/usr/include/deal.II/lac/full_matrix.h is in libdeal.ii-dev 8.4.2-2+b1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 | // ---------------------------------------------------------------------
//
// Copyright (C) 1999 - 2016 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef dealii__full_matrix_h
#define dealii__full_matrix_h
#include <deal.II/base/config.h>
#include <deal.II/base/numbers.h>
#include <deal.II/base/table.h>
#include <deal.II/lac/exceptions.h>
#include <deal.II/lac/identity_matrix.h>
#include <deal.II/base/tensor.h>
#include <vector>
#include <iomanip>
#include <cstring>
DEAL_II_NAMESPACE_OPEN
// forward declarations
template <typename number> class Vector;
template <typename number> class LAPACKFullMatrix;
/*! @addtogroup Matrix1
*@{
*/
/**
* Implementation of a classical rectangular scheme of numbers. The data type
* of the entries is provided in the template argument <tt>number</tt>. The
* interface is quite fat and in fact has grown every time a new feature was
* needed. So, a lot of functions are provided.
*
* Internal calculations are usually done with the accuracy of the vector
* argument to functions. If there is no argument with a number type, the
* matrix number type is used.
*
* @note Instantiations for this template are provided for <tt>@<float@>,
* @<double@>, @<long double@>, @<std::complex@<float@>@>,
* @<std::complex@<double@>@>, @<std::complex@<long double@>@></tt>; others
* can be generated in application programs (see the section on
* @ref Instantiations
* in the manual).
*
* @author Guido Kanschat, Franz-Theo Suttmeier, Wolfgang Bangerth, 1993-2004
*/
template <typename number>
class FullMatrix : public Table<2,number>
{
public:
/**
* A type of used to index into this container. Because we can not expect to
* store matrices bigger than what can be indexed by a regular unsigned
* integer, <code>unsigned int</code> is completely sufficient as an index
* type.
*/
typedef unsigned int size_type;
/**
* Type of matrix entries. This typedef is analogous to <tt>value_type</tt>
* in the standard library containers.
*/
typedef number value_type;
/**
* Declare a type that has holds real-valued numbers with the same precision
* as the template argument to this class. If the template argument of this
* class is a real data type, then real_type equals the template argument.
* If the template argument is a std::complex type then real_type equals the
* type underlying the complex numbers.
*
* This typedef is used to represent the return type of norms.
*/
typedef typename numbers::NumberTraits<number>::real_type real_type;
class const_iterator;
/**
* Accessor class for iterators
*/
class Accessor
{
public:
/**
* Constructor. Since we use accessors only for read access, a const
* matrix pointer is sufficient.
*/
Accessor (const FullMatrix<number> *matrix,
const size_type row,
const size_type col);
/**
* Row number of the element represented by this object.
*/
size_type row() const;
/**
* Column number of the element represented by this object.
*/
size_type column() const;
/**
* Value of this matrix entry.
*/
number value() const;
protected:
/**
* The matrix accessed.
*/
const FullMatrix<number> *matrix;
/**
* Current row number.
*/
size_type a_row;
/**
* Current column number.
*/
unsigned short a_col;
/*
* Make enclosing class a friend.
*/
friend class const_iterator;
};
/**
* Standard-conforming iterator.
*/
class const_iterator
{
public:
/**
* Constructor.
*/
const_iterator(const FullMatrix<number> *matrix,
const size_type row,
const size_type col);
/**
* Prefix increment.
*/
const_iterator &operator++ ();
/**
* Postfix increment.
*/
const_iterator &operator++ (int);
/**
* Dereferencing operator.
*/
const Accessor &operator* () const;
/**
* Dereferencing operator.
*/
const Accessor *operator-> () const;
/**
* Comparison. True, if both iterators point to the same matrix position.
*/
bool operator == (const const_iterator &) const;
/**
* Inverse of <tt>==</tt>.
*/
bool operator != (const const_iterator &) const;
/**
* Comparison operator. Result is true if either the first row number is
* smaller or if the row numbers are equal and the first index is smaller.
*/
bool operator < (const const_iterator &) const;
/**
* Comparison operator. Compares just the other way around than the
* operator above.
*/
bool operator > (const const_iterator &) const;
private:
/**
* Store an object of the accessor class.
*/
Accessor accessor;
};
/**
* @name Constructors and initalization. See also the base class Table.
*/
//@{
/**
* Constructor. Initialize the matrix as a square matrix with dimension
* <tt>n</tt>.
*
* In order to avoid the implicit conversion of integers and other types to
* a matrix, this constructor is declared <tt>explicit</tt>.
*
* By default, no memory is allocated.
*/
explicit FullMatrix (const size_type n = 0);
/**
* Constructor. Initialize the matrix as a rectangular matrix.
*/
FullMatrix (const size_type rows,
const size_type cols);
/**
* Copy constructor. This constructor does a deep copy of the matrix.
* Therefore, it poses a possible efficiency problem, if for example,
* function arguments are passed by value rather than by reference.
* Unfortunately, we can't mark this copy constructor <tt>explicit</tt>,
* since that prevents the use of this class in containers, such as
* <tt>std::vector</tt>. The responsibility to check performance of programs
* must therefore remain with the user of this class.
