## Class FieldQRDecomposition<T extends CalculusFieldElement<T>>

• Type Parameters:
T - type of the underlying field elements

public class FieldQRDecomposition<T extends CalculusFieldElement<T>>
extends Object
Calculates the QR-decomposition of a field matrix.

The QR-decomposition of a matrix A consists of two matrices Q and R that satisfy: A = QR, Q is orthogonal (QTQ = I), and R is upper triangular. If A is m×n, Q is m×m and R m×n.

This class compute the decomposition using Householder reflectors.

For efficiency purposes, the decomposition in packed form is transposed. This allows inner loop to iterate inside rows, which is much more cache-efficient in Java.

This class is based on the class QRDecomposition.

MathWorld, Wikipedia
• ### Constructor Summary

Constructors
Constructor Description
FieldQRDecomposition​(FieldMatrix<T> matrix)
Calculates the QR-decomposition of the given matrix.
FieldQRDecomposition​(FieldMatrix<T> matrix, T threshold)
Calculates the QR-decomposition of the given matrix.
FieldQRDecomposition​(FieldMatrix<T> matrix, T threshold, Predicate<T> zeroChecker)
Calculates the QR-decomposition of the given matrix.
• ### Method Summary

All Methods
Modifier and Type Method Description
protected void decompose​(T[][] matrix)
Decompose matrix.
FieldMatrix<T> getH()
Returns the Householder reflector vectors.
FieldMatrix<T> getQ()
Returns the matrix Q of the decomposition.
FieldMatrix<T> getQT()
Returns the transpose of the matrix Q of the decomposition.
FieldMatrix<T> getR()
Returns the matrix R of the decomposition.
FieldDecompositionSolver<T> getSolver()
Get a solver for finding the A × X = B solution in least square sense.
protected void performHouseholderReflection​(int minor, T[][] matrix)
Perform Householder reflection for a minor A(minor, minor) of A.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### FieldQRDecomposition

public FieldQRDecomposition​(FieldMatrix<T> matrix)
Calculates the QR-decomposition of the given matrix. The singularity threshold defaults to zero.
Parameters:
matrix - The matrix to decompose.
FieldQRDecomposition(FieldMatrix, CalculusFieldElement)
• #### FieldQRDecomposition

public FieldQRDecomposition​(FieldMatrix<T> matrix,
T threshold)
Calculates the QR-decomposition of the given matrix.
Parameters:
matrix - The matrix to decompose.
threshold - Singularity threshold.
• #### FieldQRDecomposition

public FieldQRDecomposition​(FieldMatrix<T> matrix,
T threshold,
Predicate<T> zeroChecker)
Calculates the QR-decomposition of the given matrix.
Parameters:
matrix - The matrix to decompose.
threshold - Singularity threshold.
zeroChecker - checker for zero
• ### Method Detail

• #### decompose

protected void decompose​(T[][] matrix)
Decompose matrix.
Parameters:
matrix - transposed matrix
• #### performHouseholderReflection

protected void performHouseholderReflection​(int minor,
T[][] matrix)
Perform Householder reflection for a minor A(minor, minor) of A.
Parameters:
minor - minor index
matrix - transposed matrix
• #### getR

public FieldMatrix<T> getR()
Returns the matrix R of the decomposition.

R is an upper-triangular matrix

Returns:
the R matrix
• #### getQ

public FieldMatrix<T> getQ()
Returns the matrix Q of the decomposition.

Q is an orthogonal matrix

Returns:
the Q matrix
• #### getQT

public FieldMatrix<T> getQT()
Returns the transpose of the matrix Q of the decomposition.

Q is an orthogonal matrix

Returns:
the transpose of the Q matrix, QT
• #### getH

public FieldMatrix<T> 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:
a matrix containing the Householder reflector vectors
• #### getSolver

public FieldDecompositionSolver<T> 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:
a solver