update svn to r733 (4)

This commit is contained in:
Razvan Mihalyi 2012-10-24 11:40:56 +02:00
parent b03c2a3e0c
commit 5255e4f9f0
3 changed files with 0 additions and 745 deletions

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IF(WITH_NORMALS)
FIND_PACKAGE(OpenCV REQUIRED)
add_executable(calculateNormals calculate_normals.cc ../slam6d/fbr/fbr_global.cc ../slam6d/fbr/panorama.cc)
target_link_libraries(calculateNormals scan ANN newmat fbr_panorama fbr_cv_io ${Boost_LIBRARIES} ${OpenCV_LIBS})
ENDIF(WITH_NORMALS)

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/*
* calculateNormals implementation
*
* Copyright (C) Johannes Schauer, Razvan Mihaly
*
* Released under the GPL version 3
*
*/
#include "ANN/ANN.h"
#include "newmat/newmat.h"
#include "newmat/newmatap.h"
#include "newmat/newmatio.h"
using namespace NEWMAT;
#include "slam6d/point.h"
#include "normals/pointNeighbor.h"
#include "slam6d/scan.h"
#include "slam6d/globals.icc"
#include "slam6d/fbr/panorama.h"
#include "normals/point.h"
#include "normals/SRI.h"
#include <string>
using std::string;
#include <iostream>
using std::cout;
using std::endl;
using std::vector;
#include <algorithm>
#include <boost/program_options.hpp>
#include <boost/filesystem/operations.hpp>
#include <boost/filesystem/fstream.hpp>
namespace po = boost::program_options;
enum normal_method {KNN_PCA, AKNN_PCA, PANO_PCA, PANO_SRI};
void normal_option_dependency(const po::variables_map & vm, normal_method ntype, const char *option)
{
if (vm.count("normalMethod") && vm["normalMethod"].as<normal_method>() == ntype) {
if (!vm.count(option)) {
throw std::logic_error (string("this normal method needs ")+option+" to be set");
}
}
}
void normal_option_conflict(const po::variables_map & vm, normal_method ntype, const char *option)
{
if (vm.count("normalMethod") && vm["normalMethod"].as<normal_method>() == ntype) {
if (vm.count(option)) {
throw std::logic_error (string("this normal method is incompatible with ")+option);
}
}
}
/*
* validates input type specification
*/
void validate(boost::any& v, const std::vector<std::string>& values,
IOType*, int) {
if (values.size() == 0)
throw std::runtime_error("Invalid model specification");
string arg = values.at(0);
try {
v = formatname_to_io_type(arg.c_str());
} catch (...) { // runtime_error
throw std::runtime_error("Format " + arg + " unknown.");
}
}
void validate(boost::any& v, const std::vector<std::string>& values,
normal_method*, int) {
if (values.size() == 0)
throw std::runtime_error("Invalid model specification");
string arg = values.at(0);
if(strcasecmp(arg.c_str(), "KNN_PCA") == 0) v = KNN_PCA;
else if(strcasecmp(arg.c_str(), "AKNN_PCA") == 0) v = AKNN_PCA;
else if(strcasecmp(arg.c_str(), "PANO_PCA") == 0) v = PANO_PCA;
else if(strcasecmp(arg.c_str(), "PANO_SRI") == 0) v = PANO_SRI;
else throw std::runtime_error(std::string("normal method ") + arg + std::string(" is unknown"));
}
/*
* parse commandline options, fill arguments
*/
void parse_options(int argc, char **argv, int &start, int &end,
bool &scanserver, string &dir, IOType &iotype,
int &maxDist, int &minDist, normal_method &normalMethod, int &knn,
int &kmin, int &kmax, double& alpha, int &width, int &height,
bool &flipnormals, double &factor)
{
po::options_description generic("Generic options");
generic.add_options()
("help,h", "output this help message");
po::options_description input("Input options");
input.add_options()
("start,s", po::value<int>(&start)->default_value(0),
"start at scan <arg> (i.e., neglects the first <arg> scans) "
"[ATTENTION: counting naturally starts with 0]")
("end,e", po::value<int>(&end)->default_value(-1),
"end after scan <arg>")
("format,f", po::value<IOType>(&iotype)->default_value(UOS),
