Add simple example on how to compute Cholesky decomposition.

Eigen/src/Cholesky/LDLT.h

@@ -31,7 +31,7 @@ namespace internal {
 template<typename MatrixType, int UpLo> struct LDLT_Traits;
 }
 
-/** \ingroup cholesky_Module
+/** \ingroup Cholesky_Module
   *
   * \class LDLT
   *

Eigen/src/Cholesky/LLT.h

@@ -29,7 +29,7 @@ namespace internal{
 template<typename MatrixType, int UpLo> struct LLT_Traits;
 }
 
-/** \ingroup cholesky_Module
+/** \ingroup Cholesky_Module
   *
   * \class LLT
   *
@@ -49,6 +49,9 @@ template<typename MatrixType, int UpLo> struct LLT_Traits;
   * use LDLT instead for the semidefinite case. Also, do not use a Cholesky decomposition to determine whether a system of equations
   * has a solution.
   *
+  * Example: \include LLT_example.cpp
+  * Output: \verbinclude LLT_example.out
+  *    
   * \sa MatrixBase::llt(), class LDLT
   */
  /* HEY THIS DOX IS DISABLED BECAUSE THERE's A BUG EITHER HERE OR IN LDLT ABOUT THAT (OR BOTH)
@@ -333,9 +336,11 @@ template<typename MatrixType> struct LLT_Traits<MatrixType,Upper>
 } // end namespace internal
 
 /** Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of \a matrix
-  *
   *
   * \returns a reference to *this
+  *
+  * Example: \include TutorialLinAlgComputeTwice.cpp
+  * Output: \verbinclude TutorialLinAlgComputeTwice.out
   */
 template<typename MatrixType, int _UpLo>
 LLT<MatrixType,_UpLo>& LLT<MatrixType,_UpLo>::compute(const MatrixType& a)

doc/C06_TutorialLinearAlgebra.dox

@@ -144,6 +144,9 @@ You need an eigendecomposition here, see available such decompositions on \ref T
 Make sure to check if your matrix is self-adjoint, as is often the case in these problems. Here's an example using
 SelfAdjointEigenSolver, it could easily be adapted to general matrices using EigenSolver or ComplexEigenSolver.
 
+The computation of eigenvalues and eigenvectors does not necessarily converge, but such failure to converge is
+very rare. The call to info() is to check for this possibility.
+
 <table class="example">
 <tr><th>Example:</th><th>Output:</th></tr>
 <tr>

doc/snippets/LLT_example.cpp

@@ -0,0 +1,12 @@
+MatrixXd A(3,3);
+A << 4,-1,2, -1,6,0, 2,0,5;
+cout << "The matrix A is" << endl << A << endl;
+
+LLT<MatrixXd> lltOfA(A); // compute the Cholesky decomposition of A
+MatrixXd L = lltOfA.matrixL(); // retrieve factor L  in the decomposition
+// The previous two lines can also be written as "L = A.llt().matrixL()"
+
+cout << "The Cholesky factor L is" << endl << L << endl;
+cout << "To check this, let us compute L * L.transpose()" << endl;
+cout << L * L.transpose() << endl;
+cout << "This should equal the matrix A" << endl;