml version="1.0" encoding="UTF-8"?>
   2 <!--
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   4  * Copyright (C) 2008 - INRIA
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  10  * are also available at
  11  * http://www.cecill.info/licences/Licence_CeCILL_V2-en.txt
  12  *
  13  -->
  14 <refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:id="bvode" xml:lang="ru">
  15     <refnamediv>
  16         <refname>bvode</refname>
  17         <refpurpose>задачи граничных значений для ОДУ с помощью метода коллокации
  18         </refpurpose>
  19     </refnamediv>
  20     <refnamediv xml:id="bvodeS">
  21         <refname>bvodeS</refname>
  22         <refpurpose>упрощённый вызов bvode</refpurpose>
  23     </refnamediv>
  24     <refsynopsisdiv>
  25         <title>Последовательность вызова</title>
  26         <synopsis>
  27             zu=bvode(xpoints,N,m,x_low,x_up,zeta,ipar,ltol,tol,fixpnt,fsub,dfsub,gsub,dgsub,guess)
  28         </synopsis>
  29         <synopsis>
  30             zu=bvodeS(xpoints,m,N,x_low,x_up,fsub,gsub,zeta, &lt;optional_args&gt;)
  31         </synopsis>
  32     </refsynopsisdiv>
  33     <refsection>
  34         <title>Аргументы</title>
  35         <variablelist>
  36             <varlistentry>
  37                 <term>zu</term>
  38                 <listitem>
  39                     <para>
  40                         вектор-столбец длиной <literal>M</literal>. Решение ОДУ, вычисленное по сетке,
  41                         заданной точками. Она содержит <literal>z(u(x))</literal> для каждой искомой точки;
  42                     </para>
  43                 </listitem>
  44             </varlistentry>
  45             <varlistentry>
  46                 <term>xpoints</term>
  47                 <listitem>
  48                     <para>
  49                         массив, который содержит точки для которых нужно найти решение;
  50                     </para>
  51                 </listitem>
  52             </varlistentry>
  53             <varlistentry>
  54                 <term>N</term>
  55                 <listitem>
  56                     <para>
  57                         скаляр с целочисленным значением, количество дифференциальных уравнений
  58                         (<literal>N</literal> &lt;= 20).
  59                     </para>
  60                 </listitem>
  61             </varlistentry>
  62             <varlistentry>
  63                 <term>m</term>
  64                 <listitem>
  65                     <para>
  66                         вектор размером <literal>N</literal> с целочисленными элементами. Это вектор порядка
  67                         каждого из дифференциальных уравнений: <literal>m(i)</literal> указывает порядок
  68                         <literal>i</literal>-го дифференциального уравнения. Далее, <literal>M</literal>
  69                         будет представлять сумму элементов  <literal>m</literal>. <!--Порядок
  70             <literal>i</literal>-го дифференциального уравнения <literal>m(i)</literal> не
  71               может быть больше 4.-->
  72                     </para>
  73                 </listitem>
  74             </varlistentry>
  75             <varlistentry>
  76                 <term>x_low</term>
  77                 <listitem>
  78                     <para>скаляр: левый конец интервала</para>
  79                 </listitem>
  80             </varlistentry>
  81             <varlistentry>
  82                 <term>x_up</term>
  83                 <listitem>
  84                     <para>скаляр: правый конец интервала</para>
  85                 </listitem>
  86             </varlistentry>
  87             <varlistentry>
  88                 <term>zeta</term>
  89                 <listitem>
  90                     <para>
  91                         вектор размером <literal>M</literal>; <literal>zeta(j)</literal> указывает
  92                         <literal>j</literal>-тую точку граничного условия (граничную точку). Необходимо,
  93                         чтобы выполнялось условие
  94                         <literal>x_low&lt;=zeta(j)&lt;=zeta(j+1)&lt;=x_up</literal>.
  95                     </para>
  96                     <para>
  97                         Все точки граничных условий должны составлять сетку точек во всех используемых
  98                         сетках, см. ниже описание <literal>ipar(11)</literal> и  <literal>fixpnt</literal>.
  99                     </para>
 100                 </listitem>
 101             </varlistentry>
 102             <varlistentry>
 103                 <term>ipar</term>
 104                 <listitem>
 105                     <para>массив с 11-ю целочисленными элементами:</para>
 106                     <para>
 107                         <literal>[nonlin, collpnt, subint, ntol, ndimf, ndimi, iprint,
 108                             iread, iguess, rstart,nfxpnt]
 109                         </literal>
 110                     </para>
 111                     <variablelist>
 112                         <varlistentry>
 113                             <term>nonlin: ipar(1)</term>
 114                             <listitem>
 115                                 <para>0, если задача линейна; 1, если задача нелинейна.
 116                                 </para>
 117                             </listitem>
 118                         </varlistentry>
 119                         <varlistentry>
 120                             <term>collpnt: ipar(2)</term>
 121                             <listitem>
 122                                 <para>
 123                                     Задаёт количество рядом расположенных точек на подынтервале, где
 124                                     <literal>max(m(j)) &lt;= collpnt &lt;= 7</literal>.
 125                                 </para>
 126                                 <para>
 127                                     Если <literal>ipar(2)=0</literal>, то <literal>collpnt</literal> установлен равным  <literal>max( max(m(j))+1, 5-max(m(j)) )</literal>.
 128                                 </para>
 129                             </listitem>
 130                         </varlistentry>
 131                         <varlistentry>
 132                             <term>subint: ipar(3)</term>
 133                             <listitem>
 134                                 <para>
 135                                     Задаёт количество подынтервалов в исходной сетке. Если
 136                                     <literal>ipar(3) = 0</literal>, то  <literal>bvode</literal> произвольным образом устанавливается <literal>subint = 5</literal>.
 137                                 </para>
 138                             </listitem>
 139                         </varlistentry>
 140                         <varlistentry>
 141                             <term>ntol: ipar(4)</term>
 142                             <listitem>
 143                                 <para>
 144                                     Задаёт количество решений и допустимые отклонения производных.
 145                                     Требуется, чтобы <literal>0 &lt; ntol &lt;= M</literal>.
 146                                     <literal>ipar(4)</literal> должно быть установлено равным размеру аргумента
 147                                     <literal>tol</literal> или равным <literal>0</literal>. В последнем случае
 148                                     фактическое значение будет автоматически установлено равным
 149                                     <literal>size(tol,'*')</literal>.
 150                                 </para>
 151                             </listitem>
 152                         </varlistentry>
 153                         <varlistentry>
 154                             <term>ndimf: ipar(5)</term>
 155                             <listitem>
 156                                 <para>
 157                                     Задаёт размер <literal>fspace</literal> (вещественнозначный рабочий массив). Его значение
 158                                     указывает ограничение по максимальному количеству подынтервалов
 159                                     <literal>nmax</literal>.
 160                                 </para>
 161                                 <para>
 162                                     Значение <literal>ipar(5)</literal> должно соответствовать ограничению
 163                                     <literal>ipar(5)&gt;=nmax*nsizef</literal>, где
 164                                     <literal>nsizef=4 + 3*M + (5+collpnt*N)*(collpnt*N+M) + (2*M-
 165                                         nrec)*2*M
 166                                     </literal>
 167                                     (<literal>nrec</literal> - количество граничных условий
 168                                     с правой стороны).
 169                                 </para>
 170                             </listitem>
 171                         </varlistentry>
 172                         <varlistentry>
 173                             <term>ndimi: ipar(6)</term>
 174                             <listitem>
 175                                 <para>
 176                                     Задаёт размер <literal>ispace</literal> (целочисленный рабочий массив).
 177                                     Это значение указывает ограничение по <literal>nmax</literal>, максимальному количеству подынтервалов.
 178                                 </para>
 179                                 <para>
 180                                     Значение <literal>ipar(6)</literal> должно соответствовать ограничению
 181                                     <literal>ipar(6)&gt;=nmax*nsizei</literal>, где <literal>nsizei= 3 + collpnt*N + M</literal>.
 182                                 </para>
 183                             </listitem>
 184                         </varlistentry>
 185                         <varlistentry>
 186                             <term>iprint: ipar(7)</term>
 187                             <listitem>
 188                                 <para>контроль на выходе, может принимать следующие значения:</para>
 189                                 <variablelist>
 190                                     <varlistentry>
 191                                         <term>-1</term>
 192                                         <listitem>
 193                                             <para>для полной диагностической распечатки</para>
 194                                         </listitem>
 195                                     </varlistentry>
 196                                     <varlistentry>
 197                                         <term>0</term>
 198                                         <listitem>
 199                                             <para>для избранной распечатки</para>
 200                                         </listitem>
 201                                     </varlistentry>
 202                                     <varlistentry>
 203                                         <term>1</term>
 204                                         <listitem>
 205                                             <para>для отказа от распечатки</para>
 206                                         </listitem>
 207                                     </varlistentry>
 208                                 </variablelist>
 209                             </listitem>
 210                         </varlistentry>
 211                         <varlistentry>
 212                             <term>iread: ipar(8)</term>
 213                             <listitem>
 214                                 <variablelist>
 215                                     <varlistentry>
 216                                         <term>= 0</term>
 217                                         <listitem>
 218                                             <para>
 219                                                 заставляет <literal>bvode</literal> генерировать равномерную исходную
 220                                                 сетку.
 221                                             </para>
 222                                         </listitem>
 223                                     </varlistentry>
 224                                     <varlistentry>
 225                                         <term>= xx</term>
 226                                         <listitem>
 227                                             <para>
 228                                                 другие значения в Scilab'е пока не реализованы
 229                                             </para>
 230                                         </listitem>
 231                                     </varlistentry>
 232                                     <varlistentry>
 233                                         <term>= 1</term>
 234                                         <listitem>
 235                                             <para>
 236                                                 если исходная сетка указана пользователем, то она будет определена в
 237                                                 <literal>fspace</literal> следующим образом: сетка будет занимать
 238                                                 <literal>fspace(1), ..., fspace(n+1)</literal>. Пользователю нужно предоставить
 239                                                 только внутренние точки сетки <literal>fspace(j) = x(j), j = 2, ..., n</literal>.
 240                                             </para>
 241                                         </listitem>
 242                                     </varlistentry>
 243                                     <varlistentry>
 244                                         <term>= 2 </term>
 245                                         <listitem>
 246                                             <para>
 247                                                 если исходная сетка представлена пользователем как в случае
 248                                                 <literal>ipar(8)=1</literal>, и, к тому же, никакого выбора адаптивной сетки не
 249                                                 делается.
 250                                             </para>
 251                                         </listitem>
 252                                     </varlistentry>
 253                                 </variablelist>
 254                             </listitem>
 255                         </varlistentry>
 256                         <varlistentry>
 257                             <term>iguess: ipar(9)</term>
 258                             <listitem>
 259                                 <variablelist>
 260                                     <varlistentry>
 261                                         <term>= 0</term>
 262                                         <listitem>
 263                                             <para>
 264                                                 если никакого первоначального предположения для решения не
 265                                                 предоставлено.
 266                                             </para>
 267                                         </listitem>
 268                                     </varlistentry>
 269                                     <varlistentry>
 270                                         <term>= 1</term>
 271                                         <listitem>
 272                                             <para>
 273                                                 если первоначальное предположение предоставлено пользователем с
 274                                                 помощью аргумента <literal>guess</literal>.
 275                                             </para>
 276                                         </listitem>
 277                                     </varlistentry>
 278                                     <varlistentry>
 279                                         <term>= 2</term>
 280                                         <listitem>
 281                                             <para>
 282                                                 если исходная сетка и коэффициенты приближённого решения предоставлены пользователем в <literal>fspace</literal>  (прежняя и новая сетки одни и те же).
 283                                             </para>
 284                                         </listitem>
 285                                     </varlistentry>
 286                                     <varlistentry>
 287                                         <term>= 3</term>
 288                                         <listitem>
 289                                             <para>
 290                                                 если прежняя сетка и коэффициенты приближённого решения предоставлены
 291                                                 пользователем в <literal>fspace</literal>, а новая сетка взята в два раза реже, то есть каждая вторая точка из прежней сетки.
 292                                             </para>
 293                                         </listitem>
 294                                     </varlistentry>
 295                                     <varlistentry>
 296                                         <term>= 4</term>
 297                                         <listitem>
 298                                             <para>
 299                                                 если в дополнение к прежней исходной сетке и коэффициента приближённого
 300                                                 решения также предоставлена новая сетка в <literal>fspace</literal>
 301                                                 (см. описание выходных данных для более подробной информации по iguess = 2,
 302                                                 3 и 4).
 303                                             </para>
 304                                         </listitem>
 305                                     </varlistentry>
 306                                 </variablelist>
 307                             </listitem>
 308                         </varlistentry>
 309                         <varlistentry>
 310                             <term>ireg: ipar(10)</term>
 311                             <listitem>
 312                                 <variablelist>
 313                                     <varlistentry>
 314                                         <term>= 0</term>
 315                                         <listitem>
 316                                             <para>если задача является регулярной</para>
 317                                         </listitem>
 318                                     </varlistentry>
 319                                     <varlistentry>
 320                                         <term>= 1</term>
 321                                         <listitem>
 322                                             <para>
 323                                                 если первый коэффициент релаксации равен <literal>ireg</literal>, и
 324                                                 нелинейная итерация не зависит от прошлой сходимости (использовать только
 325                                                 для сверхчувствительных нелинейных задач)
 326                                                 only).
 327                                             </para>
 328                                         </listitem>
 329                                     </varlistentry>
 330                                     <varlistentry>
 331                                         <term>= 2</term>
 332                                         <listitem>
 333                                             <para>
 334                                                 если вы хотите возврата немедленно при (а) двух, следующих друг за другом,
 335                                                 несходимостях, либо (б) после получения ошибочной оценки в первый раз.
 336                                             </para>
 337                                         </listitem>
 338                                     </varlistentry>
 339                                 </variablelist>
 340                             </listitem>
 341                         </varlistentry>
 342                         <varlistentry>
 343                             <term>nfxpnt: ipar(11)</term>
 344                             <listitem>
 345                                 <para>
 346                                     указывает количество фиксированных точек в сетке, отличных от
 347                                     <literal>x_low</literal> и <literal>x_up</literal> (размерность <literal>fixpnt</literal>).
 348                                     <literal>ipar(11)</literal> должна быть установлена равной размерности аргумента
 349                                     <literal>fixpnt</literal> или равной <literal>0</literal>. В последнем случае фактическое значение будет автоматически установлено равным
 350                                     <literal>size(fixpnt,'*')</literal>.
 351                                 </para>
 352                             </listitem>
 353                         </varlistentry>
 354                     </variablelist>
 355                 </listitem>
 356             </varlistentry>
 357             <varlistentry>
 358                 <term>ltol</term>
 359                 <listitem>
 360                     <para>
 361                         массив размерности <literal>ntol=ipar(4)</literal>.
