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1 #include <iostream>
2 #include "precomp.hpp"
3 #include "epnp.h"
4 
5 namespace cv
6 {
7 
epnp(const Mat & cameraMatrix,const Mat & opoints,const Mat & ipoints)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 
~epnp()40 epnp::~epnp()
41 {
42     if (A1)
43         delete[] A1;
44     if (A2)
45         delete[] A2;
46 }
47 
choose_control_points(void)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 
compute_barycentric_coordinates(void)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 
fill_M(CvMat * M,const int row,const double * as,const double u,const double v)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 
compute_ccs(const double * betas,const double * ut)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 
compute_pcs(void)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 
compute_pose(Mat & R,Mat & t)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 
copy_R_and_t(const double R_src[3][3],const double t_src[3],double R_dst[3][3],double t_dst[3])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 
dist2(const double * p1,const double * p2)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 
dot(const double * v1,const double * v2)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 
estimate_R_and_t(double R[3][3],double t[3])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 
solve_for_sign(void)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 
compute_R_and_t(const double * ut,const double * betas,double R[3][3],double t[3])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 
reprojection_error(const double R[3][3],const double t[3])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 
find_betas_approx_1(const CvMat * L_6x10,const CvMat * Rho,double * betas)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 
find_betas_approx_2(const CvMat * L_6x10,const CvMat * Rho,double * betas)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 
find_betas_approx_3(const CvMat * L_6x10,const CvMat * Rho,double * betas)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 
compute_L_6x10(const double * ut,double * l_6x10)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 
compute_rho(double * rho)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 
compute_A_and_b_gauss_newton(const double * l_6x10,const double * rho,const double betas[4],CvMat * A,CvMat * b)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 
gauss_newton(const CvMat * L_6x10,const CvMat * Rho,double betas[4])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 
qr_solve(CvMat * A,CvMat * b,CvMat * X)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