Package org.ejml.dense.row

Class NormOps_FDRM

  • public class NormOps_FDRM
extends java.lang.Object

    Norms are a measure of the size of a vector or a matrix. One typical application is in error analysis.

    Vector norms have the following properties:
    1. ||x|| > 0 if x ≠ 0 and ||0|| = 0
    2. ||αx|| = |α| ||x||
    3. ||x+y|| ≤ ||x|| + ||y||
    Matrix norms have the following properties:
    1. ||A|| > 0 if A ≠ 0 where A ∈ ℜ m × n
    2. || α A || = |α| ||A|| where A ∈ ℜ m × n
    3. ||A+B|| ≤ ||A|| + ||B|| where A and B are ∈ ℜ m × n
    4. ||AB|| ≤ ||A|| ||B|| where A and B are ∈ ℜ m × m
    Note that the last item in the list only applies to square matrices.

    Matrix norms can be induced from vector norms as is shown below:

    ||A||M = maxx≠0||Ax||v/||x||v

    where ||.||M is the induced matrix norm for the vector norm ||.||v.

    By default implementations that try to mitigate overflow/underflow are used. If the word fast is found before a function's name that means it does not mitigate those issues, but runs a bit faster.

    • Constructor Summary

      Constructors 
      Constructor Description
      NormOps_FDRM()  

    • Method Summary

      All Methods Static Methods Concrete Methods 
      Modifier and Type Method Description
      static float conditionP​(org.ejml.data.FMatrixRMaj A, float p)
      The condition number of a matrix is used to measure the sensitivity of the linear system Ax=b.
      static float conditionP2​(org.ejml.data.FMatrixRMaj A)
      The condition p = 2 number of a matrix is used to measure the sensitivity of the linear system Ax=b.
      static float elementP​(org.ejml.data.FMatrix1Row A, float p)
      Element wise p-norm:

      norm = {∑i=1:m ∑j=1:n { |aij|p}}1/p
      static float fastElementP​(org.ejml.data.FMatrixD1 A, float p)
      Same as elementP(org.ejml.data.FMatrix1Row, float) but runs faster by not mitigating overflow/underflow related problems.
      static float fastNormF​(org.ejml.data.FMatrixD1 a)
      This implementation of the Frobenius norm is a straight forward implementation and can be susceptible for overflow/underflow issues.
      static float fastNormP​(org.ejml.data.FMatrixRMaj A, float p)
      An unsafe but faster version of normP(org.ejml.data.FMatrixRMaj, float) that calls routines which are faster but more prone to overflow/underflow problems.
      static float fastNormP2​(org.ejml.data.FMatrixRMaj A)
      Computes the p=2 norm.
      static float inducedP1​(org.ejml.data.FMatrixRMaj A)
      Computes the induced p = 1 matrix norm.

      ||A||1= max(j=1 to n; sum(i=1 to m; |aij|))
      static float inducedP2​(org.ejml.data.FMatrixRMaj A)
      Computes the induced p = 2 matrix norm, which is the largest singular value.
      static float inducedPInf​(org.ejml.data.FMatrixRMaj A)
      Induced matrix p = infinity norm.

      ||A||∞ = max(i=1 to m; sum(j=1 to n; |aij|))
      static void normalizeF​(org.ejml.data.FMatrixRMaj A)
      Normalizes the matrix such that the Frobenius norm is equal to one.
      static float normF​(org.ejml.data.FMatrixD1 a)
      Computes the Frobenius matrix norm:

      normF = Sqrt{ ∑i=1:m ∑j=1:n { aij2} }
      static float normP​(org.ejml.data.FMatrixRMaj A, float p)
      Computes either the vector p-norm or the induced matrix p-norm depending on A being a vector or a matrix respectively.
      static float normP1​(org.ejml.data.FMatrixRMaj A)
      Computes the p=1 norm.
      static float normP2​(org.ejml.data.FMatrixRMaj A)
      Computes the p=2 norm.
      static float normPInf​(org.ejml.data.FMatrixRMaj A)
      Computes the p=∞ norm.

      • Methods inherited from class java.lang.Object

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

    • Constructor Detail

      • NormOps_FDRM

        public NormOps_FDRM()

    • Method Detail

      • normalizeF

        public static void normalizeF​(org.ejml.data.FMatrixRMaj A)
        Normalizes the matrix such that the Frobenius norm is equal to one.
        Parameters:
        A - The matrix that is to be normalized.

      • conditionP

        public static float conditionP​(org.ejml.data.FMatrixRMaj A,
                               float p)

        The condition number of a matrix is used to measure the sensitivity of the linear system Ax=b. A value near one indicates that it is a well conditioned matrix.

        κp = ||A||p||A-1||p

        If the matrix is not square then the condition of either ATA or AAT is computed.

