Property
cern.colt.matrix.linalg

Class Property

  • All Implemented Interfaces:
    Serializable, Cloneable


    public class Propertyextends PersistentObject
    Tests matrices for linear algebraic properties (equality, tridiagonality, symmetry, singularity, etc).

    Except where explicitly indicated, all methods involving equality tests (==) allow for numerical instability, to a degree specified upon instance construction and returned by method tolerance().The public static final variable DEFAULT represents a default Property object with a tolerance of 1.0E-9.The public static final variable ZERO represents a Property object with a tolerance of 0.0.The public static final variable TWELVE represents a Property object with a tolerance of 1.0E-12.As long as you are happy with these tolerances, there is no need to construct Property objects.Simply use idioms like Property.DEFAULT.equals(A,B), Property.ZERO.equals(A,B), Property.TWELVE.equals(A,B).

    To work with a different tolerance (e.g. 1.0E-15 or 1.0E-5) use the constructor and/or method setTolerance(double).Note that the public static final Property objects are immutable: Is is not possible to alter their tolerance. Any attempt to do so will throw an Exception.

    Note that this implementation is not synchronized.

    Example: equals(DoubleMatrix2D A, DoubleMatrix2D B) is defined as follows

    { some other tests not related to tolerance go here }double epsilon = tolerance();for (int row=rows; --row >= 0;) {   for (int column=columns; --column >= 0;) {      //if (!(A.getQuick(row,column) == B.getQuick(row,column))) return false;      if (Math.abs(A.getQuick(row,column) - B.getQuick(row,column)) > epsilon) return false;   }}return true;
    Here are some example properties
    matrix 4 x 4 
    0 0 0 0
    0 0 0 0
    0 0 0 0
    0 0 0 0
    4 x 4
    1 0 0 0
    0 0 0 0
    0 0 0 0
    0 0 0 1
    4 x 4
    1 1 0 0
    1 1 1 0
    0 1 1 1
    0 0 1 1
    4 x 4
    0 1 1 1
    0 1 1 1
    0 0 0 1
    0 0 0 1
    4 x 4
    0 0 0 0
    1 1 0 0
    1 1 0 0
    1 1 1 1
    4 x 4
    1 1 0 0
    0 1 1 0
    0 1 0 1
    1 0 1 1
    4 x 4
    1 1 1 0
    0 1 0 0
    1 1 0 1
    0 0 1 1
    upperBandwidth
    0
    0
    1
    3 0
    1
    2
    lowerBandwidth
    0
    0
    1
    0 3
    3
    2
    semiBandwidth
    1
    1
    2
    4 4
    4
    3
    description
    zero
    diagonal
    tridiagonal
    upper triangular lower triangular
    unstructured
    unstructured
    See Also:
    Serialized Form
    • Field Detail

      • DEFAULT

        public static final Property DEFAULT
        The default Property object; currently has tolerance()==1.0E-9.
      • ZERO

        public static final Property ZERO
        A Property object with tolerance()==0.0.
      • TWELVE

        public static final Property TWELVE
        A Property object with tolerance()==1.0E-12.
    • Constructor Detail

      • Property

        public Property(double newTolerance)
        Constructs an instance with a tolerance of Math.abs(newTolerance).
    • Method Detail

      • density

        public double density(DoubleMatrix2D A)
        Returns the matrix's fraction of non-zero cells; A.cardinality() / A.size().
      • equals

        public boolean equals(DoubleMatrix1D A,             double value)
        Returns whether all cells of the given matrix A are equal to the given value. The result is true if and only if A != null and ! (Math.abs(value - A[i]) > tolerance()) holds for all coordinates.
        Parameters:
        A - the first matrix to compare.
        value - the value to compare against.
        Returns:
        true if the matrix is equal to the value; false otherwise.
      • equals

        public boolean equals(DoubleMatrix1D A,             DoubleMatrix1D B)
        Returns whether both given matrices A and B are equal. The result is true if A==B. Otherwise, the result is true if and only if both arguments are != null, have the same size and ! (Math.abs(A[i] - B[i]) > tolerance()) holds for all indexes.
        Parameters:
        A - the first matrix to compare.
        B - the second matrix to compare.
        Returns:
        true if both matrices are equal; false otherwise.
      • equals

        public boolean equals(DoubleMatrix2D A,             double value)
        Returns whether all cells of the given matrix A are equal to the given value. The result is true if and only if A != null and ! (Math.abs(value - A[row,col]) > tolerance()) holds for all coordinates.
        Parameters:
        A - the first matrix to compare.
        value - the value to compare against.
        Returns:
        true if the matrix is equal to the value; false otherwise.
      • equals

