smile.math.matrix

Class QR

java.lang.Object

smile.math.matrix.QR

public class QR
extends java.lang.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 isSingular() has to be false.

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

Field Summary

Fields

Modifier and Type	Field and Description
protected DenseMatrix	qr Array for internal storage of decomposition.
protected boolean	singular Indicate if the matrix is singular.
protected double[]	tau The diagonal of R for this implementation.

Constructor Summary

Constructors

Constructor and Description
QR(DenseMatrix qr, double[] tau, boolean singular) Constructor.

Method Summary

Modifier and Type	Method and Description
Cholesky	CholeskyOfAtA() Returns the Cholesky decomposition of A'A.
DenseMatrix	getQ() Returns the orthogonal factor.
DenseMatrix	getR() Returns the upper triangular factor.
boolean	isSingular() Returns true if the matrix is singular (not full column rank).
void	solve(DenseMatrix B) Solve the least squares A * X = B.
void	solve(double[] b, double[] x) Solve the least squares A*x = b.

Methods inherited from class java.lang.Object

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

Field Detail

qr

protected DenseMatrix qr

Array for internal storage of decomposition.

tau

protected double[] tau

The diagonal of R for this implementation. For netlib based QR, this is the scales for the reflectors.

singular

protected boolean singular

Indicate if the matrix is singular.

Constructor Detail

QR

public QR(DenseMatrix qr,
          double[] tau,
          boolean singular)

Constructor.

Method Detail

isSingular

public boolean isSingular()

Returns true if the matrix is singular (not full column rank).

CholeskyOfAtA

public Cholesky CholeskyOfAtA()

Returns the Cholesky decomposition of A'A.

getR

public DenseMatrix getR()

Returns the upper triangular factor.

getQ

public DenseMatrix getQ()

Returns the orthogonal factor.

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

Throws:

java.lang.RuntimeException - if matrix is rank deficient.

solve

public void solve(DenseMatrix 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:

java.lang.RuntimeException - Matrix is rank deficient.