/usr/include/vigra/linear_solve.hxx is in libvigraimpex-dev 1.9.0+dfsg-10+b2.
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 | /************************************************************************/
/* */
/* Copyright 2003-2008 by Gunnar Kedenburg and Ullrich Koethe */
/* */
/* This file is part of the VIGRA computer vision library. */
/* The VIGRA Website is */
/* http://hci.iwr.uni-heidelberg.de/vigra/ */
/* Please direct questions, bug reports, and contributions to */
/* ullrich.koethe@iwr.uni-heidelberg.de or */
/* vigra@informatik.uni-hamburg.de */
/* */
/* Permission is hereby granted, free of charge, to any person */
/* obtaining a copy of this software and associated documentation */
/* files (the "Software"), to deal in the Software without */
/* restriction, including without limitation the rights to use, */
/* copy, modify, merge, publish, distribute, sublicense, and/or */
/* sell copies of the Software, and to permit persons to whom the */
/* Software is furnished to do so, subject to the following */
/* conditions: */
/* */
/* The above copyright notice and this permission notice shall be */
/* included in all copies or substantial portions of the */
/* Software. */
/* */
/* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND */
/* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES */
/* OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND */
/* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT */
/* HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, */
/* WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING */
/* FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR */
/* OTHER DEALINGS IN THE SOFTWARE. */
/* */
/************************************************************************/
#ifndef VIGRA_LINEAR_SOLVE_HXX
#define VIGRA_LINEAR_SOLVE_HXX
#include <ctype.h>
#include <string>
#include "mathutil.hxx"
#include "matrix.hxx"
#include "singular_value_decomposition.hxx"
namespace vigra
{
namespace linalg
{
namespace detail {
template <class T, class C1>
T determinantByLUDecomposition(MultiArrayView<2, T, C1> const & a)
{
typedef MultiArrayShape<2>::type Shape;
MultiArrayIndex m = rowCount(a), n = columnCount(a);
vigra_precondition(n == m,
"determinant(): square matrix required.");
Matrix<T> LU(a);
T det = 1.0;
for (MultiArrayIndex j = 0; j < n; ++j)
{
// Apply previous transformations.
for (MultiArrayIndex i = 0; i < m; ++i)
{
MultiArrayIndex end = std::min(i, j);
T s = dot(rowVector(LU, Shape(i,0), end), columnVector(LU, Shape(0,j), end));
LU(i,j) = LU(i,j) -= s;
}
// Find pivot and exchange if necessary.
MultiArrayIndex p = j + argMax(abs(columnVector(LU, Shape(j,j), m)));
if (p != j)
{
rowVector(LU, p).swapData(rowVector(LU, j));
det = -det;
}
det *= LU(j,j);
// Compute multipliers.
if (LU(j,j) != 0.0)
columnVector(LU, Shape(j+1,j), m) /= LU(j,j);
else
break; // det is zero
}
return det;
}
// returns the new value of 'a' (when this Givens rotation is applied to 'a' and 'b')
// the new value of 'b' is zero, of course
template <class T>
T givensCoefficients(T a, T b, T & c, T & s)
{
if(abs(a) < abs(b))
{
T t = a/b,
r = std::sqrt(1.0 + t*t);
s = 1.0 / r;
c = t*s;
return r*b;
}
else if(a != 0.0)
{
T t = b/a,
r = std::sqrt(1.0 + t*t);
c = 1.0 / r;
s = t*c;
return r*a;
}
else // a == b == 0.0
{
c = 1.0;
s = 0.0;
return 0.0;
}
}
// see Golub, van Loan: Algorithm 5.1.3 (p. 216)
template <class T>
bool givensRotationMatrix(T a, T b, Matrix<T> & gTranspose)
{
if(b == 0.0)
return false; // no rotation needed
givensCoefficients(a, b, gTranspose(0,0), gTranspose(0,1));
gTranspose(1,1) = gTranspose(0,0);
gTranspose(1,0) = -gTranspose(0,1);
return true;
}
// reflections are symmetric matrices and can thus be applied to rows
// and columns in the same way => code simplification relative to rotations
template <class T>
inline bool
givensReflectionMatrix(T a, T b, Matrix<T> & g)
{
if(b == 0.0)
return false; // no reflection needed
givensCoefficients(a, b, g(0,0), g(0,1));
g(1,1) = -g(0,0);
g(1,0) = g(0,1);
return true;
}
// see Golub, van Loan: Algorithm 5.2.2 (p. 227) and Section 12.5.2 (p. 608)
template <class T, class C1, class C2>
bool
qrGivensStepImpl(MultiArrayIndex i, MultiArrayView<2, T, C1> r, MultiArrayView<2, T, C2> rhs)
{
typedef typename Matrix<T>::difference_type Shape;
const MultiArrayIndex m = rowCount(r);
const MultiArrayIndex n = columnCount(r);
const MultiArrayIndex rhsCount = columnCount(rhs);
vigra_precondition(m == rowCount(rhs),
"qrGivensStepImpl(): Matrix shape mismatch.");
Matrix<T> givens(2,2);
for(int k=m-1; k>(int)i; --k)
{
if(!givensReflectionMatrix(r(k-1,i), r(k,i), givens))
continue; // r(k,i) was already zero
r(k-1,i) = givens(0,0)*r(k-1,i) + givens(0,1)*r(k,i);
r(k,i) = 0.0;
r.subarray(Shape(k-1,i+1), Shape(k+1,n)) = givens*r.subarray(Shape(k-1,i+1), Shape(k+1,n));