*/
FullMatrix (const FullMatrix &);
/**
* Constructor initializing from an array of numbers. The array is arranged
* line by line. No range checking is performed.
*/
FullMatrix (const size_type rows,
const size_type cols,
const number *entries);
/**
* Construct a full matrix that equals the identity matrix of the size of
* the argument. Using this constructor, one can easily create an identity
* matrix of size <code>n</code> by saying
* @code
* FullMatrix<double> M(IdentityMatrix(n));
* @endcode
*/
FullMatrix (const IdentityMatrix &id);
/**
* @}
*/
/**
* @name Copying into and out of other matrices
*/
/**
* @{
*/
/**
* Assignment operator.
*
* @dealiiOperationIsMultithreaded
*/
FullMatrix<number> &
operator = (const FullMatrix<number> &);
/**
* Variable assignment operator.
*/
template <typename number2>
FullMatrix<number> &
operator = (const FullMatrix<number2> &);
/**
* This operator assigns a scalar to a matrix. To avoid confusion with the
* semantics of this function, zero is the only value allowed for
* <tt>d</tt>, allowing you to clear a matrix in an intuitive way.
*
* @dealiiOperationIsMultithreaded
*/
FullMatrix<number> &
operator = (const number d);
/**
* Copy operator to create a full matrix that equals the identity matrix of
* the size of the argument. This way, one can easily create an identity
* matrix of size <code>n</code> by saying
* @code
* M = IdentityMatrix(n);
* @endcode
*/
FullMatrix<number> &
operator = (const IdentityMatrix &id);
/**
* Assignment operator for a LapackFullMatrix. The calling matrix must be of
* the same size as the LAPACK matrix.
*/
template <typename number2>
FullMatrix<number> &
operator = (const LAPACKFullMatrix<number2> &);
/**
* Assignment from different matrix classes. This assignment operator uses
* iterators of the typename MatrixType. Therefore, sparse matrices are
* possible sources.
*/
template <typename MatrixType>
void copy_from (const MatrixType &);
/**
* Transposing assignment from different matrix classes. This assignment
* operator uses iterators of the typename MatrixType. Therefore, sparse
* matrices are possible sources.
*/
template <typename MatrixType>
void copy_transposed (const MatrixType &);
/**
* Fill matrix with elements extracted from a tensor, taking rows included
* between <tt>r_i</tt> and <tt>r_j</tt> and columns between <tt>c_i</tt>
* and <tt>c_j</tt>. The resulting matrix is then inserted in the
* destination matrix at position <tt>(dst_r, dst_c)</tt> Checks on the
* indices are made.
*/
template <int dim>
void
copy_from (const Tensor<2,dim> &T,
const size_type src_r_i=0,
const size_type src_r_j=dim-1,
const size_type src_c_i=0,
const size_type src_c_j=dim-1,
const size_type dst_r=0,
const size_type dst_c=0);
/**
* Insert a submatrix (also rectangular) into a tensor, putting its upper
* left element at the specified position <tt>(dst_r, dst_c)</tt> and the
* other elements consequently. Default values are chosen so that no
* parameter needs to be specified if the size of the tensor and that of the
* matrix coincide.
*/
template <int dim>
void
copy_to(Tensor<2,dim> &T,
const size_type src_r_i=0,
const size_type src_r_j=dim-1,
const size_type src_c_i=0,
const size_type src_c_j=dim-1,
const size_type dst_r=0,
const size_type dst_c=0) const;
/**
* Copy a subset of the rows and columns of another matrix into the current
* object.
*
* @param matrix The matrix from which a subset is to be taken from.
* @param row_index_set The set of rows of @p matrix from which to extract.
* @param column_index_set The set of columns of @p matrix from which to
* extract. @pre The number of elements in @p row_index_set and @p
* column_index_set shall be equal to the number of rows and columns in the
* current object. In other words, the current object is not resized for
* this operation.
*/
template <typename MatrixType, typename index_type>
void extract_submatrix_from (const MatrixType &matrix,
const std::vector<index_type> &row_index_set,
const std::vector<index_type> &column_index_set);
/**
* Copy the elements of the current matrix object into a specified set of
* rows and columns of another matrix. Thus, this is a scatter operation.
*
* @param row_index_set The rows of @p matrix into which to write.
* @param column_index_set The columns of @p matrix into which to write.
* @param matrix The matrix within which certain elements are to be
* replaced. @pre The number of elements in @p row_index_set and @p
* column_index_set shall be equal to the number of rows and columns in the
* current object. In other words, the current object is not resized for
* this operation.
*/
template <typename MatrixType, typename index_type>
void
scatter_matrix_to (const std::vector<index_type> &row_index_set,
const std::vector<index_type> &column_index_set,
MatrixType &matrix) const;
/**
* Fill rectangular block.
*
* A rectangular block of the matrix <tt>src</tt> is copied into
* <tt>this</tt>. The upper left corner of the block being copied is
* <tt>(src_offset_i,src_offset_j)</tt>. The upper left corner of the
* copied block is <tt>(dst_offset_i,dst_offset_j)</tt>. The size of the
* rectangular block being copied is the maximum size possible, determined
* either by the size of <tt>this</tt> or <tt>src</tt>.
