com.imsl.math

Class QR

java.lang.Object

com.imsl.math.QR

All Implemented Interfaces:

Serializable, Cloneable

public class QR
extends Object
implements Serializable, Cloneable

QR Decomposition of a matrix.

Class QR computes the QR decomposition of a matrix using Householder transformations. It is based on the LINPACK routine SQRDC; see Dongarra et al. (1979).

QR determines an orthogonal matrix Q, a permutation matrix P, and an upper trapezoidal matrix R with diagonal elements of nonincreasing magnitude, such that AP = QR. The Householder transformation for column k is of the form

$$I - \frac{{u_k u_k^T }}{{P_k }}$$

for \(k = 1, 2, \ldots\), min(number of rows of A, number of columns of A), where u has zeros in the first k - 1 positions. The matrix Q is not produced directly by QR. Instead the information needed to reconstruct the Householder transformations is saved. If the matrix Q is needed explicitly, the method getQ can be called after QR. This method accumulates Q from its factored form.

Before the decomposition is computed, initial columns are moved to the beginning of the array A and the final columns to the end. Both initial and final columns are frozen in place during the computation. Only free columns are pivoted. Pivoting is done on the free columns of largest reduced norm.

See Also:

Example, Serialized Form

Constructor Summary

Constructors

Constructor and Description
QR(double[][] a) Constructs the QR decomposition of a matrix with elements of type double.

Method Summary

Modifier and Type	Method and Description
int[]	getPermute() Returns an integer vector containing information about the permutation of the elements of the matrix during pivoting.
double[][]	getQ() Returns the orthogonal or unitary matrix Q.
double[][]	getR() Returns the upper trapezoidal matrix R.
int	getRank() Returns the rank of the matrix used to construct this instance.
int	rank(double tolerance) Returns the rank of the matrix given an input tolerance.
double[]	solve(double[] b) Returns the solution to the least-squares problem Ax = b.
double[]	solve(double[] b, double tol) Returns the solution to the least-squares problem Ax = b using an input tolerance.

Methods inherited from class java.lang.Object

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

Constructor Detail

QR

public QR(double[][] a)

Constructs the QR decomposition of a matrix with elements of type double.

Parameters:

a - a double matrix to be factored

Throws:

IllegalArgumentException - Thrown when the row lengths of input matrix a are not equal (i.e. the matrix edges are "jagged".)

Method Detail

getQ

public double[][] getQ()

Returns the orthogonal or unitary matrix Q.

Returns:

a double matrix containing the accumulated orthogonal matrix Q from the QR decomposition

getR

public double[][] getR()

Returns the upper trapezoidal matrix R.

Returns:

the upper trapezoidal double matrix R of the QR decomposition

getRank

public int getRank()

Returns the rank of the matrix used to construct this instance.

Returns:

an int specifying the rank of the matrix used to construct this instance

rank

public int rank(double tolerance)

Returns the rank of the matrix given an input tolerance.

Parameters:

tolerance - a double scalar value used in determining the rank of the matrix

Returns:

an int specifying the rank of the matrix

getPermute

public int[] getPermute()

Returns an integer vector containing information about the permutation of the elements of the matrix during pivoting.

Returns:

an int array containing the permutation information. The k-th element contains the index of the column of the matrix that has been interchanged into the k-th column.

solve

public double[] solve(double[] b)
               throws SingularMatrixException

Returns the solution to the least-squares problem Ax = b.

Parameters:

b - a double array to be manipulated

Returns:

a double array containing the solution vector to Ax = b with components corresponding to the unused columns set to zero

Throws:

SingularMatrixException - Thrown when the upper triangular matrix R resulting from the QR factorization is singular.

solve

public double[] solve(double[] b,
                      double tol)
               throws SingularMatrixException

Returns the solution to the least-squares problem Ax = b using an input tolerance.

Parameters:

b - a double array to be manipulated

tol - a double scalar value used in determining the rank of A

Returns:

a double array containing the solution vector to Ax = b with components corresponding to the unused columns set to zero

Throws:

SingularMatrixException - Thrown when the upper triangular matrix R resulting from the QR factorization is singular.