rhs.subarray(Shape(k-1,0), Shape(k+1,rhsCount)) = givens*rhs.subarray(Shape(k-1,0), Shape(k+1,rhsCount));
}
return r(i,i) != 0.0;
}
// see Golub, van Loan: Section 12.5.2 (p. 608)
template <class T, class C1, class C2, class Permutation>
void
upperTriangularCyclicShiftColumns(MultiArrayIndex i, MultiArrayIndex j,
MultiArrayView<2, T, C1> &r, MultiArrayView<2, T, C2> &rhs, Permutation & permutation)
{
typedef typename Matrix<T>::difference_type Shape;
const MultiArrayIndex m = rowCount(r);
const MultiArrayIndex n = columnCount(r);
const MultiArrayIndex rhsCount = columnCount(rhs);
vigra_precondition(i < n && j < n,
"upperTriangularCyclicShiftColumns(): Shift indices out of range.");
vigra_precondition(m == rowCount(rhs),
"upperTriangularCyclicShiftColumns(): Matrix shape mismatch.");
if(j == i)
return;
if(j < i)
std::swap(j,i);
Matrix<T> t = columnVector(r, i);
MultiArrayIndex ti = permutation[i];
for(MultiArrayIndex k=i; k<j;++k)
{
columnVector(r, k) = columnVector(r, k+1);
permutation[k] = permutation[k+1];
}
columnVector(r, j) = t;
permutation[j] = ti;
Matrix<T> givens(2,2);
for(MultiArrayIndex k=i; k<j; ++k)
{
if(!givensReflectionMatrix(r(k,k), r(k+1,k), givens))
continue; // r(k+1,k) was already zero
r(k,k) = givens(0,0)*r(k,k) + givens(0,1)*r(k+1,k);
r(k+1,k) = 0.0;
r.subarray(Shape(k,k+1), Shape(k+2,n)) = givens*r.subarray(Shape(k,k+1), Shape(k+2,n));
rhs.subarray(Shape(k,0), Shape(k+2,rhsCount)) = givens*rhs.subarray(Shape(k,0), Shape(k+2,rhsCount));
}
}
// see Golub, van Loan: Section 12.5.2 (p. 608)
template <class T, class C1, class C2, class Permutation>
void
upperTriangularSwapColumns(MultiArrayIndex i, MultiArrayIndex j,
MultiArrayView<2, T, C1> &r, MultiArrayView<2, T, C2> &rhs, Permutation & permutation)
{
typedef typename Matrix<T>::difference_type Shape;
const MultiArrayIndex m = rowCount(r);
const MultiArrayIndex n = columnCount(r);
const MultiArrayIndex rhsCount = columnCount(rhs);
vigra_precondition(i < n && j < n,
"upperTriangularSwapColumns(): Swap indices out of range.");
vigra_precondition(m == rowCount(rhs),
"upperTriangularSwapColumns(): Matrix shape mismatch.");
if(j == i)
return;
if(j < i)
std::swap(j,i);
columnVector(r, i).swapData(columnVector(r, j));
std::swap(permutation[i], permutation[j]);
Matrix<T> givens(2,2);
for(int k=m-1; k>(int)i; --k)
{
if(!givensReflectionMatrix(r(k-1,i), r(k,i), givens))
continue; // r(k,i) was already zero
r(k-1,i) = givens(0,0)*r(k-1,i) + givens(0,1)*r(k,i);
r(k,i) = 0.0;
r.subarray(Shape(k-1,i+1), Shape(k+1,n)) = givens*r.subarray(Shape(k-1,i+1), Shape(k+1,n));
rhs.subarray(Shape(k-1,0), Shape(k+1,rhsCount)) = givens*rhs.subarray(Shape(k-1,0), Shape(k+1,rhsCount));
}
MultiArrayIndex end = std::min(j, m-1);
for(MultiArrayIndex k=i+1; k<end; ++k)
{
if(!givensReflectionMatrix(r(k,k), r(k+1,k), givens))
continue; // r(k+1,k) was already zero
r(k,k) = givens(0,0)*r(k,k) + givens(0,1)*r(k+1,k);
r(k+1,k) = 0.0;
r.subarray(Shape(k,k+1), Shape(k+2,n)) = givens*r.subarray(Shape(k,k+1), Shape(k+2,n));
rhs.subarray(Shape(k,0), Shape(k+2,rhsCount)) = givens*rhs.subarray(Shape(k,0), Shape(k+2,rhsCount));
}
}
// see Lawson & Hanson: Algorithm H1 (p. 57)
template <class T, class C1, class C2, class U>
bool householderVector(MultiArrayView<2, T, C1> const & v, MultiArrayView<2, T, C2> & u, U & vnorm)
{
vnorm = (v(0,0) > 0.0)
? -norm(v)
: norm(v);
U f = std::sqrt(vnorm*(vnorm - v(0,0)));
if(f == NumericTraits<U>::zero())
{
u.init(NumericTraits<T>::zero());
return false;
}
else
{
u(0,0) = (v(0,0) - vnorm) / f;
for(MultiArrayIndex k=1; k<rowCount(u); ++k)
u(k,0) = v(k,0) / f;
return true;
}
}
// see Lawson & Hanson: Algorithm H1 (p. 57)
template <class T, class C1, class C2, class C3>
bool
qrHouseholderStepImpl(MultiArrayIndex i, MultiArrayView<2, T, C1> & r,
MultiArrayView<2, T, C2> & rhs, MultiArrayView<2, T, C3> & householderMatrix)
{
typedef typename Matrix<T>::difference_type Shape;
const MultiArrayIndex m = rowCount(r);
const MultiArrayIndex n = columnCount(r);
const MultiArrayIndex rhsCount = columnCount(rhs);
vigra_precondition(i < n && i < m,
"qrHouseholderStepImpl(): Index i out of range.");
Matrix<T> u(m-i,1);
T vnorm;
bool nontrivial = householderVector(columnVector(r, Shape(i,i), m), u, vnorm);
r(i,i) = vnorm;
columnVector(r, Shape(i+1,i), m).init(NumericTraits<T>::zero());
if(columnCount(householderMatrix) == n)
columnVector(householderMatrix, Shape(i,i), m) = u;
if(nontrivial)
{
for(MultiArrayIndex k=i+1; k<n; ++k)
columnVector(r, Shape(i,k), m) -= dot(columnVector(r, Shape(i,k), m), u) * u;
for(MultiArrayIndex k=0; k<rhsCount; ++k)
columnVector(rhs, Shape(i,k), m) -= dot(columnVector(rhs, Shape(i,k), m), u) * u;
}
return r(i,i) != 0.0;
}
template <class T, class C1, class C2>
bool
qrColumnHouseholderStep(MultiArrayIndex i, MultiArrayView<2, T, C1> &r, MultiArrayView<2, T, C2> &rhs)