*/
template <typename number2>
void fill (const FullMatrix<number2> &src,
const size_type dst_offset_i = 0,
const size_type dst_offset_j = 0,
const size_type src_offset_i = 0,
const size_type src_offset_j = 0);
/**
* Make function of base class available.
*/
template <typename number2>
void fill (const number2 *);
/**
* Fill with permutation of another matrix.
*
* The matrix <tt>src</tt> is copied into the target. The two permutation
* <tt>p_r</tt> and <tt>p_c</tt> operate in a way, such that <tt>result(i,j)
* = src(p_r[i], p_c[j])</tt>.
*
* The vectors may also be a selection from a larger set of integers, if the
* matrix <tt>src</tt> is bigger. It is also possible to duplicate rows or
* columns by this method.
*/
template <typename number2>
void fill_permutation (const FullMatrix<number2> &src,
const std::vector<size_type> &p_rows,
const std::vector<size_type> &p_cols);
/**
* Set a particular entry of the matrix to a value. Thus, calling
* <code>A.set(1,2,3.141);</code> is entirely equivalent to the operation
* <code>A(1,2) = 3.141;</code>. This function exists for compatibility with
* the various sparse matrix objects.
*
* @param i The row index of the element to be set.
* @param j The columns index of the element to be set.
* @param value The value to be written into the element.
*/
void set (const size_type i,
const size_type j,
const number value);
/**
* @}
*/
/**
* @name Non-modifying operators
*/
/**
* @{
*/
/**
* Comparison operator. Be careful with this thing, it may eat up huge
* amounts of computing time! It is most commonly used for internal
* consistency checks of programs.
*/
bool operator == (const FullMatrix<number> &) const;
/**
* Number of rows of this matrix. To remember: this matrix is an <i>m x
* n</i>-matrix.
*/
size_type m () const;
/**
* Number of columns of this matrix. To remember: this matrix is an <i>m x
* n</i>-matrix.
*/
size_type n () const;
/**
* Return whether the matrix contains only elements with value zero. This
* function is mainly for internal consistency checks and should seldom be
* used when not in debug mode since it uses quite some time.
*/
bool all_zero () const;
/**
* Return the square of the norm of the vector <tt>v</tt> induced by this
* matrix, i.e. <i>(v,Mv)</i>. This is useful, e.g. in the finite element
* context, where the <i>L<sup>2</sup></i> norm of a function equals the
* matrix norm with respect to the mass matrix of the vector representing
* the nodal values of the finite element function.
*
* Obviously, the matrix needs to be quadratic for this operation, and for
* the result to actually be a norm it also needs to be either real
* symmetric or complex hermitian.
*
* The underlying template types of both this matrix and the given vector
* should either both be real or complex-valued, but not mixed, for this
* function to make sense.
*/
template <typename number2>
number2 matrix_norm_square (const Vector<number2> &v) const;
/**
* Build the matrix scalar product <tt>u<sup>T</sup> M v</tt>. This function
* is mostly useful when building the cellwise scalar product of two
* functions in the finite element context.
*
* The underlying template types of both this matrix and the given vector
* should either both be real or complex-valued, but not mixed, for this
* function to make sense.
*/
template <typename number2>
number2 matrix_scalar_product (const Vector<number2> &u,
const Vector<number2> &v) const;
/**
* Return the <i>l<sub>1</sub></i>-norm of the matrix, where $||M||_1 =
* \max_j \sum_i |M_{ij}|$ (maximum of the sums over columns).
*/
real_type l1_norm () const;
/**
* Return the $l_\infty$-norm of the matrix, where $||M||_\infty = \max_i
* \sum_j |M_{ij}|$ (maximum of the sums over rows).
*/
real_type linfty_norm () const;
/**
* Compute the Frobenius norm of the matrix. Return value is the root of
* the square sum of all matrix entries.
*
* @note For the timid among us: this norm is not the norm compatible with
* the <i>l<sub>2</sub></i>-norm of the vector space.
*/
real_type frobenius_norm () const;
/**
* Compute the relative norm of the skew-symmetric part. The return value is
* the Frobenius norm of the skew-symmetric part of the matrix divided by
* that of the matrix.
*
* Main purpose of this function is to check, if a matrix is symmetric
* within a certain accuracy, or not.
*/
real_type relative_symmetry_norm2 () const;
/**
* Computes the determinant of a matrix. This is only implemented for one,
* two, and three dimensions, since for higher dimensions the numerical work
* explodes. Obviously, the matrix needs to be quadratic for this function.
*/
number determinant () const;
/**
* Return the trace of the matrix, i.e. the sum of the diagonal values
* (which happens to also equal the sum of the eigenvalues of a matrix).
* Obviously, the matrix needs to be quadratic for this function.
*/
number trace () const;
/**
* Output of the matrix in user-defined format given by the specified
* precision and width. This function saves width and precision of the
* stream before setting these given values for output, and restores the
* previous values after output.
*/
template <class StreamType>
void print (StreamType &s,
const unsigned int width=5,
const unsigned int precision=2) const;
/**
* Print the matrix and allow formatting of entries.
*
* The parameters allow for a flexible setting of the output format:
*
* @arg <tt>precision</tt> denotes the number of trailing digits.
