TooN 2.0.0-beta8
SymEigen.h
00001 // -*- c++ -*-
00002 
00003 // Copyright (C) 2005,2009 Tom Drummond (twd20@cam.ac.uk)
00004 //
00005 // This file is part of the TooN Library.   This library is free
00006 // software; you can redistribute it and/or modify it under the
00007 // terms of the GNU General Public License as published by the
00008 // Free Software Foundation; either version 2, or (at your option)
00009 // any later version.
00010 
00011 // This library is distributed in the hope that it will be useful,
00012 // but WITHOUT ANY WARRANTY; without even the implied warranty of
00013 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
00014 // GNU General Public License for more details.
00015 
00016 // You should have received a copy of the GNU General Public License along
00017 // with this library; see the file COPYING. If not, write to the Free
00018 // Software Foundation, 59 Temple Place - Suite 330, Boston, MA 02111-1307,
00019 // USA.
00020 
00021 // As a special exception, you may use this file as part of a free software
00022 // library without restriction. Specifically, if other files instantiate
00023 // templates or use macros or inline functions from this file, or you compile
00024 // this file and link it with other files to produce an executable, this
00025 // file does not by itself cause the resulting executable to be covered by
00026 // the GNU General Public License.  This exception does not however
00027 // invalidate any other reasons why the executable file might be covered by
00028 // the GNU General Public License.
00029 
00030 #ifndef __SYMEIGEN_H
00031 #define __SYMEIGEN_H
00032 
00033 #include <iostream>
00034 #include <cassert>
00035 #include <cmath>
00036 #include <complex>
00037 #include <TooN/lapack.h>
00038 
00039 #include <TooN/TooN.h>
00040 
00041 namespace TooN {
00042 static const double root3=1.73205080756887729352744634150587236694280525381038062805580;
00043 
00044 namespace Internal{
00045 
00046         using std::swap;
00047 
00048     ///Default condition number for SymEigen::backsub, SymEigen::get_pinv and SymEigen::get_inv_diag
00049     static const double symeigen_condition_no=1e9;
00050 
00051     ///@internal
00052     ///@brief Compute eigensystems for sizes > 2
00053     ///Helper struct for computing eigensystems, to allow for specialization on
00054     ///2x2 matrices.
00055     ///@ingroup gInternal
00056     template <int Size> struct ComputeSymEigen {
00057 
00058         ///@internal
00059         ///Compute an eigensystem.
00060         ///@param m Input matrix (assumed to be symmetric)
00061         ///@param evectors Eigen vector output
00062         ///@param evalues Eigen values output
00063         template<int Rows, int Cols, typename P, typename B>
00064         static inline void compute(const Matrix<Rows,Cols,P, B>& m, Matrix<Size,Size,P> & evectors, Vector<Size, P>& evalues) {
00065 
00066             SizeMismatch<Rows, Cols>::test(m.num_rows(), m.num_cols());  //m must be square
00067             SizeMismatch<Size, Rows>::test(m.num_rows(), evalues.size()); //m must be the size of the system
00068             
00069 
00070             evectors = m;
00071             FortranInteger N = evalues.size();
00072             FortranInteger lda = evalues.size();
00073             FortranInteger info;
00074             FortranInteger lwork=-1;
00075             P size;
00076 
00077             // find out how much space fortran needs
00078             syev_((char*)"V",(char*)"U",&N,&evectors[0][0],&lda,&evalues[0], &size,&lwork,&info);
00079             lwork = int(size);
00080             Vector<Dynamic, P> WORK(lwork);
00081 
00082             // now compute the decomposition
00083             syev_((char*)"V",(char*)"U",&N,&evectors[0][0],&lda,&evalues[0], &WORK[0],&lwork,&info);
00084 
00085             if(info!=0){
00086                 std::cerr << "In SymEigen<"<<Size<<">: " << info
00087                         << " off-diagonal elements of an intermediate tridiagonal form did not converge to zero." << std::endl
00088                         << "M = " << m << std::endl;
00089             }
00090         }
00091     };
00092 
00093     ///@internal
00094     ///@brief Compute 2x2 eigensystems
00095     ///Helper struct for computing eigensystems, specialized on 2x2 matrices.
