Modifier and Type	Class and Description
private static class	QRDecomposition.Solver Specialized solver.

Modifier and Type	Field and Description
private RealMatrix	cachedH Cached value of H.
private RealMatrix	cachedQ Cached value of Q.
private RealMatrix	cachedQT Cached value of QT.
private RealMatrix	cachedR Cached value of R.
private double[][]	qrt A packed TRANSPOSED representation of the QR decomposition.
private double[]	rDiag The diagonal elements of R.
private final double	threshold Singularity threshold.

Access	Constructor and Description
public	QRDecomposition(RealMatrix The matrix to decompose. matrix) Calculates the QR-decomposition of the given matrix.
public	QRDecomposition(RealMatrix The matrix to decompose. matrix, double Singularity threshold. threshold) Calculates the QR-decomposition of the given matrix.

Method Summary

Modifier and Type	Method and Description
protected void	decompose(double[][] transposed matrix matrix) Decompose matrix.
public RealMatrix	Returns: a matrix containing the Householder reflector vectors getH() Returns the Householder reflector vectors.
public RealMatrix	Returns: the Q matrix getQ() Returns the matrix Q of the decomposition.
public RealMatrix	Returns: the transpose of the Q matrix, Q^T getQT() Returns the transpose of the matrix Q of the decomposition.
public RealMatrix	Returns: the R matrix getR() Returns the matrix R of the decomposition.
public DecompositionSolver	Returns: a solver getSolver() Get a solver for finding the A × X = B solution in least square sense.
protected void	performHouseholderReflection(int minor index minor, double[][] transposed matrix matrix) Perform Householder reflection for a minor A(minor, minor) of A.

Inherited from java.lang.Object:: clone equals finalize getClass hashCode notify notifyAll toString wait wait wait

Field Detail

cachedH	back to summary
private RealMatrix cachedH Cached value of H.

cachedQ	back to summary
private RealMatrix cachedQ Cached value of Q.

cachedQT	back to summary
private RealMatrix cachedQT Cached value of QT.

cachedR	back to summary
private RealMatrix cachedR Cached value of R.

qrt	back to summary
private double[][] qrt A packed TRANSPOSED representation of the QR decomposition. The elements BELOW the diagonal are the elements of the UPPER triangular matrix R, and the rows ABOVE the diagonal are the Householder reflector vectors from which an explicit form of Q can be recomputed if desired.

qrt

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private double[][] qrt

A packed TRANSPOSED representation of the QR decomposition.

The elements BELOW the diagonal are the elements of the UPPER triangular matrix R, and the rows ABOVE the diagonal are the Householder reflector vectors from which an explicit form of Q can be recomputed if desired.

rDiag	back to summary
private double[] rDiag The diagonal elements of R.

threshold	back to summary
private final double threshold Singularity threshold.

QRDecomposition	back to summary
public QRDecomposition(RealMatrix matrix) Calculates the QR-decomposition of the given matrix. The singularity threshold defaults to zero. Parameters matrix:RealMatrix The matrix to decompose. See Also `QRDecomposition(RealMatrix, double)`

QRDecomposition	back to summary
public QRDecomposition(RealMatrix matrix, double threshold) Calculates the QR-decomposition of the given matrix. Parameters matrix:RealMatrix The matrix to decompose. threshold:double Singularity threshold.

Method Detail

decompose	back to summary
protected void decompose(double[][] matrix) Decompose matrix. Parameters matrix:double[][] transposed matrix Since 3.2

decompose

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protected void decompose(double[][] matrix)

Decompose matrix.

Parameters

matrix:double[][]: transposed matrix

Since

3.2

getH	back to summary
public RealMatrix getH() Returns the Householder reflector vectors. H is a lower trapezoidal matrix whose columns represent each successive Householder reflector vector. This matrix is used to compute Q. Returns:RealMatrix a matrix containing the Householder reflector vectors

getH

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public RealMatrix getH()

Returns the Householder reflector vectors.

H is a lower trapezoidal matrix whose columns represent each successive Householder reflector vector. This matrix is used to compute Q.

Returns:RealMatrix: a matrix containing the Householder reflector vectors

getQ	back to summary
public RealMatrix getQ() Returns the matrix Q of the decomposition. Q is an orthogonal matrix Returns:RealMatrix the Q matrix

getQ

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public RealMatrix getQ()

Returns the matrix Q of the decomposition.

Q is an orthogonal matrix

Returns:RealMatrix: the Q matrix

getQT	back to summary
public RealMatrix getQT() Returns the transpose of the matrix Q of the decomposition. Q is an orthogonal matrix Returns:RealMatrix the transpose of the Q matrix, Q^T

getQT

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public RealMatrix getQT()

Returns the transpose of the matrix Q of the decomposition.

Q is an orthogonal matrix

Returns:RealMatrix: the transpose of the Q matrix, Q^T

getR	back to summary
public RealMatrix getR() Returns the matrix R of the decomposition. R is an upper-triangular matrix Returns:RealMatrix the R matrix

getR

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public RealMatrix getR()

Returns the matrix R of the decomposition.

