1 #include <iostream> 2 #include "precomp.hpp" 3 #include "epnp.h" 4 5 namespace cv 6 { 7 8 epnp::epnp(const Mat& cameraMatrix, const Mat& opoints, const Mat& ipoints) 9 { 10 if (cameraMatrix.depth() == CV_32F) 11 init_camera_parameters<float>(cameraMatrix); 12 else 13 init_camera_parameters<double>(cameraMatrix); 14 15 number_of_correspondences = std::max(opoints.checkVector(3, CV_32F), opoints.checkVector(3, CV_64F)); 16 17 pws.resize(3 * number_of_correspondences); 18 us.resize(2 * number_of_correspondences); 19 20 if (opoints.depth() == ipoints.depth()) 21 { 22 if (opoints.depth() == CV_32F) 23 init_points<Point3f,Point2f>(opoints, ipoints); 24 else 25 init_points<Point3d,Point2d>(opoints, ipoints); 26 } 27 else if (opoints.depth() == CV_32F) 28 init_points<Point3f,Point2d>(opoints, ipoints); 29 else 30 init_points<Point3d,Point2f>(opoints, ipoints); 31 32 alphas.resize(4 * number_of_correspondences); 33 pcs.resize(3 * number_of_correspondences); 34 35 max_nr = 0; 36 A1 = NULL; 37 A2 = NULL; 38 } 39 40 epnp::~epnp() 41 { 42 if (A1) 43 delete[] A1; 44 if (A2) 45 delete[] A2; 46 } 47 48 void epnp::choose_control_points(void) 49 { 50 // Take C0 as the reference points centroid: 51 cws[0][0] = cws[0][1] = cws[0][2] = 0; 52 for(int i = 0; i < number_of_correspondences; i++) 53 for(int j = 0; j < 3; j++) 54 cws[0][j] += pws[3 * i + j]; 55 56 for(int j = 0; j < 3; j++) 57 cws[0][j] /= number_of_correspondences; 58 59 60 // Take C1, C2, and C3 from PCA on the reference points: 61 CvMat * PW0 = cvCreateMat(number_of_correspondences, 3, CV_64F); 62 63 double pw0tpw0[3 * 3], dc[3], uct[3 * 3]; 64 CvMat PW0tPW0 = cvMat(3, 3, CV_64F, pw0tpw0); 65 CvMat DC = cvMat(3, 1, CV_64F, dc); 66 CvMat UCt = cvMat(3, 3, CV_64F, uct); 67 68 for(int i = 0; i < number_of_correspondences; i++) 69 for(int j = 0; j < 3; j++) 70 PW0->data.db[3 * i + j] = pws[3 * i + j] - cws[0][j]; 71 72 cvMulTransposed(PW0, &PW0tPW0, 1); 73 cvSVD(&PW0tPW0, &DC, &UCt, 0, CV_SVD_MODIFY_A | CV_SVD_U_T); 74 75 cvReleaseMat(&PW0); 76 77 for(int i = 1; i < 4; i++) { 78 double k = sqrt(dc[i - 1] / number_of_correspondences); 79 for(int j = 0; j < 3; j++) 80 cws[i][j] = cws[0][j] + k * uct[3 * (i - 1) + j]; 81 } 82 } 83 84 void epnp::compute_barycentric_coordinates(void) 85 { 86 double cc[3 * 3], cc_inv[3 * 3]; 87 CvMat CC = cvMat(3, 3, CV_64F, cc); 88 CvMat CC_inv = cvMat(3, 3, CV_64F, cc_inv); 89 90 for(int i = 0; i < 3; i++) 91 for(int j = 1; j < 4; j++) 92 cc[3 * i + j - 1] = cws[j][i] - cws[0][i]; 93 94 cvInvert(&CC, &CC_inv, CV_SVD); 95 double * ci = cc_inv; 96 for(int i = 0; i < number_of_correspondences; i++) { 97 double * pi = &pws[0] + 3 * i; 98 double * a = &alphas[0] + 4 * i; 99 100 for(int j = 0; j < 3; j++) 101 a[1 + j] = 102 ci[3 * j ] * (pi[0] - cws[0][0]) + 103 ci[3 * j + 1] * (pi[1] - cws[0][1]) + 104 