{
Matrix<T> dontStoreHouseholderVectors; // intentionally empty
return qrHouseholderStepImpl(i, r, rhs, dontStoreHouseholderVectors);
}
template <class T, class C1, class C2>
bool
qrRowHouseholderStep(MultiArrayIndex i, MultiArrayView<2, T, C1> &r, MultiArrayView<2, T, C2> & householderMatrix)
{
Matrix<T> dontTransformRHS; // intentionally empty
MultiArrayView<2, T, StridedArrayTag> rt = transpose(r),
ht = transpose(householderMatrix);
return qrHouseholderStepImpl(i, rt, dontTransformRHS, ht);
}
// O(n) algorithm due to Bischof: Incremental Condition Estimation, 1990
template <class T, class C1, class C2, class SNType>
void
incrementalMaxSingularValueApproximation(MultiArrayView<2, T, C1> const & newColumn,
MultiArrayView<2, T, C2> & z, SNType & v)
{
typedef typename Matrix<T>::difference_type Shape;
MultiArrayIndex n = rowCount(newColumn) - 1;
SNType vneu = squaredNorm(newColumn);
T yv = dot(columnVector(newColumn, Shape(0,0),n), columnVector(z, Shape(0,0),n));
// use atan2 as it is robust against overflow/underflow
T t = 0.5*std::atan2(T(2.0*yv), T(sq(v)-vneu)),
s = std::sin(t),
c = std::cos(t);
v = std::sqrt(sq(c*v) + sq(s)*vneu + 2.0*s*c*yv);
columnVector(z, Shape(0,0),n) = c*columnVector(z, Shape(0,0),n) + s*columnVector(newColumn, Shape(0,0),n);
z(n,0) = s*newColumn(n,0);
}
// O(n) algorithm due to Bischof: Incremental Condition Estimation, 1990
template <class T, class C1, class C2, class SNType>
void
incrementalMinSingularValueApproximation(MultiArrayView<2, T, C1> const & newColumn,
MultiArrayView<2, T, C2> & z, SNType & v, double tolerance)
{
typedef typename Matrix<T>::difference_type Shape;
if(v <= tolerance)
{
v = 0.0;
return;
}
MultiArrayIndex n = rowCount(newColumn) - 1;
T gamma = newColumn(n,0);
if(gamma == 0.0)
{
v = 0.0;
return;
}
T yv = dot(columnVector(newColumn, Shape(0,0),n), columnVector(z, Shape(0,0),n));
// use atan2 as it is robust against overflow/underflow
T t = 0.5*std::atan2(T(-2.0*yv), T(squaredNorm(gamma / v) + squaredNorm(yv) - 1.0)),
s = std::sin(t),
c = std::cos(t);
columnVector(z, Shape(0,0),n) *= c;
z(n,0) = (s - c*yv) / gamma;
v *= norm(gamma) / hypot(c*gamma, v*(s - c*yv));
}
// QR algorithm with optional column pivoting
template <class T, class C1, class C2, class C3>
unsigned int
qrTransformToTriangularImpl(MultiArrayView<2, T, C1> & r, MultiArrayView<2, T, C2> & rhs, MultiArrayView<2, T, C3> & householder,
ArrayVector<MultiArrayIndex> & permutation, double epsilon)
{
typedef typename Matrix<T>::difference_type Shape;
typedef typename NormTraits<MultiArrayView<2, T, C1> >::NormType NormType;
typedef typename NormTraits<MultiArrayView<2, T, C1> >::SquaredNormType SNType;
const MultiArrayIndex m = rowCount(r);
const MultiArrayIndex n = columnCount(r);
const MultiArrayIndex maxRank = std::min(m, n);
vigra_precondition(m >= n,
"qrTransformToTriangularImpl(): Coefficient matrix with at least as many rows as columns required.");
const MultiArrayIndex rhsCount = columnCount(rhs);
bool transformRHS = rhsCount > 0;
vigra_precondition(!transformRHS || m == rowCount(rhs),
"qrTransformToTriangularImpl(): RHS matrix shape mismatch.");
bool storeHouseholderSteps = columnCount(householder) > 0;
vigra_precondition(!storeHouseholderSteps || r.shape() == householder.shape(),
"qrTransformToTriangularImpl(): Householder matrix shape mismatch.");
bool pivoting = permutation.size() > 0;
vigra_precondition(!pivoting || n == (MultiArrayIndex)permutation.size(),
"qrTransformToTriangularImpl(): Permutation array size mismatch.");
if(n == 0)
return 0; // trivial solution
Matrix<SNType> columnSquaredNorms;
if(pivoting)
{
columnSquaredNorms.reshape(Shape(1,n));
for(MultiArrayIndex k=0; k<n; ++k)
columnSquaredNorms[k] = squaredNorm(columnVector(r, k));
int pivot = argMax(columnSquaredNorms);
if(pivot != 0)
{
columnVector(r, 0).swapData(columnVector(r, pivot));
std::swap(columnSquaredNorms[0], columnSquaredNorms[pivot]);
std::swap(permutation[0], permutation[pivot]);
}
}
qrHouseholderStepImpl(0, r, rhs, householder);
MultiArrayIndex rank = 1;
NormType maxApproxSingularValue = norm(r(0,0)),
minApproxSingularValue = maxApproxSingularValue;
double tolerance = (epsilon == 0.0)
? m*maxApproxSingularValue*NumericTraits<T>::epsilon()
: epsilon;
bool simpleSingularValueApproximation = (n < 4);
Matrix<T> zmax, zmin;
if(minApproxSingularValue <= tolerance)
{
rank = 0;
pivoting = false;
simpleSingularValueApproximation = true;
}
if(!simpleSingularValueApproximation)
{
zmax.reshape(Shape(m,1));
zmin.reshape(Shape(m,1));
zmax(0,0) = r(0,0);
zmin(0,0) = 1.0 / r(0,0);
}
for(MultiArrayIndex k=1; k<maxRank; ++k)
{
if(pivoting)
{
for(MultiArrayIndex l=k; l<n; ++l)
columnSquaredNorms[l] -= squaredNorm(r(k, l));
int pivot = k + argMax(rowVector(columnSquaredNorms, Shape(0,k), n));
if(pivot != (int)k)
{
