Least-squares and damped least-squares inverse computation for rectangular, real-valued matrices. More...

#include <matrix_inverter.h>

Public Types
typedef Eigen::internal::DlsInverseReturnType < Matrix >	DlsInverseReturnType
typedef Eigen::internal::DlsSolveReturnType < Matrix >	DlsSolveReturnType
typedef Matrix::Index	Index
typedef Eigen::internal::LsInverseReturnType < Matrix >	LsInverseReturnType
typedef Eigen::internal::LsSolveReturnType < Matrix >	LsSolveReturnType
typedef MatrixType	Matrix
typedef boost::shared_ptr< Matrix >	MatrixPtr
typedef Matrix::Scalar	Scalar
typedef Eigen::JacobiSVD < Matrix, Eigen::ColPivHouseholderQRPreconditioner >	SVD
typedef Eigen::internal::plain_col_type < Matrix >::type	Vector
Public Member Functions
MatrixInverter &	compute (const Matrix &A)
DlsInverseReturnType	dlsInverse ()
	Compute the damped least-squares inverse.
DlsSolveReturnType	dlsSolve (const Vector &b)
Scalar	getDlsInverseThreshold () const
Scalar	getLsInverseThreshold () const
Scalar	getMaxDamping () const
LsInverseReturnType	inverse ()
	Convenience alias for lsInverse(const Vector&)
LsInverseReturnType	lsInverse ()
	Compute the least-squares inverse.
LsSolveReturnType	lsSolve (const Vector &b)
	MatrixInverter (const Index rows, const Index cols)
	Problem dimension constructor. Preallocates all data structures so that compute(const Matrix&), lsInverse() and dlsInverse() are heap-allocation-free.
	MatrixInverter (const Matrix &A)
	Input matrix constructor. Using this constructor, like so.
void	setDlsInverseThreshold (const Scalar dampEps)
void	setLsInverseThreshold (const Scalar eps)
void	setMaxDamping (const Scalar dampMax)
LsSolveReturnType	solve (const Vector &b)
	Convenience alias for lsSolve(const Vector&)
Private Member Functions
void	allocate (const Index rows, const Index cols)
	Initialize default parameters. This method is common to all constructors.
Scalar	computeDampingSq (const Vector &S) const
Index	nonzeroSingularValues () const
Private Attributes
Scalar	dampMax_
	Maximum damping.
Scalar	dlsEps_
	Threshold for smallest nonzero singular value used in damped least-squares inverse.
Scalar	lsEps_
	Threshold for smallest nonzero singular value used in least-squares inverse.
Matrix	pinv_
	Work matrix used when computing inverses explicitly.
Vector	solve1_
	Work vector used when solving the associated linear system.
Vector	solve2_
	Work vector used when solving the associated linear system.
SVD	svd_
	Singular Value Decomposition.
Friends
class	Eigen::internal::DlsInverseReturnType< MatrixType >
class	Eigen::internal::DlsSolveReturnType< MatrixType >
class	Eigen::internal::LsInverseReturnType< MatrixType >
class	Eigen::internal::LsSolveReturnType< MatrixType >

Detailed Description

template<typename MatrixType>
class KDL::MatrixInverter< MatrixType >

Least-squares and damped least-squares inverse computation for rectangular, real-valued matrices.

Template Parameters:

MatrixType

Type of underlying dense matrix. This class uses the Singular Value Decomposition (SVD) for computing the inverse of a matrix. Let $ A = U S V^T $

be the SVD of a n-by-p, real-valued matrix, and let m be the smaller value among n and p, its inverse can be computed as $ A^+ = V S^+ U^T $

. This class allows two ways of computing $ S^+ $

: TODO: Finish

Least squares: Yields the least-squares solution. To compute it, every nonzero entry in S is replaced by its reciprocal. A fuzzy comparison citerion is used to determine the nonzero entries (see lsInverse()).
Damped least-squares: Realizes a tradeoff between the least-squares and minimum-norm properties.