"using shared library <arg> for input. (chose F from {uos, uos_map, "
"uos_rgb, uos_frames, uos_map_frames, old, rts, rts_map, ifp, "
"riegl_txt, riegl_rgb, riegl_bin, zahn, ply})")
("max,M", po::value<int>(&maxDist)->default_value(-1),
"neglegt all data points with a distance larger than <arg> 'units")
("min,m", po::value<int>(&minDist)->default_value(-1),
"neglegt all data points with a distance smaller than <arg> 'units")
("scanserver,S", po::bool_switch(&scanserver),
"Use the scanserver as an input method and handling of scan data")
;
po::options_description normal("Normal options");
normal.add_options()
("normalMethod,N", po::value<normal_method>(&normalMethod)->default_value(KNN_PCA),
"choose the method for computing normals:\n"
"KNN_PCA -- use kNN and PCA\n"
"AKNN_PCA -- use adaptive kNN and PCA\n"
"PANO_PCA -- use panorama image neighbors and PCA\n"
"PANO_SRI -- use panorama image neighbors and spherical range image differentiation\n")
("knn,K", po::value<int>(&knn),
"select the k in kNN search")
("kmin,1", po::value<int>(&kmin),
"select k_min in adaptive kNN search")
("kmax,2", po::value<int>(&kmax),
"select k_max in adaptive kNN search")
("alpha,a", po::value<double>(&alpha),
"select the alpha parameter for detecting an ill-conditioned neighborhood")
("width,w", po::value<int>(&width),
"width of panorama")
("height,h", po::value<int>(&height),
"height of panorama")
("flipnormals,F", po::bool_switch(&flipnormals),
"flip orientation of normals towards scan pose")
("factor,c", po::value<double>(&factor),
"factor for SRI computation")
;
po::options_description hidden("Hidden options");
hidden.add_options()
("input-dir", po::value<string>(&dir), "input dir");
// all options
po::options_description all;
all.add(generic).add(input).add(normal).add(hidden);
// options visible with --help
po::options_description cmdline_options;
cmdline_options.add(generic).add(input).add(normal);
// positional argument
po::positional_options_description pd;
pd.add("input-dir", 1);
// process options
po::variables_map vm;
po::store(po::command_line_parser(argc, argv).
options(all).positional(pd).run(), vm);
po::notify(vm);
// display help
if (vm.count("help")) {
cout << cmdline_options;
exit(0);
}
normal_option_dependency(vm, KNN_PCA, "knn");
normal_option_conflict(vm, KNN_PCA, "kmin");
normal_option_conflict(vm, KNN_PCA, "kmax");
normal_option_conflict(vm, KNN_PCA, "alpha");
normal_option_conflict(vm, KNN_PCA, "width");
normal_option_conflict(vm, KNN_PCA, "height");
normal_option_conflict(vm, KNN_PCA, "factor");
normal_option_conflict(vm, AKNN_PCA, "knn");
normal_option_dependency(vm, AKNN_PCA, "kmin");
normal_option_dependency(vm, AKNN_PCA, "kmax");
normal_option_dependency(vm, AKNN_PCA, "alpha");
normal_option_conflict(vm, AKNN_PCA, "width");
normal_option_conflict(vm, AKNN_PCA, "height");
normal_option_conflict(vm, AKNN_PCA, "factor");
//normal_option_conflict(vm, PANO_PCA, "knn");
normal_option_dependency(vm, KNN_PCA, "knn");
normal_option_conflict(vm, PANO_PCA, "kmin");
normal_option_conflict(vm, PANO_PCA, "kmax");
normal_option_conflict(vm, PANO_PCA, "alpha");
normal_option_dependency(vm, PANO_PCA, "width");
normal_option_dependency(vm, PANO_PCA, "height");
normal_option_conflict(vm, PANO_PCA, "factor");
normal_option_conflict(vm, PANO_SRI, "knn");
normal_option_conflict(vm, PANO_SRI, "kmin");
normal_option_conflict(vm, PANO_SRI, "kmax");
normal_option_conflict(vm, PANO_SRI, "alpha");
normal_option_conflict(vm, PANO_SRI, "width");
normal_option_conflict(vm, PANO_SRI, "height");
normal_option_dependency(vm, PANO_SRI, "factor");
// add trailing slash to directory if not present yet
if (dir[dir.length()-1] != '/') dir = dir + "/";
}
/*
* retrieve a cv::Mat with x,y,z,r from a scan object