ci[3 * j + 2] * (pi[2] - cws[0][2]); 105 a[0] = 1.0f - a[1] - a[2] - a[3]; 106 } 107 } 108 109 void epnp::fill_M(CvMat * M, 110 const int row, const double * as, const double u, const double v) 111 { 112 double * M1 = M->data.db + row * 12; 113 double * M2 = M1 + 12; 114 115 for(int i = 0; i < 4; i++) { 116 M1[3 * i ] = as[i] * fu; 117 M1[3 * i + 1] = 0.0; 118 M1[3 * i + 2] = as[i] * (uc - u); 119 120 M2[3 * i ] = 0.0; 121 M2[3 * i + 1] = as[i] * fv; 122 M2[3 * i + 2] = as[i] * (vc - v); 123 } 124 } 125 126 void epnp::compute_ccs(const double * betas, const double * ut) 127 { 128 for(int i = 0; i < 4; i++) 129 ccs[i][0] = ccs[i][1] = ccs[i][2] = 0.0f; 130 131 for(int i = 0; i < 4; i++) { 132 const double * v = ut + 12 * (11 - i); 133 for(int j = 0; j < 4; j++) 134 for(int k = 0; k < 3; k++) 135 ccs[j][k] += betas[i] * v[3 * j + k]; 136 } 137 } 138 139 void epnp::compute_pcs(void) 140 { 141 for(int i = 0; i < number_of_correspondences; i++) { 142 double * a = &alphas[0] + 4 * i; 143 double * pc = &pcs[0] + 3 * i; 144 145 for(int j = 0; j < 3; j++) 146 pc[j] = a[0] * ccs[0][j] + a[1] * ccs[1][j] + a[2] * ccs[2][j] + a[3] * ccs[3][j]; 147 } 148 } 149 150 void epnp::compute_pose(Mat& R, Mat& t) 151 { 152 choose_control_points(); 153 compute_barycentric_coordinates(); 154 155 CvMat * M = cvCreateMat(2 * number_of_correspondences, 12, CV_64F); 156 157 for(int i = 0; i < number_of_correspondences; i++) 158 fill_M(M, 2 * i, &alphas[0] + 4 * i, us[2 * i], us[2 * i + 1]); 159 160 double mtm[12 * 12], d[12], ut[12 * 12]; 161 CvMat MtM = cvMat(12, 12, CV_64F, mtm); 162 CvMat D = cvMat(12, 1, CV_64F, d); 163 CvMat Ut = cvMat(12, 12, CV_64F, ut); 164 165 cvMulTransposed(M, &MtM, 1); 166 cvSVD(&MtM, &D, &Ut, 0, CV_SVD_MODIFY_A | CV_SVD_U_T); 167 cvReleaseMat(&M); 168 169 double l_6x10[6 * 10], rho[6]; 170 CvMat L_6x10 = cvMat(6, 10, CV_64F, l_6x10); 171 CvMat Rho = cvMat(6, 1, CV_64F, rho); 172 173 compute_L_6x10(ut, l_6x10); 174 compute_rho(rho); 175 176 double Betas[4][4], rep_errors[4]; 177 double Rs[4][3][3], ts[4][3]; 178 179 find_betas_approx_1(&L_6x10, &Rho, Betas[1]); 180 gauss_newton(&L_6x10, &Rho, Betas[1]); 181 rep_errors[1] = compute_R_and_t(ut, Betas[1], Rs[1], ts[1]); 182 183 find_betas_approx_2(&L_6x10, &Rho, Betas[2]); 184 gauss_newton(&L_6x10, &Rho, Betas[2]); 185 rep_errors[2] = compute_R_and_t(ut, Betas[2], Rs[2], ts[2]); 186 187 find_betas_approx_3(&L_6x10, &Rho, Betas[3]); 188 gauss_newton(&L_6x10, &Rho, Betas[3]); 189 rep_errors[3] = compute_R_and_t(ut, Betas[3], Rs[3], ts[3]); 190 191 int N = 1; 192 if (rep_errors[2] < rep_errors[1]) N = 2; 193 if (rep_errors[3] < rep_errors[N]) N = 3; 194 195 Mat(3, 1, CV_64F, ts[N]).copyTo(t); 196 Mat(3, 3, CV_64F, Rs[N]).copyTo(R); 197 } 198 199 void epnp::copy_R_and_t(const double R_src[3][3], const double t_src[3], 200 double