Represents a factorization of a system of equations with no exact solution such that the singular value method of least squares can be used. More...

#include <rw/lapack/lssv.h>

Inheritance diagram for RWLeastSqSV< TypeT, SVDCalc >:

Public Member Functions
	RWLeastSqSV ()

	RWLeastSqSV (const RWGenMat< TypeT > &A, double tol=0)

	RWLeastSqSV (const RWSVDecomp< TypeT, SVDCalc > &A, double tol=0)

RWMathVec< TypeT >	residual (const RWMathVec< TypeT > &b) const

rw_numeric_traits< TypeT >::norm_type	residualNorm (const RWMathVec< TypeT > &b) const

RWMathVec< TypeT >	solve (const RWMathVec< TypeT > &b) const

Public Member Functions inherited from RWSVDecomp< TypeT, SVDCalc >
	RWSVDecomp ()

	RWSVDecomp (const RWSVDecomp< TypeT, SVDCalc > &A)

	RWSVDecomp (const RWGenMat< TypeT > &A, norm_type tol=0)

unsigned	cols () const

void	factor (const RWGenMat< TypeT > &A, norm_type tol=0)

bool	fail () const

bool	good () const

const RWMathVec< TypeT >	leftVector (int i) const

const RWGenMat< TypeT >	leftVectors () const

unsigned	numLeftVectors () const

unsigned	numRightVectors () const

RWSVDecomp< TypeT, SVDCalc > &	operator= (const RWSVDecomp< TypeT, SVDCalc > &x)

unsigned	rank () const

const RWMathVec< TypeT >	rightVector (int i) const

const RWGenMat< TypeT >	rightVectors () const

unsigned	rows () const

norm_type	singularValue (int i) const

const RWMathVec< norm_type >	singularValues () const

void	truncate (norm_type tol)

Additional Inherited Members
Public Types inherited from RWSVDecomp< TypeT, SVDCalc >
typedef rw_numeric_traits< TypeT >::norm_type	norm_type

Detailed Description

template<class TypeT, class SVDCalc>
class RWLeastSqSV< TypeT, SVDCalc >

A linear system of equations has the form \( Ax = b \), where A contains more rows than columns and generally has no exact solution; it is overdetermined. The best that we can do is to find a solution vector x such that the norm of the residual vector, \( Ax-b \), is made as small as possible. The vector x is then a solution in the least squares sense, and this sort of problem is called a least squares problem.

There are three classes in the Linear Algebra Module for solving least squares problems. The approach is to construct a factorization of the matrix A that can be used to solve the problem. The classes RWLeastSqCh RWLeastSqQR and RWLeastSqSV encapsulate this factorization object. There are three methods provided:

Cholesky method or method of normal equations, encoded by Ch
Complete orthogonal factorization method, encoded by QR
Singular value method, encoded by SV.

Which method you choose depends on the trade-off between speed and robustness that you want to make. As you move down the list, the methods decrease in speed, but increase in accuracy and robustness.

A system of equations \( Ax = b \) is underdetermined if the columns of A are linearly dependent. In this case, there are infinitely many solutions. Often, the one that is desired is the one with minimum norm. The classes RWLeastSqQR and RWLeastSqSV allow you to compute this solution.

Note that the classes RWLeastSqQR and RWLeastSqSV are publicly derived from their underlying decomposition types, so that advanced users can manipulate the underlying decomposition directly.

Note: For greater flexibility with RWLeastSqQR, the user can implement this method, or the Linear Algebra Module provides two classes to perform this function - RWQRCalc and RWQRCalcP3. Please see their descriptions in this reference guide for more information.

Synopsis: #include <rw/lapack/lsch.h>

#include <rw/lapack/lsqr.h>

#include <rw/lapack/lssv.h>

#include <rw/lapack/dvddccalc.h>

RWLeastSqSV<double, RWSVDDivConqCalc<double> > SVD(A); // A is

// an RWGenMat<double>

RWMathVec<double> x = SVD.solve(b); // b is an RWMathVec<double>

Example: // Find the least squares (if overdetermined) or

// minimum norm (if underdetermined) solution to

// the linear system of equations Ax=b. Also input

// a tolerance called epsilon which represents the

// accuracy of the entries in A. This makes the

// procedure robust in the event that A has nearly

// linearly dependent columns.

#include <iostream>

#include <rw/lapack/lsch.h>

#include <rw/lapack/lsqr.h>

#include <rw/lapack/lssv.h>

#include <rw/lapack/svddccalc.h>

int main()

{

RWGenMat<double> A;

double tol;

RWMathVec<double> b;

std::cin >> A >> tol >> b;

RWLeastSqSV<double, RWSVDDivConqCalc<double>

>decomp (A,tol);

RWMathVec<double> x = decomp.solve(b);

std::cout << "Solution is " << x << std::endl;

return 0;

}