*
* @arg <tt>scientific</tt> is used to determine the number format, where
* <tt>scientific</tt> = <tt>false</tt> means fixed point notation.
*
* @arg <tt>width</tt> denotes the with of each column. A zero entry for
* <tt>width</tt> makes the function compute a width, but it may be changed
* to a positive value, if output is crude.
*
* @arg <tt>zero_string</tt> specifies a string printed for zero entries.
*
* @arg <tt>denominator</tt> Multiply the whole matrix by this common
* denominator to get nicer numbers.
*
* @arg <tt>threshold</tt>: all entries with absolute value smaller than
* this are considered zero.
*/
void print_formatted (std::ostream &out,
const unsigned int precision=3,
const bool scientific = true,
const unsigned int width = 0,
const char *zero_string = " ",
const double denominator = 1.,
const double threshold = 0.) const;
/**
* Determine an estimate for the memory consumption (in bytes) of this
* object.
*/
std::size_t memory_consumption () const;
//@}
///@name Iterator functions
//@{
/**
* Iterator starting at the first entry.
*/
const_iterator begin () const;
/**
* Final iterator.
*/
const_iterator end () const;
/**
* Iterator starting at the first entry of row <tt>r</tt>.
*/
const_iterator begin (const size_type r) const;
/**
* Final iterator of row <tt>r</tt>.
*/
const_iterator end (const size_type r) const;
//@}
///@name Modifying operators
//@{
/**
* Scale the entire matrix by a fixed factor.
*/
FullMatrix &operator *= (const number factor);
/**
* Scale the entire matrix by the inverse of the given factor.
*/
FullMatrix &operator /= (const number factor);
/**
* Simple addition of a scaled matrix, i.e. <tt>*this += a*A</tt>.
*
* The matrix <tt>A</tt> may be a full matrix over an arbitrary underlying
* scalar type, as long as its data type is convertible to the data type of
* this matrix.
*/
template <typename number2>
void add (const number a,
const FullMatrix<number2> &A);
/**
* Multiple addition of scaled matrices, i.e. <tt>*this += a*A + b*B</tt>.
*
* The matrices <tt>A</tt> and <tt>B</tt> may be a full matrix over an
* arbitrary underlying scalar type, as long as its data type is convertible
* to the data type of this matrix.
*/
template <typename number2>
void add (const number a,
const FullMatrix<number2> &A,
const number b,
const FullMatrix<number2> &B);
/**
* Multiple addition of scaled matrices, i.e. <tt>*this += a*A + b*B +
* c*C</tt>.
*
* The matrices <tt>A</tt>, <tt>B</tt> and <tt>C</tt> may be a full matrix
* over an arbitrary underlying scalar type, as long as its data type is
* convertible to the data type of this matrix.
*/
template <typename number2>
void add (const number a,
const FullMatrix<number2> &A,
const number b,
const FullMatrix<number2> &B,
const number c,
const FullMatrix<number2> &C);
/**
* Add rectangular block.
*
* A rectangular block of the matrix <tt>src</tt> is added to <tt>this</tt>.
* The upper left corner of the block being copied is
* <tt>(src_offset_i,src_offset_j)</tt>. The upper left corner of the
* copied block is <tt>(dst_offset_i,dst_offset_j)</tt>. The size of the
* rectangular block being copied is the maximum size possible, determined
* either by the size of <tt>this</tt> or <tt>src</tt> and the given
* offsets.
*/
template <typename number2>
void add (const FullMatrix<number2> &src,
const number factor,
const size_type dst_offset_i = 0,
const size_type dst_offset_j = 0,
const size_type src_offset_i = 0,
const size_type src_offset_j = 0);
/**
* Weighted addition of the transpose of <tt>B</tt> to <tt>this</tt>.
*
* <i>A += s B<sup>T</sup></i>
*/
template <typename number2>
void Tadd (const number s,
const FullMatrix<number2> &B);
/**
* Add transpose of a rectangular block.
*
* A rectangular block of the matrix <tt>src</tt> is transposed and
* addedadded to <tt>this</tt>. The upper left corner of the block being
* copied is <tt>(src_offset_i,src_offset_j)</tt> in the coordinates of the
* <b>non</b>-transposed matrix. The upper left corner of the copied block
* is <tt>(dst_offset_i,dst_offset_j)</tt>. The size of the rectangular
* block being copied is the maximum size possible, determined either by the
* size of <tt>this</tt> or <tt>src</tt>.
*/
template <typename number2>
void Tadd (const FullMatrix<number2> &src,
const number factor,
const size_type dst_offset_i = 0,
const size_type dst_offset_j = 0,
const size_type src_offset_i = 0,
const size_type src_offset_j = 0);
/**
* Add a single element at the given position.
*/
void add (const size_type row,
const size_type column,
const number value);
/**
* Add an array of values given by <tt>values</tt> in the given global
* matrix row at columns specified by col_indices in the full matrix. This
* function is present for compatibility with the various sparse matrices in
* deal.II. In particular, the two boolean fields @p elide_zero_values and
* @p col_indices_are_sorted do not impact the performance of this routine,
* as opposed to the sparse matrix case and are indeed ignored in the
* implementation.