00096     ///@ingroup gInternal
00097     template <> struct ComputeSymEigen<2> {
00098 
00099         ///@internal
00100         ///Compute an eigensystem.
00101         ///@param m Input matrix (assumed to be symmetric)
00102         ///@param eig Eigen vector output
00103         ///@param ev Eigen values output
00104         template<typename P, typename B>
00105         static inline void compute(const Matrix<2,2,P,B>& m, Matrix<2,2,P>& eig, Vector<2, P>& ev) {
00106             double trace = m[0][0] + m[1][1];
00107             double det = m[0][0]*m[1][1] - m[0][1]*m[1][0];
00108             double disc = trace*trace - 4 * det;
00109             assert(disc>=0);
00110             using std::sqrt;
00111             double root_disc = sqrt(disc);
00112             ev[0] = 0.5 * (trace - root_disc);
00113             ev[1] = 0.5 * (trace + root_disc);
00114             double a = m[0][0] - ev[0];
00115             double b = m[0][1];
00116             double magsq = a*a + b*b;
00117             if (magsq == 0) {
00118                 eig[0][0] = 1.0;
00119                 eig[0][1] = 0;
00120             } else {
00121                 eig[0][0] = -b;
00122                 eig[0][1] = a;
00123                 eig[0] *= 1.0/sqrt(magsq);
00124             }
00125             eig[1][0] = -eig[0][1];
00126             eig[1][1] = eig[0][0];
00127         }
00128     };
00129 
00130     ///@internal
00131     ///@brief Compute 3x3 eigensystems
00132     ///Helper struct for computing eigensystems, specialized on 3x3 matrices.
00133     ///@ingroup gInternal
00134     template <> struct ComputeSymEigen<3> {
00135 
00136         ///@internal
00137         ///Compute an eigensystem.
00138         ///@param m Input matrix (assumed to be symmetric)
00139         ///@param eig Eigen vector output
00140         ///@param ev Eigen values output
00141         template<typename P, typename B>
00142         static inline void compute(const Matrix<3,3,P,B>& m, Matrix<3,3,P>& eig, Vector<3, P>& ev) {
00143             //method uses closed form solution of cubic equation to obtain roots of characteristic equation.
00144             using std::sqrt;
00145             
00146             //Polynomial terms of |a - l * Identity|
00147             //l^3 + a*l^2 + b*l + c
00148 
00149             const double& a11 = m[0][0];
00150             const double& a12 = m[0][1];
00151             const double& a13 = m[0][2];
00152 
00153             const double& a22 = m[1][1];
00154             const double& a23 = m[1][2];
00155 
00156             const double& a33 = m[2][2];
00157 
00158             //From matlab:
00159             double a = -a11-a22-a33;
00160             double b = a11*a22+a11*a33+a22*a33-a12*a12-a13*a13-a23*a23;
00161             double c = a11*(a23*a23)+(a13*a13)*a22+(a12*a12)*a33-a12*a13*a23*2.0-a11*a22*a33;
00162 
00163             //Using Cardano's method:
00164             double p = b - a*a/3;
00165             double q = c + (2*a*a*a - 9*a*b)/27;
00166 
00167             double alpha = -q/2;
00168             double beta_descriminant = q*q/4 + p*p*p/27;
00169 
00170             //beta_descriminant <= 0 for real roots!