It's important to note that a single SVD can be used to compute both the least-squares and damped least-squares inverses of a matrix. The following code exemplifies typical usage scenarios:

 typedef Eigen::MatrixXd Matrix;
 typedef boost::shared_ptr<Matrix> MatrixPtr;
 typedef Eigen::VectorXd Vector;

 const int rows = 5;
 const int cols = 10;
 Matrix A = Matrix::Random(rows, cols);
 Matrix Ainv(rows, cols);

 // Different use cases of increasing complexity
 // Case 1: Use default parameters, compute least-squares inverse
 Ainv = MatrixInverter<Matrix>(A).lsInverse(); // Handy for one-off usage
 Ainv = MatrixInverter<Matrix>(A).inverse();   // Same as above, inverse() is a convenience alias for lsInverse()

 // Case 2: similar to case 1, but with resource preallocation
 KDL::MatrixInverter<Matrix> inv(rows, cols); // Preallocate resources
 Ainv = inv.compute(A).inverse();             // No allocations take place here

 // Case 3: Use custom parameters
 inv.setLsInverseThreshold(1e-5);
 Ainv = inv.compute(A).lsInverse();

 // Case 4: Compute both least-squares amd damped least-squares inverses
 inv.setDlsInverseThreshold(0.1); // Damping activation threshold
 inv.compute(A);                  // Using the same SVD...
 Ainv         = inv.lsInverse();  // Compute the least-squares...
 Matrix Adinv = inv.dlsInverse(); // and damped least-squares inverses

Definition at line 122 of file matrix_inverter.h.

Member Typedef Documentation

template<typename MatrixType>

typedef Eigen::internal::DlsInverseReturnType<Matrix> KDL::MatrixInverter< MatrixType >::DlsInverseReturnType

Definition at line 132 of file matrix_inverter.h.

template<typename MatrixType>

typedef Eigen::internal::DlsSolveReturnType<Matrix> KDL::MatrixInverter< MatrixType >::DlsSolveReturnType

Definition at line 134 of file matrix_inverter.h.

template<typename MatrixType>

typedef Matrix::Index KDL::MatrixInverter< MatrixType >::Index

Definition at line 128 of file matrix_inverter.h.

template<typename MatrixType>

typedef Eigen::internal::LsInverseReturnType<Matrix> KDL::MatrixInverter< MatrixType >::LsInverseReturnType

Definition at line 131 of file matrix_inverter.h.

template<typename MatrixType>

typedef Eigen::internal::LsSolveReturnType<Matrix> KDL::MatrixInverter< MatrixType >::LsSolveReturnType

Definition at line 133 of file matrix_inverter.h.

template<typename MatrixType>

typedef MatrixType KDL::MatrixInverter< MatrixType >::Matrix

Definition at line 125 of file matrix_inverter.h.

template<typename MatrixType>

typedef boost::shared_ptr<Matrix> KDL::MatrixInverter< MatrixType >::MatrixPtr

Definition at line 126 of file matrix_inverter.h.

template<typename MatrixType>

typedef Matrix::Scalar KDL::MatrixInverter< MatrixType >::Scalar

Definition at line 129 of file matrix_inverter.h.

template<typename MatrixType>

typedef Eigen::JacobiSVD<Matrix, Eigen::ColPivHouseholderQRPreconditioner> KDL::MatrixInverter< MatrixType >::SVD

Definition at line 130 of file matrix_inverter.h.

template<typename MatrixType>

typedef Eigen::internal::plain_col_type<Matrix>::type KDL::MatrixInverter< MatrixType >::Vector

Definition at line 127 of file matrix_inverter.h.

Constructor & Destructor Documentation

template<typename MatrixType >

KDL::MatrixInverter< MatrixType >::MatrixInverter	(	const Index	rows,
		const Index	cols
	)

Problem dimension constructor. Preallocates all data structures so that compute(const Matrix&), lsInverse() and dlsInverse() are heap-allocation-free.

Definition at line 253 of file matrix_inverter.h.

template<typename MatrixType >

KDL::MatrixInverter< MatrixType >::MatrixInverter ( const Matrix & A )

Input matrix constructor. Using this constructor, like so.

 Matrix A(rows, cols);
 MatrixInverter<Matrix> inv(A);

is a more compact and equivalent way of doing

 Matrix A(rows, cols);
 MatrixInverter<Matrix> inv(rows, cols);
 inv.compute();

See also:: MatrixInverter(const Index rows, const Index cols)

Definition at line 259 of file matrix_inverter.h.

Member Function Documentation

template<typename MatrixType >

void KDL::MatrixInverter< MatrixType >::allocate	(	const Index	rows,
		const Index	cols
	)		`[inline, private]`

Initialize default parameters. This method is common to all constructors.

