Cholesky.h

// -*- c++ -*-

//     Copyright (C) 2009 Tom Drummond (twd20@cam.ac.uk)
//
// This file is part of the TooN Library.  This library is free
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// terms of the GNU General Public License as published by the
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// any later version.

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#ifndef TOON_INCLUDE_CHOLESKY_H
#define TOON_INCLUDE_CHOLESKY_H

#include <TooN/TooN.h>

namespace TooN {


/**
Decomposes a positive-semidefinite symmetric matrix A (such as a covariance) into L*D*L^T, where L is lower-triangular and D is diagonal.
Also can compute the classic A = L*L^T, with L lower triangular.  The LDL^T form is faster to compute than the classical Cholesky decomposition. 
Use get_unscaled_L() and get_D() to access the individual matrices of L*D*L^T decomposition. Use get_L() to access the lower triangular matrix of the classic Cholesky decomposition L*L^T.
The decomposition can be used to compute A^-1*x, A^-1*M, M*A^-1*M^T, and A^-1 itself, though the latter rarely needs to be explicitly represented.
Also efficiently computes det(A) and rank(A).
It can be used as follows:
@code
// Declare some matrices.
Matrix<3> A = ...; // we'll pretend it is pos-def
Matrix<2,3> M;
Matrix<2> B;
Vector<3> y = make_Vector(2,3,4);
// create the Cholesky decomposition of A
Cholesky<3> chol(A);
// compute x = A^-1 * y
x = cholA.backsub(y);
//compute A^-1
Matrix<3> Ainv = cholA.get_inverse();
@endcode
@ingroup gDecomps

Cholesky decomposition of a symmetric matrix.
Only the lower half of the matrix is considered
This uses the non-sqrt version of the decomposition
giving symmetric M = L*D*L.T() where the diagonal of L contains ones
@param Size the size of the matrix
@param Precision the precision of the entries in the matrix and its decomposition
**/
template <int Size=Dynamic, class Precision=DefaultPrecision>
class Cholesky {
public:
    Cholesky(){}

    /// Construct the Cholesky decomposition of a matrix. This initialises the class, and
    /// performs the decomposition immediately.
    /// Run time is O(N^3)
    template<class P2, class B2>
    Cholesky(const Matrix<Size, Size, P2, B2>& m)
        : my_cholesky(m) {
        SizeMismatch<Size,Size>::test(m.num_rows(), m.num_cols());
        do_compute();
    }
    
    /// Constructor for Size=Dynamic
    Cholesky(int size) : my_cholesky(size,size) {}


    /// Compute the LDL^T decomposition of another matrix.
    /// Run time is O(N^3)
    template<class P2, class B2> void compute(const Matrix<Size, Size, P2, B2>& m){
        SizeMismatch<Size,Size>::test(m.num_rows(), m.num_cols());
        SizeMismatch<Size,Size>::test(m.num_rows(), my_cholesky.num_rows());
        my_cholesky=m;
        do_compute();
    }
    
    private:
    void do_compute() {
        int size=my_cholesky.num_rows();
        for(int col=0; col<size; col++){
            Precision inv_diag = 1;
            for(int row=col; row < size; row++){
                // correct for the parts of cholesky already computed
                Precision val = my_cholesky(row,col);
                for(int col2=0; col2<col; col2++){
                    // val-=my_cholesky(col,col2)*my_cholesky(row,col2)*my_cholesky(col2,col2);
                    val-=my_cholesky(col2,col)*my_cholesky(row,col2);
                }
                if(row==col){
                    // this is the diagonal element so don't divide
                    my_cholesky(row,col)=val;
                    if(val == 0){
                        my_rank = row;
                        return;
                    }
                    inv_diag=1/val;
                } else {
                    // cache the value without division in the upper half
                    my_cholesky(col,row)=val;
                    // divide my the diagonal element for all others
                    my_cholesky(row,col)=val*inv_diag;
                }
            }
        }
        my_rank = size;
    }

    public:

    /// Compute x = A^-1*v
    /// Run time is O(N^2)
    template<int Size2, class P2, class B2>
    Vector<Size, Precision> backsub (const Vector<Size2, P2, B2>& v) const {
        int size=my_cholesky.num_rows();
        SizeMismatch<Size,Size2>::test(size, v.size());