        public boolean equals(DoubleMatrix2D A,             DoubleMatrix2D B)
        Returns whether both given matrices A and B are equal. The result is true if A==B. Otherwise, the result is true if and only if both arguments are != null, have the same number of columns and rows and ! (Math.abs(A[row,col] - B[row,col]) > tolerance()) holds for all coordinates.
        Parameters:
        A - the first matrix to compare.
        B - the second matrix to compare.
        Returns:
        true if both matrices are equal; false otherwise.
      • equals

        public boolean equals(DoubleMatrix3D A,             double value)
        Returns whether all cells of the given matrix A are equal to the given value. The result is true if and only if A != null and ! (Math.abs(value - A[slice,row,col]) > tolerance()) holds for all coordinates.
        Parameters:
        A - the first matrix to compare.
        value - the value to compare against.
        Returns:
        true if the matrix is equal to the value; false otherwise.
      • equals

        public boolean equals(DoubleMatrix3D A,             DoubleMatrix3D B)
        Returns whether both given matrices A and B are equal. The result is true if A==B. Otherwise, the result is true if and only if both arguments are != null, have the same number of columns, rows and slices, and ! (Math.abs(A[slice,row,col] - B[slice,row,col]) > tolerance()) holds for all coordinates.
        Parameters:
        A - the first matrix to compare.
        B - the second matrix to compare.
        Returns:
        true if both matrices are equal; false otherwise.
      • generateNonSingular

        public void generateNonSingular(DoubleMatrix2D A)
        Modifies the given matrix square matrix A such that it is diagonally dominant by row and column, hence non-singular, hence invertible.For testing purposes only.
        Parameters:
        A - the square matrix to modify.
        Throws:
        IllegalArgumentException - if !isSquare(A).
      • isDiagonal

        public boolean isDiagonal(DoubleMatrix2D A)
        A matrix A is diagonal if A[i,j] == 0 whenever i != j. Matrix may but need not be square.
      • isDiagonallyDominantByColumn

        public boolean isDiagonallyDominantByColumn(DoubleMatrix2D A)
        A matrix A is diagonally dominant by column if the absolute value of each diagonal element is larger than the sum of the absolute values of the off-diagonal elements in the corresponding column. returns true if for all i: abs(A[i,i]) > Sum(abs(A[j,i])); j != i. Matrix may but need not be square.

        Note: Ignores tolerance.

      • isDiagonallyDominantByRow

        public boolean isDiagonallyDominantByRow(DoubleMatrix2D A)
        A matrix A is diagonally dominant by row if the absolute value of each diagonal element is larger than the sum of the absolute values of the off-diagonal elements in the corresponding row. returns true if for all i: abs(A[i,i]) > Sum(abs(A[i,j])); j != i. Matrix may but need not be square.

        Note: Ignores tolerance.

      • isIdentity

        public boolean isIdentity(DoubleMatrix2D A)
        A matrix A is an identity matrix if A[i,i] == 1 and all other cells are zero. Matrix may but need not be square.
      • isLowerBidiagonal

        public boolean isLowerBidiagonal(DoubleMatrix2D A)
        A matrix A is lower bidiagonal if A[i,j]==0 unless i==j || i==j+1. Matrix may but need not be square.
      • isLowerTriangular

        public boolean isLowerTriangular(DoubleMatrix2D A)
        A matrix A is lower triangular if A[i,j]==0 whenever i < j. Matrix may but need not be square.
      • isNonNegative

        public boolean isNonNegative(DoubleMatrix2D A)
        A matrix A is non-negative if A[i,j] >= 0 holds for all cells.

        Note: Ignores tolerance.

      • isPositive

        public boolean isPositive(DoubleMatrix2D A)
        A matrix A is positive if A[i,j] > 0 holds for all cells.

        Note: Ignores tolerance.

      • isSingular

        public boolean isSingular(DoubleMatrix2D A)
        A matrix A is singular if it has no inverse, that is, iff det(A)==0.
      • isSkewSymmetric

        public boolean isSkewSymmetric(DoubleMatrix2D A)
        A square matrix A is skew-symmetric if A = -transpose(A), that is A[i,j] == -A[j,i].
        Throws:
        IllegalArgumentException - if !isSquare(A).
      • isSquare

        public boolean isSquare(DoubleMatrix2D A)
        A matrix A is square if it has the same number of rows and columns.
      • isStrictlyLowerTriangular

        public boolean isStrictlyLowerTriangular(DoubleMatrix2D A)
        A matrix A is strictly lower triangular if A[i,j]==0 whenever i <= j. Matrix may but need not be square.
      • isStrictlyTriangular

        public boolean isStrictlyTriangular(DoubleMatrix2D A)
        A matrix A is strictly triangular if it is triangular and its diagonal elements all equal 0. Matrix may but need not be square.
      • isStrictlyUpperTriangular

        public boolean isStrictlyUpperTriangular(DoubleMatrix2D A)
        A matrix A is strictly upper triangular if A[i,j]==0 whenever i >= j. Matrix may but need not be square.
      • isTriangular

