A class for orthogonal matrices

[[Imported from SVN: r8554]]

dune/gfe/Makefile.am

@@ -20,6 +20,7 @@ srcinclude_HEADERS = averagedistanceassembler.hh \
                      localgeodesicfestiffness.hh \
                      localgfetestfunctionbasis.hh \
                      maxnormtrustregion.hh \
+                     orthogonalmatrix.hh \
                      pktop1mgtransfer.hh \
                      quaternion.hh \
                      realtuple.hh \

dune/gfe/orthogonalmatrix.hh

+#ifndef ORTHOGONAL_MATRIX_HH
+#define ORTHOGONAL_MATRIX_HH
+
+#include <dune/common/fmatrix.hh>
+
+/** \brief An orthogonal \f$ n \times n \f$ matrix
+ * \tparam T Type of the matrix entries
+ * \tparam N Space dimension
+ */
+template <class T, int N>
+class OrthogonalMatrix
+{
+public:
+    
+    /** \brief Constructor from a general matrix
+     * 
+     * The input matrix gets normalized to det = 1.  Since computing
+     * the determinant is expensive you may not want to use this method
+     * if you know your input matrix to be orthogonal anyways.
+     */
+    explicit OrthogonalMatrix(const Dune::FieldMatrix<T,N,N>& matrix)
+    : data_(matrix)
+    {
+        data_ /= matrix.determinant();
+    }
+
+    /** \brief Const access to the matrix entries */
+    const Dune::FieldMatrix<T,N,N>& data() const
+    {
+        return data_;
+    }
+    
+    /** \brief Project the input matrix onto the tangent space at "this"
+     * 
+     * See Absil, Mahoney, Sepulche, Example 5.3.2 for the formula
+     * \f[ P_X \xi = (I - XX^T) \xi + X \operatorname{skew} (X^T \xi)  \f]
+     */
+    Dune::FieldMatrix<T,N,N> projectOntoTangentSpace(const Dune::FieldMatrix<T,N,N>& matrix) const
+    {
+        // rename to get the same notation as Absil & Co
+        const Dune::FieldMatrix<T,N,N>& X = data_;
+        
+        // I - XX^T
+        Dune::FieldMatrix<T,N,N> IdMinusXXT;
+        
+        for (int i=0; i<N; i++)
+            for (int j=0; j<N; j++) {
+                IdMinusXXT = (i==j);
+                for (int k=0; k<N; k++)
+                    IdMinusXXT[i][j] -= X[i][k] * X[j][k];
+            }
+            
+        // (I - XX^T) \xi
+        Dune::FieldMatrix<T,N,N> result(0);
+        for (int i=0; i<N; i++)
+            for (int j=0; j<N; j++)
+                for (int k=0; k<N; k++)
+                    result[i][j] += IdMinusXXT[i][k] * matrix[k][j];
+                
+        // X^T \xi
+        Dune::FieldMatrix<T,N,N> XTxi(0);
+        for (int i=0; i<N; i++)
+            for (int j=0; j<N; j++)
+                for (int k=0; k<N; k++)
+                    XTxi[i][j] += X[k][i] * matrix[k][j];
+                
+        // skew (X^T \xi)
+        Dune::FieldMatrix<T,N,N> skewXTxi(0);
+        for (int i=0; i<N; i++)
+            for (int j=0; j<N; j++)
+                skewXTxi[i][j] = 0.5 * ( XTxi[i][j] - XTxi[j][i] );
+                
+        // X skew (X^T \xi)
+        Dune::FieldMatrix<T,N,N> XskewXTxi(0);
+        for (int i=0; i<N; i++)
+            for (int j=0; j<N; j++)
+                for (int k=0; k<N; k++)
+                    XskewXTxi[i][j] += X[i][k] * skewXTxi[k][j];
+                
+        //
+        result += XskewXTxi;
+        
+        return result;
+    }
+    
+private:
+    
+    /** \brief The actual data */
+    Dune::FieldMatrix<T,N,N> data_;
+    
+};
+
+#endif // ORTHOGONAL_MATRIX_HH
\ No newline at end of file