 362                         <literal>ltol(j) = l</literal> определяет, что <literal>j</literal>-тый допуск в массиве
 363                         <literal> tol</literal> управляет ошибкой в <literal>l</literal>-том элементе
 364                         <inlinemediaobject>
 365                             <imageobject>
 366                                 <imagedata>
 367                                     <mml:math>
 368                                         <mml:semantics>
 369                                             <mml:mi>z</mml:mi>
 370                                             <mml:mrow>
 371                                                 <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 372                                             </mml:mrow>
 373                                         </mml:semantics>
 374                                     </mml:math>
 375                                 </imagedata>
 376                             </imageobject>
 377                         </inlinemediaobject>
 378                         .
 379                         Также требуется, чтобы:
 380                     </para>
 381                     <para>
 382                         <literal>1 &lt;= ltol(1) &lt; ltol(2) &lt; ... &lt; ltol(ntol)
 383                             &lt;= M
 384                         </literal>
 385                     </para>
 386                 </listitem>
 387             </varlistentry>
 388             <varlistentry>
 389                 <term>tol</term>
 390                 <listitem>
 391                     <para>
 392                         массив размерности <literal>ntol=ipar(4)</literal>.
 393                     </para>
 394                     <para>
 395                         <literal>tol(j)</literal> допуск ошибки в
 396                         <literal>ltol(j)</literal>-том элементе
 397                         <inlinemediaobject>
 398                             <imageobject>
 399                                 <imagedata>
 400                                     <mml:math>
 401                                         <mml:semantics>
 402                                             <mml:mi>z</mml:mi>
 403                                             <mml:mrow>
 404                                                 <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 405                                             </mml:mrow>
 406                                         </mml:semantics>
 407                                     </mml:math>
 408                                 </imagedata>
 409                             </imageobject>
 410                         </inlinemediaobject>
 411                         . Таким образом код пытается удовлетворить
 412                         <inlinemediaobject>
 413                             <imageobject>
 414                                 <imagedata>
 415                                     <mml:math>
 416                                         <mml:semantics>
 417                                             <mml:mrow>
 418                                                 <mml:mrow>
 419                                                     <mml:mfenced mml:close="∣" mml:open="∣">
 420                                                         <mml:msub>
 421                                                             <mml:mrow>
 422                                                                 <mml:mo mml:stretchy="false">(</mml:mo>
 423                                                                 <mml:mrow>
 424                                                                     <mml:mi>z</mml:mi>
 425                                                                     <mml:mrow>
 426                                                                         <mml:mrow>
 427                                                                             <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 428                                                                         </mml:mrow>
 429                                                                         <mml:mo mml:stretchy="false">−</mml:mo><mml:mi>z</mml:mi>
 430                                                                     </mml:mrow>
 431                                                                     <mml:mrow>
 432                                                                         <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 433                                                                     </mml:mrow>
 434                                                                 </mml:mrow>
 435                                                                 <mml:mo mml:stretchy="false">)</mml:mo>
 436                                                             </mml:mrow>
 437                                                             <mml:mrow>
 438                                                                 <mml:mi mml:fontstyle="italic">ltol</mml:mi>
 439                                                                 <mml:mrow>
 440                                                                     <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 441                                                                 </mml:mrow>
 442                                                             </mml:mrow>
 443                                                         </mml:msub>
 444                                                     </mml:mfenced>
 445                                                     <mml:mo mml:stretchy="false">≤</mml:mo><mml:mi mml:fontstyle="italic">tol</mml:mi>
 446                                                 </mml:mrow>
 447                                                 <mml:mrow>
 448                                                     <mml:mrow>
 449                                                         <mml:mrow>
 450                                                             <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 451                                                         </mml:mrow>
 452                                                         <mml:mo mml:stretchy="false">⋅</mml:mo>
 453                                                         <mml:mfenced mml:close="∣" mml:open="∣">
 454                                                             <mml:msub>
 455                                                                 <mml:mrow>
 456                                                                     <mml:mo mml:stretchy="false">(</mml:mo>
 457                                                                     <mml:mrow>
 458                                                                         <mml:mi>z</mml:mi>
 459                                                                         <mml:mrow>
 460                                                                             <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 461                                                                         </mml:mrow>
 462                                                                     </mml:mrow>
 463                                                                     <mml:mo mml:stretchy="false">)</mml:mo>
 464                                                                 </mml:mrow>
 465                                                                 <mml:mrow>
 466                                                                     <mml:mi mml:fontstyle="italic">ltol</mml:mi>
 467                                                                     <mml:mrow>
 468                                                                         <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 469                                                                     </mml:mrow>
 470                                                                 </mml:mrow>
 471                                                             </mml:msub>
 472                                                         </mml:mfenced>
 473                                                     </mml:mrow>
 474                                                     <mml:mo mml:stretchy="false">+</mml:mo><mml:mi mml:fontstyle="italic">tol</mml:mi>
 475                                                 </mml:mrow>
 476                                                 <mml:mrow>
 477                                                     <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 478                                                 </mml:mrow>
 479                                                 <mml:mi>,</mml:mi><mml:mi mml:fontstyle="normal">for</mml:mi>
 480                                                 <mml:mrow>
 481                                                     <mml:mi>j</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>1</mml:mn>
 482                                                 </mml:mrow>
 483                                                 <mml:mi mml:fontstyle="normal">:</mml:mi><mml:mi mml:fontstyle="normal">ntol</mml:mi>
 484                                             </mml:mrow>
 485                                             <mml:annotation mml:encoding="StarMath 5.0">abs(z(v)-z(u))_{ltol(j)}
 486                                                 &lt;= tol(j) cdot abs(z(u))_{ltol(j)} + tol(j), for
 487                                                 j=1:ntol
 488                                             </mml:annotation>
 489                                         </mml:semantics>
 490                                     </mml:math>
 491                                 </imagedata>
 492                             </imageobject>
 493                         </inlinemediaobject>
 494                         в каждом подынтервале, где
 495                     </para>
 496                     <para>
 497                         <inlinemediaobject>
 498                             <imageobject>
 499                                 <imagedata>
 500                                     <mml:math>
 501                                         <mml:semantics>
 502                                             <mml:mi>v</mml:mi>
 503                                         </mml:semantics>
 504                                     </mml:math>
 505                                 </imagedata>
 506                             </imageobject>
 507                         </inlinemediaobject>
 508                         - вектор приближённого решения, а
 509                         <inlinemediaobject>
 510                             <imageobject>
 511                                 <imagedata>
 512                                     <mml:math>
 513                                         <mml:semantics>
 514                                             <mml:mi>u</mml:mi>
 515                                         </mml:semantics>
 516                                     </mml:math>
 517                                 </imagedata>
 518                             </imageobject>
 519                         </inlinemediaobject>
 520                         - точное решение (неизвестное).
 521                     </para>
 522                 </listitem>
 523             </varlistentry>
 524             <varlistentry>
 525                 <term>fixpnt</term>
 526                 <listitem>
 527                     <para>
 528                         массив размером <literal>nfxpnt=ipar(11)</literal>. Он содержит точки отличные от
 529                         <literal>x_low</literal> и <literal>x_up</literal>, которые нужно включить во все
 530                         сетки. Код требует, чтобы все точки дополнительных условий, отличные от
 531                         <literal>x_low</literal> и <literal>x_up</literal> (см. описание <literal>zeta</literal> )
 532                         были включены в качестве фиксированных точек в <literal>fixpnt</literal>.
 533                     </para>
 534                 </listitem>
 535             </varlistentry>
 536             <varlistentry>
 537                 <term>fsub</term>
 538                 <listitem>
 539                     <para>
 540                         <link linkend="external">Внешняя функция</link>, используемая для вычисления вектор-столбца <literal>f=</literal>
 541                         <inlinemediaobject>
 542                             <imageobject>
 543                                 <imagedata>
 544                                     <mml:math>
 545                                         <mml:semantics>
 546                                             <mml:mrow>
 547                                                 <mml:mo mml:stretchy="false">[</mml:mo>
 548                                                 <mml:mrow>
 549                                                     <mml:msub>
 550                                                         <mml:mi>f</mml:mi><mml:mn>1</mml:mn>
 551                                                     </mml:msub>
 552                                                     <mml:mrow>
 553                                                         <mml:mo mml:stretchy="false">(</mml:mo>
 554                                                         <mml:mrow>
 555                                                             <mml:mi>x</mml:mi><mml:mi>,</mml:mi><mml:mi>z</mml:mi>
 556                                                             <mml:mrow>
 557                                                                 <mml:mo mml:stretchy="false">(</mml:mo>
 558                                                                 <mml:mrow>
 559                                                                     <mml:mi>u</mml:mi>
 560                                                                     <mml:mrow>
 561                                                                         <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 562                                                                     </mml:mrow>
 563                                                                 </mml:mrow>
 564                                                                 <mml:mo mml:stretchy="false">)</mml:mo>
 565                                                             </mml:mrow>
 566                                                         </mml:mrow>
 567                                                         <mml:mo mml:stretchy="false">)</mml:mo>
 568                                                     </mml:mrow>
 569                                                     <mml:mi>;</mml:mi>
 570                                                     <mml:msub>
 571                                                         <mml:mi>f</mml:mi><mml:mn>2</mml:mn>
 572                                                     </mml:msub>
 573                                                     <mml:mrow>
 574                                                         <mml:mo mml:stretchy="false">(</mml:mo>
 575                                                         <mml:mrow>
 576                                                             <mml:mi>x</mml:mi><mml:mi>,</mml:mi><mml:mi>z</mml:mi>
 577                                                             <mml:mrow>
 578                                                                 <mml:mo mml:stretchy="false">(</mml:mo>
 579                                                                 <mml:mrow>
 580                                                                     <mml:mi>u</mml:mi>
 581                                                                     <mml:mrow>
 582                                                                         <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 583                                                                     </mml:mrow>
 584                                                                 </mml:mrow>
 585                                                                 <mml:mo mml:stretchy="false">)</mml:mo>
 586                                                             </mml:mrow>
 587                                                         </mml:mrow>
 588                                                         <mml:mo mml:stretchy="false">)</mml:mo>
 589                                                     </mml:mrow>
 590                                                     <mml:mi>,</mml:mi><mml:mo mml:stretchy="false">⋯</mml:mo><mml:mi>,</mml:mi>
 591                                                     <mml:msub>
 592                                                         <mml:mi>f</mml:mi><mml:mi>N</mml:mi>
 593                                                     </mml:msub>
 594                                                     <mml:mrow>
 595                                                         <mml:mo mml:stretchy="false">(</mml:mo>
 596                                                         <mml:mrow>
 597                                                             <mml:mi>x</mml:mi><mml:mi>,</mml:mi><mml:mi>z</mml:mi>
 598                                                             <mml:mrow>
 599                                                                 <mml:mo mml:stretchy="false">(</mml:mo>
 600                                                                 <mml:mrow>
 601                                                                     <mml:mi>u</mml:mi>
 602                                                                     <mml:mrow>
 603                                                                         <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 604                                                                     </mml:mrow>
 605                                                                 </mml:mrow>
 606                                                                 <mml:mo mml:stretchy="false">)</mml:mo>
 607                                                             </mml:mrow>
 608                                                         </mml:mrow>
 609                                                         <mml:mo mml:stretchy="false">)</mml:mo>
 610                                                     </mml:mrow>
 611                                                 </mml:mrow>
 612                                                 <mml:mo mml:stretchy="false">]</mml:mo>
 613                                             </mml:mrow>
 614                                             <mml:annotation mml:encoding="StarMath 5.0">[f_1(x,z(u(x)));
 615                                                 f_2(x,z(u(x))),dotsaxis,
 616                                                 f_N(x,z(u(x)))]
 617                                             </mml:annotation>
 618                                         </mml:semantics>
 619                                     </mml:math>
 620                                 </imagedata>
 621                             </imageobject>
 622                         </inlinemediaobject>
 623                         ,
 624                         для всех  <literal>x</literal> таких, что <literal>x_low</literal> &lt;= <literal>x</literal> &lt;= <literal>x_up</literal>
 625                         и для любых <literal>z=z(u(x))</literal> (см. описание ниже).