* functionality borrowed from scan_cv::convertScanToMat but this function
* does not allow a scanserver to be used, prints to stdout and can only
* handle a single scan
*/
void scan2mat(Scan* scan, cv::Mat& scan_cv) {
DataXYZ xyz = scan->get("xyz");
unsigned int nPoints = xyz.size();
scan_cv.create(nPoints,1,CV_32FC(4));
scan_cv = cv::Scalar::all(0);
cv::MatIterator_<cv::Vec3f> it = scan_cv.begin<cv::Vec3f>();
for(unsigned int i = 0; i < nPoints; i++){
(*it)[0] = xyz[i][0];
(*it)[1] = xyz[i][1];
(*it)[2] = xyz[i][2];
++it;
}
}
/**
* Helper function that maps x, y, z to R, G, B using a linear function
*/
void mapNormalToRGB(const Point& normal, Point& rgb)
{
rgb.x = 127.5 * normal.x + 127.5;
rgb.y = 127.5 * normal.y + 127.5;
rgb.z = 255.0 * fabs(normal.z);
}
/**
* Write normals to .3d files using the uos_rgb format
*/
void writeNormals(const Scan* scan, const string& dir,
const vector<Point>& points, const vector<Point>& normals)
{
stringstream ss;
ss << dir << "scan" << string(scan->getIdentifier()) << ".3d";
ofstream scan_file;
scan_file.open(ss.str().c_str());
for(size_t i = 0; i < points.size(); ++i) {
Point rgb;
mapNormalToRGB(normals[i], rgb);
scan_file
<< points[i].x << " " << points[i].y << " " << points[i].z << " "
<< (unsigned int) rgb.x << " " << (unsigned int) rgb.y << " "
<< (unsigned int) rgb.z << "\n";
}
scan_file.close();
ss.clear(); ss.str(string());
ss << dir << "scan" << string(scan->getIdentifier()) << ".pose";
ofstream pose_file;
pose_file.open(ss.str().c_str());
pose_file << 0 << " " << 0 << " " << 0 << "\n" << 0 << " " << 0 << " " << 0 << "\n";
pose_file.close();
}
/**
* Compute eigen decomposition of a point and its neighbors using the NEWMAT library
* @param point - input points with corresponding neighbors
* @param e_values - out parameter returns the eigenvalues
* @param e_vectors - out parameter returns the eigenvectors
*/
void computeEigenDecomposition(const PointNeighbor& point,
DiagonalMatrix& e_values, Matrix& e_vectors)
{
Point centroid(0, 0, 0);
vector<Point> neighbors = point.neighbors;
for (size_t j = 0; j < neighbors.size(); ++j) {
centroid.x += neighbors[j].x;
centroid.y += neighbors[j].y;
centroid.z += neighbors[j].z;
}
centroid.x /= (double) neighbors.size();
centroid.y /= (double) neighbors.size();
centroid.z /= (double) neighbors.size();
Matrix S(3, 3);
S = 0.0;
for (size_t j = 0; j < neighbors.size(); ++j) {
ColumnVector point_prime(3);
point_prime(1) = neighbors[j].x - centroid.x;
point_prime(2) = neighbors[j].y - centroid.y;
point_prime(3) = neighbors[j].z - centroid.z;
S = S + point_prime * point_prime.t();
}
// normalize S
for (int j = 0; j < 3; ++j)
for (int k = 0; k < 3; ++k)
S(j+1, k+1) /= (double) neighbors.size();
SymmetricMatrix C;
C << S;
// the decomposition
Jacobi(C, e_values, e_vectors);
#ifdef DEBUG
// Print the result
cout << "The eigenvalues matrix:" << endl;
cout << e_values << endl;
#endif
}
/**
* Compute neighbors using kNN search
* @param points - input set of points
* @param points_neighbors - output set of points with corresponding neighbors
* @param knn - k constant in kNN search
* @param kmax - to be used in adaptive knn search as the upper bound on adapting the k constant, defaults to -1 for regular kNN search
* @param alpha - to be used in adaptive knn search for detecting ill-conditioned neighborhoods
* @param eps - parameter required by the ANN library in kNN search
*/
void computeKNearestNeighbors(const vector<Point>& points,
vector<PointNeighbor>& points_neighbors, int knn, int kmax=-1,
double alpha=1000.0, double eps=1.0)
{
ANNpointArray point_array = annAllocPts(points.size(), 3);
for (size_t i = 0; i < points.size(); ++i) {
point_array[i] = new ANNcoord[3];
point_array[i][0] = points[i].x;