Definition at line 266 of file matrix_inverter.h.

template<typename MatrixType >

MatrixInverter< MatrixType > & KDL::MatrixInverter< MatrixType >::compute ( const Matrix & A ) [inline]

Parameters:

A	Matrix to invert.

Returns:: A reference to this.

Definition at line 281 of file matrix_inverter.h.

template<typename MatrixType >

MatrixInverter< MatrixType >::Scalar KDL::MatrixInverter< MatrixType >::computeDampingSq ( const Vector & S ) const [inline, private]

Parameters:

S	Vector of singular values.

Returns:: Square of damping value.

Definition at line 307 of file matrix_inverter.h.

template<typename MatrixType>

DlsInverseReturnType KDL::MatrixInverter< MatrixType >::dlsInverse ( ) [inline]

Compute the damped least-squares inverse.

Returns:: Expression object representing the damped least-squares inverse.

Precondition:: Must be called after compute(const Matrix& A).

See also:: setDlsInverseThreshold(const Scalar), setMaxDamping(const Scalar)

Definition at line 174 of file matrix_inverter.h.

template<typename MatrixType>

DlsSolveReturnType KDL::MatrixInverter< MatrixType >::dlsSolve ( const Vector & b ) [inline]

TODO: Doc!

Precondition:: Must be called after compute(const Matrix& A).

Definition at line 186 of file matrix_inverter.h.

template<typename MatrixType>

Scalar KDL::MatrixInverter< MatrixType >::getDlsInverseThreshold ( ) const [inline]

Returns:: Threshold for smallest nonzero singular value used when computing the damped least-squares inverse.

See also:: setDlsInverseThreshold(const Scalar)

Definition at line 215 of file matrix_inverter.h.

template<typename MatrixType>

Scalar KDL::MatrixInverter< MatrixType >::getLsInverseThreshold ( ) const [inline]

Returns:: Threshold for smallest nonzero singular value used when computing the least-squares inverse.

See also:: setLsInverseThreshold(const Scalar)

Definition at line 194 of file matrix_inverter.h.

template<typename MatrixType>

Scalar KDL::MatrixInverter< MatrixType >::getMaxDamping ( ) const [inline]

Returns:: Maximum damping used when computing the damped least-squares inverse.

See also:: setMaxDamping(const Scalar), setDlsInverseThreshold(const Scalar)

Definition at line 223 of file matrix_inverter.h.

template<typename MatrixType>

LsInverseReturnType KDL::MatrixInverter< MatrixType >::inverse ( ) [inline]

Convenience alias for lsInverse(const Vector&)

See also:: lsInverse(const Vector&)

Definition at line 162 of file matrix_inverter.h.

template<typename MatrixType>

LsInverseReturnType KDL::MatrixInverter< MatrixType >::lsInverse ( ) [inline]

Compute the least-squares inverse.

Returns:: Expression object representing the least-squares inverse.

Precondition:: Must be called after compute(const Matrix& A).

See also:: setLsInverseThreshold(const Scalar eps), getLsInverseThreshold()

Definition at line 168 of file matrix_inverter.h.

template<typename MatrixType>

LsSolveReturnType KDL::MatrixInverter< MatrixType >::lsSolve ( const Vector & b ) [inline]

TODO: Doc!

Precondition:: Must be called after compute(const Matrix& A).

Definition at line 182 of file matrix_inverter.h.

template<typename MatrixType >

MatrixInverter< MatrixType >::Index KDL::MatrixInverter< MatrixType >::nonzeroSingularValues ( ) const [inline, private]

Returns:: The number of singular values that are not approximately zero.

Definition at line 293 of file matrix_inverter.h.

template<typename MatrixType>

void KDL::MatrixInverter< MatrixType >::setDlsInverseThreshold ( const Scalar dampEps ) [inline]

Parameters:

dampEps Threshold for smallest nonzero singular value used when computing the damped least-squares inverse. Valid values are in the range [0,1].

Note:: The square of the damping factor $\lambda^2$ is computed as [1]:
$\lambda^2 = \left\lbrace \begin{array}{ c l } 0 & \mathrm{when} \;\; \sigma_m \geq eps \\ \left[ 1 - \left(\frac{\sigma_m}{dampEps}\right)^2 \right] \lambda_{max}^2 & \mathrm{otherwise} \end{array} \right.$
where $\sigma_m$ is the smallestsingular values of a matrix , and $\lambda_{max} = dampMax$ is the maximum damping (see setMaxDamping(const Scalar) ).