00171             double beta = sqrt(-beta_descriminant);
00172             double r2 = alpha*alpha  - beta_descriminant;
00173 
00174             ///Need A,B = cubert(alpha +- beta)
00175             ///Turn in to r, theta
00176             /// r^(1/3) * e^(i * theta/3)
00177 
00178             double cuberoot_r = pow(r2, 1./6);
00179             double theta3 = atan2(beta, alpha)/3;
00180 
00181             double A_plus_B = 2*cuberoot_r*cos(theta3);
00182             double A_minus_B = -2*cuberoot_r*sin(theta3);
00183 
00184             //calculate eigenvalues
00185             ev =  makeVector(A_plus_B, -A_plus_B/2 + A_minus_B * sqrt(3)/2, -A_plus_B/2 - A_minus_B * sqrt(3)/2) - Ones * a/3;
00186 
00187             if(ev[0] > ev[1])
00188                 swap(ev[0], ev[1]);
00189             if(ev[1] > ev[2])
00190                 swap(ev[1], ev[2]);
00191             if(ev[0] > ev[1])
00192                 swap(ev[0], ev[1]);
00193 
00194             //calculate the eigenvectors
00195             eig[0][0]=a12 * a23 - a13 * (a22 - ev[0]);
00196             eig[0][1]=a12 * a13 - a23 * (a11 - ev[0]);
00197             eig[0][2]=(a11-ev[0])*(a22-ev[0]) - a12*a12;
00198             normalize(eig[0]);
00199             eig[1][0]=a12 * a23 - a13 * (a22 - ev[1]);
00200             eig[1][1]=a12 * a13 - a23 * (a11 - ev[1]);
00201             eig[1][2]=(a11-ev[1])*(a22-ev[1]) - a12*a12;
00202             normalize(eig[1]);
00203             eig[2][0]=a12 * a23 - a13 * (a22 - ev[2]);
00204             eig[2][1]=a12 * a13 - a23 * (a11 - ev[2]);
00205             eig[2][2]=(a11-ev[2])*(a22-ev[2]) - a12*a12;
00206             normalize(eig[2]);
00207         }
00208     };
00209 
00210 };
00211 
00212 /**
00213 Performs eigen decomposition of a matrix.
00214 Real symmetric (and hence square matrices) can be decomposed into
00215 \f[M = U \times \Lambda \times U^T\f]
00216 where \f$U\f$ is an orthogonal matrix (and hence \f$U^T = U^{-1}\f$) whose columns
00217 are the eigenvectors of \f$M\f$ and \f$\Lambda\f$ is a diagonal matrix whose entries
00218 are the eigenvalues of \f$M\f$. These quantities are often of use directly, and can
00219 be obtained as follows:
00220 @code
00221 // construct M
00222 Matrix<3> M(3,3);
00223 M[0]=makeVector(4,0,2);
00224 M[1]=makeVector(0,5,3);
00225 M[2]=makeVector(2,3,6);
00226 
00227 // create the eigen decomposition of M
00228 SymEigen<3> eigM(M);
00229 cout << "A=" << M << endl;
00230 cout << "(E,v)=eig(A)" << endl;
00231 // print the smallest eigenvalue
00232 cout << "v[0]=" << eigM.get_evalues()[0] << endl;
00233 // print the associated eigenvector
00234 cout << "E[0]=" << eigM.get_evectors()[0] << endl;
00235 @endcode
00236 
00237 Further, provided the eigenvalues are nonnegative, the square root of
00238 a matrix and its inverse can also be obtained,
00239 @code
00240 // print the square root of the matrix.
00241 cout << "R=sqrtm(A)=" << eigM.get_sqrtm() << endl;
00242 // print the square root of the matrix squared.
00243 cout << "(should equal A), R^T*R="
00244      << eigM.get_sqrtm().T() * eigM.get_sqrtm() << endl;
00245 // print the inverse of the matrix.
00246 cout << "A^-1=" << eigM.get_pinv() << endl;
00247 // print the inverse square root of the matrix.
00248 cout << "C=isqrtm(A)=" << eigM.get_isqrtm() << endl;
00249 // print the inverse square root of the matrix squared.
00250 cout << "(should equal A^-1), C^T*C="
00251      << eigM.get_isqrtm().T() * eigM.get_isqrtm() << endl;
00252 @endcode
00253 
00254 This decomposition is very similar to the SVD (q.v.), and can be used to solve
00255 equations using backsub() or get_pinv(), with the same treatment of condition numbers.
00256 
00257 SymEigen<> (= SymEigen<-1>) can be used to create an eigen decomposition whose size is determined at run-time.
00258 @ingroup gDecomps
00259 **/
00260 template <int Size=Dynamic, typename Precision = double>
00261 class SymEigen {
00262 public:
00263     inline SymEigen(){}
00264 
00265         /// Initialise this eigen decomposition but do no immediately
00266         /// perform a decomposition.