point_array[i][1] = points[i].y;
point_array[i][2] = points[i].z;
}
ANNkd_tree t(point_array, points.size(), 3);
ANNidxArray n;
ANNdistArray d;
if (kmax < 0) {
/// regular kNN search, allocate memory for knn
n = new ANNidx[knn];
d = new ANNdist[knn];
} else {
/// adaptive kNN search, allocate memory for kmax
n = new ANNidx[kmax];
d = new ANNdist[kmax];
}
for (size_t i = 0; i < points.size(); ++i) {
vector<Point> neighbors;
ANNpoint p = point_array[i];
t.annkSearch(p, knn, n, d, eps);
neighbors.push_back(points[i]);
for (int j = 0; j < knn; ++j) {
if ( n[j] != (int)i )
neighbors.push_back(points[n[j]]);
}
PointNeighbor current_point(points[i], neighbors);
points_neighbors.push_back( current_point );
Matrix e_vectors(3,3); e_vectors = 0.0;
DiagonalMatrix e_values(3); e_values = 0.0;
computeEigenDecomposition( current_point, e_values, e_vectors );
if (kmax > 0) {
/// detecting an ill-conditioned neighborhood
if (e_values(3) / e_values(2) > alpha && e_values(2) > 0.0) {
if (knn < kmax)
cout << "Increasing kmin to " << ++knn << endl;
}
}
}
delete[] n;
delete[] d;
}
/**
* Compute neighbors using kNN search
* @param point - input point with neighbors
* @param new_point - output point with new neighbors
* @param knn - k constant in kNN search
* @param eps - parameter required by the ANN library in kNN search
*/
void computeKNearestNeighbors(const PointNeighbor& point,
PointNeighbor& new_point, int knn,
double eps=1.0)
{
/// allocate memory for all neighbors of point plus the point itself
ANNpointArray point_array = annAllocPts(point.neighbors.size()+1, 3);
for (size_t i = 0; i < point.neighbors.size(); ++i) {
point_array[i] = new ANNcoord[3];
point_array[i][0] = point.neighbors[i].x;
point_array[i][1] = point.neighbors[i].y;
point_array[i][2] = point.neighbors[i].z;
}
int last = point.neighbors.size();
point_array[last] = new ANNcoord[3];
point_array[last][0] = point.point.x;
point_array[last][1] = point.point.y;
point_array[last][2] = point.point.z;
ANNkd_tree t(point_array, point.neighbors.size()+1, 3);
ANNidxArray n;
ANNdistArray d;
/// regular kNN search, allocate memory for knn
n = new ANNidx[knn];
d = new ANNdist[knn];
vector<Point> new_neighbors;
/// last point in the array is the current point
ANNpoint p = point_array[point.neighbors.size()];
t.annkSearch(p, knn, n, d, eps);
new_neighbors.push_back(point.point);
for (int j = 0; j < knn; ++j) {
if ( n[j] != (int) point.neighbors.size() )
new_neighbors.push_back(point.neighbors[n[j]]);
}
new_point.point = point.point;
new_point.neighbors = new_neighbors;
delete[] n;
delete[] d;
}
/**
* Compute neighbors using panorama images
* @param fPanorama - input panorama image created from the current scan
* @param points_neighbors - output set of points with corresponding neighbors
*/
void computePanoramaNeighbors(Scan* scan,
vector<PointNeighbor>& points_neighbors, int width, int height, int knn)
{
cv::Mat scan_cv;
scan2mat(scan, scan_cv);
fbr::panorama fPanorama(width, height, fbr::EQUIRECTANGULAR, 1, 0, fbr::EXTENDED);
fPanorama.createPanorama(scan_cv);
cv::Mat img = fPanorama.getRangeImage();
vector<vector<vector<cv::Vec3f> > > extended_map = fPanorama.getExtendedMap();
for (int row = 0; row < height; ++row) {
for (int col = 0; col < width; ++col) {
vector<cv::Vec3f> points_panorama = extended_map[row][col];
/// if no points found, skip pixel
if (points_panorama.size() < 1) continue;
/// foreach point from panorama consider all points in the bucket as its neighbors
for (size_t point_idx = 0; point_idx < points_panorama.size(); ++point_idx) {
Point point;
point.x = points_panorama[point_idx][0];
point.y = points_panorama[point_idx][1];
point.z = points_panorama[point_idx][2];
vector<Point> neighbors;
for (size_t i = 0; i < points_panorama.size(); ++i) {
