org.ejml.alg.dense.linsol.qr
Class BaseLinearSolverQrp

java.lang.Object
  org.ejml.alg.dense.linsol.LinearSolverAbstract
      org.ejml.alg.dense.linsol.qr.BaseLinearSolverQrp

All Implemented Interfaces:

LinearSolver<DenseMatrix64F>

Direct Known Subclasses:

LinearSolverQrpHouseCol, SolvePseudoInverseQrp

public abstract class BaseLinearSolverQrp
extends LinearSolverAbstract

Base class for QR pivot based pseudo inverse classes. It will return either the basic of minimal 2-norm solution. See [1] for details. The minimal 2-norm solution refers to the solution 'x' whose 2-norm is the smallest making it unique, not some other error function.

 R = [ R12  R12 ] r      P^T*x = [ y ] r       Q^T*b = [ c ] r
     [  0    0  ] m-r            [ z ] n -r            [ d ] m-r
        r   n-r

 where r is the rank of the matrix and (m,n) is the dimension of the linear system.
 

 The solution 'x' is found by solving the system below.  The basic solution is found by setting z=0

     [ R_11^-1*(c - R12*z) ]
 x = [          z          ]
 

NOTE: The matrix rank is determined using the provided QR decomposition. [1] mentions that this will not always work and could cause some problems.

[1] See page 258-259 in Gene H. Golub and Charles F. Van Loan "Matrix Computations" 3rd Ed, 1996

Author:

Peter Abeles

Field Summary

protected DenseMatrix64F	I
protected LinearSolver<DenseMatrix64F>	internalSolver
protected boolean	norm2Solution
protected DenseMatrix64F	R
protected DenseMatrix64F	R11
protected int	rank
protected DenseMatrix64F	Y

Fields inherited from class org.ejml.alg.dense.linsol.LinearSolverAbstract

A, numCols, numRows

Constructor Summary

protected

BaseLinearSolverQrp(QRPDecomposition<DenseMatrix64F> decomposition, boolean norm2Solution) Configures internal parameters.

Method Summary

void	invert(DenseMatrix64F A_inv) Computes the inverse of of the 'A' matrix passed into LinearSolver.setA(org.ejml.data.Matrix64F) and writes the results to the provided matrix.
double	quality() Returns a very quick to compute measure of how singular the system is.
boolean	setA(DenseMatrix64F A) Specifies the A matrix in the linear equation.
protected void	upgradeSolution(DenseMatrix64F X) Upgrades the basic solution to the optimal 2-norm solution.

Methods inherited from class org.ejml.alg.dense.linsol.LinearSolverAbstract

_setA, getA

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface org.ejml.alg.dense.linsol.LinearSolver

modifiesA, modifiesB, solve

Field Detail

norm2Solution

protected boolean norm2Solution

Y

protected DenseMatrix64F Y

R

protected DenseMatrix64F R

R11

protected DenseMatrix64F R11

I

protected DenseMatrix64F I

rank

protected int rank

internalSolver

protected LinearSolver<DenseMatrix64F> internalSolver

Constructor Detail

BaseLinearSolverQrp

protected BaseLinearSolverQrp(QRPDecomposition<DenseMatrix64F> decomposition,
                              boolean norm2Solution)

Configures internal parameters.

Parameters:

decomposition - Used to solve the linear system.

norm2Solution - If true then the optimal 2-norm solution will be computed for degenerate systems.

Method Detail

setA

public boolean setA(DenseMatrix64F A)

Description copied from interface: LinearSolver

Specifies the A matrix in the linear equation. A reference might be saved and it might also be modified depending on the implementation. If it is modified then LinearSolver.modifiesA() will return true.

If this value returns true that does not guarantee a valid solution was generated. This is because some decompositions don't detect singular matrices.

Parameters:

A - The 'A' matrix in the linear equation. Might be modified or save the reference.

Returns:

true if it can be processed.

quality

public double quality()

Description copied from interface: LinearSolver

Returns a very quick to compute measure of how singular the system is. This measure will be invariant to the scale of the matrix and always be positive, with larger values indicating it is less singular. If not supported by the solver then the runtime exception IllegalArgumentException is thrown. This is NOT the matrix's condition.

How this function is implemented is not specified. One possible implementation is the following: In many decompositions a triangular matrix is extracted. The determinant of a triangular matrix is easily computed and once normalized to be scale invariant and its absolute value taken it will provide functionality described above.

Returns:

The quality of the linear system.

upgradeSolution

protected void upgradeSolution(DenseMatrix64F X)

Upgrades the basic solution to the optimal 2-norm solution.

 First solves for 'z'

       || x_b - P*[ R_11^-1 * R_12 ] * z ||2
 min z ||         [ - I_{n-r}      ]     ||

 

Parameters:

X - basic solution, also output solution

invert

public void invert(DenseMatrix64F A_inv)

Description copied from interface: LinearSolver

Computes the inverse of of the 'A' matrix passed into LinearSolver.setA(org.ejml.data.Matrix64F) and writes the results to the provided matrix. If 'A_inv' needs to be different from 'A' is implementation dependent.

Specified by:

invert in interface LinearSolver<DenseMatrix64F>

Overrides:

invert in class LinearSolverAbstract

Parameters:

A_inv - Where the inverted matrix saved. Modified.