TooN 2.1

Performs SVD and back substitute to solve equations. More...
#include <TooN/SVD.h>
Public Member Functions  
SVD ()  
SVD (int rows, int cols)  
template<int R2, int C2, typename P2 , typename B2 >  
SVD (const Matrix< R2, C2, P2, B2 > &m)  
template<int R2, int C2, typename P2 , typename B2 >  
void  compute (const Matrix< R2, C2, P2, B2 > &m) 
template<int Rows2, int Cols2, typename P2 , typename B2 >  
Matrix< Cols, Cols2, typename Internal::MultiplyType < Precision, P2 >::type >  backsub (const Matrix< Rows2, Cols2, P2, B2 > &rhs, const Precision condition=condition_no) 
template<int Size, typename P2 , typename B2 >  
Vector< Cols, typename Internal::MultiplyType < Precision, P2 >::type >  backsub (const Vector< Size, P2, B2 > &rhs, const Precision condition=condition_no) 
Matrix< Cols, Rows >  get_pinv (const Precision condition=condition_no) 
Precision  determinant () 
int  rank (const Precision condition=condition_no) 
Matrix< Rows, Min_Dim, Precision, Reference::RowMajor >  get_U () 
Vector< Min_Dim, Precision > &  get_diagonal () 
Matrix< Min_Dim, Cols, Precision, Reference::RowMajor >  get_VT () 
void  get_inv_diag (Vector< Min_Dim > &inv_diag, const Precision condition) 
Performs SVD and back substitute to solve equations.
Singular value decompositions are more robust than LU decompositions in the face of singular or nearly singular matrices. They decompose a matrix (of any shape) into:
where is a diagonal matrix of positive numbers whose dimension is the minimum of the dimensions of . If is tall and thin (more rows than columns) then has the same shape as and is square (viceversa if is short and fat). The columns of and the rows of are orthogonal and of unit norm (so one of them lies in SO(N)). The inverse of (or pseudoinverse if is not square) is then given by
If is nearly singular then the diagonal matrix has some small values (relative to its largest value) and these terms dominate . To deal with this problem, the inverse is conditioned by setting a maximum ratio between the largest and smallest values in (passed as the condition
parameter to the various functions). Any values which are too small are set to zero in the inverse (rather than a large number)
It can be used as follows to solve the problem as follows:
// construct M Matrix<3> M; M[0] = makeVector(1,2,3); M[1] = makeVector(4,5,6); M[2] = makeVector(7,8.10); // construct c Vector<3> c; c = 2,3,4; // create the SVD decomposition of M SVD<3> svdM(M); // compute x = M^1 * c Vector<3> x = svdM.backsub(c);
SVD<> (= SVD<1>) can be used to create an SVD whose size is determined at runtime.
Construct the SVD decomposition of a matrix.
This initialises the class, and performs the decomposition immediately.
Matrix<Cols,Cols2, typename Internal::MultiplyType<Precision,P2>::type > backsub  (  const Matrix< Rows2, Cols2, P2, B2 > &  rhs, 
const Precision  condition = condition_no 

) 
Calculate result of multiplying the (pseudo)inverse of M by another matrix.
For a matrix , this calculates by back substitution (i.e. without explictly calculating the (pseudo)inverse). See the detailed description for a description of condition variables.
Vector<Cols, typename Internal::MultiplyType<Precision,P2>::type > backsub  (  const Vector< Size, P2, B2 > &  rhs, 
const Precision  condition = condition_no 

) 
Calculate result of multiplying the (pseudo)inverse of M by a vector.
For a vector , this calculates by back substitution (i.e. without explictly calculating the (pseudo)inverse). See the detailed description for a description of condition variables.
Matrix<Cols,Rows> get_pinv  (  const Precision  condition = condition_no  ) 
Calculate (pseudo)inverse of the matrix.
This is not usually needed: if you need the inverse just to multiply it by a matrix or a vector, use one of the backsub() functions, which will be faster. See the detailed description of the pseudoinverse and condition variables.
int rank  (  const Precision  condition = condition_no  ) 
Calculate the rank of the matrix.
See the detailed description of the pseudoinverse and condition variables.
void get_inv_diag  (  Vector< Min_Dim > &  inv_diag, 
const Precision  condition  
) 
Return the pesudoinverse diagonal.
The reciprocal of the diagonal elements is returned if the elements are well scaled with respect to the largest element, otherwise 0 is returned.
inv_diag  Vector in which to return the inverse diagonal. 
condition  Elements must be larger than this factor times the largest diagonal element to be considered well scaled. 
Referenced by SVD< Size, Size, Precision >::backsub(), and SVD< Size, Size, Precision >::get_pinv().