*/
template <typename number2, typename index_type>
void add (const size_type row,
const unsigned int n_cols,
const index_type *col_indices,
const number2 *values,
const bool elide_zero_values = true,
const bool col_indices_are_sorted = false);
/**
* <i>A(i,1...n) += s*A(j,1...n)</i>. Simple addition of rows of this
*/
void add_row (const size_type i,
const number s,
const size_type j);
/**
* <i>A(i,1...n) += s*A(j,1...n) + t*A(k,1...n)</i>. Multiple addition of
* rows of this.
*/
void add_row (const size_type i,
const number s, const size_type j,
const number t, const size_type k);
/**
* <i>A(1...n,i) += s*A(1...n,j)</i>. Simple addition of columns of this.
*/
void add_col (const size_type i,
const number s,
const size_type j);
/**
* <i>A(1...n,i) += s*A(1...n,j) + t*A(1...n,k)</i>. Multiple addition of
* columns of this.
*/
void add_col (const size_type i,
const number s, const size_type j,
const number t, const size_type k);
/**
* Swap <i>A(i,1...n) <-> A(j,1...n)</i>. Swap rows i and j of this
*/
void swap_row (const size_type i,
const size_type j);
/**
* Swap <i>A(1...n,i) <-> A(1...n,j)</i>. Swap columns i and j of this
*/
void swap_col (const size_type i,
const size_type j);
/**
* Add constant to diagonal elements of this, i.e. add a multiple of the
* identity matrix.
*/
void diagadd (const number s);
/**
* Assignment <tt>*this = a*A</tt>.
*/
template <typename number2>
void equ (const number a,
const FullMatrix<number2> &A);
/**
* Assignment <tt>*this = a*A + b*B</tt>.
*/
template <typename number2>
void equ (const number a,
const FullMatrix<number2> &A,
const number b,
const FullMatrix<number2> &B);
/**
* Assignment <tt>*this = a*A + b*B + c*C</tt>.
*/
template <typename number2>
void equ (const number a,
const FullMatrix<number2> &A,
const number b,
const FullMatrix<number2> &B,
const number c,
const FullMatrix<number2> &C);
/**
* Symmetrize the matrix by forming the mean value between the existing
* matrix and its transpose, <i>A = 1/2(A+A<sup>T</sup>)</i>.
*
* Obviously the matrix must be quadratic for this operation.
*/
void symmetrize ();
/**
* A=Inverse(A). A must be a square matrix. Inversion of this matrix by
* Gauss-Jordan algorithm with partial pivoting. This process is well-
* behaved for positive definite matrices, but be aware of round-off errors
* in the indefinite case.
*
* In case deal.II was configured with LAPACK, the functions Xgetrf and
* Xgetri build an LU factorization and invert the matrix upon that
* factorization, providing best performance up to matrices with a few
* hundreds rows and columns.
*
* The numerical effort to invert an <tt>n x n</tt> matrix is of the order
* <tt>n**3</tt>.
*/
void gauss_jordan ();
/**
* Assign the inverse of the given matrix to <tt>*this</tt>. This function
* is hardcoded for quadratic matrices of dimension one to four. However,
* since the amount of code needed grows quickly, the method gauss_jordan()
* is invoked implicitly if the dimension is larger.
*/
template <typename number2>
void invert (const FullMatrix<number2> &M);
/**
* Assign the Cholesky decomposition of the given matrix to <tt>*this</tt>.
* The given matrix must be symmetric positive definite.
*
* ExcMatrixNotPositiveDefinite will be thrown in the case that the matrix
* is not positive definite.
*/
template <typename number2>
void cholesky (const FullMatrix<number2> &A);
/**
* <tt>*this(i,j)</tt> = $V(i) W(j)$ where $V,W$ are vectors of the same
* length.
*/
template <typename number2>
void outer_product (const Vector<number2> &V,
const Vector<number2> &W);
/**
* Assign the left_inverse of the given matrix to <tt>*this</tt>. The
* calculation being performed is <i>(A<sup>T</sup>*A)<sup>-1</sup>
* *A<sup>T</sup></i>.
*/
template <typename number2>
void left_invert (const FullMatrix<number2> &M);
/**
* Assign the right_inverse of the given matrix to <tt>*this</tt>. The
* calculation being performed is <i>A<sup>T</sup>*(A*A<sup>T</sup>)
* <sup>-1</sup></i>.
*/
template <typename number2>
void right_invert (const FullMatrix<number2> &M);
//@}
///@name Multiplications
//@{
/**
* Matrix-matrix-multiplication.
*
* The optional parameter <tt>adding</tt> determines, whether the result is
* stored in <tt>C</tt> or added to <tt>C</tt>.
*
* if (adding) <i>C += A*B</i>
*
* if (!adding) <i>C = A*B</i>
*
* Assumes that <tt>A</tt> and <tt>B</tt> have compatible sizes and that
* <tt>C</tt> already has the right size.
*
* This function uses the BLAS function Xgemm if the product of the three
* matrix dimensions is larger than 300 and BLAS was detected during
* configuration. Using BLAS usually results in considerable performance
* gains.