R is an upper-triangular matrix

Returns:RealMatrix: the R matrix

getSolver back to summary

getSolver	back to summary
public DecompositionSolver getSolver() Get a solver for finding the A × X = B solution in least square sense. Least Square sense means a solver can be computed for an overdetermined system, (i.e. a system with more equations than unknowns, which corresponds to a tall A matrix with more rows than columns). In any case, if the matrix is singular within the tolerance set at `construction`, an error will be triggered when the `solve` method will be called. Returns:DecompositionSolver a solver

public DecompositionSolver getSolver()

Get a solver for finding the A × X = B solution in least square sense.

Least Square sense means a solver can be computed for an overdetermined system, (i.e. a system with more equations than unknowns, which corresponds to a tall A matrix with more rows than columns). In any case, if the matrix is singular within the tolerance set at construction, an error will be triggered when the solve method will be called.

Returns:DecompositionSolver: a solver

performHouseholderReflection	back to summary
protected void performHouseholderReflection(int minor, double[][] matrix) Perform Householder reflection for a minor A(minor, minor) of A. Parameters minor:int minor index matrix:double[][] transposed matrix Since 3.2

performHouseholderReflection

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protected void performHouseholderReflection(int minor, double[][] matrix)

Perform Householder reflection for a minor A(minor, minor) of A.

Parameters

minor:int: minor index
matrix:double[][]: transposed matrix

Since

3.2

org.apache.commons.math3.linear back to summary

private Class QRDecomposition.Solver

Class Inheritance

extends Object
implements DecompositionSolver

java.lang.Object
org.apache.commons.math3.linear.QRDecomposition.Solver

All Implemented Interfaces

org.apache.commons.math3.linear.DecompositionSolver

Specialized solver.

Field Summary

Modifier and Type	Field and Description
private final double[][]	qrt A packed TRANSPOSED representation of the QR decomposition.
private final double[]	rDiag The diagonal elements of R.
private final double	threshold Singularity threshold.

Constructor Summary

Access	Constructor and Description
private	Solver(final double[][] Packed TRANSPOSED representation of the QR decomposition. qrt, final double[] Diagonal elements of R. rDiag, final double Singularity threshold. threshold) Build a solver from decomposed matrix.

Method Summary

Modifier and Type	Method and Description
public RealMatrix	getInverse() Implements org.apache.commons.math3.linear.DecompositionSolver.getInverse. Get the pseudo-inverse of the decomposed matrix.
public boolean	isNonSingular() Implements org.apache.commons.math3.linear.DecompositionSolver.isNonSingular. Check if the decomposed matrix is non-singular.
public RealVector	solve(RealVector right-hand side of the equation A × X = B b) Implements org.apache.commons.math3.linear.DecompositionSolver.solve. Solve the linear equation A × X = B for matrices A.
public RealMatrix	solve(RealMatrix right-hand side of the equation A × X = B b) Implements org.apache.commons.math3.linear.DecompositionSolver.solve. Solve the linear equation A × X = B for matrices A.

Inherited from java.lang.Object:: clone equals finalize getClass hashCode notify notifyAll toString wait wait wait

Field Detail

qrt	back to summary
private final double[][] qrt A packed TRANSPOSED representation of the QR decomposition. The elements BELOW the diagonal are the elements of the UPPER triangular matrix R, and the rows ABOVE the diagonal are the Householder reflector vectors from which an explicit form of Q can be recomputed if desired.

qrt

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private final double[][] qrt

A packed TRANSPOSED representation of the QR decomposition.

rDiag	back to summary
private final double[] rDiag The diagonal elements of R.

threshold	back to summary
private final double threshold Singularity threshold.

Constructor Detail

Solver	back to summary
private Solver(final double[][] qrt, final double[] rDiag, final double threshold) Build a solver from decomposed matrix. Parameters qrt:double[][] Packed TRANSPOSED representation of the QR decomposition. rDiag:double[] Diagonal elements of R. threshold:double Singularity threshold.

Solver

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private Solver(final double[][] qrt, final double[] rDiag, final double threshold)

Build a solver from decomposed matrix.

Parameters

qrt:double[][]: Packed TRANSPOSED representation of the QR decomposition.
rDiag:double[]: Diagonal elements of R.
threshold:double: Singularity threshold.

