cern.colt.matrix.linalg
Class Property

java.lang.Object
  cern.colt.PersistentObject
      cern.colt.matrix.linalg.Property

All Implemented Interfaces:

Cloneable, Serializable

public class Property

extends 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;

Version:

1.1, 28/May/2000 (fixed strange bugs involving NaN, -inf, inf)

See Also:

Serialized Form

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.

Property

public Property(double newTolerance)

Constructs an instance with a tolerance of Math.abs(newTolerance).

checkRectangular

public void checkRectangular(DoubleMatrix2D A)

Checks whether the given matrix A is rectangular.

Throws:

IllegalArgumentException - if A.rows() < A.columns().

checkSquare

public void checkSquare(DoubleMatrix2D A)

Checks whether the given matrix A is square.

Throws:

IllegalArgumentException - if A.rows() != A.columns().

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.

isOrthogonal

public boolean isOrthogonal(DoubleMatrix2D A)

A square matrix A is orthogonal if A*transpose(A) = I.

Throws:

IllegalArgumentException - if !isSquare(A).

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.

isSymmetric

public boolean isSymmetric(DoubleMatrix2D A)

A matrix A is symmetric if A = tranpose(A), that is A[i,j] == A[j,i].

Throws:

IllegalArgumentException - if !isSquare(A).

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.

lowerBandwidth

public int lowerBandwidth(DoubleMatrix2D A)

The lower bandwidth of a square matrix A is the maximum i-j for which A[i,j] is nonzero and i > j. A banded matrix has a "band" about the diagonal. Diagonal, tridiagonal and triangular matrices are special cases.

Parameters:

A - the square matrix to analyze.

Returns:

the lower bandwith.

Throws:

IllegalArgumentException - if !isSquare(A).

See Also:

semiBandwidth(DoubleMatrix2D), upperBandwidth(DoubleMatrix2D)

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                      : 0.9
isDiagonal                   : false
isDiagonallyDominantByRow    : false
isDiagonallyDominantByColumn : false
isIdentity                   : false
isLowerBidiagonal            : false
isLowerTriangular            : false
isNonNegative                : true
isOrthogonal                 : Illegal operation or error: Matrix must be square.
isPositive                   : true
isSingular                   : Illegal operation or error: Matrix must be square.
isSkewSymmetric              : Illegal operation or error: Matrix must be square.
isSquare                     : false
isStrictlyLowerTriangular    : false
isStrictlyTriangular         : false
isStrictlyUpperTriangular    : false
isSymmetric                  : Illegal operation or error: Matrix must be square.
isTriangular                 : false
isTridiagonal                : false
isUnitTriangular             : false
isUpperBidiagonal            : false
isUpperTriangular            : false
isZero                       : false
lowerBandwidth               : 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.

upperBandwidth

public int upperBandwidth(DoubleMatrix2D A)

The upper bandwidth of a square matrix A is the maximum j-i for which A[i,j] is nonzero and j > i. A banded matrix has a "band" about the diagonal. Diagonal, tridiagonal and triangular matrices are special cases.

Parameters:

A - the square matrix to analyze.

Returns:

the upper bandwith.

Throws:

IllegalArgumentException - if !isSquare(A).

See Also:

semiBandwidth(DoubleMatrix2D), lowerBandwidth(DoubleMatrix2D)