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ops_matrices_eigen.h

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#ifndef  mrpt_ops_matrices_eigen_H
#define  mrpt_ops_matrices_eigen_H

#include <mrpt/math/math_frwds.h>  // Fordward declarations
#include <mrpt/math/matrices_metaprogramming.h>  // TMatrixProductType, ...


/** \file ops_matrices_eigen.h
  * This file implements eigenvalues/eigenvectors for any kind of matrix-like object.
  */

namespace mrpt
{
        namespace math
        {
        namespace detail
        {
                /** @name Matrix eigenvectors
                   @{ */

                template <class MAT,class ARR>  void  tred2(MAT &a, const size_t nn, ARR &d, ARR &e);
                template <class MAT,class ARR>  void  tqli(ARR &d, ARR &e, const size_t nn, MAT &z);

                /** eigenVectorsMatrix
                  * \param eVecs if set to NULL, only compute eigenvalues
                  */
                template <class MATRIX1,class MATRIX2,class VECTOR1> void
                eigenVectorsMatrix(const MATRIX1 &M,MATRIX2 *eVecs,VECTOR1 &eVals )
                {
                        ASSERTMSG_(M.isSquare(), "eigenvalues can be computed for square matrices only.")
                        bool doEVecs = (eVecs!=NULL);
                        const size_t N = M.getColCount();

                        // TODO: (JL) Rewrite this whole thing, please!
                        typedef typename MATRIX1::value_type T;
                        MAT_TYPE_SAMESIZE_OF(MATRIX1)  a; // Auxiliary matrix a, of the same size than "M"
                        a.setSize(M.getRowCount(),M.getColCount());

                        // d, e: Auxiliary vectors of the same length than the matrix dims:
                        ARRAY_TYPE_SAMESIZE_ROWS_OF(MATRIX1) d,e;
                        d.resize(M.getRowCount());
                        e.resize(M.getRowCount());

                        MRPT_START

                        // Check for symmetry
#ifdef _DEBUG
                        for (size_t i=0;i<N;i++)
                                for (size_t j=i;j<N;j++)
                                        if (M.get_unsafe(i,j)!=M.get_unsafe(j,i))
                                        {
                                                THROW_EXCEPTION(format("eigenVectors: The matrix is not symmetric! m(%lu,%lu)=%.16e != m(%lu,%lu)=%.16e\n",
                                                        static_cast<unsigned long>(i),static_cast<unsigned long>(j), static_cast<double> ( M.get_unsafe(i,j) ),
                                                        static_cast<unsigned long>(j),static_cast<unsigned long>(i), static_cast<double> ( M.get_unsafe(j,i) )) )
                                        }
#endif

                        // Copy the matrix content to "a":
                        // --------------------------------------
                        a = M;

                        // Algorithm
                        // --------------------------------------
                        tred2( a, N, d, e);
                        tqli(d,e,N,a);

                        // "d" -> eigenvalues
                        // "a" -> eigenvectors as columns:

                        // SORT: Build a list of the N index in
                        //   ascending order of eigenvalue:
                        // --------------------------------------
                        std::vector<unsigned int>       indxs(N+1);
                        std::vector<bool>       already(N+1, false);

                        for (size_t i=0;i<N;i++)
                        {
                                size_t  minIndx = std::numeric_limits<size_t>::max();
                                for (size_t j=0;j<N;j++)
                                        if (!already[j])
                                        {
                                                if (minIndx==std::numeric_limits<size_t>::max())                minIndx = j;
                                                else
                                                        if (d[j]<d[minIndx])    minIndx = j;
                                        }

                                // The i'th sorted element:
                                indxs[i] = static_cast<unsigned int> ( minIndx );
                                already[minIndx] = true;
                        }

                        for (size_t i=0;i<N;i++)
                                ASSERT_(already[i]);

                        // Prepare output:
                        eVals.resize(N);
                        if (doEVecs) eVecs->setSize(N,N);

