cern.colt.matrix.tint.algo
Class IntProperty

java.lang.Object
  cern.colt.PersistentObject
      cern.colt.matrix.tint.algo.IntProperty

All Implemented Interfaces:

Serializable, Cloneable

public class IntProperty
extends PersistentObject

Tests matrices for linear algebraic properties (equality, tridiagonality, symmetry, singularity, etc).

Note that this implementation is not synchronized.

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

Author:

Piotr Wendykier (piotr.wendykier@gmail.com)

See Also:

Serialized Form

Field Summary

static IntProperty

DEFAULT The default Property object;.

Constructor Summary

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

Method Summary

void	checkRectangular(IntMatrix2D A) Checks whether the given matrix A is rectangular.
void	checkSquare(IntMatrix2D A) Checks whether the given matrix A is square.
int	density(IntMatrix2D A) Returns the matrix's fraction of non-zero cells; A.cardinality() / A.size().
boolean	equals(IntMatrix1D A, int value) Returns whether all cells of the given matrix A are equal to the given value.
boolean	equals(IntMatrix1D A, IntMatrix1D B) Returns whether both given matrices A and B are equal.
boolean	equals(IntMatrix2D A, int value) Returns whether all cells of the given matrix A are equal to the given value.
boolean	equals(IntMatrix2D A, IntMatrix2D B) Returns whether both given matrices A and B are equal.
boolean	equals(IntMatrix3D A, int value) Returns whether all cells of the given matrix A are equal to the given value.
boolean	equals(IntMatrix3D A, IntMatrix3D B) Returns whether both given matrices A and B are equal.
void	generateNonSingular(IntMatrix2D A) Modifies the given matrix square matrix A such that it is diagonally dominant by row and column, hence non-singular, hence invertible.
boolean	isDiagonal(IntMatrix2D A) A matrix A is diagonal if A[i,j] == 0 whenever i != j.
boolean	isDiagonallyDominantByColumn(IntMatrix2D 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.
boolean	isDiagonallyDominantByRow(IntMatrix2D 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.
boolean	isIdentity(IntMatrix2D A) A matrix A is an identity matrix if A[i,i] == 1 and all other cells are zero.
boolean	isLowerBidiagonal(IntMatrix2D A) A matrix A is lower bidiagonal if A[i,j]==0 unless i==j || i==j+1.
boolean	isLowerTriangular(IntMatrix2D A) A matrix A is lower triangular if A[i,j]==0 whenever i < j.
boolean	isNonNegative(IntMatrix2D A) A matrix A is non-negative if A[i,j] >= 0 holds for all cells.
boolean	isOrthogonal(IntMatrix2D A) A square matrix A is orthogonal if A*transpose(A) = I.
boolean	isPositive(IntMatrix2D A) A matrix A is positive if A[i,j] > 0 holds for all cells.
boolean	isSkewSymmetric(IntMatrix2D A) A square matrix A is skew-symmetric if A = -transpose(A), that is A[i,j] == -A[j,i].
boolean	isSquare(IntMatrix2D A) A matrix A is square if it has the same number of rows and columns.
boolean	isStrictlyLowerTriangular(IntMatrix2D A) A matrix A is strictly lower triangular if A[i,j]==0 whenever i <= j.
boolean	isStrictlyTriangular(IntMatrix2D A) A matrix A is strictly triangular if it is triangular and its diagonal elements all equal 0.
boolean	isStrictlyUpperTriangular(IntMatrix2D A) A matrix A is strictly upper triangular if A[i,j]==0 whenever i >= j.
boolean	isSymmetric(IntMatrix2D A) A matrix A is symmetric if A = tranpose(A), that is A[i,j] == A[j,i].
boolean	isTriangular(IntMatrix2D A) A matrix A is triangular iff it is either upper or lower triangular.
boolean	isTridiagonal(IntMatrix2D A) A matrix A is tridiagonal if A[i,j]==0 whenever Math.abs(i-j) > 1.
boolean	isUnitTriangular(IntMatrix2D A) A matrix A is unit triangular if it is triangular and its diagonal elements all equal 1.
boolean	isUpperBidiagonal(IntMatrix2D A) A matrix A is upper bidiagonal if A[i,j]==0 unless i==j || i==j-1.
boolean	isUpperTriangular(IntMatrix2D A) A matrix A is upper triangular if A[i,j]==0 whenever i > j.
boolean	isZero(IntMatrix2D A) A matrix A is zero if all its cells are zero.
int	lowerBandwidth(IntMatrix2D A) The lower bandwidth of a square matrix A is the maximum i-j for which A[i,j] is nonzero and i > j.
int	semiBandwidth(IntMatrix2D A) Returns the semi-bandwidth of the given square matrix A.
String	toString(IntMatrix2D A) Returns summary information about the given matrix A.
int	upperBandwidth(IntMatrix2D A) The upper bandwidth of a square matrix A is the maximum j-i for which A[i,j] is nonzero and j > i.

Methods inherited from class cern.colt.PersistentObject

clone

Methods inherited from class java.lang.Object

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

Field Detail

DEFAULT

public static final IntProperty DEFAULT

The default Property object;.

Constructor Detail

IntProperty

public IntProperty()

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

Method Detail

checkRectangular

public void checkRectangular(IntMatrix2D A)

Checks whether the given matrix A is rectangular.

Throws:

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

checkSquare

public void checkSquare(IntMatrix2D A)

Checks whether the given matrix A is square.

Throws:

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

density

public int density(IntMatrix2D A)

Returns the matrix's fraction of non-zero cells; A.cardinality() / A.size().

equals

public boolean equals(IntMatrix1D A,
                      int 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(IntMatrix1D A,
                      IntMatrix1D 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(IntMatrix2D A,
                      int 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(IntMatrix2D A,
                      IntMatrix2D 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(IntMatrix3D A,
                      int 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(IntMatrix3D A,
                      IntMatrix3D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D A)

A matrix A is non-negative if A[i,j] >= 0 holds for all cells.

Note: Ignores tolerance.

isOrthogonal

public boolean isOrthogonal(IntMatrix2D A)

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

Throws:

IllegalArgumentException - if !isSquare(A).

isPositive

public boolean isPositive(IntMatrix2D A)

A matrix A is positive if A[i,j] > 0 holds for all cells.

Note: Ignores tolerance.

isSkewSymmetric

public boolean isSkewSymmetric(IntMatrix2D 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(IntMatrix2D A)

A matrix A is square if it has the same number of rows and columns.

isStrictlyLowerTriangular

public boolean isStrictlyLowerTriangular(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D A)

A matrix A is zero if all its cells are zero.

lowerBandwidth

public int lowerBandwidth(IntMatrix2D 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(IntMatrix2D), upperBandwidth(IntMatrix2D)

semiBandwidth

public int semiBandwidth(IntMatrix2D 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(IntMatrix2D), upperBandwidth(IntMatrix2D)

toString

public String toString(IntMatrix2D 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(IntMatrix2D 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(IntMatrix2D), lowerBandwidth(IntMatrix2D)