 626                     </para>
 627                     <para>Внешняя функция должна иметь заголовки:</para>
 628                     <itemizedlist>
 629                         <listitem>
 630                             <para>В Fortran последовательность вызова должна быть:</para>
 631                             <programlisting role="no-scilab-exec"><![CDATA[
 632 subroutine fsub(x,zu,f)
 633 double precision zu(*), f(*),x
 634  ]]></programlisting>
 635                         </listitem>
 636                         <listitem>
 637                             <para>В C прототип функции должен быть:</para>
 638                             <programlisting role="no-scilab-exec"><![CDATA[
 639 void fsub(double *x, double *zu, double *f)
 640  ]]></programlisting>
 641                         </listitem>
 642                         <listitem>
 643                             <para>А в Scilab'е:</para>
 644                             <programlisting role="no-scilab-exec"><![CDATA[
 645 function f=fsub(x,zu,parameters)
 646  ]]></programlisting>
 647                         </listitem>
 648                     </itemizedlist>
 649                 </listitem>
 650             </varlistentry>
 651             <varlistentry>
 652                 <term>dfsub</term>
 653                 <listitem>
 654                     <para>
 655                         <link linkend="external">Внешняя функция</link>, используемая для вычисления
 656                         якобиана от <literal>f(x,z(u))</literal> в точке <literal>x</literal>, где
 657                         <literal>z(u(x))</literal> определена как для <literal>fsub</literal>, и массив
 658                         <literal>df</literal> размером <literal>N</literal> на <literal>M</literal> должен быть заполнен частными производными от <literal>f</literal>:
 659                     </para>
 660                     <informalequation>
 661                         <mediaobject>
 662                             <imageobject>
 663                                 <imagedata align="center">
 664                                     <mml:math scilab:localized="true">
 665                                         <mml:semantics>
 666                                             <mml:mrow>
 667                                                 <mml:mi mml:fontstyle="italic">df</mml:mi>
 668                                                 <mml:mrow>
 669                                                     <mml:mrow>
 670                                                         <mml:mo mml:stretchy="false">(</mml:mo>
 671                                                         <mml:mrow>
 672                                                             <mml:mi>i</mml:mi>
 673                                                             <mml:mi>,</mml:mi>
 674                                                             <mml:mi>j</mml:mi>
 675                                                         </mml:mrow>
 676                                                         <mml:mo mml:stretchy="false">)</mml:mo>
 677                                                     </mml:mrow>
 678                                                     <mml:mo mml:stretchy="false">=</mml:mo>
 679                                                     <mml:mfrac>
 680                                                         <mml:mrow>
 681                                                             <mml:mo mml:stretchy="false">∂</mml:mo>
 682                                                             <mml:msub>
 683                                                                 <mml:mi>f</mml:mi>
 684                                                                 <mml:mi>i</mml:mi>
 685                                                             </mml:msub>
 686                                                         </mml:mrow>
 687                                                         <mml:mrow>
 688                                                             <mml:mo mml:stretchy="false">∂</mml:mo>
 689                                                             <mml:msub>
 690                                                                 <mml:mi>z</mml:mi>
 691                                                                 <mml:mi>j</mml:mi>
 692                                                             </mml:msub>
 693                                                         </mml:mrow>
 694                                                     </mml:mfrac>
 695                                                 </mml:mrow>
 696                                                 <mml:mtext> для
 697                                                 </mml:mtext>
 698                                                 <mml:mrow>
 699                                                     <mml:mo mml:stretchy="true">{</mml:mo>
 700                                                     <mml:mtable>
 701                                                         <mml:mtr>
 702                                                             <mml:mrow>
 703                                                                 <mml:mrow>
 704                                                                     <mml:mi>i</mml:mi>
 705                                                                     <mml:mo mml:stretchy="false">=</mml:mo>
 706                                                                     <mml:mn>1</mml:mn>
 707                                                                 </mml:mrow>
 708                                                                 <mml:mi mml:fontstyle="normal">:</mml:mi>
 709                                                                 <mml:mi mml:fontstyle="italic">N</mml:mi>
 710                                                             </mml:mrow>
 711                                                         </mml:mtr>
 712                                                         <mml:mtr>
 713                                                             <mml:mrow>
 714                                                                 <mml:mrow>
 715                                                                     <mml:mi>j</mml:mi>
 716                                                                     <mml:mo mml:stretchy="false">=</mml:mo>
 717                                                                     <mml:mn>1</mml:mn>
 718                                                                 </mml:mrow>
 719                                                                 <mml:mi mml:fontstyle="normal">:</mml:mi>
 720                                                                 <mml:mi mml:fontstyle="italic">M</mml:mi>
 721                                                             </mml:mrow>
 722                                                         </mml:mtr>
 723                                                     </mml:mtable>
 724                                                 </mml:mrow>
 725                                             </mml:mrow>
 726                                             <mml:annotation mml:encoding="StarMath 5.0">df(i,j)= {partial{f_i}} over
 727                                                 {partial{z_j}} ~ for ~ left lbrace binom{i=1:N}{j=1:M} right
 728                                                 none
 729                                             </mml:annotation>
 730                                         </mml:semantics>
 731                                     </mml:math>
 732                                 </imagedata>
 733                             </imageobject>
 734                         </mediaobject>
 735                     </informalequation>
 736                     <para>Внешняя функция должна иметь заголовки:</para>
 737                     <itemizedlist>
 738                         <listitem>
 739                             <para>В Fortran вызывающая последовательность должна быть:</para>
 740                             <programlisting role="no-scilab-exec"><![CDATA[
 741 subroutine dfsub(x,zu,df)
 742 double precision zu(*), df(*),x
 743  ]]></programlisting>
 744                         </listitem>
 745                         <listitem>
 746                             <para>В C прототип функции должен быть:</para>
 747                             <programlisting role="no-scilab-exec"><![CDATA[
 748 void dfsub(double *x, double *zu, double *df)
 749  ]]></programlisting>
 750                         </listitem>
 751                         <listitem>
 752                             <para>И в Scilab'е:</para>
 753                             <programlisting role="no-scilab-exec"><![CDATA[
 754 function df=dfsub(x,zu,parameters)
 755  ]]></programlisting>
 756                         </listitem>
 757                     </itemizedlist>
 758                 </listitem>
 759             </varlistentry>
 760             <varlistentry>
 761                 <term>gsub</term>
 762                 <listitem>
 763                     <para>
 764                         <link linkend="external">Внешняя функция</link>, используемая для вычисления
 765                         <inlinemediaobject>
 766                             <imageobject>
 767                                 <imagedata>
 768                                     <mml:math>
 769                                         <mml:semantics>
 770                                             <mml:mrow>
 771                                                 <mml:msub>
 772                                                     <mml:mi>g</mml:mi><mml:mi>i</mml:mi>
 773                                                 </mml:msub>
 774                                                 <mml:mrow>
 775                                                     <mml:mo mml:stretchy="false">(</mml:mo>
 776                                                     <mml:mrow>
 777                                                         <mml:msub>
 778                                                             <mml:mo mml:stretchy="false">ζ</mml:mo><mml:mi>i</mml:mi>
 779                                                         </mml:msub>
 780                                                         <mml:mi>,</mml:mi><mml:mi>z</mml:mi>
 781                                                         <mml:mrow>
 782                                                             <mml:mo mml:stretchy="false">(</mml:mo>
 783                                                             <mml:mrow>
 784                                                                 <mml:mi>u</mml:mi>
 785                                                                 <mml:mrow>
 786                                                                     <mml:mo mml:stretchy="false">(</mml:mo>
 787                                                                     <mml:msub>
 788                                                                         <mml:mo mml:stretchy="false">ζ</mml:mo><mml:mi>i</mml:mi>
 789                                                                     </mml:msub>
 790                                                                     <mml:mo mml:stretchy="false">)</mml:mo>
 791                                                                 </mml:mrow>
 792                                                             </mml:mrow>
 793                                                             <mml:mo mml:stretchy="false">)</mml:mo>
 794                                                         </mml:mrow>
 795                                                     </mml:mrow>
 796                                                     <mml:mo mml:stretchy="false">)</mml:mo>
 797                                                 </mml:mrow>
 798                                             </mml:mrow>
 799                                             <mml:annotation mml:encoding="StarMath 5.0">g_i(%zeta_i,z(u(%zeta_i)))</mml:annotation>
 800                                         </mml:semantics>
 801                                     </mml:math>
 802                                 </imagedata>
 803                             </imageobject>
 804                         </inlinemediaobject>
 805                         задавая z=
 806                         <inlinemediaobject>
 807                             <imageobject>
 808                                 <imagedata>
 809                                     <mml:math>
 810                                         <mml:semantics>
 811                                             <mml:mrow>
 812                                                 <mml:mi>z</mml:mi>
 813                                                 <mml:mrow>
 814                                                     <mml:mo mml:stretchy="false">(</mml:mo>
 815                                                     <mml:mrow>
 816                                                         <mml:mi>u</mml:mi>
 817                                                         <mml:mrow>
 818                                                             <mml:mo mml:stretchy="false">(</mml:mo>
 819                                                             <mml:msub>
 820                                                                 <mml:mo mml:stretchy="false">ζ</mml:mo><mml:mi>i</mml:mi>
 821                                                             </mml:msub>
 822                                                             <mml:mo mml:stretchy="false">)</mml:mo>
 823                                                         </mml:mrow>
 824                                                     </mml:mrow>
 825                                                     <mml:mo mml:stretchy="false">)</mml:mo>
 826                                                 </mml:mrow>
 827                                             </mml:mrow>
 828                                             <mml:annotation mml:encoding="StarMath 5.0">z(u(%zeta_i))</mml:annotation>
 829                                         </mml:semantics>
 830                                     </mml:math>
 831                                 </imagedata>
 832                             </imageobject>
 833                         </inlinemediaobject>
 834                         <literal>z = zeta(i)</literal> для
 835                         <literal>1&lt;=i&lt;=M.</literal>
 836                     </para>
 837                     <para>Внешняя функция должна иметь заголовки:</para>
 838                     <itemizedlist>
 839                         <listitem>
 840                             <para>В Fortran вызывающая последовательность должна быть:</para>
 841                             <programlisting role="no-scilab-exec"><![CDATA[
 842 subroutine gsub(i,zu,g)
 843 double precision zu(*), g(*)
 844 integer i
 845  ]]></programlisting>
 846                         </listitem>
 847                         <listitem>
 848                             <para>В C прототип функции должен быть:</para>
 849                             <programlisting role="no-scilab-exec"><![CDATA[
 850 void gsub(int *i, double *zu, double *g)
 851  ]]></programlisting>
 852                         </listitem>
 853                         <listitem>
 854                             <para>А в Scilab'е:</para>
 855                             <programlisting role="no-scilab-exec"><![CDATA[
 856 function g=gsub(i,zu,parameters)
 857  ]]></programlisting>
 858                             <para>
 859                                 Заметьте, что в отличие от <literal>f</literal> в <literal>fsub</literal>, здесь только одно значение за вызов возвращается в <literal>g</literal>.
 860                             </para>
 861                         </listitem>
 862                     </itemizedlist>
 863                 </listitem>
 864             </varlistentry>
 865             <varlistentry>
 866                 <term>dgsub</term>
 867                 <listitem>
 868                     <para>
 869                         <link linkend="external">Внешняя функция</link>, используемая для вычисления
 870                         <literal>i</literal>-той строки якобиана от <literal>g(x,u(x))</literal>, где
 871                         <literal>z(u)</literal> такая, как для  <literal>fsub</literal>, <literal>i</literal> как для <literal>gsub</literal> а <literal>M</literal>-вектор <literal>dg</literal> должен быть заполнен частными производными от <literal>g</literal>, то есть, для отдельного вызова вычисляется
 872                     </para>
 873                     <informalequation>
 874                         <mediaobject>
 875                             <imageobject>
 876                                 <imagedata align="center">
 877                                     <mml:math scilab:localized="true">
 878                                         <mml:semantics>
 879                                             <mml:mrow>
 880                                                 <mml:mi mml:fontstyle="italic">dg</mml:mi>
 881                                                 <mml:mrow>
 882                                                     <mml:mrow>
 883                                                         <mml:mo mml:stretchy="false">(</mml:mo>
 884                                                         <mml:mrow>
 885                                                             <mml:mi>i</mml:mi>
 886                                                             <mml:mi>,</mml:mi>
 887                                                             <mml:mi>j</mml:mi>
 888                                                         </mml:mrow>
 889                                                         <mml:mo mml:stretchy="false">)</mml:mo>
 890                                                     </mml:mrow>
 891                                                     <mml:mo mml:stretchy="false">=</mml:mo>
 892                                                     <mml:mfrac>
 893                                                         <mml:mrow>
 894                                                             <mml:mo mml:stretchy="false">∂</mml:mo>
 895                                                             <mml:msub>
 896                                                                 <mml:mi>g</mml:mi>
 897                                                                 <mml:mi>i</mml:mi>
 898                                                             </mml:msub>
 899                                                         </mml:mrow>
 900                                                         <mml:mrow>
 901                                                             <mml:mo mml:stretchy="false">∂</mml:mo>
 902                                                             <mml:msub>
 903                                                                 <mml:mi>z</mml:mi>
 904                                                                 <mml:mi>j</mml:mi>
 905                                                             </mml:msub>
 906                                                         </mml:mrow>
 907                                                     </mml:mfrac>
 908                                                 </mml:mrow>
 909                                                 <mml:mtext> для
 910                                                 </mml:mtext>
 911                                                 <mml:mrow>
 912                                                     <mml:mo mml:stretchy="true">{</mml:mo>
 913                                                     <mml:mtable>
 914                                                         <mml:mtr>
 915                                                             <mml:mrow>
 916                                                                 <mml:mrow>
 917                                                                     <mml:mi>i</mml:mi>
 918                                                                     <mml:mo mml:stretchy="false">=</mml:mo>
 919                                                                     <mml:mn>1</mml:mn>
 920                                                                 </mml:mrow>
 921                                                                 <mml:mi mml:fontstyle="normal">:</mml:mi>
 922                                                                 <mml:mi mml:fontstyle="italic">M</mml:mi>
 923                                                             </mml:mrow>
 924                                                         </mml:mtr>
 925                                                         <mml:mtr>
 926                                                             <mml:mrow>
 927                                                                 <mml:mrow>
 928                                                                     <mml:mi>j</mml:mi>
 929                                                                     <mml:mo mml:stretchy="false">=</mml:mo>
 930                                                                     <mml:mn>1</mml:mn>
 931                                                                 </mml:mrow>
 932                                                                 <mml:mi mml:fontstyle="normal">:</mml:mi>
 933                                                                 <mml:mi mml:fontstyle="italic">M</mml:mi>
 934                                                             </mml:mrow>
 935                                                         </mml:mtr>
 936                                                     </mml:mtable>
 937                                                 </mml:mrow>
 938                                             </mml:mrow>
 939                                             <mml:annotation mml:encoding="StarMath 5.0">dg(i,j)= {partial{g_i}} over
 940                                                 {partial{z_j}} ~ for ~ left lbrace binom{i=1:M}{j=1:M} right
 941                                                 none
 942                                             </mml:annotation>
 943                                         </mml:semantics>
 944                                     </mml:math>
 945                                 </imagedata>
 946                             </imageobject>
 947                         </mediaobject>
 948                     </informalequation>
 949                     <para>Внешняя функция должна иметь заголовки:</para>
 950                     <itemizedlist>
 951                         <listitem>
 952                             <para>В Fortran вызывающая последовательность должна быть:</para>
 953                             <programlisting role="no-scilab-exec"><![CDATA[
 954 subroutine dgsub(i,zu,dg)
 955 double precision zu(*), dg(*)
 956  ]]></programlisting>
 957                         </listitem>
 958                         <listitem>
 959                             <para>В C прототип функции должен быть:</para>
 960                             <programlisting role="no-scilab-exec"><![CDATA[
 961 void dgsub(int *i, double *zu, double *dg)
 962  ]]></programlisting>
 963                         </listitem>
 964                         <listitem>
 965                             <para>А в Scilab'е: </para>
 966                             <programlisting role="no-scilab-exec"><![CDATA[
 967 function dg=dgsub(i,zu,parameters)
 968  ]]></programlisting>
 969                         </listitem>
 970                     </itemizedlist>
 971                 </listitem>
 972             </varlistentry>
 973             <varlistentry>
 974                 <term>guess</term>
 975                 <listitem>
 976                     <para>
 977                         <link linkend="external">Внешняя функция</link>, используемая для вычисления
 978                         исходной аппроксимации для <literal>z(u(x))</literal> и
 979                         <literal>dmval(u(x))</literal>, вектора <literal>mj</literal>-тых производных от
 980                         <literal>u(x)</literal>. Заметьте, что эта процедура используется только если
 981                         <literal>ipar(9) = 1</literal>, и тогда все <literal>M</literal>
 982                         элементы <literal>zu</literal> и <literal>N</literal> элементы <literal>dmval</literal> должны быть вычислены для любого <literal>x</literal> такого, что <literal>x_low</literal> &lt;= <literal>x</literal> &lt;= <literal>x_up</literal>.