                        // Copy results to matrices classes
                        for (size_t i=0;i<N;i++)
                        {
                                eVals[i]=d[indxs[i]]; // Copy eigenvalue
                                if (doEVecs)
                                        for (size_t j=0;j<N;j++)
                                                eVecs->get_unsafe(i,j) = a.get_unsafe(i,indxs[j]);
                        }

                        MRPT_END_WITH_CLEAN_UP( std::cout << "[eigenVectors] The matrix leading to exception is:" << std::endl << M << std::endl; )
                }


                // Special case for 2x2 matrices:
                template <class MATRIX1,class MATRIX2,class VECTOR1> void
                eigenVectorsMatrix_special_2x2(const MATRIX1 &M,MATRIX2 *eVecs,VECTOR1 &eVals )
                {
                        typedef typename MATRIX1::value_type T;
                        const T t = M.get_unsafe(0,0) + M.get_unsafe(1,1);
                        const T de = M.get_unsafe(0,0)*M.get_unsafe(1,1)-M.get_unsafe(1,0)*M.get_unsafe(0,1);
                        eVals.resize(2);
                        // Eigenvalues, in ascending order:
                        const T disc = ::sqrt(0.25*t*t-de);
                        eVals[0] =0.5*t - disc;
                        eVals[1] =0.5*t + disc;
                        if (!eVecs) return;
                        eVecs->setSize(2,2);
                        const T c = M.get_unsafe(1,0);
                        const T b = M.get_unsafe(0,1);
                        if (c!=0)
                        {
                                eVecs->get_unsafe(0,0)=eVals[0]-M.get_unsafe(1,1);
                                const T ev1n = 1.0/hypot( eVecs->get_unsafe(0,0), c );
                                eVecs->get_unsafe(0,0)*=ev1n;
                                eVecs->get_unsafe(1,0)=c*ev1n;

                                eVecs->get_unsafe(0,1)=eVals[1]-M.get_unsafe(1,1);
                                const T ev2n = 1.0/hypot( eVecs->get_unsafe(0,1), c );
                                eVecs->get_unsafe(0,1)*=ev2n;
                                eVecs->get_unsafe(1,1)=c*ev2n;
                        }
                        else if (b!=0)
                        {
                                eVecs->get_unsafe(1,0)=eVals[0]-M.get_unsafe(0,0);
                                const T ev1n = 1.0/hypot( eVecs->get_unsafe(1,0), b );
                                eVecs->get_unsafe(1,0)*=ev1n;
                                eVecs->get_unsafe(0,0)=b*ev1n;

                                eVecs->get_unsafe(1,1)=eVals[1]-M.get_unsafe(0,0);
                                const T ev2n = 1.0/hypot( eVecs->get_unsafe(1,1), b );
                                eVecs->get_unsafe(1,1)*=ev2n;
                                eVecs->get_unsafe(0,1)=b*ev2n;
                                // Normalize eigenvectors:
                        } else
                        {
                                eVecs->get_unsafe(0,0)=1;
                                eVecs->get_unsafe(1,0)=0;
                                eVecs->get_unsafe(0,1)=0;
                                eVecs->get_unsafe(1,1)=1;
                        }
                }


                // Auxiliary funcs:
                template <class T> inline T  pythag(const T a, const T b) {
                        T at, bt, ct;
                        return static_cast<T>(( ((at = fabs(a)) > (bt = fabs(b)) ?
                                        (ct = bt/at, at*(::sqrt(1.0+ct*ct))) :
                                        (bt ? (ct = at/bt, bt*(::sqrt(1.0+ct*ct))): 0)) ) );
                }

                template <class T> inline T  SIGN(T a,T b) { return static_cast<T>(( ((b) >= 0 ? fabs(a) : -fabs(a)) )); }

                /**
                        Householder reduction of a real, symmetric matrix a[1..n][1..n]. On output, a is replaced
                        by the orthogonal matrix Q reflecting the transformation. d[1..n] returns the diagonal elements
                        of the tridiagonal matrix, and e[1..n] the off-diagonal elements, with e[1]=0. Several
                        statements, as noted in comments, can be omitted if only eigenvalues are to be found, in which
                        case a contains no useful information on output. Otherwise they are to be included.
                */
                template <class MAT,class ARR>
                void  tred2(MAT &a, const size_t n, ARR &d, ARR &e)
                {
                        typename MAT::value_type scale,hh,h,g,f;