Method Detail

getInverse back to summary

getInverse	back to summary
public RealMatrix getInverse() Implements org.apache.commons.math3.linear.DecompositionSolver.getInverse. Doc from org.apache.commons.math3.linear.DecompositionSolver.getInverse. Get the pseudo-inverse of the decomposed matrix. This is equal to the inverse of the decomposed matrix, if such an inverse exists. If no such inverse exists, then the result has properties that resemble that of an inverse. In particular, in this case, if the decomposed matrix is A, then the system of equations \( A x = b \) may have no solutions, or many. If it has no solutions, then the pseudo-inverse \( A^+ \) gives the "closest" solution \( z = A^+ b \), meaning \( \left \\| A z - b \right \\|_2 \) is minimized. If there are many solutions, then \( z = A^+ b \) is the smallest solution, meaning \( \left \\| z \right \\|_2 \) is minimized. Note however that some decompositions cannot compute a pseudo-inverse for all matrices. For example, the `LUDecomposition` is not defined for non-square matrices to begin with. The `QRDecomposition` can operate on non-square matrices, but will throw `SingularMatrixException` if the decomposed matrix is singular. Refer to the javadoc of specific decomposition implementations for more details. Returns:RealMatrix pseudo-inverse matrix (which is the inverse, if it exists), if the decomposition can pseudo-invert the decomposed matrix Exceptions SingularMatrixException: if the decomposed matrix is singular.

public RealMatrix getInverse()

Implements org.apache.commons.math3.linear.DecompositionSolver.getInverse.

Doc from org.apache.commons.math3.linear.DecompositionSolver.getInverse.

Get the pseudo-inverse of the decomposed matrix.

This is equal to the inverse of the decomposed matrix, if such an inverse exists.

If no such inverse exists, then the result has properties that resemble that of an inverse.

In particular, in this case, if the decomposed matrix is A, then the system of equations \( A x = b \) may have no solutions, or many. If it has no solutions, then the pseudo-inverse \( A^+ \) gives the "closest" solution \( z = A^+ b \), meaning \( \left \| A z - b \right \|_2 \) is minimized. If there are many solutions, then \( z = A^+ b \) is the smallest solution, meaning \( \left \| z \right \|_2 \) is minimized.

Note however that some decompositions cannot compute a pseudo-inverse for all matrices. For example, the LUDecomposition is not defined for non-square matrices to begin with. The QRDecomposition can operate on non-square matrices, but will throw SingularMatrixException if the decomposed matrix is singular. Refer to the javadoc of specific decomposition implementations for more details.

Returns:RealMatrix

pseudo-inverse matrix (which is the inverse, if it exists), if the decomposition can pseudo-invert the decomposed matrix

Exceptions

SingularMatrixException:: if the decomposed matrix is singular.

isNonSingular	back to summary
public boolean isNonSingular() Implements org.apache.commons.math3.linear.DecompositionSolver.isNonSingular. Doc from org.apache.commons.math3.linear.DecompositionSolver.isNonSingular. Check if the decomposed matrix is non-singular. Returns:boolean true if the decomposed matrix is non-singular.

isNonSingular

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public boolean isNonSingular()

Implements org.apache.commons.math3.linear.DecompositionSolver.isNonSingular.

Doc from org.apache.commons.math3.linear.DecompositionSolver.isNonSingular.

Check if the decomposed matrix is non-singular.

Returns:boolean: true if the decomposed matrix is non-singular.

solve	back to summary
public RealVector solve(RealVector b) Implements org.apache.commons.math3.linear.DecompositionSolver.solve. Doc from org.apache.commons.math3.linear.DecompositionSolver.solve. Solve the linear equation A × X = B for matrices A. The A matrix is implicit, it is provided by the underlying decomposition algorithm. Parameters b:RealVector right-hand side of the equation A × X = B Returns:RealVector a vector X that minimizes the two norm of A × X - B

solve

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public RealVector solve(RealVector b)

Implements org.apache.commons.math3.linear.DecompositionSolver.solve.

Doc from org.apache.commons.math3.linear.DecompositionSolver.solve.

Solve the linear equation A × X = B for matrices A.

The A matrix is implicit, it is provided by the underlying decomposition algorithm.

Parameters

b:RealVector: right-hand side of the equation A × X = B

Returns:RealVector

a vector X that minimizes the two norm of A × X - B

solve	back to summary
public RealMatrix solve(RealMatrix b) Implements org.apache.commons.math3.linear.DecompositionSolver.solve. Doc from org.apache.commons.math3.linear.DecompositionSolver.solve. Solve the linear equation A × X = B for matrices A. The A matrix is implicit, it is provided by the underlying decomposition algorithm. Parameters b:RealMatrix right-hand side of the equation A × X = B Returns:RealMatrix a matrix X that minimizes the two norm of A × X - B

solve

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public RealMatrix solve(RealMatrix b)

Implements org.apache.commons.math3.linear.DecompositionSolver.solve.

Doc from org.apache.commons.math3.linear.DecompositionSolver.solve.

Solve the linear equation A × X = B for matrices A.

The A matrix is implicit, it is provided by the underlying decomposition algorithm.

Parameters

b:RealMatrix: right-hand side of the equation A × X = B

Returns:RealMatrix

a matrix X that minimizes the two norm of A × X - B

public Class QRDecomposition

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Method Detail

private Class QRDecomposition.Solver

Field Summary

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