        Parameters:
        A - The matrix.
        p - p-norm
        Returns:
        The condition number.

      • conditionP2

        public static float conditionP2​(org.ejml.data.FMatrixRMaj A)

        The condition p = 2 number of a matrix is used to measure the sensitivity of the linear system Ax=b. A value near one indicates that it is a well conditioned matrix.

        κ2 = ||A||2||A-1||2

        This is also known as the spectral condition number.

        Parameters:
        A - The matrix.
        Returns:
        The condition number.

      • fastNormF

        public static float fastNormF​(org.ejml.data.FMatrixD1 a)

        This implementation of the Frobenius norm is a straight forward implementation and can be susceptible for overflow/underflow issues. A more resilient implementation is normF(org.ejml.data.FMatrixD1).

        Parameters:
        a - The matrix whose norm is computed. Not modified.

      • normF

        public static float normF​(org.ejml.data.FMatrixD1 a)

        Computes the Frobenius matrix norm:

        normF = Sqrt{ ∑i=1:m ∑j=1:n { aij2} }

        This is equivalent to the element wise p=2 norm. See fastNormF(org.ejml.data.FMatrixD1) for another implementation that is faster, but more prone to underflow/overflow errors.

        Parameters:
        a - The matrix whose norm is computed. Not modified.
        Returns:
        The norm's value.

      • elementP

        public static float elementP​(org.ejml.data.FMatrix1Row A,
                             float p)

        Element wise p-norm:

        norm = {∑i=1:m ∑j=1:n { |aij|p}}1/p

        This is not the same as the induced p-norm used on matrices, but is the same as the vector p-norm.

        Parameters:
        A - Matrix. Not modified.
        p - p value.
        Returns:
        The norm's value.

      • fastElementP

        public static float fastElementP​(org.ejml.data.FMatrixD1 A,
                                 float p)
        Same as elementP(org.ejml.data.FMatrix1Row, float) but runs faster by not mitigating overflow/underflow related problems.
        Parameters:
        A - Matrix. Not modified.
        p - p value.
        Returns:
        The norm's value.

      • normP

        public static float normP​(org.ejml.data.FMatrixRMaj A,
                          float p)
        Computes either the vector p-norm or the induced matrix p-norm depending on A being a vector or a matrix respectively.
        Parameters:
        A - Vector or matrix whose norm is to be computed.
        p - The p value of the p-norm.
        Returns:
        The computed norm.

      • fastNormP

        public static float fastNormP​(org.ejml.data.FMatrixRMaj A,
                              float p)
        An unsafe but faster version of normP(org.ejml.data.FMatrixRMaj, float) that calls routines which are faster but more prone to overflow/underflow problems.
        Parameters:
        A - Vector or matrix whose norm is to be computed.
        p - The p value of the p-norm.
        Returns:
        The computed norm.

      • normP1

        public static float normP1​(org.ejml.data.FMatrixRMaj A)
        Computes the p=1 norm. If A is a matrix then the induced norm is computed.
        Parameters:
        A - Matrix or vector.
        Returns:
        The norm.

      • normP2

        public static float normP2​(org.ejml.data.FMatrixRMaj A)
        Computes the p=2 norm. If A is a matrix then the induced norm is computed.
        Parameters:
        A - Matrix or vector.
        Returns:
        The norm.

      • fastNormP2

        public static float fastNormP2​(org.ejml.data.FMatrixRMaj A)
        Computes the p=2 norm. If A is a matrix then the induced norm is computed. This implementation is faster, but more prone to buffer overflow or underflow problems.
        Parameters:
        A - Matrix or vector.
        Returns:
        The norm.

      • normPInf

        public static float normPInf​(org.ejml.data.FMatrixRMaj A)
        Computes the p=∞ norm. If A is a matrix then the induced norm is computed.
        Parameters:
        A - Matrix or vector.
        Returns:
        The norm.

      • inducedP1

        public static float inducedP1​(org.ejml.data.FMatrixRMaj A)

        Computes the induced p = 1 matrix norm.

        ||A||1= max(j=1 to n; sum(i=1 to m; |aij|))

        Parameters:
        A - Matrix. Not modified.
        Returns:
        The norm.

      • inducedP2

        public static float inducedP2​(org.ejml.data.FMatrixRMaj A)

        Computes the induced p = 2 matrix norm, which is the largest singular value.

        Parameters:
        A - Matrix. Not modified.
        Returns:
        The norm.

      • inducedPInf

        public static float inducedPInf​(org.ejml.data.FMatrixRMaj A)

        Induced matrix p = infinity norm.

        ||A||∞ = max(i=1 to m; sum(j=1 to n; |aij|))

        Parameters:
        A - A matrix.
        Returns:
        the norm.