                        for (size_t i=n-1;i!=0;i--)
                        {
                                //size_t l=i-1;
                                h=scale=0.0;
                                if (i > 1)
                                {
                                        for (size_t k=0;k<i;k++)
                                                scale += fabs(a.get_unsafe(i,k));
                                        if (scale == 0)                         // Skip transformation.
                                                e[i]=a.get_unsafe(i,i-1);
                                        else
                                        {
                                                for (size_t k=0;k<i;k++)
                                                {
                                                        a.get_unsafe(i,k) /= scale;             // Use scaled a's for transformation.
                                                        h += a.get_unsafe(i,k)*a.get_unsafe(i,k);       // Form sigma in h.
                                                }
                                                f=a.get_unsafe(i,i-1);
                                                g=(f >= 0 ? -::sqrt(h) : (::sqrt(h)));
                                                e[i]=scale*g;
                                                h -= f*g;                                               // Now h is equation (11.2.4).
                                                a.get_unsafe(i,i-1)=f-g;                                        // Store u in the ith row of a.
                                                f=0;
                                                for (size_t j=0;j<i;j++)
                                                {
                                                        a.get_unsafe(j,i)=a.get_unsafe(i,j)/h;                  // Store u=H in ith column of a.
                                                        g=0.0;                                          // Form an element of A . u in g.
                                                        for (size_t k=0;k<=j;k++)
                                                                g += a.get_unsafe(j,k)*a.get_unsafe(i,k);
                                                        for (size_t k=j+1;k<i;k++)
                                                                g += a.get_unsafe(k,j)*a.get_unsafe(i,k);
                                                        e[j]=g/h;                                       // Form element of p in temporarily unused element of e.
                                                        f += e[j]*a.get_unsafe(i,j);
                                                }
                                                hh=f/(h+h);                                             // Form K, equation (11.2.11).
                                                for (size_t j=0;j<i;j++)
                                                {                                                               // Form q and store in e overwriting p.
                                                        f=a.get_unsafe(i,j);
                                                        e[j]=g=e[j]-hh*f;
                                                        for (size_t k=0;k<=j;k++)                       // Reduce a, equation (11.2.13).
                                                                a.get_unsafe(j,k) -= (f*e[k]+g*a.get_unsafe(i,k));
                                                }
                                        }
                                } else
                                        e[i]=a.get_unsafe(i,i-1);
                                d[i]=h;
                        }

                        /* Next statement can be omitted if eigenvectors not wanted */
                        d[0]=0;
                        e[0]=0;

                        /* Contents of this loop can be omitted if eigenvectors not
                        wanted except for statement d[i]=a[i][i]; */
                        for (size_t i=0;i<n;i++)        // Begin accumulation of transformationmatrices
                        {
                                if (d[i]) // This block skipped when i=1.
                                        for (size_t j=0;j<i;j++)
                                        {
                                                g=0.0;
                                                for (size_t k=0;k<i;k++) // Use u and u=H stored in a to form P.Q.
                                                        g += a.get_unsafe(i,k)*a.get_unsafe(k,j);
                                                for (size_t k=0;k<i;k++)
                                                        a.get_unsafe(k,j) -= g*a.get_unsafe(k,i);
                                        }
                                d[i]=a.get_unsafe(i,i);                 // This statement remains.
                                a.get_unsafe(i,i)=1;                            // Reset row and column of a to identity
                                for (size_t j=0;j<i;j++)
                                        a.get_unsafe(j,i)=a.get_unsafe(i,j)= 0; // matrix for next iteration.
                        }
                }

