de.lmu.ifi.dbs.elki.math.linearalgebra

Class QRDecomposition

java.lang.Object

de.lmu.ifi.dbs.elki.math.linearalgebra.QRDecomposition

All Implemented Interfaces:

java.io.Serializable

public class QRDecomposition
extends java.lang.Object
implements java.io.Serializable

QR Decomposition. For an m-by-n matrix A with m >= n, the QR decomposition is an m-by-n orthogonal matrix Q and an n-by-n upper triangular matrix R so that A = Q*R. The QR decompostion always exists, even if the matrix does not have full rank, so the constructor will never fail. The primary use of the QR decomposition is in the least squares solution of nonsquare systems of simultaneous linear equations. This will fail if isFullRank() returns false.

Since:

0.1

Author:

Arthur Zimek

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
protected static java.lang.String	ERR_MATRIX_RANK_DEFICIENT When a matrix is rank deficient.
private int	m Row and column dimensions.
private int	n Row and column dimensions.
private double[][]	QR Array for internal storage of decomposition.
private double[]	Rdiag Array for internal storage of diagonal of R.
private static long	serialVersionUID Serial version

Constructor Summary

Constructors

Constructor and Description
QRDecomposition(double[][] A) QR Decomposition, computed by Householder reflections.
QRDecomposition(double[][] A, int m, int n) QR Decomposition, computed by Householder reflections.

Method Summary

Modifier and Type	Method and Description
double[][]	getH() Return the Householder vectors
double[][]	getQ() Generate and return the (economy-sized, m by n) orthogonal factor
double[][]	getR() Return the upper triangular factor
double[][]	inverse() Find the inverse matrix.
boolean	isFullRank() Is the matrix full rank?
int	rank(double t) Get the matrix rank?
double[]	solve(double[] b) Least squares solution of A*X = b
double[][]	solve(double[][] B) Least squares solution of A*X = B
double[]	solveInplace(double[] b) Least squares solution of A*X = b
private double[][]	solveInplace(double[][] B) Least squares solution of A*X = B

Methods inherited from class java.lang.Object

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

Field Detail

ERR_MATRIX_RANK_DEFICIENT

protected static final java.lang.String ERR_MATRIX_RANK_DEFICIENT

When a matrix is rank deficient.

See Also:

Constant Field Values

serialVersionUID

private static final long serialVersionUID

Serial version

See Also:

Constant Field Values

QR

private double[][] QR

Array for internal storage of decomposition.

m

private int m

Row and column dimensions.

n

private int n

Row and column dimensions.

Rdiag

private double[] Rdiag

Array for internal storage of diagonal of R.

Constructor Detail

QRDecomposition

public QRDecomposition(double[][] A)

QR Decomposition, computed by Householder reflections.

Parameters:

A - Rectangular matrix

QRDecomposition

public QRDecomposition(double[][] A,
                       int m,
                       int n)

QR Decomposition, computed by Householder reflections.

Parameters:

A - Rectangular matrix

m - row dimensionality

n - column dimensionality

Method Detail

isFullRank

public boolean isFullRank()

Is the matrix full rank?

Returns:

true if R, and hence A, has full rank.

rank

public int rank(double t)

Get the matrix rank?

Parameters:

t - Tolerance threshold

Returns:

Rank of R

getH

public double[][] getH()

Return the Householder vectors

Returns:

Lower trapezoidal matrix whose columns define the reflections

getR

public double[][] getR()

Return the upper triangular factor

Returns:

R

getQ

public double[][] getQ()

Generate and return the (economy-sized, m by n) orthogonal factor

Returns:

Q

solve

public double[][] solve(double[][] B)

Least squares solution of A*X = B

Parameters:

B - The matrix B with as many rows as A and any number of columns.

Returns:

X that minimizes the two norm of Q*R*X-B.

Throws:

java.lang.IllegalArgumentException - Matrix row dimensions must agree.

java.lang.ArithmeticException - Matrix is rank deficient.

solveInplace

private double[][] solveInplace(double[][] B)

Least squares solution of A*X = B

Parameters:

B - The matrix B with as many rows as A and any number of columns (will be overwritten).

Returns:

X that minimizes the two norm of Q*R*X-B.

Throws:

java.lang.IllegalArgumentException - Matrix row dimensions must agree.

java.lang.ArithmeticException - Matrix is rank deficient.

solve

public double[] solve(double[] b)

Least squares solution of A*X = b

Parameters:

b - A column vector with as many rows as A.

Returns:

X that minimizes the two norm of Q*R*X-b.

Throws:

java.lang.IllegalArgumentException - Matrix row dimensions must agree.

java.lang.ArithmeticException - Matrix is rank deficient.

solveInplace

public double[] solveInplace(double[] b)

Least squares solution of A*X = b

Parameters:

b - A column vector b with as many rows as A.

Returns:

X that minimizes the two norm of Q*R*X-b.

Throws:

java.lang.IllegalArgumentException - Matrix row dimensions must agree.

java.lang.ArithmeticException - Matrix is rank deficient.

inverse

public double[][] inverse()

Find the inverse matrix.

Returns:

Inverse matrix

Throws:

java.lang.ArithmeticException - Matrix is rank deficient.