*/
template <typename number2>
void mmult (FullMatrix<number2> &C,
const FullMatrix<number2> &B,
const bool adding=false) const;
/**
* Matrix-matrix-multiplication using transpose of <tt>this</tt>.
*
* The optional parameter <tt>adding</tt> determines, whether the result is
* stored in <tt>C</tt> or added to <tt>C</tt>.
*
* if (adding) <i>C += A<sup>T</sup>*B</i>
*
* if (!adding) <i>C = A<sup>T</sup>*B</i>
*
* Assumes that <tt>A</tt> and <tt>B</tt> have compatible sizes and that
* <tt>C</tt> already has the right size.
*
* This function uses the BLAS function Xgemm if the product of the three
* matrix dimensions is larger than 300 and BLAS was detected during
* configuration. Using BLAS usually results in considerable performance
* gains.
*/
template <typename number2>
void Tmmult (FullMatrix<number2> &C,
const FullMatrix<number2> &B,
const bool adding=false) const;
/**
* Matrix-matrix-multiplication using transpose of <tt>B</tt>.
*
* The optional parameter <tt>adding</tt> determines, whether the result is
* stored in <tt>C</tt> or added to <tt>C</tt>.
*
* if (adding) <i>C += A*B<sup>T</sup></i>
*
* if (!adding) <i>C = A*B<sup>T</sup></i>
*
* Assumes that <tt>A</tt> and <tt>B</tt> have compatible sizes and that
* <tt>C</tt> already has the right size.
*
* This function uses the BLAS function Xgemm if the product of the three
* matrix dimensions is larger than 300 and BLAS was detected during
* configuration. Using BLAS usually results in considerable performance
* gains.
*/
template <typename number2>
void mTmult (FullMatrix<number2> &C,
const FullMatrix<number2> &B,
const bool adding=false) const;
/**
* Matrix-matrix-multiplication using transpose of <tt>this</tt> and
* <tt>B</tt>.
*
* The optional parameter <tt>adding</tt> determines, whether the result is
* stored in <tt>C</tt> or added to <tt>C</tt>.
*
* if (adding) <i>C += A<sup>T</sup>*B<sup>T</sup></i>
*
* if (!adding) <i>C = A<sup>T</sup>*B<sup>T</sup></i>
*
* Assumes that <tt>A</tt> and <tt>B</tt> have compatible sizes and that
* <tt>C</tt> already has the right size.
*
* This function uses the BLAS function Xgemm if the product of the three
* matrix dimensions is larger than 300 and BLAS was detected during
* configuration. Using BLAS usually results in considerable performance
* gains.
*/
template <typename number2>
void TmTmult (FullMatrix<number2> &C,
const FullMatrix<number2> &B,
const bool adding=false) const;
/**
* Add to the current matrix the triple product <b>B A D</b>. Optionally,
* use the transposes of the matrices <b>B</b> and <b>D</b>. The scaling
* factor scales the whole product, which is helpful when adding a multiple
* of the triple product to the matrix.
*
* This product was written with the Schur complement <b>B<sup>T</sup>
* A<sup>-1</sup> D</b> in mind. Note that in this case the argument for
* <tt>A</tt> must be the inverse of the matrix <b>A</b>.
*/
void triple_product(const FullMatrix<number> &A,
const FullMatrix<number> &B,
const FullMatrix<number> &D,
const bool transpose_B = false,
const bool transpose_D = false,
const number scaling = number(1.));
/**
* Matrix-vector-multiplication.
*
* The optional parameter <tt>adding</tt> determines, whether the result is
* stored in <tt>w</tt> or added to <tt>w</tt>.
*
* if (adding) <i>w += A*v</i>
*
* if (!adding) <i>w = A*v</i>
*
* Source and destination must not be the same vector.
*/
template <typename number2>
void vmult (Vector<number2> &w,
const Vector<number2> &v,
const bool adding=false) const;
/**
* Adding Matrix-vector-multiplication. <i>w += A*v</i>
*
* Source and destination must not be the same vector.
*/
template <typename number2>
void vmult_add (Vector<number2> &w,
const Vector<number2> &v) const;
/**
* Transpose matrix-vector-multiplication.
*
* The optional parameter <tt>adding</tt> determines, whether the result is
* stored in <tt>w</tt> or added to <tt>w</tt>.
*
* if (adding) <i>w += A<sup>T</sup>*v</i>
*
* if (!adding) <i>w = A<sup>T</sup>*v</i>
*
*
* Source and destination must not be the same vector.
*/
template <typename number2>
void Tvmult (Vector<number2> &w,
const Vector<number2> &v,
const bool adding=false) const;
/**
* Adding transpose matrix-vector-multiplication. <i>w +=
* A<sup>T</sup>*v</i>
*
* Source and destination must not be the same vector.
*/
template <typename number2>
void Tvmult_add (Vector<number2> &w,
const Vector<number2> &v) const;
/**
* Apply the Jacobi preconditioner, which multiplies every element of the
* <tt>src</tt> vector by the inverse of the respective diagonal element and
* multiplies the result with the damping factor <tt>omega</tt>.