if (i != point_idx)
neighbors.push_back(Point (points_panorama[i][0], points_panorama[i][1], points_panorama[i][2]) );
}
/// add neighbors from adjacent pixels and buckets
for (int i = -1; i <= 1; ++i) {
for (int j = -1; j <= 1; ++j) {
if (!(i==0 && j==0) && !(row+i < 0 || col+j < 0)
&& !(row+i >= height || col+j >= width) ) {
vector<cv::Vec3f> neighbors_panorama = extended_map[row+i][col+j];
for (size_t k = 0; k < neighbors_panorama.size(); ++k)
neighbors.push_back(Point (neighbors_panorama[k][0],
neighbors_panorama[k][1],
neighbors_panorama[k][2]) );
}
}
}
/// filter the point by kNN search
PointNeighbor current_point(point, neighbors);
if (knn > 0) {
PointNeighbor filtered_point;
computeKNearestNeighbors(current_point, filtered_point, knn);
points_neighbors.push_back(filtered_point);
} else {
points_neighbors.push_back(current_point);
}
}
}
}
}
/**
* Compute normals using PCA given a set of points and their neighbors
* @param scan - pointer to current scan, used to compute the position vectors
* @param points - input set of points with corresponding neighbors
* @param normals - output set of normals
*/
void computePCA(const Scan* scan, const vector<PointNeighbor>& points,
vector<Point>& normals, bool flipnormals)
{
ColumnVector origin(3);
const double *scan_pose = scan->get_rPos();
for (int i = 0; i < 3; ++i)
origin(i+1) = scan_pose[i];
for(size_t i = 0; i < points.size(); ++i) {
vector<Point> neighbors = points[i].neighbors;
if (points[i].neighbors.size() < 2) {
normals.push_back( Point(0,0,0) );
continue;
}
ColumnVector point_vector(3);
point_vector(1) = points[i].point.x - origin(1);
point_vector(2) = points[i].point.y - origin(2);
point_vector(3) = points[i].point.z - origin(3);
point_vector = point_vector / point_vector.NormFrobenius();
Matrix e_vectors(3,3); e_vectors = 0.0;
DiagonalMatrix e_values(3); e_values = 0.0;
computeEigenDecomposition(points[i], e_values, e_vectors);
ColumnVector v1(3);
v1(1) = e_vectors(1,1);
v1(2) = e_vectors(2,1);
v1(3) = e_vectors(3,1);
// consider first (smallest) eigenvector as the normal
Real angle = (v1.t() * point_vector).AsScalar();
// orient towards scan pose
// works better when orientation is not flipped
if (flipnormals && angle < 0) {
v1 *= -1.0;
}
normals.push_back( Point(v1(1), v1(2), v1(3)) );
}
}
void computeSRI(int factor, vector<Point>& points, vector<Point>& normals)
{
SRI *sri2 = new SRI(0, factor);
for (size_t i = 0; i < points.size(); i++) {
sri2->addPoint(points[i].x, points[i].y, points[i].z);
}
points.clear();
for (unsigned int i = 0; i < sri2->points.size(); i++) {
double rgbN[3], x, y, z;
PointN* p = sri2->points[i];
p->getCartesian(x, y, z);
sri2->getNormalSRI(p, rgbN);
normals.push_back(Point(rgbN[0], rgbN[1], rgbN[2]));
points.push_back(Point(x, z, y));
}
}
void scan2points(Scan* scan, vector<Point> &points)
{
DataXYZ xyz = scan->get("xyz");
unsigned int nPoints = xyz.size();
for(unsigned int i = 0; i < nPoints; ++i) {
points.push_back(Point(xyz[i][0], xyz[i][1], xyz[i][2]));
}
}
int main(int argc, char **argv)
{
// commandline arguments
int start, end;
bool scanserver;
int maxDist, minDist;
string dir;
IOType iotype;
normal_method normalMethod;
int knn, kmin, kmax;
double alpha;
int width, height;
bool flipnormals;
double factor;
parse_options(argc, argv, start, end, scanserver, dir, iotype, maxDist,
minDist, normalMethod, knn, kmin, kmax, alpha, width, height,
flipnormals, factor);
Scan::openDirectory(scanserver, dir, iotype, start, end);
if(Scan::allScans.size() == 0) {
cerr << "No scans found. Did you use the correct format?" << endl;
exit(-1);
}
boost::filesystem::path boost_dir(dir + "normals/");
boost::filesystem::create_directory(boost_dir);