 983                     </para>
 984                     <para>Внешняя функция должна иметь заголовки:</para>
 985                     <itemizedlist>
 986                         <listitem>
 987                             <para>В Fortran вызывающая последовательность должна быть:</para>
 988                             <programlisting role="no-scilab-exec"><![CDATA[
 989 subroutine guess(x,zu,dmval)
 990 double precision x,z(*), dmval(*)
 991  ]]></programlisting>
 992                         </listitem>
 993                         <listitem>
 994                             <para>В C прототип функции должне быть:</para>
 995                             <programlisting role="no-scilab-exec"><![CDATA[
 996 void fsub(double *x, double *zu, double *dmval)
 997  ]]></programlisting>
 998                         </listitem>
 999                         <listitem>
1000                             <para>А в Scilab'е: </para>
1001                             <programlisting role="no-scilab-exec"><![CDATA[
1002 function [dmval,zu]=fsub(x,parameters)
1003  ]]></programlisting>
1004                         </listitem>
1005                     </itemizedlist>
1006                 </listitem>
1007             </varlistentry>
1008             <varlistentry>
1009                 <term>&lt;optional_args&gt;</term>
1010                 <listitem>
1011                     <para>Необязательные аргументы, должны быть либо:</para>
1012                     <itemizedlist>
1013                         <listitem>
1014                             <para>любой левой частью упорядоченной последовательности значений:
1015                                 <literal>guess, dfsub, dgsub, fixpnt, ndimf, ndimi, ltol, tol,
1016                                     ntol,nonlin, collpnt, subint, iprint, ireg, ifail
1017                                 </literal>
1018                             </para>
1019                         </listitem>
1020                         <listitem>
1021                             <para>
1022                                 либо любой последовательностью <literal>arg_name=argvalue</literal>
1023                                 с <literal>arg_name</literal> из: <literal>guess</literal>,
1024                                 <literal>dfsub</literal>, <literal>dgsub</literal>,
1025                                 <literal>fixpnt</literal>, <literal>ndimf</literal>,
1026                                 <literal>ndimi</literal>, <literal>ltol</literal>,
1027                                 <literal>tol</literal>, <literal>ntol</literal>,
1028                                 <literal>nonlin</literal>, <literal>collpnt</literal>,
1029                                 <literal>subint</literal>, <literal>iprint</literal>,
1030                                 <literal>ireg</literal>, <literal>ifail</literal>.
1031                             </para>
1032                         </listitem>
1033                     </itemizedlist>
1034                     <para>
1035                         Все эти аргументы за исключением <literal>ifail</literal> описаны выше.
1036                         <literal>ifail</literal> может использоваться для отображения вызова
1037                         bvode в соответствии с выбранными необязательными аргументами. Если
1038                         <literal>guess</literal>задано, то <literal>iguess</literal> равно 1.
1039                     </para>
1040                 </listitem>
1041             </varlistentry>
1042         </variablelist>
1043     </refsection>
1044     <refsection>
1045         <title>Описание</title>
1046         <para>
1047             Эти функции решают задачу многоточечных граничных значений для системы ОДУ смешанного порядка, заданной как:
1048         </para>
1049         <informalequation>
1050             <mediaobject>
1051                 <imageobject>
1052                     <imagedata align="left">
1053                         <mml:math>
1054                             <mml:semantics>
1055                                 <mml:mrow>
1056                                     <mml:mo mml:stretchy="true">{</mml:mo>
1057                                     <mml:mtable>
1058                                         <mml:mtr>
1059                                             <mml:mrow>
1060                                                 <mml:mrow>
1061                                                     <mml:msubsup>
1062                                                         <mml:mi>u</mml:mi>
1063                                                         <mml:mi>i</mml:mi>
1064                                                         <mml:mrow>
1065                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1066                                                             <mml:msub>
1067                                                                 <mml:mi>m</mml:mi>
1068                                                                 <mml:mi>i</mml:mi>
1069                                                             </mml:msub>
1070                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1071                                                         </mml:mrow>
1072                                                     </mml:msubsup>
1073                                                     <mml:mo mml:stretchy="false">=</mml:mo>
1074                                                     <mml:msub>
1075                                                         <mml:mi>f</mml:mi>
1076                                                         <mml:mi>i</mml:mi>
1077                                                     </mml:msub>
1078                                                 </mml:mrow>
1079                                                 <mml:mrow>
1080                                                     <mml:mo mml:stretchy="false">(</mml:mo>
1081                                                     <mml:mrow>
1082                                                         <mml:mi>x</mml:mi>
1083                                                         <mml:mi>,</mml:mi>
1084                                                         <mml:mi>z</mml:mi>
1085                                                         <mml:mrow>
1086                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1087                                                             <mml:mrow>
1088                                                                 <mml:mi>u</mml:mi>
1089                                                                 <mml:mrow>
1090                                                                     <mml:mo mml:stretchy="false">(</mml:mo>
1091                                                                     <mml:mi>x</mml:mi>
1092                                                                     <mml:mo mml:stretchy="false">)</mml:mo>
1093                                                                 </mml:mrow>
1094                                                             </mml:mrow>
1095                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1096                                                         </mml:mrow>
1097                                                     </mml:mrow>
1098                                                     <mml:mo mml:stretchy="false">)</mml:mo>
1099                                                 </mml:mrow>
1100                                                 <mml:mi/>
1101                                                 <mml:mrow>
1102                                                     <mml:mo mml:stretchy="true">{</mml:mo>
1103                                                     <mml:mtable>
1104                                                         <mml:mtr>
1105                                                             <mml:mrow>
1106                                                                 <mml:mrow>
1107                                                                     <mml:mi>i</mml:mi>
1108                                                                     <mml:mo mml:stretchy="false">=</mml:mo>
1109                                                                     <mml:mn>1</mml:mn>
1110                                                                 </mml:mrow>
1111                                                                 <mml:mi mml:fontstyle="normal">:</mml:mi>
1112                                                                 <mml:mi>N</mml:mi>
1113                                                             </mml:mrow>
1114                                                         </mml:mtr>
1115                                                         <mml:mtr>
1116                                                             <mml:mrow>
1117                                                                 <mml:mi>x</mml:mi>
1118                                                                 <mml:mo mml:stretchy="false">∈</mml:mo>
1119                                                                 <mml:mfenced mml:close="[" mml:open="]">
1120                                                                     <mml:mrow>
1121                                                                         <mml:msub>
1122                                                                             <mml:mi>x</mml:mi>
1123                                                                             <mml:mi>l</mml:mi>
1124                                                                         </mml:msub>
1125                                                                         <mml:mi>,</mml:mi>
1126                                                                         <mml:msub>
1127                                                                             <mml:mi>x</mml:mi>
1128                                                                             <mml:mi>u</mml:mi>
1129                                                                         </mml:msub>
1130                                                                     </mml:mrow>
1131                                                                 </mml:mfenced>
1132                                                             </mml:mrow>
1133                                                         </mml:mtr>
1134                                                     </mml:mtable>
1135                                                 </mml:mrow>
1136                                             </mml:mrow>
1137                                         </mml:mtr>
1138                                         <mml:mtr>
1139                                             <mml:mi/>
1140                                         </mml:mtr>
1141                                         <mml:mtr>
1142                                             <mml:mrow>
1143                                                 <mml:msub>
1144                                                     <mml:mi>g</mml:mi>
1145                                                     <mml:mi>j</mml:mi>
1146                                                 </mml:msub>
1147                                                 <mml:mrow>
1148                                                     <mml:mrow>
1149                                                         <mml:mo mml:stretchy="false">(</mml:mo>
1150                                                         <mml:mrow>
1151                                                             <mml:msub>
1152                                                                 <mml:mo mml:stretchy="false">ζ</mml:mo>
1153                                                                 <mml:mi>j</mml:mi>
1154                                                             </mml:msub>
1155                                                             <mml:mi>,</mml:mi>
1156                                                             <mml:mi>z</mml:mi>
1157                                                             <mml:mrow>
1158                                                                 <mml:mo mml:stretchy="false">(</mml:mo>
1159                                                                 <mml:mrow>
1160                                                                     <mml:mi>u</mml:mi>
1161                                                                     <mml:mrow>
1162                                                                         <mml:mo mml:stretchy="false">(</mml:mo>
1163                                                                         <mml:msub>
1164                                                                             <mml:mo mml:stretchy="false">ζ</mml:mo>
1165                                                                             <mml:mi>j</mml:mi>
1166                                                                         </mml:msub>
1167                                                                         <mml:mo mml:stretchy="false">)</mml:mo>
1168                                                                     </mml:mrow>
1169                                                                 </mml:mrow>
1170                                                                 <mml:mo mml:stretchy="false">)</mml:mo>
1171                                                             </mml:mrow>
1172                                                         </mml:mrow>
1173                                                         <mml:mo mml:stretchy="false">)</mml:mo>
1174                                                     </mml:mrow>
1175                                                     <mml:mo mml:stretchy="false">=</mml:mo>
1176                                                     <mml:mn>0</mml:mn>
1177                                                 </mml:mrow>
1178                                                 <mml:mi/>
1179                                                 <mml:mrow>
1180                                                     <mml:mtext>,</mml:mtext>
1181                                                     <mml:mi>j</mml:mi>
1182                                                     <mml:mo mml:stretchy="false">=</mml:mo>
1183                                                     <mml:mn>1</mml:mn>
1184                                                 </mml:mrow>
1185                                                 <mml:mi mml:fontstyle="normal">:</mml:mi>
1186                                                 <mml:mi>M</mml:mi>
1187                                             </mml:mrow>
1188                                         </mml:mtr>
1189                                     </mml:mtable>
1190                                 </mml:mrow>
1191                                 <mml:annotation mml:encoding="StarMath 5.0">left lbrace
1192                                     stack{u_i^(m_i)=f_i(x,z(u(x))) ~~left lbrace { binom{i=1:N}{x
1193                                     in left ] a_l, a_u right [} } right none# `#
1194                                     g_j(%zeta_j,z(u(%zeta_j))) = 0~~j=1:M} right
1195                                     none
1196                                 </mml:annotation>
1197                             </mml:semantics>
1198                         </mml:math>
1199                     </imagedata>
1200                 </imageobject>
1201             </mediaobject>
1202         </informalequation>
1203         <para>где:</para>
1204         <mediaobject>
1205             <imageobject>
1206                 <imagedata align="left">
1207                     <mml:math>
1208                         <mml:semantics>
1209                             <mml:mrow>
1210                                 <mml:mo mml:stretchy="true">{</mml:mo>
1211                                 <mml:mtable>
1212                                     <mml:mtr>
1213                                         <mml:mrow>
1214                                             <mml:mi>M</mml:mi>
1215                                             <mml:mo mml:stretchy="false">=</mml:mo>
1216                                             <mml:mrow>
1217                                                 <mml:munderover>
1218                                                     <mml:mo mml:stretchy="false">∑</mml:mo>
1219                                                     <mml:mrow>
1220                                                         <mml:mi>i</mml:mi>
1221                                                         <mml:mo mml:stretchy="false">=</mml:mo>
1222                                                         <mml:mn>1</mml:mn>
1223                                                     </mml:mrow>
1224                                                     <mml:mi>N</mml:mi>
1225                                                 </mml:munderover>
1226                                                 <mml:msub>
1227                                                     <mml:mi>m</mml:mi>
1228                                                     <mml:mi>i</mml:mi>
1229                                                 </mml:msub>
1230                                             </mml:mrow>
1231                                         </mml:mrow>
1232                                     </mml:mtr>
1233                                     <mml:mtr>
1234                                         <mml:mi/>
1235                                     </mml:mtr>
1236                                     <mml:mtr>
1237                                         <mml:mrow>
1238                                             <mml:mi>z</mml:mi>
1239                                             <mml:mrow>
1240                                                 <mml:mrow>
1241                                                     <mml:mo mml:stretchy="false">(</mml:mo>
1242                                                     <mml:mrow>
1243                                                         <mml:mi>u</mml:mi>
1244                                                         <mml:mrow>
1245                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1246                                                             <mml:mi>x</mml:mi>
1247                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1248                                                         </mml:mrow>
1249                                                     </mml:mrow>
1250                                                     <mml:mo mml:stretchy="false">)</mml:mo>
1251                                                 </mml:mrow>
1252                                                 <mml:mo mml:stretchy="false">=</mml:mo>
1253                                                 <mml:mrow>
1254                                                     <mml:mo mml:stretchy="false">[</mml:mo>
1255                                                     <mml:mrow>
1256                                                         <mml:msub>
1257                                                             <mml:mi>u</mml:mi>
1258                                                             <mml:mn>1</mml:mn>
1259                                                         </mml:msub>
1260                                                         <mml:mrow>
1261                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1262                                                             <mml:mi>x</mml:mi>
1263                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1264                                                         </mml:mrow>
1265                                                         <mml:mi>;</mml:mi>
1266                                                         <mml:msubsup>
1267                                                             <mml:mi>u</mml:mi>
1268                                                             <mml:mn>1</mml:mn>
1269                                                             <mml:mn>1</mml:mn>
1270                                                         </mml:msubsup>
1271                                                         <mml:mrow>
1272                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1273                                                             <mml:mi>x</mml:mi>
1274                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1275                                                         </mml:mrow>
1276                                                         <mml:mi>;</mml:mi>
1277                                                         <mml:mo mml:stretchy="false">⋯</mml:mo>
1278                                                         <mml:msubsup>
1279                                                             <mml:mi>u</mml:mi>
1280                                                             <mml:mn>1</mml:mn>
1281                                                             <mml:mrow>
1282                                                                 <mml:mo mml:stretchy="false">(</mml:mo>
1283                                                                 <mml:mrow>
1284                                                                     <mml:msub>
1285                                                                         <mml:mi>m</mml:mi>
1286                                                                         <mml:mn>1</mml:mn>
1287                                                                     </mml:msub>
1288                                                                     <mml:mo mml:stretchy="false">−</mml:mo>
1289                                                                     <mml:mn>1</mml:mn>
1290                                                                 </mml:mrow>
1291                                                                 <mml:mo mml:stretchy="false">)</mml:mo>
1292                                                             </mml:mrow>
1293                                                         </mml:msubsup>
1294                                                         <mml:mrow>
1295                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1296                                                             <mml:mi>x</mml:mi>
1297                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1298                                                         </mml:mrow>
1299                                                         <mml:mi>;</mml:mi>
1300                                                         <mml:mi/>
1301                                                         <mml:mo mml:stretchy="false">⋯</mml:mo>
1302                                                         <mml:mi>;</mml:mi>
1303                                                         <mml:mi/>
1304                                                         <mml:msub>
1305                                                             <mml:mi>u</mml:mi>
1306                                                             <mml:mi>N</mml:mi>
1307                                                         </mml:msub>
1308                                                         <mml:mrow>
1309                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1310                                                             <mml:mi>x</mml:mi>
1311                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1312                                                         </mml:mrow>
1313                                                         <mml:mi>;</mml:mi>
1314                                                         <mml:msubsup>
1315                                                             <mml:mi>u</mml:mi>
1316                                                             <mml:mi>N</mml:mi>
1317                                                             <mml:mn>1</mml:mn>
1318                                                         </mml:msubsup>
1319                                                         <mml:mrow>
1320                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1321                                                             <mml:mi>x</mml:mi>
1322                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1323                                                         </mml:mrow>
1324                                                         <mml:mi>;</mml:mi>
1325                                                         <mml:mo mml:stretchy="false">⋯</mml:mo>
1326                                                         <mml:mi>;</mml:mi>
1327                                                         <mml:msubsup>
1328                                                             <mml:mi>u</mml:mi>
1329                                                             <mml:mi>N</mml:mi>
1330                                                             <mml:mrow>
1331                                                                 <mml:mo mml:stretchy="false">(</mml:mo>
1332                                                                 <mml:mrow>
1333                                                                     <mml:msub>
1334                                                                         <mml:mi>m</mml:mi>
1335                                                                         <mml:mi>N</mml:mi>
1336                                                                     </mml:msub>
1337                                                                     <mml:mo mml:stretchy="false">−</mml:mo>
1338                                                                     <mml:mn>1</mml:mn>
1339                                                                 </mml:mrow>
1340                                                                 <mml:mo mml:stretchy="false">)</mml:mo>
1341                                                             </mml:mrow>
1342                                                         </mml:msubsup>
1343                                                         <mml:mrow>
1344                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1345                                                             <mml:mi>x</mml:mi>
1346                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1347                                                         </mml:mrow>
1348                                                     </mml:mrow>
1349                                                     <mml:mo mml:stretchy="false">]</mml:mo>
1350                                                 </mml:mrow>
1351                                             </mml:mrow>
1352                                         </mml:mrow>
1353                                     </mml:mtr>
1354                                     <mml:mtr>
1355                                         <mml:mi/>
1356                                     </mml:mtr>
1357                                     <mml:mtr>
1358                                         <mml:mrow>
1359                                             <mml:mrow>
1360                                                 <mml:mrow>
1361                                                     <mml:mrow>
1362                                                         <mml:msub>
1363                                                             <mml:mi>x</mml:mi>
1364                                                             <mml:mi>l</mml:mi>
1365                                                         </mml:msub>
1366                                                         <mml:mo mml:stretchy="false">≤</mml:mo>
1367                                                         <mml:msub>
1368                                                             <mml:mo mml:stretchy="false">ζ</mml:mo>
1369                                                             <mml:mn>1</mml:mn>
1370                                                         </mml:msub>
1371                                                     </mml:mrow>
1372                                                     <mml:mo mml:stretchy="false">≤</mml:mo>
1373                                                     <mml:mo mml:stretchy="false">⋯</mml:mo>
1374                                                 </mml:mrow>
1375                                                 <mml:mo mml:stretchy="false">≤</mml:mo>
1376                                                 <mml:msub>
1377                                                     <mml:mo mml:stretchy="false">ζ</mml:mo>
1378                                                     <mml:mi>M</mml:mi>
1379                                                 </mml:msub>
1380                                             </mml:mrow>
1381                                             <mml:mo mml:stretchy="false">≤</mml:mo>
1382                                             <mml:msub>
1383                                                 <mml:mi>x</mml:mi>
1384                                                 <mml:mi>u</mml:mi>
1385                                             </mml:msub>
1386                                         </mml:mrow>
1387                                     </mml:mtr>
1388                                 </mml:mtable>
1389                             </mml:mrow>
1390                             <mml:annotation mml:encoding="StarMath 5.0">left lbrace stack
1391                                 { M= sum from {i=1} to {N} m_i # ~# z(u(x)) =
1392                                 [u_1(x);u_1^{1}(x);dotsaxis u_1^(m_1-1)(x);~ dotsaxis;
1393                                 ~u_N(x);u_N^{1}(x);dotsaxis; u_N^(m_N-1)(x)]# ~# x_l &lt;=
1394                                 %zeta_1 &lt;= dotsaxis &lt;= %zeta_M &lt;=x_u}right
1395                                 none
1396                             </mml:annotation>
1397                         </mml:semantics>
1398                     </mml:math>
1399                 </imagedata>
1400             </imageobject>
1401         </mediaobject>
1402         <para>
1403             Аргумент <literal>zu</literal>, используемый внешними функциями и возвращаемый
1404             <literal>bvode</literal>, является вектор-столбцом, сформированным элементами
1405             <literal>z(u(x))</literal> для заданных <literal>x</literal>.
