smile.math.matrix

Class QRDecomposition

java.lang.Object

smile.math.matrix.QRDecomposition

public class QRDecomposition
extends Object

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 such that A = Q*R.

The QR decomposition always exists, even if the matrix does not have full rank. The primary use of the QR decomposition is in the least squares solution of non-square systems of simultaneous linear equations, where isFullColumnRank() has to be true.

QR decomposition is also the basis for a particular eigenvalue algorithm, the QR algorithm.

Author:

Haifeng Li

Constructor Summary

Constructors

Constructor and Description
QRDecomposition(double[][] A) Constructor.
QRDecomposition(double[][] A, boolean overwrite) Constructor.

Method Summary

Modifier and Type	Method and Description
double[][]	getH() Returns the Householder vectors.
double[][]	getQ() Returns the orthogonal factor.
double[][]	getR() Returns the upper triangular factor.
double[][]	inverse() Returns the matrix pseudo inverse.
boolean	isFullColumnRank() Returns true if the matrix is full column rank.
boolean	isSingular() Returns true if the matrix is singular.
void	solve(double[] b) Solve the least squares A * x = b.
void	solve(double[][] B) Solve the least squares A * X = B.
void	solve(double[][] B, double[][] X) Solve the least squares A * X = B.
void	solve(double[] b, double[] x) Solve the least squares A*x = b.
CholeskyDecomposition	toCholesky() Returns the Cholesky decomposition of A'A.
void	update(double[] u, double[] v) Rank-1 update of the QR decomposition for A = A + u * v.

Methods inherited from class java.lang.Object

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

Constructor Detail

QRDecomposition

public QRDecomposition(double[][] A)

Constructor. QR Decomposition is computed by Householder reflections. The decomposition will be stored in a new create matrix. The input matrix will not be modified.

Parameters:

A - rectangular matrix

QRDecomposition

public QRDecomposition(double[][] A,
                       boolean overwrite)

Constructor. QR Decomposition is computed by Householder reflections. The user can specify if the decomposition takes in place, i.e. if the decomposition will be stored in the input matrix. Otherwise, a new matrix will be allocated to store the decomposition.

Parameters:

A - rectangular matrix

overwrite - true if the decomposition will be taken in place. If true, the decomposition will be stored in the input matrix to save space. It is very useful in practice if the matrix is huge. Otherwise, a new matrix will created to store the decomposition.

Method Detail

isFullColumnRank

public boolean isFullColumnRank()

Returns true if the matrix is full column rank.

isSingular

public boolean isSingular()

Returns true if the matrix is singular.

getH

public double[][] getH()

Returns the Householder vectors.

Returns:

Lower trapezoidal matrix whose columns define the reflections

toCholesky

public CholeskyDecomposition toCholesky()

Returns the Cholesky decomposition of A'A.

getR

public double[][] getR()

Returns the upper triangular factor.

getQ

public double[][] getQ()

Returns the orthogonal factor.

inverse

public double[][] inverse()

Returns the matrix pseudo inverse.

solve

public void solve(double[] b)

Solve the least squares A * x = b. On output, b will be overwritten with the solution vector.

Parameters:

b - right hand side of linear system. On output, b will be overwritten with the solution vector.

Throws:

RuntimeException - if matrix is rank deficient.

solve

public void solve(double[] b,
                  double[] x)

Solve the least squares A*x = b.

Parameters:

b - right hand side of linear system.

x - the solution vector that minimizes the L2 norm of Q*R*x - b.

Throws:

RuntimeException - if matrix is rank deficient.

solve

public void solve(double[][] B)

Solve the least squares A * X = B. B will be overwritten with the solution matrix on output.

Parameters:

B - right hand side of linear system. B will be overwritten with the solution matrix on output.

Throws:

RuntimeException - Matrix is rank deficient.

solve

public void solve(double[][] B,
                  double[][] X)

Solve the least squares A * X = B.

Parameters:

B - right hand side of linear system.

X - the solution matrix that minimizes the L2 norm of Q*R*X - B.

Throws:

RuntimeException - Matrix is rank deficient.

update

public void update(double[] u,
                   double[] v)

Rank-1 update of the QR decomposition for A = A + u * v. Instead of a full decomposition from scratch in O(N³), we can update the QR factorization in O(N²). Note that u is modified during the update.