for(ScanVector::iterator it = Scan::allScans.begin(); it != Scan::allScans.end(); ++it) {
Scan* scan = *it;
// apply optional filtering
scan->setRangeFilter(maxDist, minDist);
vector<PointNeighbor> points_neighbors;
vector<Point> normals;
vector<Point> points;
scan2points(scan, points);
switch (normalMethod) {
case KNN_PCA:
computeKNearestNeighbors(points, points_neighbors, knn);
computePCA(scan, points_neighbors, normals, flipnormals);
break;
case AKNN_PCA:
computeKNearestNeighbors(points, points_neighbors, kmin, kmax, alpha);
computePCA(scan, points_neighbors, normals, flipnormals);
break;
case PANO_PCA:
computePanoramaNeighbors(scan, points_neighbors, width, height, knn);
computePCA(scan, points_neighbors, normals, flipnormals);
break;
case PANO_SRI:
computeSRI(factor, points, normals);
break;
default:
cerr << "unknown normal method" << endl;
return 1;
break;
}
if (points.size() != normals.size()) {
cerr << "got " << points.size() << " points but " << normals.size() << " normals" << endl;
return 1;
}
writeNormals(scan, dir + "normals/", points, normals);
}
Scan::closeDirectory();
return 0;
}

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/*******************************************************
A simple program that demonstrates NewMat10 library.
The program defines a random symmetric matrix
and computes its eigendecomposition.
For further details read the NewMat10 Reference Manual
********************************************************/
#include <stdlib.h>
#include <time.h>
#include <string.h>
// the following two are needed for printing
#include <iostream.h>
#include <iomanip.h>
/**************************************
/* The NewMat10 include files */
#include <include.h>
#include <newmat.h>
#include <newmatap.h>
#include <newmatio.h>
/***************************************/
int main(int argc, char **argv) {
int M = 3, N = 5;
Matrix X(M,N); // Define an M x N general matrix
// Fill X by random numbers between 0 and 9
// Note that indexing into matrices in NewMat is 1-based!
srand(time(NULL));
for (int i = 1; i <= M; ++i) {
for (int j = 1; j <= N; ++j) {
X(i,j) = rand() % 10;
}
}
SymmetricMatrix C;
C << X * X.t(); // fill in C by X * X^t.
// Works because we *know* that the result is symmetric
cout << "The symmetrix matrix C" << endl;
cout << setw(5) << setprecision(0) << C << endl;
// compute eigendecomposition of C
Matrix V(3,3); // for eigenvectors
DiagonalMatrix D(3); // for eigenvalues
// the decomposition
Jacobi(C, D, V);
// Print the result
cout << "The eigenvalues matrix:" << endl;
cout << setw(10) << setprecision(5) << D << endl;
cout << "The eigenvectors matrix:" << endl;
cout << setw(10) << setprecision(5) << V << endl;
// Check that the first eigenvector indeed has the eigenvector property
ColumnVector v1(3);
v1(1) = V(1,1);
v1(2) = V(2,1);
v1(3) = V(3,1);
ColumnVector Cv1 = C * v1;
ColumnVector lambda1_v1 = D(1) * v1;
cout << "The max-norm of the difference between C*v1 and lambda1*v1 is " <<
NormInfinity(Cv1 - lambda1_v1) << endl << endl;
// Build the inverse and check the result
Matrix Ci = C.i();
Matrix I = Ci * C;
cout << "The inverse of C is" << endl;
cout << setw(10) << setprecision(5) << Ci << endl;
cout << "And the inverse times C is identity" << endl;
cout << setw(10) << setprecision(5) << I << endl;
// Example for multiple solves (see NewMat documentation)
ColumnVector r1(3), r2(3);
for (i = 1; i <= 3; ++i) {
r1(i) = rand() % 10;
r2(i) = rand() % 10;
}
LinearEquationSolver CLU = C; // decomposes C
ColumnVector s1 = CLU.i() * r1;
ColumnVector s2 = CLU.i() * r2;
cout << "solution for right hand side r1" << endl;
cout << setw(10) << setprecision(5) << s1 << endl;
cout << "solution for right hand side r2" << endl;
cout << setw(10) << setprecision(5) << s2 << endl;
return 0;
}