1406         </para>
1407         <para>
1408             Метод, используемый для аппроксимации решения, <literal>u</literal> является
1409             коллокацией в гауссовских точках, требующих <literal>m(i)-1</literal> непрерывных
1410             производных в <literal>i</literal>-том элементе, <literal>i = 1:N</literal>. Здесь,
1411             <literal>k</literal> - количество точек коллокации (этапов) на подынтервале, и оно
1412             выбирается так, чтобы <literal>k .ge. max m(i)</literal>. Используется представление
1413             решения одночленного решения Рунге-Кутты.
1414         </para>
1415     </refsection>
1416     <refsection>
1417         <title>Примеры</title>
1418         <para>Первые две задачи взяты из [1] раздела Литература.
1419         </para>
1420         <itemizedlist>
1421             <listitem>
1422                 <para>
1423                     <emphasis role="bold">Задача 1</emphasis> описывает однородно нагруженную балку переменной жёсткости просто поддерживаемую на обоих концах.
1424                 </para>
1425                 <para>Она может быть определена следующим образом.</para>
1426                 <para>Решим дифференциальное уравнение четвёртого порядка:</para>
1427                 <mediaobject>
1428                     <imageobject>
1429                         <imagedata align="left">
1430                             <mml:math>
1431                                 <mml:semantics>
1432                                     <mml:mrow>
1433                                         <mml:mfrac>
1434                                             <mml:msup>
1435                                                 <mml:mi>d</mml:mi>
1436                                                 <mml:mn>4</mml:mn>
1437                                             </mml:msup>
1438                                             <mml:mrow>
1439                                                 <mml:mi>d</mml:mi>
1440                                                 <mml:msup>
1441                                                     <mml:mi>x</mml:mi>
1442                                                     <mml:mn>4</mml:mn>
1443                                                 </mml:msup>
1444                                             </mml:mrow>
1445                                         </mml:mfrac>
1446                                         <mml:mi>u</mml:mi>
1447                                         <mml:mrow>
1448                                             <mml:mrow>
1449                                                 <mml:mo mml:stretchy="false">(</mml:mo>
1450                                                 <mml:mi>x</mml:mi>
1451                                                 <mml:mo mml:stretchy="false">)</mml:mo>
1452                                             </mml:mrow>
1453                                             <mml:mo mml:stretchy="false">=</mml:mo>
1454                                             <mml:mfrac>
1455                                                 <mml:mrow>
1456                                                     <mml:mrow>
1457                                                         <mml:mn>1</mml:mn>
1458                                                         <mml:mo mml:stretchy="false">−</mml:mo>
1459                                                         <mml:mrow>
1460                                                             <mml:mrow>
1461                                                                 <mml:mn>6</mml:mn>
1462                                                                 <mml:mo mml:stretchy="false">.</mml:mo>
1463                                                                 <mml:msup>
1464                                                                     <mml:mi>x</mml:mi>
1465                                                                     <mml:mn>2</mml:mn>
1466                                                                 </mml:msup>
1467                                                             </mml:mrow>
1468                                                             <mml:mo mml:stretchy="false">.</mml:mo>
1469                                                             <mml:mfrac>
1470                                                                 <mml:msup>
1471                                                                     <mml:mi>d</mml:mi>
1472                                                                     <mml:mn>3</mml:mn>
1473                                                                 </mml:msup>
1474                                                                 <mml:mrow>
1475                                                                     <mml:mi>d</mml:mi>
1476                                                                     <mml:msup>
1477                                                                         <mml:mi>x</mml:mi>
1478                                                                         <mml:mn>3</mml:mn>
1479                                                                     </mml:msup>
1480                                                                 </mml:mrow>
1481                                                             </mml:mfrac>
1482                                                         </mml:mrow>
1483                                                     </mml:mrow>
1484                                                     <mml:mi>u</mml:mi>
1485                                                     <mml:mrow>
1486                                                         <mml:mrow>
1487                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1488                                                             <mml:mi>x</mml:mi>
1489                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1490                                                         </mml:mrow>
1491                                                         <mml:mo mml:stretchy="false">−</mml:mo>
1492                                                         <mml:mrow>
1493                                                             <mml:mrow>
1494                                                                 <mml:mn>6</mml:mn>
1495                                                                 <mml:mo mml:stretchy="false">.</mml:mo>
1496                                                                 <mml:mi>x</mml:mi>
1497                                                             </mml:mrow>
1498                                                             <mml:mo mml:stretchy="false">.</mml:mo>
1499                                                             <mml:mfrac>
1500                                                                 <mml:msup>
1501                                                                     <mml:mi>d</mml:mi>
1502                                                                     <mml:mn>2</mml:mn>
1503                                                                 </mml:msup>
1504                                                                 <mml:mrow>
1505                                                                     <mml:mi>d</mml:mi>
1506                                                                     <mml:msup>
1507                                                                         <mml:mi>x</mml:mi>
1508                                                                         <mml:mn>2</mml:mn>
1509                                                                     </mml:msup>
1510                                                                 </mml:mrow>
1511                                                             </mml:mfrac>
1512                                                         </mml:mrow>
1513                                                     </mml:mrow>
1514                                                     <mml:mi>u</mml:mi>
1515                                                     <mml:mrow>
1516                                                         <mml:mo mml:stretchy="false">(</mml:mo>
1517                                                         <mml:mi>x</mml:mi>
1518                                                         <mml:mo mml:stretchy="false">)</mml:mo>
1519                                                     </mml:mrow>
1520                                                 </mml:mrow>
1521                                                 <mml:msup>
1522                                                     <mml:mi>x</mml:mi>
1523                                                     <mml:mn>3</mml:mn>
1524                                                 </mml:msup>
1525                                             </mml:mfrac>
1526                                         </mml:mrow>
1527                                     </mml:mrow>
1528                                     <mml:annotation mml:encoding="StarMath 5.0">d^4 over {d
1529                                         x^4} u(x)={1 -6*x^2*{{d^3} over {d x^3}} u(x)-6*x*{{d^2}
1530                                         over {d x^2}} u(x)} over x^3
1531                                     </mml:annotation>
1532                                 </mml:semantics>
1533                             </mml:math>
1534                         </imagedata>
1535                     </imageobject>
1536                 </mediaobject>
1537                 <para>Налагаемые граничные условия:</para>
1538                 <mediaobject>
1539                     <imageobject>
1540                         <imagedata align="left">
1541                             <mml:math>
1542                                 <mml:semantics>
1543                                     <mml:mrow>
1544                                         <mml:mo mml:stretchy="true">{</mml:mo>
1545                                         <mml:mtable>
1546                                             <mml:mtr>
1547                                                 <mml:mrow>
1548                                                     <mml:mi>u</mml:mi>
1549                                                     <mml:mrow>
1550                                                         <mml:mrow>
1551                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1552                                                             <mml:mn>1</mml:mn>
1553                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1554                                                         </mml:mrow>
1555                                                         <mml:mo mml:stretchy="false">=</mml:mo>
1556                                                         <mml:mn>0</mml:mn>
1557                                                     </mml:mrow>
1558                                                 </mml:mrow>
1559                                             </mml:mtr>
1560                                             <mml:mtr>
1561                                                 <mml:mrow>
1562                                                     <mml:mfrac>
1563                                                         <mml:msup>
1564                                                             <mml:mi>d</mml:mi>
1565                                                             <mml:mn>2</mml:mn>
1566                                                         </mml:msup>
1567                                                         <mml:mrow>
1568                                                             <mml:mi>d</mml:mi>
1569                                                             <mml:msup>
1570                                                                 <mml:mi>x</mml:mi>
1571                                                                 <mml:mn>2</mml:mn>
1572                                                             </mml:msup>
1573                                                         </mml:mrow>
1574                                                     </mml:mfrac>
1575                                                     <mml:mi>u</mml:mi>
1576                                                     <mml:mrow>
1577                                                         <mml:mo mml:stretchy="false">(</mml:mo>
1578                                                         <mml:mi>x</mml:mi>
1579                                                         <mml:mo mml:stretchy="false">)</mml:mo>
1580                                                     </mml:mrow>
1581                                                     <mml:mrow>
1582                                                         <mml:mrow>
1583                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1584                                                             <mml:mn>1</mml:mn>
1585                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1586                                                         </mml:mrow>
1587                                                         <mml:mo mml:stretchy="false">=</mml:mo>
1588                                                         <mml:mn>0</mml:mn>
1589                                                     </mml:mrow>
1590                                                 </mml:mrow>
1591                                             </mml:mtr>
1592                                             <mml:mtr>
1593                                                 <mml:mrow>
1594                                                     <mml:mi>u</mml:mi>
1595                                                     <mml:mrow>
1596                                                         <mml:mrow>
1597                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1598                                                             <mml:mn>2</mml:mn>
1599                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1600                                                         </mml:mrow>
1601                                                         <mml:mo mml:stretchy="false">=</mml:mo>
1602                                                         <mml:mn>0</mml:mn>
1603                                                     </mml:mrow>
1604                                                 </mml:mrow>
1605                                             </mml:mtr>
1606                                             <mml:mtr>
1607                                                 <mml:mrow>
1608                                                     <mml:mfrac>
1609                                                         <mml:msup>
1610                                                             <mml:mi>d</mml:mi>
1611                                                             <mml:mn>2</mml:mn>
1612                                                         </mml:msup>
1613                                                         <mml:mrow>
1614                                                             <mml:mi>d</mml:mi>
1615                                                             <mml:msup>
1616                                                                 <mml:mi>x</mml:mi>
1617                                                                 <mml:mn>2</mml:mn>
1618                                                             </mml:msup>
1619                                                         </mml:mrow>
1620                                                     </mml:mfrac>
1621                                                     <mml:mi>u</mml:mi>
1622                                                     <mml:mrow>
1623                                                         <mml:mo mml:stretchy="false">(</mml:mo>
1624                                                         <mml:mi>x</mml:mi>
1625                                                         <mml:mo mml:stretchy="false">)</mml:mo>
1626                                                     </mml:mrow>
1627                                                     <mml:mrow>
1628                                                         <mml:mrow>
1629                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1630                                                             <mml:mn>2</mml:mn>
1631                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1632                                                         </mml:mrow>
1633                                                         <mml:mo mml:stretchy="false">=</mml:mo>
1634                                                         <mml:mn>0</mml:mn>
1635                                                     </mml:mrow>
1636                                                 </mml:mrow>
1637                                             </mml:mtr>
1638                                         </mml:mtable>
1639                                     </mml:mrow>
1640                                     <mml:annotation mml:encoding="StarMath 5.0">left lbrace
1641                                         stack{ u(1)=0# {{d^2} over {d x^2}} u(x)(1)=0# u(2)=0#
1642                                         {{d^2} over {d x^2}} u(x)(2)=0} right
1643                                         none
1644                                     </mml:annotation>
1645                                 </mml:semantics>
1646                             </mml:math>
1647                         </imagedata>
1648                     </imageobject>
1649                 </mediaobject>
1650                 <para>Известно точное решение этой задачи:</para>
1651                 <mediaobject>
1652                     <imageobject>
1653                         <imagedata align="left">
1654                             <mml:math>
1655                                 <mml:semantics>
1656                                     <mml:mrow>
