nearly complete page 6 / linear algebra + examples

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doc/examples/TutorialLinAlgComputeTwice.cpp

@@ -0,0 +1,23 @@
+#include <iostream>
+#include <Eigen/Dense>
+
+using namespace std;
+using namespace Eigen;
+
+int main()
+{
+   Matrix2f A, b;
+   LLT<Matrix2f> llt;
+   A << 2, -1, -1, 3;
+   b << 1, 2, 3, 1;
+   cout << "Here is the matrix A:\n" << A << endl;
+   cout << "Here is the right hand side b:\n" << b << endl;
+   cout << "Computing LLT decomposition..." << endl;
+   llt.compute(A);
+   cout << "The solution is:\n" << llt.solve(b) << endl;
+   A(1,1)++;
+   cout << "The matrix A is now:\n" << A << endl;
+   cout << "Computing LLT decomposition..." << endl;
+   llt.compute(A);
+   cout << "The solution is now:\n" << llt.solve(b) << endl;
+}

doc/examples/TutorialLinAlgExComputeSolveError.cpp

@@ -0,0 +1,14 @@
+#include <iostream>
+#include <Eigen/Dense>
+
+using namespace std;
+using namespace Eigen;
+
+int main()
+{
+   MatrixXd A = MatrixXd::Random(100,100);
+   MatrixXd b = MatrixXd::Random(100,50);
+   MatrixXd x = A.fullPivLu().solve(b);
+   double relative_error = (A*x - b).norm() / b.norm(); // norm() is L2 norm
+   cout << "The relative error is:\n" << relative_error << endl;
+}

doc/examples/TutorialLinAlgExSolveColPivHouseholderQR.cpp

@@ -10,8 +10,8 @@ int main()
    Vector3f b;
    A << 1,2,3,  4,5,6,  7,8,10;
    b << 3, 3, 4;
-   cout << "Here is the matrix A:" << endl << A << endl;
-   cout << "Here is the vector b:" << endl << b << endl;
+   cout << "Here is the matrix A:\n" << A << endl;
+   cout << "Here is the vector b:\n" << b << endl;
    Vector3f x = A.colPivHouseholderQr().solve(b);
-   cout << "The solution is:" << endl << x << endl;
+   cout << "The solution is:\n" << x << endl;
 }

doc/examples/TutorialLinAlgExSolveLDLT.cpp

@@ -0,0 +1,16 @@
+#include <iostream>
+#include <Eigen/Dense>
+
+using namespace std;
+using namespace Eigen;
+
+int main()
+{
+   Matrix2f A, b;
+   A << 2, -1, -1, 3;
+   b << 1, 2, 3, 1;
+   cout << "Here is the matrix A:\n" << A << endl;
+   cout << "Here is the right hand side b:\n" << b << endl;
+   Matrix2f x = A.ldlt().solve(b);
+   cout << "The solution is:\n" << x << endl;
+}

doc/examples/TutorialLinAlgInverseDeterminant.cpp

@@ -0,0 +1,16 @@
+#include <iostream>
+#include <Eigen/Dense>
+
+using namespace std;
+using namespace Eigen;
+
+int main()
+{
+   Matrix3f A;
+   A << 1, 2, 1,
+        2, 1, 0,
+        -1, 1, 2;
+   cout << "Here is the matrix A:\n" << A << endl;
+   cout << "The determinant of A is " << A.determinant() << endl;
+   cout << "The inverse of A is:\n" << A.inverse() << endl;
+}

doc/examples/TutorialLinAlgRankRevealing.cpp

@@ -0,0 +1,20 @@
+#include <iostream>
+#include <Eigen/Dense>
+
+using namespace std;
+using namespace Eigen;
+
+int main()
+{
+   Matrix3f A;
+   A << 1, 2, 5,
+        2, 1, 4,
+        3, 0, 3;
+   cout << "Here is the matrix A:\n" << A << endl;
+   FullPivLU<Matrix3f> lu_decomp(A);
+   cout << "The rank of A is " << lu_decomp.rank() << endl;
+   cout << "Here is a matrix whose columns form a basis of the null-space of A:\n"
+        << lu_decomp.kernel() << endl;
+   cout << "Here is a matrix whose columns form a basis of the column-space of A:\n"
+        << lu_decomp.image(A) << endl; // yes, have to pass the original A
+}

doc/examples/TutorialLinAlgSelfAdjointEigenSolver.cpp

@@ -0,0 +1,17 @@
+#include <iostream>
+#include <Eigen/Dense>
+
+using namespace std;
+using namespace Eigen;
+
+int main()
+{
+   Matrix2f A;
+   A << 1, 2, 2, 3;
+   cout << "Here is the matrix A:\n" << A << endl;
+   SelfAdjointEigenSolver<Matrix2f> eigensolver(A);
+   cout << "The eigenvalues of A are:\n" << eigensolver.eigenvalues() << endl;
+   cout << "Here's a matrix whose columns are eigenvectors of A "
+        << "corresponding to these eigenvalues:\n"
+        << eigensolver.eigenvectors() << endl;
+}

doc/examples/TutorialLinAlgSetThreshold.cpp

@@ -0,0 +1,19 @@
+#include <iostream>
+#include <Eigen/Dense>
+
+using namespace std;
+using namespace Eigen;
+
+int main()
+{
+   Matrix2d A;
+   FullPivLU<Matrix2d> lu;
+   A << 2, 1,
+        2, 0.9999999999;
+   lu.compute(A);
+   cout << "By default, the rank of A is found to be " << lu.rank() << endl;
+   cout << "Now recomputing the LU decomposition with threshold 1e-5" << endl;
+   lu.setThreshold(1e-5);
+   lu.compute(A);
+   cout << "The rank of A is found to be " << lu.rank() << endl;
+}