R_dst[3][3], double t_dst[3]) 201 { 202 for(int i = 0; i < 3; i++) { 203 for(int j = 0; j < 3; j++) 204 R_dst[i][j] = R_src[i][j]; 205 t_dst[i] = t_src[i]; 206 } 207 } 208 209 double epnp::dist2(const double * p1, const double * p2) 210 { 211 return 212 (p1[0] - p2[0]) * (p1[0] - p2[0]) + 213 (p1[1] - p2[1]) * (p1[1] - p2[1]) + 214 (p1[2] - p2[2]) * (p1[2] - p2[2]); 215 } 216 217 double epnp::dot(const double * v1, const double * v2) 218 { 219 return v1[0] * v2[0] + v1[1] * v2[1] + v1[2] * v2[2]; 220 } 221 222 void epnp::estimate_R_and_t(double R[3][3], double t[3]) 223 { 224 double pc0[3], pw0[3]; 225 226 pc0[0] = pc0[1] = pc0[2] = 0.0; 227 pw0[0] = pw0[1] = pw0[2] = 0.0; 228 229 for(int i = 0; i < number_of_correspondences; i++) { 230 const double * pc = &pcs[3 * i]; 231 const double * pw = &pws[3 * i]; 232 233 for(int j = 0; j < 3; j++) { 234 pc0[j] += pc[j]; 235 pw0[j] += pw[j]; 236 } 237 } 238 for(int j = 0; j < 3; j++) { 239 pc0[j] /= number_of_correspondences; 240 pw0[j] /= number_of_correspondences; 241 } 242 243 double abt[3 * 3], abt_d[3], abt_u[3 * 3], abt_v[3 * 3]; 244 CvMat ABt = cvMat(3, 3, CV_64F, abt); 245 CvMat ABt_D = cvMat(3, 1, CV_64F, abt_d); 246 CvMat ABt_U = cvMat(3, 3, CV_64F, abt_u); 247 CvMat ABt_V = cvMat(3, 3, CV_64F, abt_v); 248 249 cvSetZero(&ABt); 250 for(int i = 0; i < number_of_correspondences; i++) { 251 double * pc = &pcs[3 * i]; 252 double * pw = &pws[3 * i]; 253 254 for(int j = 0; j < 3; j++) { 255 abt[3 * j ] += (pc[j] - pc0[j]) * (pw[0] - pw0[0]); 256 abt[3 * j + 1] += (pc[j] - pc0[j]) * (pw[1] - pw0[1]); 257 abt[3 * j + 2] += (pc[j] - pc0[j]) * (pw[2] - pw0[2]); 258 } 259 } 260 261 cvSVD(&ABt, &ABt_D, &ABt_U, &ABt_V, CV_SVD_MODIFY_A); 262 263 for(int i = 0; i < 3; i++) 264 for(int j = 0; j < 3; j++) 265 R[i][j] = dot(abt_u + 3 * i, abt_v + 3 * j); 266 267 const double det = 268 R[0][0] * R[1][1] * R[2][2] + R[0][1] * R[1][2] * R[2][0] + R[0][2] * R[1][0] * R[2][1] - 269 R[0][2] * R[1][1] * R[2][0] - R[0][1] * R[1][0] * R[2][2] - R[0][0] * R[1][2] * R[2][1]; 270 271 if (det < 0) { 272 R[2][0] = -R[2][0]; 273 R[2][1] = -R[2][1]; 274 R[2][2] = -R[2][2]; 275 } 276 277 t[0] = pc0[0] - dot(R[0], pw0); 278 t[1] = pc0[1] - dot(R[1], pw0); 279 t[2] = pc0[2] - dot(R[2], pw0); 280 } 281 282 void epnp::solve_for_sign(void) 283 { 284 if (pcs[2] < 0.0) { 285 for(int i = 0; i < 4; i++) 286 for(int j = 0; j < 3; j++) 287 ccs[i][j] = -ccs[i][j]; 288 289 for(int i = 0; i < number_of_correspondences; i++) { 290 pcs[3 * i ] = -pcs[3 * i]; 291 pcs[3 * i + 1] = -pcs[3 * i + 1]; 292 pcs[3 * i + 2] = -pcs[3 * i + 2]; 293 } 294 } 295 } 296 297 double epnp::compute_R_and_t(const double * ut, const double * betas, 298 double R[3][3], double t[3]) 299 { 300 compute_ccs(betas, ut); 301 compute_pcs(); 302 303 solve_for_sign(); 304 305 estimate_R_and_t(R, t); 