1657                                         <mml:mi>u</mml:mi>
1658                                         <mml:mrow>
1659                                             <mml:mrow>
1660                                                 <mml:mo mml:stretchy="false">(</mml:mo>
1661                                                 <mml:mi>x</mml:mi>
1662                                                 <mml:mo mml:stretchy="false">)</mml:mo>
1663                                             </mml:mrow>
1664                                             <mml:mo mml:stretchy="false">=</mml:mo>
1665                                             <mml:mfrac>
1666                                                 <mml:mn>1</mml:mn>
1667                                                 <mml:mn>4</mml:mn>
1668                                             </mml:mfrac>
1669                                         </mml:mrow>
1670                                         <mml:mrow>
1671                                             <mml:mo mml:stretchy="false">(</mml:mo>
1672                                             <mml:mrow>
1673                                                 <mml:mn>10</mml:mn>
1674                                                 <mml:mi/>
1675                                                 <mml:mi>log</mml:mi>
1676                                                 <mml:mrow>
1677                                                     <mml:mrow>
1678                                                         <mml:mo mml:stretchy="false">(</mml:mo>
1679                                                         <mml:mn>2</mml:mn>
1680                                                         <mml:mo mml:stretchy="false">)</mml:mo>
1681                                                     </mml:mrow>
1682                                                     <mml:mo mml:stretchy="false">−</mml:mo>
1683                                                     <mml:mn>3</mml:mn>
1684                                                 </mml:mrow>
1685                                             </mml:mrow>
1686                                             <mml:mo mml:stretchy="false">)</mml:mo>
1687                                         </mml:mrow>
1688                                         <mml:mi/>
1689                                         <mml:mrow>
1690                                             <mml:mrow>
1691                                                 <mml:mo mml:stretchy="false">(</mml:mo>
1692                                                 <mml:mrow>
1693                                                     <mml:mn>1</mml:mn>
1694                                                     <mml:mo mml:stretchy="false">−</mml:mo>
1695                                                     <mml:mi>x</mml:mi>
1696                                                 </mml:mrow>
1697                                                 <mml:mo mml:stretchy="false">)</mml:mo>
1698                                             </mml:mrow>
1699                                             <mml:mo mml:stretchy="false">+</mml:mo>
1700                                             <mml:mfrac>
1701                                                 <mml:mn>1</mml:mn>
1702                                                 <mml:mn>2</mml:mn>
1703                                             </mml:mfrac>
1704                                         </mml:mrow>
1705                                         <mml:mrow>
1706                                             <mml:mo mml:stretchy="false">[</mml:mo>
1707                                             <mml:mrow>
1708                                                 <mml:mfrac>
1709                                                     <mml:mn>1</mml:mn>
1710                                                     <mml:mi>x</mml:mi>
1711                                                 </mml:mfrac>
1712                                                 <mml:mo mml:stretchy="false">+</mml:mo>
1713                                                 <mml:mrow>
1714                                                     <mml:mrow>
1715                                                         <mml:mo mml:stretchy="false">(</mml:mo>
1716                                                         <mml:mrow>
1717                                                             <mml:mn>3</mml:mn>
1718                                                             <mml:mo mml:stretchy="false">+</mml:mo>
1719                                                             <mml:mi>x</mml:mi>
1720                                                         </mml:mrow>
1721                                                         <mml:mo mml:stretchy="false">)</mml:mo>
1722                                                     </mml:mrow>
1723                                                     <mml:mi/>
1724                                                     <mml:mi>log</mml:mi>
1725                                                     <mml:mrow>
1726                                                         <mml:mrow>
1727                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1728                                                             <mml:mi>x</mml:mi>
1729                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1730                                                         </mml:mrow>
1731                                                         <mml:mo mml:stretchy="false">−</mml:mo>
1732                                                         <mml:mi>x</mml:mi>
1733                                                     </mml:mrow>
1734                                                 </mml:mrow>
1735                                             </mml:mrow>
1736                                             <mml:mo mml:stretchy="false">]</mml:mo>
1737                                         </mml:mrow>
1738                                     </mml:mrow>
1739                                     <mml:annotation mml:encoding="StarMath 5.0">u(x)=1 over 4
1740                                         (10`log(2)-3)`(1-x)+ {1 over 2} [ {1 over x} + {(3+x) `
1741                                         log(x)-x}]
1742                                     </mml:annotation>
1743                                 </mml:semantics>
1744                             </mml:math>
1745                         </imagedata>
1746                     </imageobject>
1747                 </mediaobject>
1748                 <programlisting role="example"><![CDATA[
1749 N=1;// только одно дифференциальное уравнение
1750 m=4;//дифференциальное уравнение четвёртого порядка
1751 M=sum(m);
1752
1753 x_low=1;
1754 x_up=2; // пределы x
1755 zeta=[x_low,x_low,x_up,x_up]; // два ограничения (на значение u и его вторую производную) на каждой границе.
1756
1757 //Внешние функции
1758 //Эти функции вызываются программой решения с zu=[u(x);u'(x);u''(x);u'''(x)]
1759
1760 // - Функция, которая вычисляет правую сторону дифференциального уравнения
1761 function f=fsub(x,zu)
1762     f=(1-6*x^2*zu(4)-6*x*zu(3))/x^3
1763 endfunction
1764
1765 // - Функция, которая вычисляет производную fsub по zu
1766 function df=dfsub(x,zu)
1767     df=[0,0,-6/x^2,-6/x]
1768 endfunction
1769
1770 // - Функция, которая вычисляет i-тое ограничение для заданной i
1771 function g=gsub(i,zu),
1772   select i
1773   case 1 then  //x=zeta(1)=1
1774     g=zu(1) //u(1)=0
1775   case 2 then //x=zeta(2)=1
1776     g=zu(3) //u''(1)=0
1777   case 3 then //x=zeta(3)=2
1778     g=zu(1) //u(2)=0
1779   case 4 then  //x=zeta(4)=2
1780     g=zu(3) //u''(2)=0
1781   end
1782 endfunction
1783
1784 // - Функция, которая вычисляет производную gsub по z
1785 function dg=dgsub(i,z)
1786   select i
1787   case 1 then  //x=zeta(1)=1
1788     dg=[1,0,0,0]
1789   case 2 then //x=zeta(2)=1
1790     dg=[0,0,1,0]
1791   case 3 then //x=zeta(3)=2
1792      dg=[1,0,0,0]
1793   case 4 then  //x=zeta(4)=2
1794     dg=[0,0,1,0]
1795   end
1796 endfunction
1797
1798 // - Функция, которая вычисляет начальное предположение, здесь не используется
1799 function [zu,mpar]=guess(x)
1800     zu=0;
1801     mpar=0;
1802 endfunction
1803
1804  //определим функцию, которая вычисляет точное значение u для заданного x (в целях проверки)
1805 function zu=trusol(x)
1806   zu=0*ones(4,1)
1807   zu(1) =  0.25*(10*log(2)-3)*(1-x) + 0.5 *( 1/x   + (3+x)*log(x) - x)
1808   zu(2) = -0.25*(10*log(2)-3)       + 0.5 *(-1/x^2 + (3+x)/x      + log(x) - 1)
1809   zu(3) = 0.5*( 2/x^3 + 1/x   - 3/x^2)
1810   zu(4) = 0.5*(-6/x^4 - 1/x/x + 6/x^3)
1811 endfunction
1812
1813 fixpnt=[ ];//Все граничные условия находятся в x_low и x_up
1814
1815 //    nonlin  collpnt n  ntol  ndimf        ndimi       iprint iread iguess rstart nfxpnt
1816 +ipar=[1       collpnt 10  4   nmax*nsizef  nmax*nsizei   -1      0     0      0      0     ]
1817
1818 ltol=[1,3];//установка контроля допуска на zu(1) и zu(3)
1819 tol=[1.e-11,1.e-11];//установка значений допуска для контроля допуска
1820 xpoints=x_low:0.01:x_up;
1821
1822 zu=bvode(xpoints,N,m,x_low,x_up,zeta,ipar,ltol,tol,fixpnt,...
1823         fsub,dfsub,gsub,dgsub,guess)
1824 //проверка ограничений
1825 zu([1,3],[1 $]) // должно быть нулевым
1826 plot(xpoints,zu(1,:)) // процесс решения u
1827 zu1=[];
1828 for x=xpoints
1829     zu1=[zu1,trusol(x)];
1830 end
1831 norm(zu-zu1)
1832  ]]></programlisting>
1833             </listitem>
1834             <listitem>
1835                 <para>
1836                     Та же задача с использованием <literal>bvodeS</literal> и начальным предположением.
1837                 </para>
1838                 <programlisting role="no-scilab-exec"><![CDATA[
1839 function [z,lhS]=zstart(x)
1840   z=zeros(5,1);z(5)=1;
1841   lhS=[0;1];
1842 endfunction
1843 zu=bvode(xpoints,N,m,x_low,x_up,zeta,ltol=ltol,tol=tol,guess=zstart)
1844   ]]></programlisting>
1845             </listitem>
1846             <listitem>
1847                 <para>
1848                     <emphasis role="bold">Задача 2</emphasis> описывает малую конечную деформацию тонкой мелкой сферической чаши постоянной толщины, подверженной квадратично меняющемуся асимметричному распределению внешнего давления. Здесь <latex>$\varphi$</latex> описывает изменение угла по меридиану деформированной стенки, а <latex>$\psi$</latex> является функцией напряжений. Для <latex>$\varepsilon=\mu=10^{-3}$</latex> можно найти два различных решения в зависимости от точки начала.
1849                 </para>
1850                 <mediaobject>
1851                     <imageobject>
1852                         <imagedata align="left">
1853                             <mml:math>
1854                                 <mml:semantics>
1855                                     <mml:mrow>
1856                                         <mml:mo mml:stretchy="true">{</mml:mo>
1857                                         <mml:mtable>
1858                                             <mml:mtr>
1859                                                 <mml:mrow>
1860                                                     <mml:mfrac>
1861                                                         <mml:msup>
1862                                                             <mml:mo mml:stretchy="false">ε</mml:mo>
1863                                                             <mml:mn>4</mml:mn>
1864                                                         </mml:msup>
1865                                                         <mml:mo mml:stretchy="false">μ</mml:mo>
1866                                                     </mml:mfrac>
1867                                                     <mml:mrow>
1868                                                         <mml:mrow>
1869                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1870                                                             <mml:mrow>
1871                                                                 <mml:mo mml:stretchy="false">φ</mml:mo>
1872                                                                 <mml:mi>'</mml:mi>
1873                                                                 <mml:mrow>
1874                                                                     <mml:mrow>
1875                                                                         <mml:mi>'</mml:mi>
1876                                                                         <mml:mo mml:stretchy="false">+</mml:mo>
1877                                                                         <mml:mfrac>
1878                                                                             <mml:mrow>
1879                                                                                 <mml:mo mml:stretchy="false">φ</mml:mo>
1880                                                                                 <mml:mi>'</mml:mi>
1881                                                                             </mml:mrow>
1882                                                                             <mml:mi>x</mml:mi>
1883                                                                         </mml:mfrac>
1884                                                                     </mml:mrow>
1885                                                                     <mml:mo mml:stretchy="false">−</mml:mo>
1886                                                                     <mml:mfrac>
1887                                                                         <mml:mo mml:stretchy="false">φ</mml:mo>
1888                                                                         <mml:msup>
1889                                                                             <mml:mi>x</mml:mi>
1890                                                                             <mml:mn>2</mml:mn>
1891                                                                         </mml:msup>
1892                                                                     </mml:mfrac>
1893                                                                 </mml:mrow>
1894                                                             </mml:mrow>
1895                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1896                                                         </mml:mrow>
1897                                                         <mml:mo mml:stretchy="false">+</mml:mo>
1898                                                         <mml:mo mml:stretchy="false">ψ</mml:mo>
1899                                                     </mml:mrow>
1900                                                     <mml:mrow>
1901                                                         <mml:mrow>
1902                                                             <mml:mrow>
1903                                                                 <mml:mo mml:stretchy="false">(</mml:mo>
1904                                                                 <mml:mrow>
1905                                                                     <mml:mn>1</mml:mn>
1906                                                                     <mml:mo mml:stretchy="false">−</mml:mo>
1907                                                                     <mml:mfrac>
1908                                                                         <mml:mo mml:stretchy="false">φ</mml:mo>
1909                                                                         <mml:mi>x</mml:mi>
1910                                                                     </mml:mfrac>
1911                                                                 </mml:mrow>
1912                                                                 <mml:mo mml:stretchy="false">)</mml:mo>
1913                                                             </mml:mrow>
1914                                                             <mml:mo mml:stretchy="false">−</mml:mo>
1915                                                             <mml:mo mml:stretchy="false">φ</mml:mo>
1916                                                         </mml:mrow>
1917                                                         <mml:mo mml:stretchy="false">+</mml:mo>
1918                                                         <mml:mo mml:stretchy="false">γ</mml:mo>
1919                                                     </mml:mrow>
1920                                                     <mml:mi>x</mml:mi>
1921                                                     <mml:mrow>
1922                                                         <mml:mrow>
1923                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1924                                                             <mml:mrow>
1925                                                                 <mml:mn>1</mml:mn>
1926                                                                 <mml:mo mml:stretchy="false">−</mml:mo>
1927                                                                 <mml:mfrac>
1928                                                                     <mml:msup>
1929                                                                         <mml:mi>x</mml:mi>
1930                                                                         <mml:mn>2</mml:mn>
1931                                                                     </mml:msup>
1932                                                                     <mml:mn>2</mml:mn>
1933                                                                 </mml:mfrac>
1934                                                             </mml:mrow>
1935                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1936                                                         </mml:mrow>
1937                                                         <mml:mo mml:stretchy="false">=</mml:mo>