columnVector(r, k).swapData(columnVector(r, pivot));
std::swap(columnSquaredNorms[k], columnSquaredNorms[pivot]);
std::swap(permutation[k], permutation[pivot]);
}
}
qrHouseholderStepImpl(k, r, rhs, householder);
if(simpleSingularValueApproximation)
{
NormType nv = norm(r(k,k));
maxApproxSingularValue = std::max(nv, maxApproxSingularValue);
minApproxSingularValue = std::min(nv, minApproxSingularValue);
}
else
{
incrementalMaxSingularValueApproximation(columnVector(r, Shape(0,k),k+1), zmax, maxApproxSingularValue);
incrementalMinSingularValueApproximation(columnVector(r, Shape(0,k),k+1), zmin, minApproxSingularValue, tolerance);
}
#if 0
Matrix<T> u(k+1,k+1), s(k+1, 1), v(k+1,k+1);
singularValueDecomposition(r.subarray(Shape(0,0), Shape(k+1,k+1)), u, s, v);
std::cerr << "estimate, svd " << k << ": " << minApproxSingularValue << " " << s(k,0) << "\n";
#endif
if(epsilon == 0.0)
tolerance = m*maxApproxSingularValue*NumericTraits<T>::epsilon();
if(minApproxSingularValue > tolerance)
++rank;
else
pivoting = false; // matrix doesn't have full rank, triangulize the rest without pivoting
}
return (unsigned int)rank;
}
template <class T, class C1, class C2>
unsigned int
qrTransformToUpperTriangular(MultiArrayView<2, T, C1> & r, MultiArrayView<2, T, C2> & rhs,
ArrayVector<MultiArrayIndex> & permutation, double epsilon = 0.0)
{
Matrix<T> dontStoreHouseholderVectors; // intentionally empty
return qrTransformToTriangularImpl(r, rhs, dontStoreHouseholderVectors, permutation, epsilon);
}
// QR algorithm with optional row pivoting
template <class T, class C1, class C2, class C3>
unsigned int
qrTransformToLowerTriangular(MultiArrayView<2, T, C1> & r, MultiArrayView<2, T, C2> & rhs, MultiArrayView<2, T, C3> & householderMatrix,
double epsilon = 0.0)
{
ArrayVector<MultiArrayIndex> permutation((unsigned int)rowCount(rhs));
for(MultiArrayIndex k=0; k<(MultiArrayIndex)permutation.size(); ++k)
permutation[k] = k;
Matrix<T> dontTransformRHS; // intentionally empty
MultiArrayView<2, T, StridedArrayTag> rt = transpose(r),
ht = transpose(householderMatrix);
unsigned int rank = qrTransformToTriangularImpl(rt, dontTransformRHS, ht, permutation, epsilon);
// apply row permutation to RHS
Matrix<T> tempRHS(rhs);
for(MultiArrayIndex k=0; k<(MultiArrayIndex)permutation.size(); ++k)
rowVector(rhs, k) = rowVector(tempRHS, permutation[k]);
return rank;
}
// QR algorithm without column pivoting
template <class T, class C1, class C2>
inline bool
qrTransformToUpperTriangular(MultiArrayView<2, T, C1> & r, MultiArrayView<2, T, C2> & rhs,
double epsilon = 0.0)
{
ArrayVector<MultiArrayIndex> noPivoting; // intentionally empty
return (qrTransformToUpperTriangular(r, rhs, noPivoting, epsilon) ==
(unsigned int)columnCount(r));
}
// QR algorithm without row pivoting
template <class T, class C1, class C2>
inline bool
qrTransformToLowerTriangular(MultiArrayView<2, T, C1> & r, MultiArrayView<2, T, C2> & householder,
double epsilon = 0.0)
{
Matrix<T> noPivoting; // intentionally empty
return (qrTransformToLowerTriangular(r, noPivoting, householder, epsilon) ==
(unsigned int)rowCount(r));
}
// restore ordering of result vector elements after QR solution with column pivoting
template <class T, class C1, class C2, class Permutation>
void inverseRowPermutation(MultiArrayView<2, T, C1> &permuted, MultiArrayView<2, T, C2> &res,
Permutation const & permutation)
{
for(MultiArrayIndex k=0; k<columnCount(permuted); ++k)
for(MultiArrayIndex l=0; l<rowCount(permuted); ++l)
res(permutation[l], k) = permuted(l,k);
}
template <class T, class C1, class C2>
void applyHouseholderColumnReflections(MultiArrayView<2, T, C1> const &householder, MultiArrayView<2, T, C2> &res)
{
typedef typename Matrix<T>::difference_type Shape;
MultiArrayIndex n = rowCount(householder);
MultiArrayIndex m = columnCount(householder);
MultiArrayIndex rhsCount = columnCount(res);
for(int k = m-1; k >= 0; --k)
{
MultiArrayView<2, T, C1> u = columnVector(householder, Shape(k,k), n);
for(MultiArrayIndex l=0; l<rhsCount; ++l)
columnVector(res, Shape(k,l), n) -= dot(columnVector(res, Shape(k,l), n), u) * u;
}
}
} // namespace detail
template <class T, class C1, class C2, class C3>
unsigned int
linearSolveQRReplace(MultiArrayView<2, T, C1> &A, MultiArrayView<2, T, C2> &b,
MultiArrayView<2, T, C3> & res,
double epsilon = 0.0)
{
typedef typename Matrix<T>::difference_type Shape;
MultiArrayIndex n = columnCount(A);
MultiArrayIndex m = rowCount(A);
MultiArrayIndex rhsCount = columnCount(res);
MultiArrayIndex rank = std::min(m,n);
Shape ul(MultiArrayIndex(0), MultiArrayIndex(0));
vigra_precondition(rhsCount == columnCount(b),
"linearSolveQR(): RHS and solution must have the same number of columns.");
vigra_precondition(m == rowCount(b),
"linearSolveQR(): Coefficient matrix and RHS must have the same number of rows.");
