Package org.ejml.dense.row

Class NormOps_DDRM

java.lang.Object
org.ejml.dense.row.NormOps_DDRM
public class NormOps_DDRM extends 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.

  • Method Details

    • normalizeF

      public static void normalizeF(DMatrixRMaj 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 double conditionP(DMatrixRMaj A, double 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 double conditionP2(DMatrixRMaj 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 double fastNormF(DMatrixD1 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.DMatrixD1).

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

    • normF

      public static double normF(DMatrixD1 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.DMatrixD1) 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 double elementP(DMatrix1Row A, double 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 double fastElementP(DMatrixD1 A, double p)
      Same as elementP(org.ejml.data.DMatrix1Row, double) 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 double normP(DMatrixRMaj A, double 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 double fastNormP(DMatrixRMaj A, double p)
      An unsafe but faster version of normP(org.ejml.data.DMatrixRMaj, double) 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 double normP1(DMatrixRMaj 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 double normP2(DMatrixRMaj 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 double fastNormP2(DMatrixRMaj 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 double normPInf(DMatrixRMaj 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 double inducedP1(DMatrixRMaj 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 double inducedP2(DMatrixRMaj 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 double inducedPInf(DMatrixRMaj 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.