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
| Modifier and Type | Field and Description |
|---|---|
protected DenseMatrix64F |
I |
protected LinearSolver<DenseMatrix64F> |
internalSolver |
protected boolean |
norm2Solution |
protected DenseMatrix64F |
R |
protected DenseMatrix64F |
R11 |
protected int |
rank |
protected DenseMatrix64F |
Y |
A, numCols, numRows| Modifier | Constructor and Description |
|---|---|
protected |
BaseLinearSolverQrp(QRPDecomposition<DenseMatrix64F> decomposition,
boolean norm2Solution)
Configures internal parameters.
|
| Modifier and Type | Method and Description |
|---|---|
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.
|
_setA, getAclone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, waitmodifiesA, modifiesB, solveprotected boolean norm2Solution
protected DenseMatrix64F Y
protected DenseMatrix64F R
protected DenseMatrix64F R11
protected DenseMatrix64F I
protected int rank
protected LinearSolver<DenseMatrix64F> internalSolver
protected BaseLinearSolverQrp(QRPDecomposition<DenseMatrix64F> decomposition, boolean norm2Solution)
decomposition - Used to solve the linear system.norm2Solution - If true then the optimal 2-norm solution will be computed for degenerate systems.public boolean setA(DenseMatrix64F A)
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.
A - The 'A' matrix in the linear equation. Might be modified or save the reference.public double quality()
LinearSolverReturns 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.
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} ] ||
X - basic solution, also output solutionpublic void invert(DenseMatrix64F A_inv)
LinearSolverLinearSolver.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.invert in interface LinearSolver<DenseMatrix64F>invert in class LinearSolverAbstractA_inv - Where the inverted matrix saved. Modified.Copyright © 2013. All Rights Reserved.