Package 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:

  • Constructor Summary

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

  • Method Summary

    Modifier and Type
    Method
    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 Details

    • 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 Details

    • 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.