306 307 return reprojection_error(R, t); 308 } 309 310 double epnp::reprojection_error(const double R[3][3], const double t[3]) 311 { 312 double sum2 = 0.0; 313 314 for(int i = 0; i < number_of_correspondences; i++) { 315 double * pw = &pws[3 * i]; 316 double Xc = dot(R[0], pw) + t[0]; 317 double Yc = dot(R[1], pw) + t[1]; 318 double inv_Zc = 1.0 / (dot(R[2], pw) + t[2]); 319 double ue = uc + fu * Xc * inv_Zc; 320 double ve = vc + fv * Yc * inv_Zc; 321 double u = us[2 * i], v = us[2 * i + 1]; 322 323 sum2 += sqrt( (u - ue) * (u - ue) + (v - ve) * (v - ve) ); 324 } 325 326 return sum2 / number_of_correspondences; 327 } 328 329 // betas10 = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44] 330 // betas_approx_1 = [B11 B12 B13 B14] 331 332 void epnp::find_betas_approx_1(const CvMat * L_6x10, const CvMat * Rho, 333 double * betas) 334 { 335 double l_6x4[6 * 4], b4[4]; 336 CvMat L_6x4 = cvMat(6, 4, CV_64F, l_6x4); 337 CvMat B4 = cvMat(4, 1, CV_64F, b4); 338 339 for(int i = 0; i < 6; i++) { 340 cvmSet(&L_6x4, i, 0, cvmGet(L_6x10, i, 0)); 341 cvmSet(&L_6x4, i, 1, cvmGet(L_6x10, i, 1)); 342 cvmSet(&L_6x4, i, 2, cvmGet(L_6x10, i, 3)); 343 cvmSet(&L_6x4, i, 3, cvmGet(L_6x10, i, 6)); 344 } 345 346 cvSolve(&L_6x4, Rho, &B4, CV_SVD); 347 348 if (b4[0] < 0) { 349 betas[0] = sqrt(-b4[0]); 350 betas[1] = -b4[1] / betas[0]; 351 betas[2] = -b4[2] / betas[0]; 352 betas[3] = -b4[3] / betas[0]; 353 } else { 354 betas[0] = sqrt(b4[0]); 355 betas[1] = b4[1] / betas[0]; 356 betas[2] = b4[2] / betas[0]; 357 betas[3] = b4[3] / betas[0]; 358 } 359 } 360 361 // betas10 = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44] 362 // betas_approx_2 = [B11 B12 B22 ] 363 364 void epnp::find_betas_approx_2(const CvMat * L_6x10, const CvMat * Rho, 365 double * betas) 366 { 367 double l_6x3[6 * 3], b3[3]; 368 CvMat L_6x3 = cvMat(6, 3, CV_64F, l_6x3); 369 CvMat B3 = cvMat(3, 1, CV_64F, b3); 370 371 for(int i = 0; i < 6; i++) { 372 cvmSet(&L_6x3, i, 0, cvmGet(L_6x10, i, 0)); 373 cvmSet(&L_6x3, i, 1, cvmGet(L_6x10, i, 1)); 374 cvmSet(&L_6x3, i, 2, cvmGet(L_6x10, i, 2)); 375 } 376 377 cvSolve(&L_6x3, Rho, &B3, CV_SVD); 378 379 if (b3[0] < 0) { 380 betas[0] = sqrt(-b3[0]); 381 betas[1] = (b3[2] < 0) ? sqrt(-b3[2]) : 0.0; 382 } else { 383 betas[0] = sqrt(b3[0]); 384 betas[1] = (b3[2] > 0) ? sqrt(b3[2]) : 0.0; 385 } 386 387 if (b3[1] < 0) betas[0] = -betas[0]; 388 389 betas[2] = 0.0; 390 betas[3] = 0.0; 391 } 392 393 // betas10 = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44] 394 // betas_approx_3 = [B11 B12 B22 B13 B23 ] 395 396 void epnp::find_betas_approx_3(const CvMat * L_6x10, const CvMat * Rho, 397 double * betas) 398 { 399 double l_6x5[6 * 5], b5[5]; 400 CvMat L_6x5 = cvMat(6, 5, CV_64F, l_6x5); 401 CvMat B5 = cvMat(5, 1, CV_64F, b5); 402 403 for(int i = 0; i < 6; i++) { 404 cvmSet(&L_6x5, i, 