LeastSquares.hpp File Reference

Least Squares Algorithm. More...

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Classes
struct	NAV::LeastSquaresResult< SolutionVector, VarianceMatrix >
	Least Squares Uncertainties return value. More...

Functions
template<typename DerivedA , typename DerivedB >
Eigen::Vector< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime >	NAV::solveLinearLeastSquares (const Eigen::MatrixBase< DerivedA > &H, const Eigen::MatrixBase< DerivedB > &dz)
	Finds the "least squares" solution for the equation \( \mathbf{v} = \mathbf{dz} - \mathbf{H} \mathbf{x} \).
template<typename DerivedA , typename DerivedB >
LeastSquaresResult< Eigen::Vector< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime >, Eigen::Matrix< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime, DerivedA::ColsAtCompileTime > >	NAV::solveLinearLeastSquaresUncertainties (const Eigen::MatrixBase< DerivedA > &H, const Eigen::MatrixBase< DerivedB > &dz)
	Finds the "least squares" solution for the equation \( \mathbf{v} = \mathbf{dz} - \mathbf{H} \mathbf{x} \).
template<typename DerivedA , typename DerivedW , typename DerivedB >
Eigen::Vector< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime >	NAV::solveWeightedLinearLeastSquares (const Eigen::MatrixBase< DerivedA > &H, const Eigen::MatrixBase< DerivedW > &W, const Eigen::MatrixBase< DerivedB > &dz)
	Finds the "weighted least squares" solution.
template<typename DerivedA , typename DerivedW , typename DerivedB >
LeastSquaresResult< Eigen::Vector< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime >, Eigen::Matrix< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime, DerivedA::ColsAtCompileTime > >	NAV::solveWeightedLinearLeastSquaresUncertainties (const Eigen::MatrixBase< DerivedA > &H, const Eigen::MatrixBase< DerivedW > &W, const Eigen::MatrixBase< DerivedB > &dz)
	Finds the "weighted least squares" solution.

Detailed Description

Least Squares Algorithm.

Author

T. Topp (topp@.nosp@m.ins..nosp@m.uni-s.nosp@m.tutt.nosp@m.gart..nosp@m.de)

Date

2022-05-04

Function Documentation

solveLinearLeastSquares()

template<typename DerivedA , typename DerivedB >

Eigen::Vector< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime > NAV::solveLinearLeastSquares	(	const Eigen::MatrixBase< DerivedA > &	H,
		const Eigen::MatrixBase< DerivedB > &	dz )

Finds the "least squares" solution for the equation \( \mathbf{v} = \mathbf{dz} - \mathbf{H} \mathbf{x} \).

Minimizes the functional

\begin{equation} \label{eq:eq-LinearLeastSquares-functional} J(\mathbf{x}) \equiv \sum_{i=1}^m v_i^2 = \mathbf{v}^T \mathbf{v} = (\mathbf{dz} - \mathbf{H} \mathbf{x})^T (\mathbf{dz} - \mathbf{H} \mathbf{x}) \end{equation}

which has the solution (assuming that the inverse to \( \mathbf{H}^T \mathbf{H} \) exists)

\begin{equation} \label{eq:eq-LinearLeastSquares-solution} \mathbf{x} = \left(\mathbf{H}^T \mathbf{H} \right)^{-1} \mathbf{H}^T \mathbf{dz} \end{equation}

Parameters

[in]	H	Design Matrix
[in]	dz	Residual vector

Returns

Least squares solution

solveLinearLeastSquaresUncertainties()

template<typename DerivedA , typename DerivedB >

LeastSquaresResult< Eigen::Vector< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime >, Eigen::Matrix< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime, DerivedA::ColsAtCompileTime > > NAV::solveLinearLeastSquaresUncertainties	(	const Eigen::MatrixBase< DerivedA > &	H,
		const Eigen::MatrixBase< DerivedB > &	dz )

Finds the "least squares" solution for the equation \( \mathbf{v} = \mathbf{dz} - \mathbf{H} \mathbf{x} \).

Parameters

[in]	H	Design Matrix
[in]	dz	Residual vector

Returns

Least squares solution and variance

solveWeightedLinearLeastSquares()

template<typename DerivedA , typename DerivedW , typename DerivedB >

Eigen::Vector< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime > NAV::solveWeightedLinearLeastSquares	(	const Eigen::MatrixBase< DerivedA > &	H,
		const Eigen::MatrixBase< DerivedW > &	W,
		const Eigen::MatrixBase< DerivedB > &	dz )

Finds the "weighted least squares" solution.

\begin{equation} \label{eq:eq-WeightedLinearLeastSquares} \mathbf{x} = \left(\mathbf{H}^T \mathbf{W} \mathbf{H} \right)^{-1} \mathbf{H}^T \mathbf{W} \mathbf{dz} \end{equation}

Parameters

[in]	H	Design Matrix
[in]	W	Weight matrix
[in]	dz	Residual vector

Returns

Least squares solution

solveWeightedLinearLeastSquaresUncertainties()

template<typename DerivedA , typename DerivedW , typename DerivedB >

LeastSquaresResult< Eigen::Vector< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime >, Eigen::Matrix< typename DerivedA::Scalar, DerivedA::ColsAtCompileTime, DerivedA::ColsAtCompileTime > > NAV::solveWeightedLinearLeastSquaresUncertainties	(	const Eigen::MatrixBase< DerivedA > &	H,
		const Eigen::MatrixBase< DerivedW > &	W,
		const Eigen::MatrixBase< DerivedB > &	dz )

Finds the "weighted least squares" solution.

Parameters

[in]	H	Design Matrix
[in]	W	Weight matrix
[in]	dz	Residual vector

Returns

Weighted least squares solution and variance