00267         ///
00268         /// @param m The size of the matrix to perform the eigen decomposition on.
00269         inline SymEigen(int m) : my_evectors(m,m), my_evalues(m) {}
00270 
00271     /// Construct the eigen decomposition of a matrix. This initialises the class, and
00272     /// performs the decomposition immediately.
00273     template<int R, int C, typename B>
00274     inline SymEigen(const Matrix<R, C, Precision, B>& m) : my_evectors(m.num_rows(), m.num_cols()), my_evalues(m.num_rows()) {
00275         compute(m);
00276     }
00277 
00278     /// Perform the eigen decomposition of a matrix.
00279     template<int R, int C, typename B>
00280     inline void compute(const Matrix<R,C,Precision,B>& m){
00281         SizeMismatch<R, C>::test(m.num_rows(), m.num_cols());
00282         SizeMismatch<R, Size>::test(m.num_rows(), my_evectors.num_rows());
00283         Internal::ComputeSymEigen<Size>::compute(m, my_evectors, my_evalues);
00284     }
00285 
00286     /// Calculate result of multiplying the (pseudo-)inverse of M by a vector.
00287     /// For a vector \f$b\f$, this calculates \f$M^{\dagger}b\f$ by back substitution
00288     /// (i.e. without explictly calculating the (pseudo-)inverse).
00289     /// See the SVD detailed description for a description of condition variables.
00290     template <int S, typename P, typename B>
00291     Vector<Size, Precision> backsub(const Vector<S,P,B>& rhs) const {
00292         return (my_evectors.T() * diagmult(get_inv_diag(Internal::symeigen_condition_no),(my_evectors * rhs)));
00293     }
00294 
00295     /// Calculate result of multiplying the (pseudo-)inverse of M by another matrix.
00296     /// For a matrix \f$A\f$, this calculates \f$M^{\dagger}A\f$ by back substitution
00297     /// (i.e. without explictly calculating the (pseudo-)inverse).
00298     /// See the SVD detailed description for a description of condition variables.
00299     template <int R, int C, typename P, typename B>
00300     Matrix<Size,C, Precision> backsub(const Matrix<R,C,P,B>& rhs) const {
00301         return (my_evectors.T() * diagmult(get_inv_diag(Internal::symeigen_condition_no),(my_evectors * rhs)));
00302     }
00303 
00304     /// Calculate (pseudo-)inverse of the matrix. This is not usually needed:
00305     /// if you need the inverse just to multiply it by a matrix or a vector, use
00306     /// one of the backsub() functions, which will be faster.
00307     /// See the SVD detailed description for a description of the pseudo-inverse
00308     /// and condition variables.
00309     Matrix<Size, Size, Precision> get_pinv(const double condition=Internal::symeigen_condition_no) const {
00310         return my_evectors.T() * diagmult(get_inv_diag(condition),my_evectors);
00311     }
00312 
00313     /// Calculates the reciprocals of the eigenvalues of the matrix.
00314     /// The vector <code>invdiag</code> lists the eigenvalues in order, from
00315     /// the largest (i.e. smallest reciprocal) to the smallest.
00316     /// These are also the diagonal values of the matrix \f$Lambda^{-1}\f$.
00317     /// Any eigenvalues which are too small are set to zero (see the SVD
00318     /// detailed description for a description of the and condition variables).
00319     Vector<Size, Precision> get_inv_diag(const double condition) const {
00320         Precision max_diag = -my_evalues[0] > my_evalues[my_evalues.size()-1] ? -my_evalues[0]:my_evalues[my_evalues.size()-1];
00321         Vector<Size, Precision> invdiag(my_evalues.size());
00322         for(int i=0; i<my_evalues.size(); i++){
00323             if(fabs(my_evalues[i]) * condition > max_diag) {
00324                 invdiag[i] = 1/my_evalues[i];
00325             } else {
00326                 invdiag[i]=0;
00327             }
00328         }
00329         return invdiag;
00330     }
00331 
00332     /// Returns the eigenvectors of the matrix.