vigra_precondition(n == rowCount(res),
"linearSolveQR(): Mismatch between column count of coefficient matrix and row count of solution.");
vigra_precondition(epsilon >= 0.0,
"linearSolveQR(): 'epsilon' must be non-negative.");
if(m < n)
{
// minimum norm solution of underdetermined system
Matrix<T> householderMatrix(n, m);
MultiArrayView<2, T, StridedArrayTag> ht = transpose(householderMatrix);
rank = (MultiArrayIndex)detail::qrTransformToLowerTriangular(A, b, ht, epsilon);
res.subarray(Shape(rank,0), Shape(n, rhsCount)).init(NumericTraits<T>::zero());
if(rank < m)
{
// system is also rank-deficient => compute minimum norm least squares solution
MultiArrayView<2, T, C1> Asub = A.subarray(ul, Shape(m,rank));
detail::qrTransformToUpperTriangular(Asub, b, epsilon);
linearSolveUpperTriangular(A.subarray(ul, Shape(rank,rank)),
b.subarray(ul, Shape(rank,rhsCount)),
res.subarray(ul, Shape(rank, rhsCount)));
}
else
{
// system has full rank => compute minimum norm solution
linearSolveLowerTriangular(A.subarray(ul, Shape(rank,rank)),
b.subarray(ul, Shape(rank, rhsCount)),
res.subarray(ul, Shape(rank, rhsCount)));
}
detail::applyHouseholderColumnReflections(householderMatrix.subarray(ul, Shape(n, rank)), res);
}
else
{
// solution of well-determined or overdetermined system
ArrayVector<MultiArrayIndex> permutation((unsigned int)n);
for(MultiArrayIndex k=0; k<n; ++k)
permutation[k] = k;
rank = (MultiArrayIndex)detail::qrTransformToUpperTriangular(A, b, permutation, epsilon);
Matrix<T> permutedSolution(n, rhsCount);
if(rank < n)
{
// system is rank-deficient => compute minimum norm solution
Matrix<T> householderMatrix(n, rank);
MultiArrayView<2, T, StridedArrayTag> ht = transpose(householderMatrix);
MultiArrayView<2, T, C1> Asub = A.subarray(ul, Shape(rank,n));
detail::qrTransformToLowerTriangular(Asub, ht, epsilon);
linearSolveLowerTriangular(A.subarray(ul, Shape(rank,rank)),
b.subarray(ul, Shape(rank, rhsCount)),
permutedSolution.subarray(ul, Shape(rank, rhsCount)));
detail::applyHouseholderColumnReflections(householderMatrix, permutedSolution);
}
else
{
// system has full rank => compute exact or least squares solution
linearSolveUpperTriangular(A.subarray(ul, Shape(rank,rank)),
b.subarray(ul, Shape(rank,rhsCount)),
permutedSolution);
}
detail::inverseRowPermutation(permutedSolution, res, permutation);
}
return (unsigned int)rank;
}
template <class T, class C1, class C2, class C3>
unsigned int linearSolveQR(MultiArrayView<2, T, C1> const & A, MultiArrayView<2, T, C2> const & b,
MultiArrayView<2, T, C3> & res)
{
Matrix<T> r(A), rhs(b);
return linearSolveQRReplace(r, rhs, res);
}
/** \defgroup MatrixAlgebra Advanced Matrix Algebra
\brief Solution of linear systems, eigen systems, linear least squares etc.
\ingroup LinearAlgebraModule
*/
//@{
/** Create the inverse or pseudo-inverse of matrix \a v.
If the matrix \a v is square, \a res must have the same shape and will contain the
inverse of \a v. If \a v is rectangular, \a res must have the transposed shape
of \a v. The inverse is then computed in the least-squares
sense, i.e. \a res will be the pseudo-inverse (Moore-Penrose inverse).
The function returns <tt>true</tt> upon success, and <tt>false</tt> if \a v
is not invertible (has not full rank). The inverse is computed by means of QR
decomposition. This function can be applied in-place.
<b>\#include</b> \<vigra/linear_solve.hxx\> or<br>
<b>\#include</b> \<vigra/linear_algebra.hxx\><br>
Namespaces: vigra and vigra::linalg
*/
template <class T, class C1, class C2>
bool inverse(const MultiArrayView<2, T, C1> &v, MultiArrayView<2, T, C2> &res)
{
typedef typename MultiArrayShape<2>::type Shape;
const MultiArrayIndex n = columnCount(v);
const MultiArrayIndex m = rowCount(v);
vigra_precondition(n == rowCount(res) && m == columnCount(res),
"inverse(): shape of output matrix must be the transpose of the input matrix' shape.");
if(m < n)
{
MultiArrayView<2, T, StridedArrayTag> vt = transpose(v);
Matrix<T> r(vt.shape()), q(n, n);
if(!qrDecomposition(vt, q, r))
return false; // a didn't have full rank
linearSolveUpperTriangular(r.subarray(Shape(0,0), Shape(m,m)),
transpose(q).subarray(Shape(0,0), Shape(m,n)),
transpose(res));
}
else
{
Matrix<T> r(v.shape()), q(m, m);
if(!qrDecomposition(v, q, r))
return false; // a didn't have full rank
linearSolveUpperTriangular(r.subarray(Shape(0,0), Shape(n,n)),
transpose(q).subarray(Shape(0,0), Shape(n,m)),
res);
}
return true;
}
/** Create the inverse or pseudo-inverse of matrix \a v.
The result is returned as a temporary matrix. If the matrix \a v is square,
the result will have the same shape and contains the inverse of \a v.
If \a v is rectangular, the result will have the transposed shape of \a v.
The inverse is then computed in the least-squares
sense, i.e. \a res will be the pseudo-inverse (Moore-Penrose inverse).