1938                                                         <mml:mn>0</mml:mn>
1939                                                     </mml:mrow>
1940                                                 </mml:mrow>
1941                                             </mml:mtr>
1942                                             <mml:mtr>
1943                                                 <mml:mrow>
1944                                                     <mml:mo mml:stretchy="false">μ</mml:mo>
1945                                                     <mml:mrow>
1946                                                         <mml:mrow>
1947                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1948                                                             <mml:mrow>
1949                                                                 <mml:mo mml:stretchy="false">ψ</mml:mo>
1950                                                                 <mml:mi>'</mml:mi>
1951                                                                 <mml:mrow>
1952                                                                     <mml:mrow>
1953                                                                         <mml:mi>'</mml:mi>
1954                                                                         <mml:mo mml:stretchy="false">+</mml:mo>
1955                                                                         <mml:mfrac>
1956                                                                             <mml:mrow>
1957                                                                                 <mml:mo mml:stretchy="false">ψ</mml:mo>
1958                                                                                 <mml:mi>'</mml:mi>
1959                                                                             </mml:mrow>
1960                                                                             <mml:mi>x</mml:mi>
1961                                                                         </mml:mfrac>
1962                                                                     </mml:mrow>
1963                                                                     <mml:mo mml:stretchy="false">−</mml:mo>
1964                                                                     <mml:mfrac>
1965                                                                         <mml:mo mml:stretchy="false">ψ</mml:mo>
1966                                                                         <mml:msup>
1967                                                                             <mml:mi>x</mml:mi>
1968                                                                             <mml:mn>2</mml:mn>
1969                                                                         </mml:msup>
1970                                                                     </mml:mfrac>
1971                                                                 </mml:mrow>
1972                                                             </mml:mrow>
1973                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1974                                                         </mml:mrow>
1975                                                         <mml:mo mml:stretchy="false">−</mml:mo>
1976                                                         <mml:mo mml:stretchy="false">φ</mml:mo>
1977                                                     </mml:mrow>
1978                                                     <mml:mrow>
1979                                                         <mml:mrow>
1980                                                             <mml:mo mml:stretchy="false">(</mml:mo>
1981                                                             <mml:mrow>
1982                                                                 <mml:mn>1</mml:mn>
1983                                                                 <mml:mo mml:stretchy="false">−</mml:mo>
1984                                                                 <mml:mfrac>
1985                                                                     <mml:mo mml:stretchy="false">φ</mml:mo>
1986                                                                     <mml:mn>2x</mml:mn>
1987                                                                 </mml:mfrac>
1988                                                             </mml:mrow>
1989                                                             <mml:mo mml:stretchy="false">)</mml:mo>
1990                                                         </mml:mrow>
1991                                                         <mml:mo mml:stretchy="false">=</mml:mo>
1992                                                         <mml:mn>0</mml:mn>
1993                                                     </mml:mrow>
1994                                                 </mml:mrow>
1995                                             </mml:mtr>
1996                                             <mml:mtr>
1997                                                 <mml:mrow>
1998                                                     <mml:mrow>
1999                                                         <mml:mn>0</mml:mn>
2000                                                         <mml:mo mml:stretchy="false">&lt;</mml:mo>
2001                                                         <mml:mi>x</mml:mi>
2002                                                     </mml:mrow>
2003                                                     <mml:mo mml:stretchy="false">&lt;</mml:mo>
2004                                                     <mml:mn>1</mml:mn>
2005                                                 </mml:mrow>
2006                                             </mml:mtr>
2007                                         </mml:mtable>
2008                                     </mml:mrow>
2009                                     <mml:annotation mml:encoding="StarMath 5.0">left lbrace
2010                                         stack{{%epsilon^4} over %mu (%phi''+ {%phi'} over x
2011                                         -{%phi} over {x^2} ) +%psi (1- %phi over x ) -%phi+%gamma
2012                                         x(1-{x^2} over 2)=0# %mu(%psi''+{%psi'} over x-%psi over
2013                                         {x^2})-%phi(1-%phi over{2x})=0 # 0&lt;x&lt;1} right
2014                                         none
2015                                     </mml:annotation>
2016                                 </mml:semantics>
2017                             </mml:math>
2018                         </imagedata>
2019                     </imageobject>
2020                 </mediaobject>
2021                 <para>Граничные условия:</para>
2022                 <mediaobject>
2023                     <imageobject>
2024                         <imagedata align="left">
2025                             <mml:math scilab:localized="true">
2026                                 <mml:semantics>
2027                                     <mml:mrow>
2028                                         <mml:mo mml:stretchy="true">{</mml:mo>
2029                                         <mml:mtable>
2030                                             <mml:mtr>
2031                                                 <mml:mrow>
2032                                                     <mml:mo mml:stretchy="false">φ</mml:mo>
2033                                                     <mml:mo mml:stretchy="false">=</mml:mo>
2034                                                     <mml:mn>0</mml:mn>
2035                                                 </mml:mrow>
2036                                             </mml:mtr>
2037                                             <mml:mtr>
2038                                                 <mml:mrow>
2039                                                     <mml:mi>x</mml:mi>
2040                                                     <mml:mi/>
2041                                                     <mml:mo mml:stretchy="false">ψ</mml:mo>
2042                                                     <mml:mrow>
2043                                                         <mml:mi>'</mml:mi>
2044                                                         <mml:mo mml:stretchy="false">−</mml:mo>
2045                                                         <mml:mn>0,3</mml:mn>
2046                                                     </mml:mrow>
2047                                                     <mml:mrow>
2048                                                         <mml:mrow>
2049                                                             <mml:mo mml:stretchy="false">ψ</mml:mo>
2050                                                             <mml:mo mml:stretchy="false">+</mml:mo>
2051                                                             <mml:mn>0,7x</mml:mn>
2052                                                         </mml:mrow>
2053                                                         <mml:mo mml:stretchy="false">=</mml:mo>
2054                                                         <mml:mn>0</mml:mn>
2055                                                     </mml:mrow>
2056                                                 </mml:mrow>
2057                                             </mml:mtr>
2058                                         </mml:mtable>
2059                                     </mml:mrow>
2060                                     <mml:annotation mml:encoding="StarMath 5.0">left lbrace
2061                                         stack{ %phi=0# x`%psi'-0.3%psi+0.7x=0} right
2062                                         none
2063                                     </mml:annotation>
2064                                 </mml:semantics>
2065                             </mml:math>
2066                         </imagedata>
2067                     </imageobject>
2068                 </mediaobject>
2069                 <para>
2070                     для <literal>x=0</literal> и <literal>x=1</literal>
2071                 </para>
2072                 <programlisting role="example"><![CDATA[
2073 N=2;// два дифференциальных уравнения
2074 m=[2 2];// каждое дифференциальное уравнение второго порядка
2075 M=sum(m);
2076
2077 x_low=0;x_up=1; // пределы по x
2078 zeta=[x_low,x_low, x_up x_up]; //два ограничения на каждой границе
2079
2080 //Внешние функции
2081 //Эти функции вызываются программой решения с zu=[u1(x);u1'(x);u2(x);u2'(x)]
2082
2083 // - Функция, которая вычисляет правую сторону дифференциального уравнения
2084 function f=fsub2(x,zu,eps,dmu,eps4mu,gam,xt),
2085    f=[zu(1)/x^2-zu(2)/x+(zu(1)-zu(3)*(1-zu(1)/x)-gam*x*(1-x^2/2))/eps4mu //phi''
2086       zu(3)/x^2-zu(4)/x+zu(1)*(1-zu(1)/(2*x))/dmu];//psi''
2087 endfunction
2088
2089 // - Функция, которая вычисляет производную от fsub по zu
2090 function df=dfsub2(x,zu,eps,dmu,eps4mu,gam,xt),
2091   df=[1/x^2+(1+zu(3)/x)/eps4mu, -1/x, -(1-zu(1)/x)/eps4mu, 0
2092       (1-zu(1)/x)/dmu             0    1/x^2              -1/x];
2093 endfunction
2094
2095 // - Функция, которая вычисляет i-тое ограничение для заданного i
2096 function g=gsub2(i,zu),
2097   select i
2098   case 1 then  //x=zeta(1)=0
2099     g=zu(1) //u(0)=0
2100   case 2 then //x=zeta(2)=0
2101     g=-0.3*zu(3) //x*psi'-0.3*psi+0.7x=0
2102   case 3 then //x=zeta(3)=1
2103     g=zu(1) //u(1)=0
2104   case 4 then  //x=zeta(4)=1
2105     g=1*zu(4)-0.3*zu(3)+0.7*1 //x*psi'-0.3*psi+0.7x=0
2106   end
2107 endfunction
2108
2109 // - Функция, которая вычисляет производную от gsub по z
2110 function dg=dgsub2(i,z)
2111   select i
2112   case 1 then  //x=zeta(1)=1
2113     dg=[1,0,0,0]
2114   case 2 then //x=zeta(2)=1
2115     dg=[0,0,-0.3,0]
2116   case 3 then //x=zeta(3)=2
2117      dg=[1,0,0,0]
2118   case 4 then  //x=zeta(4)=2
2119     dg=[0,0,-0.3,1]
2120   end
2121 endfunction
2122
2123 gam=1.1
2124 eps=1d-3
2125 dmu=eps
2126 eps4mu=eps^4/dmu
2127 xt=sqrt(2*(gam-1)/gam)
2128
2129 fixpnt=[ ];// все граничные условия размещены в x_low и x_up
2130 collpnt=4;
2131 nsizef=4+3*M+(5+collpnt*N)*(collpnt*N+M)+(2*M-2)*2*M ;
2132 nsizei=3 + collpnt*N+M;;
2133 nmax=200;
2134 //    nonlin  collpnt n  ntol ndimf        ndimi        iprint iread iguess rstart nfxpnt
2135 ipar=[1       k       10 4    nmax*nsizef  nmax*nsizei   -1      0     0      0      0     ]
2136
2137 ltol=1:4;//установка контроля допусков на zu(1), zu(2), zu(3) и zu(4)
2138 tol=[1.e-5,1.e-5,1.e-5,1.e-5];//установка значений для контроля допусков
2139 xpoints=x_low:0.01:x_up;
2140
2141 // - Функция, которая вычисляет начальное предположение, здесь не используется
2142 function [zu,dmval]=guess2(x,gam),
2143    cons=gam*x*(1-x^2/2)
2144    dcons=gam*(1-3*x^2/2)
2145    d2cons=-3*gam*x
2146    dmval=zeros(2,1)
2147    if x>xt then
2148      zu=[0 0 -cons -dcons]
2149      dmval(2)=-d2cons
2150    else
2151      zu=[2*x;2;-2*x+cons;-2*dcons]
2152      dmval(2)=d2cons
2153    end
2154 endfunction
2155
2156 zu=bvode(xpoints,N,m,x_low,x_up,zeta,ipar,ltol,tol,fixpnt,...
2157         fsub2,dfsub2,gsub2,dgsub2,guess2);
2158 scf(1);clf();plot(xpoints,zu([1 3],:)) // процесс решения phi и psi
2159
2160 //использование начального предположения
2161 ipar(9)=1;//iguess
2162
2163 zu2=bvode(xpoints,N,m,x_low,x_up,zeta,ipar,ltol,tol,fixpnt,...
2164         fsub2,dfsub2,gsub2,dgsub2,guess2);
2165 scf(2);clf();plot(xpoints,zu2([1 3],:)) // процесс решения phi и psi
2166  ]]></programlisting>
2167             </listitem>
2168             <listitem>
2169                 <para>
2170                     <emphasis role="bold">Задача нахождения собственных значений</emphasis>
2171                 </para>
2172                 <programlisting role="example"><![CDATA[
2173 // y''(x)=-la*y(x)
2174 // BV: y(0)=y'(0); y(1)=0
2175 // Собственные функции и собственные значения y(x,n)=sin(s(n)*(1-x)), la(n)=s(n)^2,
2176 // где s(n) - нули f(s,n)=s+atan(s)-(n+1)*pi, n=0,1,2,...
2177 // Чтобы получить третье граничное условие, мы выбираем y(0)=1
2178 // (с y(x) также c*y(x) является решением для каждой константы c.)
2179 // Решим следующую систему ОДУ:
2180 // y''=-la*y
2181 // la'=0
2182 // BV: y(0)=y'(0), y(0)=1; y(1)=0
2183 // z=[y(x) ; y'(x) ; la]
2184
2185 function rhs=fsub(x,z)
2186   rhs=[-z(3)*z(1);0]
2187 endfunction
2188
2189 function g=gsub(i,z)
2190   g=[z(1)-z(2) z(1)-1 z(1)]
2191   g=g(i)
2192 endfunction
2193
2194 // Следующая начальная функция годится для первых 8 собственных функций.
2195 function [z,lhs]=ystart(x,z,la0)
2196   z=[1;0;la0]
2197   lhs=[0;0]
2198 endfunction
2199
2200 a=0;b=1;
2201 m=[2;1];
2202 n=2;
2203 zeta=[a a b];
2204 N=101;
2205 x=linspace(a,b,N)';
2206
2207 //Имеем s(n)-(n+1/2)*pi -> 0 для n, стремящимся к бесконечности.
2208 la0=input('n-ное собственное значение: n= ?');la0=(%pi/2+la0*%pi)^2;
2209
2210 z=bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,la0));
2211 // Тот же вызов без вывода на экран
2212 z=bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,la0),iprint=1);
2213 // Тот же вызов с подробным выводом на экран
2214 z=bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,la0),iprint=-1);
2215
2216 clf()
2217 plot(x,[z(1,:)' z(2,:)'])
2218 xtitle(['Начальное значение =  '+string(la0);'Собственное значение = '+string(z(3,1))],'x',' ')
2219 legend(['y(x)';'y''(x)'])
2220  ]]></programlisting>
2221                  <scilab:image localized="true">
2222                      function rhs=fsub(x,z)
2223                      rhs=[-z(3)*z(1);0]
2224                      endfunction
2225
2226                      function g=gsub(i,z)
2227                      g=[z(1)-z(2) z(1)-1 z(1)]
2228                      g=g(i)
2229                      endfunction
2230
2231                      function [z,lhs]=ystart(x,z,la0)
2232                      z=[1;0;la0]
2233                      lhs=[0;0]
2234                      endfunction
2235
2236                      a=0;b=1;
2237                      m=[2;1];
2238                      n=2;
2239                      zeta=[a a b];
2240                      N=101;
2241                      x=linspace(a,b,N)';
2242
2243                      la0=10;
2244                      la0=(%pi/2+la0*%pi)^2;
2245
2246                      z=bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,la0),iprint=1);
2247
2248                      plot(x,[z(1,:)' z(2,:)']);
2249                      xtitle(['Начальное значение =  '+string(la0);'Собственное значение = '+string(z(3,1))],'x',' ');
2250                      legend(['y(x)';'y''(x)']);
2251                  </scilab:image>
2252             </listitem>
2253             <listitem>
2254                 <para>
2255                     <emphasis role="bold">Краевая задача, у которой решений больше одного</emphasis>
2256                 </para>
2257                 <programlisting role="example"><![CDATA[
2258 // DE: y''(x)=-exp(y(x))
2259 // BV: y(0)=0; y(1)=0
2260 // Эта краевая задача имеет более одного решения.
2261 // Показано как найти два из них с помощью некоторой
2262 // предварительной информации о решениях y(x) для построения функции ystart.
2263 // z=[y(x);y'(x)]
2264
2265 a=0;
2266 b=1;
2267 m=2;
2268 n=1;
2269 zeta=[a b];
2270 N=101;
2271 tol=1e-8*[1 1];
2272 x=linspace(a,b,N);
2273
2274 function rhs=fsub(x,z),rhs=-exp(z(1));endfunction
2275
2276 function g=gsub(i,z)
2277   g=[z(1) z(1)]
2278   g=g(i)
2279 endfunction
2280
2281 function [z,lhs]=ystart(x,z,M)
2282   //z=[4*x*(1-x)*M ; 4*(1-2*x)*M]
2283   z=[M;0]
2284   //lhs=[-exp(4*x*(1-x)*M)]
2285   lhs=0
2286 endfunction
2287
2288 for M=[1 4]
2289    if M==1
2290       z=bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,M),tol=tol);
2291    else
2292       z1=bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,M),tol=tol);
2293    end
2294 end
2295
2296 // Интегрирование ОДУ даёт, например, два решения yex и yex1.