*/
template <typename somenumber>
void precondition_Jacobi (Vector<somenumber> &dst,
const Vector<somenumber> &src,
const number omega = 1.) const;
/**
* <i>dst=b-A*x</i>. Residual calculation, returns the
* <i>l<sub>2</sub></i>-norm |<i>dst</i>|.
*
* Source <i>x</i> and destination <i>dst</i> must not be the same vector.
*/
template <typename number2, typename number3>
number residual (Vector<number2> &dst,
const Vector<number2> &x,
const Vector<number3> &b) const;
/**
* Forward elimination of lower triangle. Inverts the lower triangle of a
* rectangular matrix for a given right hand side.
*
* If the matrix has more columns than rows, this function only operates on
* the left quadratic submatrix. If there are more rows, the upper quadratic
* part of the matrix is considered.
*
* @note It is safe to use the same object for @p dst and @p src.
*/
template <typename number2>
void forward (Vector<number2> &dst,
const Vector<number2> &src) const;
/**
* Backward elimination of upper triangle.
*
* See forward()
*
* @note It is safe to use the same object for @p dst and @p src.
*/
template <typename number2>
void backward (Vector<number2> &dst,
const Vector<number2> &src) const;
//@}
/**
* @addtogroup Exceptions
* @{
*/
/**
* Exception
*/
DeclException0 (ExcEmptyMatrix);
/**
* Exception
*/
DeclException1 (ExcNotRegular,
number,
<< "The maximal pivot is " << arg1
<< ", which is below the threshold. The matrix may be singular.");
/**
* Exception
*/
DeclException3 (ExcInvalidDestination,
size_type, size_type, size_type,
<< "Target region not in matrix: size in this direction="
<< arg1 << ", size of new matrix=" << arg2
<< ", offset=" << arg3);
/**
* Exception
*/
DeclException0 (ExcSourceEqualsDestination);
/**
* Exception
*/
DeclException0 (ExcMatrixNotPositiveDefinite);
//@}
friend class Accessor;
};
/**@}*/
#ifndef DOXYGEN
/*-------------------------Inline functions -------------------------------*/
template <typename number>
inline
typename FullMatrix<number>::size_type
FullMatrix<number>::m() const
{
return this->n_rows();
}
template <typename number>
inline
typename FullMatrix<number>::size_type
FullMatrix<number>::n() const
{
return this->n_cols();
}
template <typename number>
FullMatrix<number> &
FullMatrix<number>::operator = (const number d)
{
Assert (d==number(0), ExcScalarAssignmentOnlyForZeroValue());
(void)d; // removes -Wunused-parameter warning in optimized mode
if (this->n_elements() != 0)
this->reset_values();
return *this;
}
template <typename number>
template <typename number2>
inline
void FullMatrix<number>::fill (const number2 *src)
{
Table<2,number>::fill(src);
}
template <typename number>
template <typename MatrixType>
void
FullMatrix<number>::copy_from (const MatrixType &M)
{
this->reinit (M.m(), M.n());
// loop over the elements of the argument matrix row by row, as suggested
// in the documentation of the sparse matrix iterator class, and
// copy them into the current object
for (size_type row = 0; row < M.m(); ++row)
{
const typename MatrixType::const_iterator end_row = M.end(row);
for (typename MatrixType::const_iterator entry = M.begin(row);
entry != end_row; ++entry)
this->el(row, entry->column()) = entry->value();
}
}
template <typename number>
template <typename MatrixType>
void
FullMatrix<number>::copy_transposed (const MatrixType &M)
{
this->reinit (M.n(), M.m());
// loop over the elements of the argument matrix row by row, as suggested
// in the documentation of the sparse matrix iterator class, and
// copy them into the current object
for (size_type row = 0; row < M.m(); ++row)
{
const typename MatrixType::const_iterator end_row = M.end(row);
for (typename MatrixType::const_iterator entry = M.begin(row);
entry != end_row; ++entry)
this->el(entry->column(), row) = entry->value();
}
}
template <typename number>
template <typename MatrixType, typename index_type>
inline
void
FullMatrix<number>::extract_submatrix_from (const MatrixType &matrix,
const std::vector<index_type> &row_index_set,
const std::vector<index_type> &column_index_set)
{
AssertDimension(row_index_set.size(), this->n_rows());
AssertDimension(column_index_set.size(), this->n_cols());
const size_type n_rows_submatrix = row_index_set.size();
const size_type n_cols_submatrix = column_index_set.size();
for (size_type sub_row = 0; sub_row < n_rows_submatrix; ++sub_row)
for (size_type sub_col = 0; sub_col < n_cols_submatrix; ++sub_col)
(*this)(sub_row, sub_col) = matrix.el(row_index_set[sub_row], column_index_set[sub_col]);
}
template <typename number>
template <typename MatrixType, typename index_type>
inline
void
FullMatrix<number>::scatter_matrix_to (const std::vector<index_type> &row_index_set,
const std::vector<index_type> &column_index_set,
MatrixType &matrix) const
{
AssertDimension(row_index_set.size(), this->n_rows());
AssertDimension(column_index_set.size(), this->n_cols());
const size_type n_rows_submatrix = row_index_set.size();
const size_type n_cols_submatrix = column_index_set.size();