        // first backsub through L
        Vector<Size, Precision> y(size);
        for(int i=0; i<size; i++){
            Precision val = v[i];
            for(int j=0; j<i; j++){
                val -= my_cholesky(i,j)*y[j];
            }
            y[i]=val;
        }
        
        // backsub through diagonal
        for(int i=0; i<size; i++){
            y[i]/=my_cholesky(i,i);
        }

        // backsub through L.T()
        Vector<Size,Precision> result(size);
        for(int i=size-1; i>=0; i--){
            Precision val = y[i];
            for(int j=i+1; j<size; j++){
                val -= my_cholesky(j,i)*result[j];
            }
            result[i]=val;
        }

        return result;
    }

    /**overload
    */
    template<int Size2, int C2, class P2, class B2>
    Matrix<Size, C2, Precision> backsub (const Matrix<Size2, C2, P2, B2>& m) const {
        int size=my_cholesky.num_rows();
        SizeMismatch<Size,Size2>::test(size, m.num_rows());

        // first backsub through L
        Matrix<Size, C2, Precision> y(size, m.num_cols());
        for(int i=0; i<size; i++){
            Vector<C2, Precision> val = m[i];
            for(int j=0; j<i; j++){
                val -= my_cholesky(i,j)*y[j];
            }
            y[i]=val;
        }
        
        // backsub through diagonal
        for(int i=0; i<size; i++){
            y[i]*=(1/my_cholesky(i,i));
        }

        // backsub through L.T()
        Matrix<Size,C2,Precision> result(size, m.num_cols());
        for(int i=size-1; i>=0; i--){
            Vector<C2,Precision> val = y[i];
            for(int j=i+1; j<size; j++){
                val -= my_cholesky(j,i)*result[j];
            }
            result[i]=val;
        }
        return result;
    }


    /// Compute A^-1 and store in M
    /// Run time is O(N^3)
    // easy way to get inverse - could be made more efficient
    Matrix<Size,Size,Precision> get_inverse(){
        Matrix<Size,Size,Precision>I(Identity(my_cholesky.num_rows()));
        return backsub(I);
    }
    
    ///Compute the determinant.
    Precision determinant(){
        Precision answer=my_cholesky(0,0);
        for(int i=1; i<my_cholesky.num_rows(); i++){
            answer*=my_cholesky(i,i);
        }
        return answer;
    }

    template <int Size2, typename P2, typename B2>
    Precision mahalanobis(const Vector<Size2, P2, B2>& v) const {
        return v * backsub(v);
    }

    Matrix<Size,Size,Precision> get_unscaled_L() const {
        Matrix<Size,Size,Precision> m(my_cholesky.num_rows(),
                          my_cholesky.num_rows());
        m=Identity;
        for (int i=1;i<my_cholesky.num_rows();i++) {
            for (int j=0;j<i;j++) {
                m(i,j)=my_cholesky(i,j);
            }
        }
        return m;
    }
            
    Matrix<Size,Size,Precision> get_D() const {
        Matrix<Size,Size,Precision> m(my_cholesky.num_rows(),
                          my_cholesky.num_rows());
        m=Zeros;
        for (int i=0;i<my_cholesky.num_rows();i++) {
            m(i,i)=my_cholesky(i,i);
        }
        return m;
    }
    
    Matrix<Size,Size,Precision> get_L() const {
        using std::sqrt;
        Matrix<Size,Size,Precision> m(my_cholesky.num_rows(),
                          my_cholesky.num_rows());
        m=Zeros;
        for (int j=0;j<my_cholesky.num_cols();j++) {
            Precision sqrtd=sqrt(my_cholesky(j,j));
            m(j,j)=sqrtd;
            for (int i=j+1;i<my_cholesky.num_rows();i++) {
                m(i,j)=my_cholesky(i,j)*sqrtd;
            }
        }
        return m;
    }

    int rank() const { return my_rank; }

private:
    Matrix<Size,Size,Precision> my_cholesky;
    int my_rank;
};


}

#endif