2297
2298 function y=f(c)
2299     y=c.*(1-tanh(sqrt(c)/4).^2)-2;
2300 endfunction
2301 c=fsolve(2,f);
2302
2303 function y=yex(x,c)
2304   y=log(c/2*(1-tanh(sqrt(c)*(1/4-x/2)).^2))
2305 endfunction
2306
2307 function y=f1(c1)
2308     y=2*c1^2+tanh(1/4/c1)^2-1;
2309 endfunction
2310 c1=fsolve(0.1,f1);
2311
2312 function y=yex1(x,c1)
2313   y=log((1-tanh((2*x-1)/4/c1).^2)/2/c1/c1)
2314 endfunction
2315
2316 disp(norm(z(1,:)-yex(x)),'norm(yex(x)-z(1,:))= ')
2317 disp(norm(z1(1,:)-yex1(x)),'norm(yex1(x)-z1(1,:))= ')
2318 clf();
2319 subplot(2,1,1)
2320 plot2d(x,z(1,:),style=[5])
2321 xtitle('Два различных решения','x',' ')
2322 subplot(2,1,2)
2323 plot2d(x,z1(1,:),style=[5])
2324 xtitle(' ','x',' ')
2325  ]]></programlisting>
2326                 <scilab:image localized="true">
2327                     a=0;
2328                     b=1;
2329                     m=2;
2330                     n=1;
2331                     zeta=[a b];
2332                     N=101;
2333                     tol=1e-8*[1 1];
2334                     x=linspace(a,b,N);
2335
2336                     function rhs=fsub(x,z)
2337                     rhs=-exp(z(1));
2338                     endfunction
2339
2340                     function g=gsub(i,z)
2341                     g=[z(1) z(1)]
2342                     g=g(i)
2343                     endfunction
2344
2345                     function [z,lhs]=ystart(x,z,M)
2346                     //z=[4*x*(1-x)*M ; 4*(1-2*x)*M]
2347                     z=[M;0]
2348                     //lhs=[-exp(4*x*(1-x)*M)]
2349                     lhs=0
2350                     endfunction
2351
2352                     for M=[1 4]
2353                     if M==1
2354                     z=bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,M),tol=tol);
2355                     else
2356                     z1=bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,M),tol=tol);
2357                     end
2358                     end
2359
2360                     //  Интегрирование ОДУ даёт, например, два решения yex и yex1.
2361
2362                     function y=f(c)
2363                     y=c.*(1-tanh(sqrt(c)/4).^2)-2;
2364                     endfunction
2365                     c=fsolve(2,f);
2366
2367                     function y=yex(x,c)
2368                     y=log(c/2*(1-tanh(sqrt(c)*(1/4-x/2)).^2))
2369                     endfunction
2370
2371                     function y=f1(c1)
2372                         y=2*c1^2+tanh(1/4/c1)^2-1
2373                     endfunction
2374                     c1=fsolve(0.1,f1);
2375
2376                     function y=yex1(x,c1)
2377                     y=log((1-tanh((2*x-1)/4/c1).^2)/2/c1/c1)
2378                     endfunction
2379
2380                     disp(norm(z(1,:)-yex(x)),'norm(yex(x)-z(1,:))= ')
2381                     disp(norm(z1(1,:)-yex1(x)),'norm(yex1(x)-z1(1,:))= ')
2382                     clf();
2383                     subplot(2,1,1)
2384                     plot2d(x,z(1,:),style=[5])
2385                     xtitle('Два различных решения','x',' ')
2386                     subplot(2,1,2)
2387                     plot2d(x,z1(1,:),style=[5])
2388                     xtitle(' ','x',' ')
2389                 </scilab:image>
2390             </listitem>
2391             <listitem>
2392                 <para>
2393                     <emphasis role="bold">Многоточечная краевая задача</emphasis>
2394                 </para>
2395                 <programlisting role="example"><![CDATA[
2396 // DE y'''(x)=1
2397 // z=[y(x);y'(x);y''(x)]
2398 // BV: y(-1)=2 y(1)=2
2399 // Дополнительное условие: y(0)=1
2400
2401 a=-1;b=1;c=0;
2402 // Точка дополнительного условия c должна быть включена в массив fixpnt.
2403 n=1;
2404 m=[3];
2405
2406 function rhs=fsub(x,z)
2407   rhs=1
2408 endfunction
2409
2410 function g=gsub(i,z)
2411   g=[z(1)-2 z(1)-1 z(1)-2]
2412   g=g(i)
2413 endfunction
2414
2415 N=10;
2416 zeta=[a c b];
2417 x=linspace(a,b,N);
2418
2419 z=bvodeS(x,m,n,a,b,fsub,gsub,zeta,fixpnt=c);
2420
2421 function y=yex(x)
2422     y=x.^3/6+x.^2-x./6+1
2423 endfunction
2424
2425 disp(norm(yex(x)-z(1,:)),'norm(yex(x)-z(1,:))= ')
2426  ]]></programlisting>
2427             </listitem>
2428             <listitem>
2429                 <para>
2430                     <emphasis role="bold">
2431                         Квантовое уравнение Неймана с 2 "собственными значениями"
2432                         (c_1 и c2). Используется продолжение.
2433                     </emphasis>
2434                 </para>
2435                 <programlisting role="example"><![CDATA[
2436
2437 // Квантовое уравнение Неймана с 2 "собственными значениями" c_1 и c_2
2438 // (c_1=v-c_2-c_3, где v - параметр, используемый в продолжении)
2439 //
2440 // diff(f,x,2) + (1/2)*(1/x + 1/(x-1) + 1/(x-y))*diff(f,x)
2441 //      - (c_1/x + c_2/(x-1) + c_3/(x-y))* f(x) = 0
2442 // diff(c_2,x)=0,  diff(c_3,x) = 0
2443 //
2444 // и 4 "граничных" условия: diff(f,x)(a_k)=2*c_k*f(a_k) для
2445 // k=1,2,3, a_k=(0, 1 , y)  и нормировка f(1) = 1
2446 //
2447 // Вектор z - это z_1=f, z_2=diff(f,x), z_3=c_2 и z_4=c_3
2448 // Предположение выбирается так, чтобы иметь один узел в интервале
2449 // [0,1],  f(x)=2*x-1 такая, что f(1)=1, c_2 и c_3 выбираются так,
2450 // чтобы исключить полюсы в дифференциальном уравнении в 1.0 и
2451 // y, z_3=1, z_4=1/(2*y-1)
2452 // См.: http://arxiv.org/pdf/hep-th/0407005
2453
2454
2455
2456
2457 y= 1.9d0;
2458 eigens=zeros(3,40); // Для хранения результатов
2459
2460 // Общая настройка bvode
2461
2462 // Количество дифференциальных уравнений
2463 ncomp = 3;
2464
2465 // Порядки уравнений
2466 m = [2, 1, 1];
2467
2468 // Нелинейная задача
2469 ipar(1) = 1;
2470
2471 // Количество точек коллокации
2472 ipar(2) = 3;
2473
2474 // Исходная сетка из 4 подынтервалов
2475 ipar(3) = 4;
2476 ipar(8) = 0;
2477
2478 // Размер fspace, ispace, см. colnew.f для изменения размера
2479 ipar(5) =  30000;
2480 ipar(6) =  2000;
2481
2482 // Средний вывод данных
2483 ipar(7) = 0;
2484
2485 // Есть исходное приближение
2486 ipar(9) = 1;
2487
2488 // fixpnt - это массив, содержащий все фиксированные точки в сетке, в
2489 // частности, "граничные" точки, за исключением aleft и aright, ipar[11]
2490 // её размер, здесь только одна внутренняя "граничная точка"
2491 ipar(11) = 1;
2492 fixpnt = [1.0d0];
2493
2494 // Допустимые отклонения для всех элементов z_1, z_2, z_3, z_4
2495 ipar(4) = 4;
2496
2497 // Проверка допустимого отклонения для f и diff(f,x) и для c_2 и c_3
2498 ltol = [1, 2, 3, 4];
2499 tol = [1d-5, 1d-5, 1d-5, 1d-5];
2500
2501 // Определение дифференциальных уравнений
2502
2503 function [f]=fsub(x,z)
2504     f = [ -.5*(1/x+1/(x-1)+1/(x-y))*z(2) +...
2505      z(1) * ((v-z(3)-z(4))/x + z(3)/(x-1) + z(4)/(x-y)),...
2506     0,0];
2507 endfunction
2508 function [df] = dfsub(x,z)
2509     df = [(v-z(3)-z(4))/x + z(3)/(x-1) + z(4)/(x-y),...
2510     -.5*(1/x+1/(x-1)+1/(x-y)),z(1)/(x*(x-1)),z(1)*y/(x*(x-y));...
2511     0,0,0,0;0,0,0,0];
2512 endfunction
2513
2514 // Граничные условия
2515
2516 function [g]=gsub(i,z)
2517     select i
2518     case 1, g = z(2) - 2*z(1)*(v-z(3)-z(4))
2519     case 2, g = z(2) - 2*z(1)*z(3)
2520     case 3, g = z(1)-1.
2521     case 4, g = z(2) - 2*z(1)*z(4)
2522     end
2523 endfunction
2524 function [dg]=dgsub(i,z)
2525         select i
2526         case 1, dg = [-2*(v-z(3)-z(4)),1.,2*z(1),2*z(1)]
2527         case 2, dg = [-2*z(3),1.,-2*z(1),0]
2528         case 3, dg = [1,0,0,0]
2529         case 4, dg = [-2*z(4),1.,0,-2*z(1)]
2530         end
2531 endfunction
2532
2533 // Начало вычисления
2534
2535 // Положения краевых условий, отсортированные
2536 zeta = [0.0d0, 1.0d0, 1.0d0, y];
2537 // Края интервала
2538 aleft = 0.0d0;
2539 aright = y;
2540
2541 // Массив из 40 значений v, исследуемых продолжением, и массив из 202
2542 // точек, в которых следует вычислить функцию f.
2543 valv = [linspace(0,.9,10) logspace(0,2,30)];
2544 res = [linspace(0,.99,100) linspace(1,y,101)];
2545
2546 // Собственные состояния характеризуются количеством узлов на интервале
2547 // [0,1] и на [1,y], здесь предполжение выбирает один узел (ноль) на
2548 // интервале [0,1] с линейной функцией f(x)=2*x-1 и константами c_2, c_3,
2549 // так что dmval=0. Заметьте, что вектор z имеет mstar = 4 элементов,
2550 // в то время как dmval имеет ncomp = 3 элемента.
2551
2552 function [z,dmval]=guess(x)
2553      z=[2*x-1, 2., 1., 1/(2*y-1)]
2554      dmval=[0,0,0]
2555 endfunction
2556
2557 // Первое выполнение имеет ipar(9)=1 и использует предположение
2558 // Последующие выполнения имеют ipar(9)=3 и используют продолжение.
2559 // Это выполняется в плотно закрытом цикле, чтобы не нарушить стек
2560
2561 for i=1:40
2562 v=valv(i);
2563 sol=bvode(res,ncomp,m,aleft,aright,zeta,ipar,ltol,tol,fixpnt,...
2564  fsub,dfsub,gsub,dgsub,guess);
2565 eigens(:,i)=[v;sol(3,101);sol(4,101)];  // c_2 и c_3 являются константами!
2566 ipar(9)=3;
2567 end
2568
2569
2570 // Чтобы увидеть изменение собственных значений с v, disp(eigens)
2571 // Обратите внимание на то, что они изменяются гладко.
2572 // Чтобы увидеть решение f для v=40, disp(sol(1,:)).
2573 // Обратите внимание на то, что оно исчезает точно один раз в
2574 // [0,1] в x близком  0.98, и становится очень малым, когда
2575 // x -> 0 и очень большим, когда x -> y.
2576 // Это явно отличается от случая при малом v.
2577 // Процедура продолжения позволяет исследовать эти экспоненциальные
2578 // поведения без перехода в другие собственные состояния.
2579
2580  ]]></programlisting>
2581             </listitem>
2582         </itemizedlist>
2583     </refsection>
2584     <refsection role="see also">
2585         <title>Смотрите также</title>
2586         <simplelist type="inline">
2587             <member>
2588                 <link linkend="link">link</link>
2589             </member>
2590             <member>
2591                 <link linkend="external">external</link>
2592             </member>
2593             <member>
2594                 <link linkend="ode">ode</link>
2595             </member>
2596             <member>
2597                 <link linkend="dassl">dassl</link>
2598             </member>
2599         </simplelist>
2600     </refsection>
2601     <refsection>
2602         <title>Используемые функции</title>
2603         <para>Эта функция основана на процедуре Fortran
2604             <literal>colnew</literal>, разработанной
2605         </para>
2606         <para>U. Ascher, Department of Computer Science, University of British
2607             Columbia, Vancouver, B.C. V6T 1W5, Canada
2608         </para>
2609         <para>G. Bader, institut f. Angewandte mathematik university of
2610             Heidelberg; im Neuenheimer feld 294d-6900 Heidelberg 1
2611         </para>
2612     </refsection>
2613     <refsection>
2614         <title>Литература</title>
2615         <orderedlist>
2616             <listitem>
2617                 <para>U. Ascher, J. Christiansen and R.D. Russell, collocation
2618                     software for boundary-value ODEs, acm trans. math software 7 (1981),
2619                     209-222. this paper contains EXAMPLES where use of the code is
2620                     demonstrated.
2621                 </para>
2622             </listitem>
2623             <listitem>
2624                 <para>G. Bader and U. Ascher, a new basis implementation for a mixed
2625                     order boundary value ode solver, siam j. scient. stat. comput.
2626                     (1987).
2627                 </para>
2628             </listitem>
2629             <listitem>
2630                 <para>U. Ascher, J. Christiansen and R.D. Russell, a collocation
2631                     solver for mixed order systems of boundary value problems, math. comp.
2632                     33 (1979), 659-679.
2633                 </para>
2634             </listitem>
2635             <listitem>
2636                 <para>U. Ascher, J. Christiansen and R.D. russell, colsys - a
2637                     collocation code for boundary value problems, lecture notes comp.sc.
2638                     76, springer verlag, b. children et. al. (eds.) (1979), 164-185.
2639                 </para>
2640             </listitem>
2641             <listitem>
2642                 <para>C. Deboor and R. Weiss, solveblok: a package for solving almost
2643                     block diagonal linear systems, acm trans. math. software 6 (1980),
2644                     80-87.
2645                 </para>
2646             </listitem>
2647         </orderedlist>
2648     </refsection>
2649 </refentry>