00333     /// This returns \f$U^T\f$, so that the rows of the matrix are the eigenvectors,
00334     /// which can be extracted using usual Matrix::operator[]() subscript operator.
00335     /// They are returned in order of the size of the corresponding eigenvalue, i.e.
00336     /// the vector with the largest eigenvalue is first.
00337     Matrix<Size,Size,Precision>& get_evectors() {return my_evectors;}
00338 
00339     /**\overload
00340     */
00341     const Matrix<Size,Size,Precision>& get_evectors() const {return my_evectors;}
00342 
00343 
00344     /// Returns the eigenvalues of the matrix.
00345     /// The eigenvalues are listed in order, from the smallest to the largest.
00346     /// These are also the diagonal values of the matrix \f$\Lambda\f$.
00347     Vector<Size, Precision>& get_evalues() {return my_evalues;}
00348     /**\overload
00349     */
00350     const Vector<Size, Precision>& get_evalues() const {return my_evalues;}
00351 
00352     /// Is the matrix positive definite?
00353     bool is_posdef() const {
00354         for (int i = 0; i < my_evalues.size(); ++i) {
00355             if (my_evalues[i] <= 0.0)
00356                 return false;
00357         }
00358         return true;
00359     }
00360 
00361     /// Is the matrix negative definite?
00362     bool is_negdef() const {
00363         for (int i = 0; i < my_evalues.size(); ++i) {
00364             if (my_evalues[i] >= 0.0)
00365                 return false;
00366         }
00367         return true;
00368     }
00369 
00370     /// Get the determinant of the matrix
00371     Precision get_determinant () const {
00372         Precision det = 1.0;
00373         for (int i = 0; i < my_evalues.size(); ++i) {
00374             det *= my_evalues[i];
00375         }
00376         return det;
00377     }
00378 
00379     /// Calculate the square root of a matrix which is a matrix M
00380     /// such that M.T*M=A.
00381     Matrix<Size, Size, Precision> get_sqrtm () const {
00382         Vector<Size, Precision> diag_sqrt(my_evalues.size());
00383         // In the future, maybe throw an exception if an
00384         // eigenvalue is negative?
00385         for (int i = 0; i < my_evalues.size(); ++i) {
00386             diag_sqrt[i] = std::sqrt(my_evalues[i]);
00387         }
00388         return my_evectors.T() * diagmult(diag_sqrt, my_evectors);
00389     }
00390 
00391     /// Calculate the inverse square root of a matrix which is a
00392     /// matrix M such that M.T*M=A^-1.
00393         ///
00394         /// Any square-rooted eigenvalues which are too small are set
00395         /// to zero (see the SVD detailed description for a
00396         /// description of the condition variables).
00397     Matrix<Size, Size, Precision> get_isqrtm (const double condition=Internal::symeigen_condition_no) const {
00398         Vector<Size, Precision> diag_isqrt(my_evalues.size());
00399 
00400         // Because sqrt is a monotonic-preserving transformation,
00401         Precision max_diag = -my_evalues[0] > my_evalues[my_evalues.size()-1] ? (-std::sqrt(my_evalues[0])):(std::sqrt(my_evalues[my_evalues.size()-1]));
00402         // In the future, maybe throw an exception if an
00403         // eigenvalue is negative?
00404         for (int i = 0; i < my_evalues.size(); ++i) {
00405             diag_isqrt[i] = std::sqrt(my_evalues[i]);
00406             if(fabs(diag_isqrt[i]) * condition > max_diag) {
00407                 diag_isqrt[i] = 1/diag_isqrt[i];
00408             } else {
00409                 diag_isqrt[i] = 0;
00410             }
00411         }
00412         return my_evectors.T() * diagmult(diag_isqrt, my_evectors);
00413     }
00414 
00415 private:
00416     // eigen vectors laid out row-wise so evectors[i] is the ith evector
00417     Matrix<Size,Size,Precision> my_evectors;
00418 
00419     Vector<Size, Precision> my_evalues;
00420 };
00421 
00422 }
00423 
00424 #endif
00425