The inverse is computed by means of QR decomposition. If \a v
is not invertible, <tt>vigra::PreconditionViolation</tt> exception is thrown.
Usage:
\code
vigra::Matrix<double> v(n, n);
v = ...;
vigra::Matrix<double> m = inverse(v);
\endcode
<b>\#include</b> \<vigra/linear_solve.hxx\> or<br>
<b>\#include</b> \<vigra/linear_algebra.hxx\><br>
Namespaces: vigra and vigra::linalg
*/
template <class T, class C>
TemporaryMatrix<T> inverse(const MultiArrayView<2, T, C> &v)
{
TemporaryMatrix<T> ret(columnCount(v), rowCount(v)); // transpose shape
vigra_precondition(inverse(v, ret),
"inverse(): matrix is not invertible.");
return ret;
}
template <class T>
TemporaryMatrix<T> inverse(const TemporaryMatrix<T> &v)
{
if(columnCount(v) == rowCount(v))
{
vigra_precondition(inverse(v, const_cast<TemporaryMatrix<T> &>(v)),
"inverse(): matrix is not invertible.");
return v;
}
else
{
TemporaryMatrix<T> ret(columnCount(v), rowCount(v)); // transpose shape
vigra_precondition(inverse(v, ret),
"inverse(): matrix is not invertible.");
return ret;
}
}
/** Compute the determinant of a square matrix.
\a method must be one of the following:
<DL>
<DT>"Cholesky"<DD> Compute the solution by means of Cholesky decomposition. This
method is faster than "LU", but requires the matrix \a a
to be symmetric positive definite. If this is
not the case, a <tt>ContractViolation</tt> exception is thrown.
<DT>"LU"<DD> (default) Compute the solution by means of LU decomposition.
</DL>
<b>\#include</b> \<vigra/linear_solve.hxx\> or<br>
<b>\#include</b> \<vigra/linear_algebra.hxx\><br>
Namespaces: vigra and vigra::linalg
*/
template <class T, class C1>
T determinant(MultiArrayView<2, T, C1> const & a, std::string method = "LU")
{
MultiArrayIndex n = columnCount(a);
vigra_precondition(rowCount(a) == n,
"determinant(): Square matrix required.");
method = tolower(method);
if(n == 1)
return a(0,0);
if(n == 2)
return a(0,0)*a(1,1) - a(0,1)*a(1,0);
if(method == "lu")
{
return detail::determinantByLUDecomposition(a);
}
else if(method == "cholesky")
{
Matrix<T> L(a.shape());
vigra_precondition(choleskyDecomposition(a, L),
"determinant(): Cholesky method requires symmetric positive definite matrix.");
T det = L(0,0);
for(MultiArrayIndex k=1; k<n; ++k)
det *= L(k,k);
return sq(det);
}
else
{
vigra_precondition(false, "determinant(): Unknown solution method.");
}
return T();
}
/** Compute the logarithm of the determinant of a symmetric positive definite matrix.
This is useful to avoid multiplication of very large numbers in big matrices.
It is implemented by means of Cholesky decomposition.
<b>\#include</b> \<vigra/linear_solve.hxx\> or<br>
<b>\#include</b> \<vigra/linear_algebra.hxx\><br>
Namespaces: vigra and vigra::linalg
*/
template <class T, class C1>
T logDeterminant(MultiArrayView<2, T, C1> const & a)
{
MultiArrayIndex n = columnCount(a);
vigra_precondition(rowCount(a) == n,
"logDeterminant(): Square matrix required.");
if(n == 1)
{
vigra_precondition(a(0,0) > 0.0,
"logDeterminant(): Matrix not positive definite.");
return std::log(a(0,0));
}
if(n == 2)
{
T det = a(0,0)*a(1,1) - a(0,1)*a(1,0);
vigra_precondition(det > 0.0,
"logDeterminant(): Matrix not positive definite.");
return std::log(det);
}
else
{
Matrix<T> L(a.shape());
vigra_precondition(choleskyDecomposition(a, L),
"logDeterminant(): Matrix not positive definite.");
T logdet = std::log(L(0,0));
for(MultiArrayIndex k=1; k<n; ++k)
logdet += std::log(L(k,k)); // L(k,k) is guaranteed to be positive
return 2.0*logdet;
}
}
/** Cholesky decomposition.
\a A must be a symmetric positive definite matrix, and \a L will be a lower
triangular matrix, such that (up to round-off errors):
\code
A == L * transpose(L);
\endcode
This implementation cannot be applied in-place, i.e. <tt>&L == &A</tt> is an error.
If \a A is not symmetric, a <tt>ContractViolation</tt> exception is thrown. If it
is not positive definite, the function returns <tt>false</tt>.
<b>\#include</b> \<vigra/linear_solve.hxx\> or<br>
<b>\#include</b> \<vigra/linear_algebra.hxx\><br>
Namespaces: vigra and vigra::linalg
*/
template <class T, class C1, class C2>
bool choleskyDecomposition(MultiArrayView<2, T, C1> const & A,
MultiArrayView<2, T, C2> &L)
{
MultiArrayIndex n = columnCount(A);
vigra_precondition(rowCount(A) == n,
"choleskyDecomposition(): Input matrix must be square.");
vigra_precondition(n == columnCount(L) && n == rowCount(L),
"choleskyDecomposition(): Output matrix must have same shape as input matrix.");
vigra_precondition(isSymmetric(A),
"choleskyDecomposition(): Input matrix must be symmetric.");
for (MultiArrayIndex j = 0; j < n; ++j)
{
T d(0.0);
for (MultiArrayIndex k = 0; k < j; ++k)
{
T s(0.0);
for (MultiArrayIndex i = 0; i < k; ++i)
{
s += L(k, i)*L(j, i);
}
L(j, k) = s = (A(j, k) - s)/L(k, k);
d = d + s*s;
}
d = A(j, j) - d;
if(d <= 0.0)
return false; // A is not positive definite
L(j, j) = std::sqrt(d);
for (MultiArrayIndex k = j+1; k < n; ++k)
{
L(j, k) = 0.0;
}
}
return true;
}
/** QR decomposition.
