Move eigen to third party (#282)

* remove useless statement

* Add eigen to third_party dir

* remove reducdant lines

third_party/eigen/doc/examples/TutorialInplaceLU.cpp

@@ -0,0 +1,66 @@
+#include <iostream>
+struct init {
+  init() {
+    std::cout << "["
+              << "init"
+              << "]" << std::endl;
+  }
+};
+init init_obj;
+// [init]
+#include <Eigen/Dense>
+#include <iostream>
+
+using namespace std;
+using namespace Eigen;
+
+int main() {
+  MatrixXd A(2, 2);
+  A << 2, -1, 1, 3;
+  cout << "Here is the input matrix A before decomposition:\n" << A << endl;
+  cout << "[init]" << endl;
+
+  cout << "[declaration]" << endl;
+  PartialPivLU<Ref<MatrixXd> > lu(A);
+  cout << "Here is the input matrix A after decomposition:\n" << A << endl;
+  cout << "[declaration]" << endl;
+
+  cout << "[matrixLU]" << endl;
+  cout << "Here is the matrix storing the L and U factors:\n"
+       << lu.matrixLU() << endl;
+  cout << "[matrixLU]" << endl;
+
+  cout << "[solve]" << endl;
+  MatrixXd A0(2, 2);
+  A0 << 2, -1, 1, 3;
+  VectorXd b(2);
+  b << 1, 2;
+  VectorXd x = lu.solve(b);
+  cout << "Residual: " << (A0 * x - b).norm() << endl;
+  cout << "[solve]" << endl;
+
+  cout << "[modifyA]" << endl;
+  A << 3, 4, -2, 1;
+  x = lu.solve(b);
+  cout << "Residual: " << (A0 * x - b).norm() << endl;
+  cout << "[modifyA]" << endl;
+
+  cout << "[recompute]" << endl;
+  A0 = A;  // save A
+  lu.compute(A);
+  x = lu.solve(b);
+  cout << "Residual: " << (A0 * x - b).norm() << endl;
+  cout << "[recompute]" << endl;
+
+  cout << "[recompute_bis0]" << endl;
+  MatrixXd A1(2, 2);
+  A1 << 5, -2, 3, 4;
+  lu.compute(A1);
+  cout << "Here is the input matrix A1 after decomposition:\n" << A1 << endl;
+  cout << "[recompute_bis0]" << endl;
+
+  cout << "[recompute_bis1]" << endl;
+  x = lu.solve(b);
+  cout << "Residual: " << (A1 * x - b).norm() << endl;
+  cout << "[recompute_bis1]" << endl;
+}