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) |
void | do_compute () |
bool | is_vertical () |
int | min_dim () |
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) |
Static Public Attributes | |
static const int | Min_Dim = Rows<Cols?Rows:Cols |
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 (vice-versa 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 pseudo-inverse 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 run-time.
TooN::SVD< Rows, Cols, Precision >::SVD | ( | ) |
default constructor for Rows>0 and Cols>0
TooN::SVD< Rows, Cols, Precision >::SVD | ( | int | rows, | |
int | cols | |||
) |
constructor for Rows=-1 or Cols=-1 (or both)
TooN::SVD< Rows, Cols, Precision >::SVD | ( | const Matrix< R2, C2, P2, B2 > & | m | ) |
Construct the SVD decomposition of a matrix.
This initialises the class, and performs the decomposition immediately.
void TooN::SVD< Rows, Cols, Precision >::compute | ( | const Matrix< R2, C2, P2, B2 > & | m | ) |
Compute the SVD decomposition of M, typically used after the default constructor.
Matrix<Cols,Cols2, typename Internal::MultiplyType<Precision,P2>::type > TooN::SVD< Rows, Cols, Precision >::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 > TooN::SVD< Rows, Cols, Precision >::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> TooN::SVD< Rows, Cols, Precision >::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 pseudo-inverse and condition variables.
Precision TooN::SVD< Rows, Cols, Precision >::determinant | ( | ) |
Calculate the product of the singular values for square matrices this is the determinant.
int TooN::SVD< Rows, Cols, Precision >::rank | ( | const Precision | condition = condition_no |
) |
Calculate the rank of the matrix.
See the detailed description of the pseudo-inverse and condition variables.
Matrix<Rows,Min_Dim,Precision,Reference::RowMajor> TooN::SVD< Rows, Cols, Precision >::get_U | ( | ) |
Return the U matrix from the decomposition The size of this depends on the shape of the original matrix it is square if the original matrix is wide or tall if the original matrix is tall.
Vector<Min_Dim,Precision>& TooN::SVD< Rows, Cols, Precision >::get_diagonal | ( | ) |
Return the singular values as a vector.
Matrix<Min_Dim,Cols,Precision,Reference::RowMajor> TooN::SVD< Rows, Cols, Precision >::get_VT | ( | ) |
Return the VT matrix from the decomposition The size of this depends on the shape of the original matrix it is square if the original matrix is tall or wide if the original matrix is wide.