\a a contains the original matrix, results are returned in \a q and \a r, where
\a q is a orthogonal matrix, and \a r is an upper triangular matrix, such that
(up to round-off errors):
\code
a == q * r;
\endcode
If \a a doesn't have full rank, the function returns <tt>false</tt>.
The decomposition is computed by householder transformations. It can be applied in-place,
i.e. <tt>&a == &q</tt> or <tt>&a == &r</tt> are allowed.
<b>\#include</b> \<vigra/linear_solve.hxx\> or<br>
<b>\#include</b> \<vigra/linear_algebra.hxx\><br>
Namespaces: vigra and vigra::linalg
*/
template <class T, class C1, class C2, class C3>
bool qrDecomposition(MultiArrayView<2, T, C1> const & a,
MultiArrayView<2, T, C2> &q, MultiArrayView<2, T, C3> &r,
double epsilon = 0.0)
{
const MultiArrayIndex m = rowCount(a);
const MultiArrayIndex n = columnCount(a);
vigra_precondition(n == columnCount(r) && m == rowCount(r) &&
m == columnCount(q) && m == rowCount(q),
"qrDecomposition(): Matrix shape mismatch.");
q = identityMatrix<T>(m);
MultiArrayView<2,T, StridedArrayTag> tq = transpose(q);
r = a;
ArrayVector<MultiArrayIndex> noPivoting; // intentionally empty
return ((MultiArrayIndex)detail::qrTransformToUpperTriangular(r, tq, noPivoting, epsilon) == std::min(m,n));
}
/** Deprecated, use \ref linearSolveUpperTriangular().
*/
template <class T, class C1, class C2, class C3>
inline
bool reverseElimination(const MultiArrayView<2, T, C1> &r, const MultiArrayView<2, T, C2> &b,
MultiArrayView<2, T, C3> x)
{
return linearSolveUpperTriangular(r, b, x);
}
/** Solve a linear system with upper-triangular coefficient matrix.
The square matrix \a r must be an upper-triangular coefficient matrix as can,
for example, be obtained by means of QR decomposition. If \a r doesn't have full rank
the function fails and returns <tt>false</tt>, otherwise it returns <tt>true</tt>. The
lower triangular part of matrix \a r will not be touched, so it doesn't need to contain zeros.
The column vectors of matrix \a b are the right-hand sides of the equation (several equations
with the same coefficients can thus be solved in one go). The result is returned
int \a x, whose columns contain the solutions for the corresponding
columns of \a b. This implementation can be applied in-place, i.e. <tt>&b == &x</tt> is allowed.
The following size requirements apply:
\code
rowCount(r) == columnCount(r);
rowCount(r) == rowCount(b);
columnCount(r) == rowCount(x);
columnCount(b) == columnCount(x);
\endcode
<b>\#include</b> \<vigra/linear_solve.hxx\> or<br>
<b>\#include</b> \<vigra/linear_algebra.hxx\><br>
Namespaces: vigra and vigra::linalg
*/
template <class T, class C1, class C2, class C3>
bool linearSolveUpperTriangular(const MultiArrayView<2, T, C1> &r, const MultiArrayView<2, T, C2> &b,
MultiArrayView<2, T, C3> x)
{
typedef MultiArrayShape<2>::type Shape;
MultiArrayIndex m = rowCount(r);
MultiArrayIndex rhsCount = columnCount(b);
vigra_precondition(m == columnCount(r),
"linearSolveUpperTriangular(): square coefficient matrix required.");
vigra_precondition(m == rowCount(b) && m == rowCount(x) && rhsCount == columnCount(x),
"linearSolveUpperTriangular(): matrix shape mismatch.");
for(MultiArrayIndex k = 0; k < rhsCount; ++k)
{
for(int i=m-1; i>=0; --i)
{
if(r(i,i) == NumericTraits<T>::zero())
return false; // r doesn't have full rank
T sum = b(i, k);
for(MultiArrayIndex j=i+1; j<m; ++j)
sum -= r(i, j) * x(j, k);
x(i, k) = sum / r(i, i);
}
}
return true;
}
/** Solve a linear system with lower-triangular coefficient matrix.
The square matrix \a l must be a lower-triangular coefficient matrix. If \a l
doesn't have full rank the function fails and returns <tt>false</tt>,
otherwise it returns <tt>true</tt>. The upper triangular part of matrix \a l will not be touched,
so it doesn't need to contain zeros.
The column vectors of matrix \a b are the right-hand sides of the equation (several equations
with the same coefficients can thus be solved in one go). The result is returned
in \a x, whose columns contain the solutions for the corresponding
columns of \a b. This implementation can be applied in-place, i.e. <tt>&b == &x</tt> is allowed.
The following size requirements apply:
\code
rowCount(l) == columnCount(l);
rowCount(l) == rowCount(b);
columnCount(l) == rowCount(x);
columnCount(b) == columnCount(x);
\endcode
<b>\#include</b> \<vigra/linear_solve.hxx\> or<br>
<b>\#include</b> \<vigra/linear_algebra.hxx\><br>
Namespaces: vigra and vigra::linalg
*/
template <class T, class C1, class C2, class C3>
bool linearSolveLowerTriangular(const MultiArrayView<2, T, C1> &l, const MultiArrayView<2, T, C2> &b,
MultiArrayView<2, T, C3> x)
{
MultiArrayIndex m = columnCount(l);
MultiArrayIndex n = columnCount(b);
vigra_precondition(m == rowCount(l),
"linearSolveLowerTriangular(): square coefficient matrix required.");
vigra_precondition(m == rowCount(b) && m == rowCount(x) && n == columnCount(x),
"linearSolveLowerTriangular(): matrix shape mismatch.");
for(MultiArrayIndex k = 0; k < n; ++k)
{
for(MultiArrayIndex i=0; i<m; ++i)
{
if(l(i,i) == NumericTraits<T>::zero())
return false; // l doesn't have full rank
T sum = b(i, k);
for(MultiArrayIndex j=0; j<i; ++j)
sum -= l(i, j) * x(j, k);
x(i, k) = sum / l(i, i);
}
}
return true;
}
/** Solve a linear system when the Cholesky decomposition of the left hand side is given.