                /** QL algorithm with implicit shifts, to determine the eigenvalues and eigenvectors of a real, symmetric,
                        tridiagonal matrix, or of a real, symmetric matrix previously reduced by tred2 x11.2. On
                        input, d[1..n] contains the diagonal elements of the tridiagonal matrix. On output, it returns
                        the eigenvalues. The vector e[1..n] inputs the subdiagonal elements of the tridiagonal matrix,
                        with e[1] arbitrary. On output e is destroyed. When finding only the eigenvalues, several lines
                        may be omitted, as noted in the comments. If the eigenvectors of a tridiagonal matrix are desired,
                        the matrix z[1..n][1..n] is input as the identity matrix. If the eigenvectors of a matrix
                        that has been reduced by tred2 are required, then z is input as the matrix output by tred2.
                        In either case, the kth column of z returns the normalized eigenvector corresponding to d[k].
                */
                template <class MAT,class ARR>
                void  tqli(ARR &d, ARR &e, const size_t n, MAT &z)
                {
                        typedef typename MAT::value_type T;

                        T                       s,r,p,g,f,dd,c,b;
                        const T         EPS= std::numeric_limits<T>::epsilon();

                        for (size_t i=1;i<n;i++)
                                e[i-1]=e[i];                            // Convenient to renumber the elements of e.

                        e[n-1]=0.0;
                        for (size_t l=0;l<n;l++)
                        {
                                size_t iter=0;
                                size_t m;
                                do
                                {
                                        for (m=l;m<(n-1);m++)   // Look for a single small subdiagonal element to split the matrix.
                                        {
                                                dd=static_cast<T>(( fabs(d[m])+fabs(d[m+1]) ) );
                                                // ----------------------------------------------------------------------
                                                // Fix added 14/DEC/2006: Avoid a convergence problem due to compiler
                                                //  optimization. See http://www.nr.com/forum/showthread.php?p=1803
                                                // ----------------------------------------------------------------------
                                                // Fixed again 25/JAN/2008: Using EPS is a better approach!
                                                // ----------------------------------------------------------------------
                                                //temp = fabs(e[m])+dd;
                                                //if (fabs(e[m])+dd == dd) break;
                                                if (fabs(e[m]) <= EPS*dd)  break;
                                        }
                                        if (m != l)
                                        {
                                                if (iter++ == 60) THROW_EXCEPTION("tqli: Too many iterations in tqli!")

                                                g=static_cast<T>(( (d[l+1]-d[l])/(2.0*e[l])));                  // Form shift.
                                                r=pythag(g,static_cast<T>(1.0));
                                                g=d[m]-d[l]+e[l]/(g+SIGN(r,g));         // This is dm - ks.
                                                s = c = 1;
                                                p = 0;
                                                int i;
                                                for (i=int(m)-1;i>=int(l);i--)                          // A plane rotation as in the original QL, followed by Givens rotations to restore tridiagonal  form.
                                                {
                                                        f=s*e[i];
                                                        b=c*e[i];
                                                        e[i+1]=(r=pythag(f,g));
                                                        if (r == 0.0)                                           // Recover from underflow.
                                                        {
                                                                d[i+1] -= p;
                                                                e[m]=0.0;
                                                                break;
                                                        }
                                                        s=f/r;
                                                        c=g/r;
                                                        g=d[i+1]-p;
                                                        r=(d[i]-g)*s+2.0f*c*b;
                                                        d[i+1]=g+(p=s*r);
                                                        g=c*r-b;
                                                        /* Next loop can be omitted if eigenvectors not wanted*/
                                                        for (size_t k=0;k<n;k++)  // Form eigenvectors.
                                                        {
                                                                f=z.get_unsafe(k,i+1);
                                                                z.get_unsafe(k,i+1)=s*z.get_unsafe(k,i)+c*f;
                                                                z.get_unsafe(k,i)=c*z.get_unsafe(k,i)-s*f;
                                                        }
                                                }

                                                if (r == 0.0 && i >= int(l)) continue;
                                                d[l] -= p;
                                                e[l]=g;
                                                e[m]=0.0;
                                        }
                                } while (m != l);
                        }
                }



                /** @} */  // END OF MATRIX EIGENVECTORS

        } // end namespaces
        }
}

#endif

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