0, cvmGet(L_6x10, i, 0)); 405 cvmSet(&L_6x5, i, 1, cvmGet(L_6x10, i, 1)); 406 cvmSet(&L_6x5, i, 2, cvmGet(L_6x10, i, 2)); 407 cvmSet(&L_6x5, i, 3, cvmGet(L_6x10, i, 3)); 408 cvmSet(&L_6x5, i, 4, cvmGet(L_6x10, i, 4)); 409 } 410 411 cvSolve(&L_6x5, Rho, &B5, CV_SVD); 412 413 if (b5[0] < 0) { 414 betas[0] = sqrt(-b5[0]); 415 betas[1] = (b5[2] < 0) ? sqrt(-b5[2]) : 0.0; 416 } else { 417 betas[0] = sqrt(b5[0]); 418 betas[1] = (b5[2] > 0) ? sqrt(b5[2]) : 0.0; 419 } 420 if (b5[1] < 0) betas[0] = -betas[0]; 421 betas[2] = b5[3] / betas[0]; 422 betas[3] = 0.0; 423 } 424 425 void epnp::compute_L_6x10(const double * ut, double * l_6x10) 426 { 427 const double * v[4]; 428 429 v[0] = ut + 12 * 11; 430 v[1] = ut + 12 * 10; 431 v[2] = ut + 12 * 9; 432 v[3] = ut + 12 * 8; 433 434 double dv[4][6][3]; 435 436 for(int i = 0; i < 4; i++) { 437 int a = 0, b = 1; 438 for(int j = 0; j < 6; j++) { 439 dv[i][j][0] = v[i][3 * a ] - v[i][3 * b]; 440 dv[i][j][1] = v[i][3 * a + 1] - v[i][3 * b + 1]; 441 dv[i][j][2] = v[i][3 * a + 2] - v[i][3 * b + 2]; 442 443 b++; 444 if (b > 3) { 445 a++; 446 b = a + 1; 447 } 448 } 449 } 450 451 for(int i = 0; i < 6; i++) { 452 double * row = l_6x10 + 10 * i; 453 454 row[0] = dot(dv[0][i], dv[0][i]); 455 row[1] = 2.0f * dot(dv[0][i], dv[1][i]); 456 row[2] = dot(dv[1][i], dv[1][i]); 457 row[3] = 2.0f * dot(dv[0][i], dv[2][i]); 458 row[4] = 2.0f * dot(dv[1][i], dv[2][i]); 459 row[5] = dot(dv[2][i], dv[2][i]); 460 row[6] = 2.0f * dot(dv[0][i], dv[3][i]); 461 row[7] = 2.0f * dot(dv[1][i], dv[3][i]); 462 row[8] = 2.0f * dot(dv[2][i], dv[3][i]); 463 row[9] = dot(dv[3][i], dv[3][i]); 464 } 465 } 466 467 void epnp::compute_rho(double * rho) 468 { 469 rho[0] = dist2(cws[0], cws[1]); 470 rho[1] = dist2(cws[0], cws[2]); 471 rho[2] = dist2(cws[0], cws[3]); 472 rho[3] = dist2(cws[1], cws[2]); 473 rho[4] = dist2(cws[1], cws[3]); 474 rho[5] = dist2(cws[2], cws[3]); 475 } 476 477 void epnp::compute_A_and_b_gauss_newton(const double * l_6x10, const double * rho, 478 const double betas[4], CvMat * A, CvMat * b) 479 { 480 for(int i = 0; i < 6; i++) { 481 const double * rowL = l_6x10 + i * 10; 482 double * rowA = A->data.db + i * 4; 483 484 rowA[0] = 2 * rowL[0] * betas[0] + rowL[1] * betas[1] + rowL[3] * betas[2] + rowL[6] * betas[3]; 485 rowA[1] = rowL[1] * betas[0] + 2 * rowL[2] * betas[1] + rowL[4] * betas[2] + rowL[7] * betas[3]; 486 rowA[2] = rowL[3] * betas[0] + rowL[4] * betas[1] + 2 * rowL[5] * betas[2] + rowL[8] * betas[3]; 487 rowA[3] = rowL[6] * betas[0] + rowL[7] * betas[1] + rowL[8] * betas[2] + 2 * rowL[9] * betas[3]; 488 489 cvmSet(b, i, 0, rho[i] - 490 ( 491 rowL[0] * betas[0] * betas[0] + 492 rowL[1] * betas[0] * betas[1] + 493 rowL[2] * betas[1] * betas[1] + 494 rowL[3] * betas[0] * betas[2] + 495 rowL[4] * betas[1] * betas[2] + 496 rowL[5] * betas[2] * betas[2] + 