The square matrix \a L must be a lower-triangular matrix resulting from Cholesky
decomposition of some positive definite coefficient matrix.
The column vectors of matrix \a b are the right-hand sides of the equation (several equations
with the same matrix \a L can thus be solved in one go). The result is returned
in \a x, whose columns contain the solutions for the corresponding
columns of \a b. This implementation can be applied in-place, i.e. <tt>&b == &x</tt> is allowed.
The following size requirements apply:
\code
rowCount(L) == columnCount(L);
rowCount(L) == rowCount(b);
columnCount(L) == rowCount(x);
columnCount(b) == columnCount(x);
\endcode
<b>\#include</b> \<vigra/linear_solve.hxx\> or<br>
<b>\#include</b> \<vigra/linear_algebra.hxx\><br>
Namespaces: vigra and vigra::linalg
*/
template <class T, class C1, class C2, class C3>
inline
void choleskySolve(MultiArrayView<2, T, C1> & L, MultiArrayView<2, T, C2> const & b, MultiArrayView<2, T, C3> & x)
{
/* Solve L * y = b */
linearSolveLowerTriangular(L, b, x);
/* Solve L^T * x = y */
linearSolveUpperTriangular(transpose(L), x, x);
}
/** Solve a linear system.
\a A is the coefficient matrix, and the column vectors
in \a b are the right-hand sides of the equation (so, several equations
with the same coefficients can be solved in one go). The result is returned
in \a res, whose columns contain the solutions for the corresponding
columns of \a b. The number of columns of \a A must equal the number of rows of
both \a b and \a res, and the number of columns of \a b and \a res must match.
\a method must be one of the following:
<DL>
<DT>"Cholesky"<DD> Compute the solution by means of Cholesky decomposition. The
coefficient matrix \a A must by symmetric positive definite. If
this is not the case, the function returns <tt>false</tt>.
<DT>"QR"<DD> (default) Compute the solution by means of QR decomposition. The
coefficient matrix \a A can be square or rectangular. In the latter case,
it must have more rows than columns, and the solution will be computed in the
least squares sense. If \a A doesn't have full rank, the function
returns <tt>false</tt>.
<DT>"SVD"<DD> Compute the solution by means of singular value decomposition. The
coefficient matrix \a A can be square or rectangular. In the latter case,
it must have more rows than columns, and the solution will be computed in the
least squares sense. If \a A doesn't have full rank, the function
returns <tt>false</tt>.
<DT>"NE"<DD> Compute the solution by means of the normal equations, i.e. by applying Cholesky
decomposition to the equivalent problem <tt>A'*A*x = A'*b</tt>. This only makes sense
when the equation is to be solved in the least squares sense, i.e. when \a A is a
rectangular matrix with more rows than columns. If \a A doesn't have full column rank,
the function returns <tt>false</tt>.
</DL>
This function can be applied in-place, i.e. <tt>&b == &res</tt> or <tt>&A == &res</tt> are allowed
(provided they have the required shapes).
The following size requirements apply:
\code
rowCount(r) == rowCount(b);
columnCount(r) == rowCount(x);
columnCount(b) == columnCount(x);
\endcode
<b>\#include</b> \<vigra/linear_solve.hxx\> or<br>
<b>\#include</b> \<vigra/linear_algebra.hxx\><br>
Namespaces: vigra and vigra::linalg
*/
template <class T, class C1, class C2, class C3>
bool linearSolve(const MultiArrayView<2, T, C1> &A, const MultiArrayView<2, T, C2> &b,
MultiArrayView<2, T, C3> & res, std::string method = "QR")
{
typedef typename Matrix<T>::difference_type Shape;
typedef typename Matrix<T>::view_type SubMatrix;
const MultiArrayIndex n = columnCount(A);
const MultiArrayIndex m = rowCount(A);
vigra_precondition(n <= m,
"linearSolve(): Coefficient matrix A must have at least as many rows as columns.");
vigra_precondition(n == rowCount(res) &&
m == rowCount(b) && columnCount(b) == columnCount(res),
"linearSolve(): matrix shape mismatch.");
method = tolower(method);
if(method == "cholesky")
{
vigra_precondition(columnCount(A) == rowCount(A),
"linearSolve(): Cholesky method requires square coefficient matrix.");
Matrix<T> L(A.shape());
if(!choleskyDecomposition(A, L))
return false; // false if A wasn't symmetric positive definite
choleskySolve(L, b, res);
}
else if(method == "qr")
{
return (MultiArrayIndex)linearSolveQR(A, b, res) == n;
}
else if(method == "ne")
{
return linearSolve(transpose(A)*A, transpose(A)*b, res, "Cholesky");
}
else if(method == "svd")
{
MultiArrayIndex rhsCount = columnCount(b);
Matrix<T> u(A.shape()), s(n, 1), v(n, n);
MultiArrayIndex rank = (MultiArrayIndex)singularValueDecomposition(A, u, s, v);
Matrix<T> t = transpose(u)*b;
for(MultiArrayIndex l=0; l<rhsCount; ++l)
{
for(MultiArrayIndex k=0; k<rank; ++k)
t(k,l) /= s(k,0);
for(MultiArrayIndex k=rank; k<n; ++k)
t(k,l) = NumericTraits<T>::zero();
}
res = v*t;
return (rank == n);
}
else
{
vigra_precondition(false, "linearSolve(): Unknown solution method.");
}
return true;
}
//@}
} // namespace linalg
using linalg::inverse;
using linalg::determinant;
using linalg::logDeterminant;
using linalg::linearSolve;
using linalg::choleskySolve;
using linalg::choleskyDecomposition;
using linalg::qrDecomposition;
using linalg::linearSolveUpperTriangular;
using linalg::linearSolveLowerTriangular;
} // namespace vigra
#endif // VIGRA_LINEAR_SOLVE_HXX
|