for (size_type sub_row = 0; sub_row < n_rows_submatrix; ++sub_row)
for (size_type sub_col = 0; sub_col < n_cols_submatrix; ++sub_col)
matrix.set(row_index_set[sub_row],
column_index_set[sub_col],
(*this)(sub_row, sub_col));
}
template <typename number>
inline
void
FullMatrix<number>::set (const size_type i,
const size_type j,
const number value)
{
(*this)(i,j) = value;
}
template <typename number>
template <typename number2>
void
FullMatrix<number>::vmult_add (Vector<number2> &w,
const Vector<number2> &v) const
{
vmult(w, v, true);
}
template <typename number>
template <typename number2>
void
FullMatrix<number>::Tvmult_add (Vector<number2> &w,
const Vector<number2> &v) const
{
Tvmult(w, v, true);
}
//---------------------------------------------------------------------------
template <typename number>
inline
FullMatrix<number>::Accessor::
Accessor (const FullMatrix<number> *matrix,
const size_type r,
const size_type c)
:
matrix(matrix),
a_row(r),
a_col(c)
{}
template <typename number>
inline
typename FullMatrix<number>::size_type
FullMatrix<number>::Accessor::row() const
{
return a_row;
}
template <typename number>
inline
typename FullMatrix<number>::size_type
FullMatrix<number>::Accessor::column() const
{
return a_col;
}
template <typename number>
inline
number
FullMatrix<number>::Accessor::value() const
{
AssertIsFinite(matrix->el(a_row, a_col));
return matrix->el(a_row, a_col);
}
template <typename number>
inline
FullMatrix<number>::const_iterator::
const_iterator(const FullMatrix<number> *matrix,
const size_type r,
const size_type c)
:
accessor(matrix, r, c)
{}
template <typename number>
inline
typename FullMatrix<number>::const_iterator &
FullMatrix<number>::const_iterator::operator++ ()
{
Assert (accessor.a_row < accessor.matrix->m(), ExcIteratorPastEnd());
++accessor.a_col;
if (accessor.a_col >= accessor.matrix->n())
{
accessor.a_col = 0;
accessor.a_row++;
}
return *this;
}
template <typename number>
inline
const typename FullMatrix<number>::Accessor &
FullMatrix<number>::const_iterator::operator* () const
{
return accessor;
}
template <typename number>
inline
const typename FullMatrix<number>::Accessor *
FullMatrix<number>::const_iterator::operator-> () const
{
return &accessor;
}
template <typename number>
inline
bool
FullMatrix<number>::const_iterator::
operator == (const const_iterator &other) const
{
return (accessor.row() == other.accessor.row() &&
accessor.column() == other.accessor.column());
}
template <typename number>
inline
bool
FullMatrix<number>::const_iterator::
operator != (const const_iterator &other) const
{
return ! (*this == other);
}
template <typename number>
inline
bool
FullMatrix<number>::const_iterator::
operator < (const const_iterator &other) const
{
return (accessor.row() < other.accessor.row() ||
(accessor.row() == other.accessor.row() &&
accessor.column() < other.accessor.column()));
}
template <typename number>
inline
bool
FullMatrix<number>::const_iterator::
operator > (const const_iterator &other) const
{
return (other < *this);
}
template <typename number>
inline
typename FullMatrix<number>::const_iterator
FullMatrix<number>::begin () const
{
return const_iterator(this, 0, 0);
}
template <typename number>
inline
typename FullMatrix<number>::const_iterator
FullMatrix<number>::end () const
{
return const_iterator(this, m(), 0);
}
template <typename number>
inline
typename FullMatrix<number>::const_iterator
FullMatrix<number>::begin (const size_type r) const
{
Assert (r<m(), ExcIndexRange(r,0,m()));
return const_iterator(this, r, 0);
}
template <typename number>
inline
typename FullMatrix<number>::const_iterator
FullMatrix<number>::end (const size_type r) const
{
Assert (r<m(), ExcIndexRange(r,0,m()));
return const_iterator(this, r+1, 0);
}
template <typename number>
inline
void
FullMatrix<number>::add (const size_type r, const size_type c, const number v)
{
AssertIndexRange(r, this->m());
AssertIndexRange(c, this->n());
this->operator()(r,c) += v;
}
template <typename number>
template <typename number2, typename index_type>
inline
void
FullMatrix<number>::add (const size_type row,
const unsigned int n_cols,
const index_type *col_indices,
const number2 *values,
const bool,
const bool)
{
AssertIndexRange(row, this->m());
for (size_type col=0; col<n_cols; ++col)
{
AssertIndexRange(col_indices[col], this->n());
this->operator()(row,col_indices[col]) += values[col];
}
}
template <typename number>
template <class StreamType>
inline
void
FullMatrix<number>::print (StreamType &s,
const unsigned int w,
const unsigned int p) const
{
Assert (!this->empty(), ExcEmptyMatrix());
// save the state of out stream
const unsigned int old_precision = s.precision (p);
const unsigned int old_width = s.width (w);
for (size_type i=0; i<this->m(); ++i)
{
for (size_type j=0; j<this->n(); ++j)
{
s.width(w);
s.precision(p);
s << this->el(i,j);
}
s << std::endl;
}
// reset output format
s.precision(old_precision);
s.width(old_width);
}
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
#endif
|