Documentation of 'smile.math.matrix.PowerIteration' Java class
PowerIteration
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smile.math.matrix

Class PowerIteration


  • public class PowerIteration
extends java.lang.Object
    The power iteration (also known as power method) is an eigenvalue algorithm that will produce the greatest (in absolute value) eigenvalue and a nonzero vector the corresponding eigenvector.

    • Constructor Summary

      Constructors 
      Constructor and Description
      PowerIteration() 

    • Method Summary

      All Methods Static Methods Concrete Methods 
      Modifier and Type Method and Description
      static double eigen(Matrix A, double[] v)
      Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
      static double eigen(Matrix A, double[] v, double tol)
      Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
      static double eigen(Matrix A, double[] v, double p, double tol)
      Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
      static double eigen(Matrix A, double[] v, double p, double tol, int maxIter)
      Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
      static double eigen(Matrix A, double[] v, double tol, int maxIter)
      Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.

      • Methods inherited from class java.lang.Object

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

    • Constructor Detail

      • PowerIteration

        public PowerIteration()

    • Method Detail

      • eigen

        public static double eigen(Matrix A,
                           double[] v)
        Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
        Parameters:
        A - the matrix supporting matrix vector multiplication operation.
        v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.
        Returns:
        the largest eigen value.

      • eigen

        public static double eigen(Matrix A,
                           double[] v,
                           double tol)
        Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
        Parameters:
        A - the matrix supporting matrix vector multiplication operation.
        v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.
        tol - the desired convergence tolerance.
        Returns:
        the largest eigen value.

      • eigen

        public static double eigen(Matrix A,
                           double[] v,
                           double tol,
                           int maxIter)
        Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
        Parameters:
        A - the matrix supporting matrix vector multiplication operation.
        v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.
        tol - the desired convergence tolerance.
        maxIter - the maximum number of iterations in case that the algorithm does not converge.
        Returns:
        the largest eigen value.

      • eigen

        public static double eigen(Matrix A,
                           double[] v,
                           double p,
                           double tol)
        Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
        Parameters:
        A - the matrix supporting matrix vector multiplication operation.
        v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.
        p - the origin in the shifting power method. A - pI will be used in the iteration to accelerate the method. p should be such that |(λ2 - p) / (λ1 - p)| < |λ2 / λ1|, where λ2 is the second largest eigenvalue in magnitude. If we known the eigenvalue spectrum of A, (λ2 + λn)/2 is the optimal choice of p, where λn is the smallest eigenvalue in magnitude. Good estimates of λ2 are more difficult to compute. However, if μ is an approximation to largest eigenvector, then using any x0 such that x0*μ = 0 as the initial vector for a few iterations may yield a reasonable estimate of λ2.
        tol - the desired convergence tolerance.
        Returns:
        the largest eigen value.

      • eigen

        public static double eigen(Matrix A,
                           double[] v,
                           double p,
                           double tol,
                           int maxIter)
        Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
        Parameters:
        A - the matrix supporting matrix vector multiplication operation.
        v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.
        p - the origin in the shifting power method. A - pI will be used in the iteration to accelerate the method. p should be such that |(λ2 - p) / (λ1 - p)| < |λ2 / λ1|, where λ2 is the second largest eigenvalue in magnitude. If we known the eigenvalue spectrum of A, (λ2 + λn)/2 is the optimal choice of p, where λn is the smallest eigenvalue in magnitude. Good estimates of λ2 are more difficult to compute. However, if μ is an approximation to largest eigenvector, then using any x0 such that x0*μ = 0 as the initial vector for a few iterations may yield a reasonable estimate of λ2.
        tol - the desired convergence tolerance.
        maxIter - the maximum number of iterations in case that the algorithm does not converge.
        Returns:
        the largest eigen value.
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