        public boolean isTriangular(DoubleMatrix2D A)
        A matrix A is triangular iff it is either upper or lower triangular. Matrix may but need not be square.
      • isTridiagonal

        public boolean isTridiagonal(DoubleMatrix2D A)
        A matrix A is tridiagonal if A[i,j]==0 whenever Math.abs(i-j) > 1. Matrix may but need not be square.
      • isUnitTriangular

        public boolean isUnitTriangular(DoubleMatrix2D A)
        A matrix A is unit triangular if it is triangular and its diagonal elements all equal 1. Matrix may but need not be square.
      • isUpperBidiagonal

        public boolean isUpperBidiagonal(DoubleMatrix2D A)
        A matrix A is upper bidiagonal if A[i,j]==0 unless i==j || i==j-1. Matrix may but need not be square.
      • isUpperTriangular

        public boolean isUpperTriangular(DoubleMatrix2D A)
        A matrix A is upper triangular if A[i,j]==0 whenever i > j. Matrix may but need not be square.
      • isZero

        public boolean isZero(DoubleMatrix2D A)
        A matrix A is zero if all its cells are zero.
      • semiBandwidth

        public int semiBandwidth(DoubleMatrix2D A)
        Returns the semi-bandwidth of the given square matrix A.A banded matrix has a "band" about the diagonal.It is a matrix with all cells equal to zero, with the possible exception of the cells along the diagonal line,the k diagonal lines above the diagonal, and the k diagonal lines below the diagonal.The semi-bandwith l is the number k+1.The bandwidth p is the number 2*k + 1.For example, a tridiagonal matrix corresponds to k=1, l=2, p=3, a diagonal or zero matrix corresponds to k=0, l=1, p=1,

        The upper bandwidth is the maximum j-i for which A[i,j] is nonzero and j > i.The lower bandwidth is the maximum i-j for which A[i,j] is nonzero and i > j. Diagonal, tridiagonal and triangular matrices are special cases.

        Examples:

        matrix 4 x 4 
        0 0 0 0
        0 0 0 0
        0 0 0 0
        0 0 0 0
        4 x 4
        1 0 0 0
        0 0 0 0
        0 0 0 0
        0 0 0 1
        4 x 4
        1 1 0 0
        1 1 1 0
        0 1 1 1
        0 0 1 1
        4 x 4
        0 1 1 1
        0 1 1 1
        0 0 0 1
        0 0 0 1
        4 x 4
        0 0 0 0
        1 1 0 0
        1 1 0 0
        1 1 1 1
        4 x 4
        1 1 0 0
        0 1 1 0
        0 1 0 1
        1 0 1 1
        4 x 4
        1 1 1 0
        0 1 0 0
        1 1 0 1
        0 0 1 1
        upperBandwidth
        0
        0
        1
        3 0
        1
        2
        lowerBandwidth
        0
        0
        1
        0 3
        3
        2
        semiBandwidth
        1
        1
        2
        4 4
        4
        3
        description
        zero
        diagonal
        tridiagonal
        upper triangular lower triangular
        unstructured
        unstructured
        Parameters:
        A - the square matrix to analyze.
        Returns:
        the semi-bandwith l.
        Throws:
        IllegalArgumentException - if !isSquare(A).
        See Also:
        lowerBandwidth(DoubleMatrix2D), upperBandwidth(DoubleMatrix2D)
      • setTolerance

        public void setTolerance(double newTolerance)
        Sets the tolerance to Math.abs(newTolerance).
        Throws:
        UnsupportedOperationException - if this==DEFAULT || this==ZERO || this==TWELVE.
      • tolerance

        public double tolerance()
        Returns the current tolerance.
      • toString

        public String toString(DoubleMatrix2D A)
        Returns summary information about the given matrix A.That is a String with (propertyName, propertyValue) pairs.Useful for debugging or to quickly get the rough picture of a matrix.For example,
        density                      : 0.9isDiagonal                   : falseisDiagonallyDominantByRow    : falseisDiagonallyDominantByColumn : falseisIdentity                   : falseisLowerBidiagonal            : falseisLowerTriangular            : falseisNonNegative                : trueisOrthogonal                 : Illegal operation or error: Matrix must be square.isPositive                   : trueisSingular                   : Illegal operation or error: Matrix must be square.isSkewSymmetric              : Illegal operation or error: Matrix must be square.isSquare                     : falseisStrictlyLowerTriangular    : falseisStrictlyTriangular         : falseisStrictlyUpperTriangular    : falseisSymmetric                  : Illegal operation or error: Matrix must be square.isTriangular                 : falseisTridiagonal                : falseisUnitTriangular             : falseisUpperBidiagonal            : falseisUpperTriangular            : falseisZero                       : falselowerBandwidth               : Illegal operation or error: Matrix must be square.semiBandwidth                : Illegal operation or error: Matrix must be square.upperBandwidth               : Illegal operation or error: Matrix must be square.

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