497 rowL[6] * betas[0] * betas[3] + 498 rowL[7] * betas[1] * betas[3] + 499 rowL[8] * betas[2] * betas[3] + 500 rowL[9] * betas[3] * betas[3] 501 )); 502 } 503 } 504 505 void epnp::gauss_newton(const CvMat * L_6x10, const CvMat * Rho, double betas[4]) 506 { 507 const int iterations_number = 5; 508 509 double a[6*4], b[6], x[4]; 510 CvMat A = cvMat(6, 4, CV_64F, a); 511 CvMat B = cvMat(6, 1, CV_64F, b); 512 CvMat X = cvMat(4, 1, CV_64F, x); 513 514 for(int k = 0; k < iterations_number; k++) 515 { 516 compute_A_and_b_gauss_newton(L_6x10->data.db, Rho->data.db, 517 betas, &A, &B); 518 qr_solve(&A, &B, &X); 519 for(int i = 0; i < 4; i++) 520 betas[i] += x[i]; 521 } 522 } 523 524 void epnp::qr_solve(CvMat * A, CvMat * b, CvMat * X) 525 { 526 const int nr = A->rows; 527 const int nc = A->cols; 528 529 if (max_nr != 0 && max_nr < nr) 530 { 531 delete [] A1; 532 delete [] A2; 533 } 534 if (max_nr < nr) 535 { 536 max_nr = nr; 537 A1 = new double[nr]; 538 A2 = new double[nr]; 539 } 540 541 double * pA = A->data.db, * ppAkk = pA; 542 for(int k = 0; k < nc; k++) 543 { 544 double * ppAik1 = ppAkk, eta = fabs(*ppAik1); 545 for(int i = k + 1; i < nr; i++) 546 { 547 double elt = fabs(*ppAik1); 548 if (eta < elt) eta = elt; 549 ppAik1 += nc; 550 } 551 if (eta == 0) 552 { 553 A1[k] = A2[k] = 0.0; 554 //cerr << "God damnit, A is singular, this shouldn't happen." << endl; 555 return; 556 } 557 else 558 { 559 double * ppAik2 = ppAkk, sum2 = 0.0, inv_eta = 1. / eta; 560 for(int i = k; i < nr; i++) 561 { 562 *ppAik2 *= inv_eta; 563 sum2 += *ppAik2 * *ppAik2; 564 ppAik2 += nc; 565 } 566 double sigma = sqrt(sum2); 567 if (*ppAkk < 0) 568 sigma = -sigma; 569 *ppAkk += sigma; 570 A1[k] = sigma * *ppAkk; 571 A2[k] = -eta * sigma; 572 for(int j = k + 1; j < nc; j++) 573 { 574 double * ppAik = ppAkk, sum = 0; 575 for(int i = k; i < nr; i++) 576 { 577 sum += *ppAik * ppAik[j - k]; 578 ppAik += nc; 579 } 580 double tau = sum / A1[k]; 581 ppAik = ppAkk; 582 for(int i = k; i < nr; i++) 583 { 584 ppAik[j - k] -= tau * *ppAik; 585 ppAik += nc; 586 } 587 } 588 } 589 ppAkk += nc + 1; 590 } 591 592 // b <- Qt b 593 double * ppAjj = pA, * pb = b->data.db; 594 for(int j = 0; j < nc; j++) 595 { 596 double * ppAij = ppAjj, tau = 0; 597 for(int i = j; i < nr; i++) 598 { 599 tau += *ppAij * pb[i]; 600 ppAij += nc; 601 } 602 tau /= A1[j]; 603 ppAij = ppAjj; 604 for(int i = j; i < nr; i++) 605 { 606 pb[i] -= tau * *ppAij; 607 ppAij += nc; 608 } 609 ppAjj += nc + 1; 610 } 611 612 // X = R-1 b 613 double * pX = X->data.db; 614 pX[nc - 1] = pb[nc - 1] / A2[nc - 1]; 615 for(int i = nc - 2; i >= 0; i--) 616 { 617 double * ppAij = pA + i * nc + (i + 1), sum = 0; 618 619 for(int j = i + 1; j < nc; j++) 620 { 621 sum += *ppAij * pX[j]; 622 ppAij++; 623 } 624 pX[i] = (pb[i] - sum) / A2[i]; 625 } 626 } 627 628 } 629
