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java.lang.Objectcern.colt.PersistentObject
cern.colt.matrix.linalg.Property
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;
 | 
| 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 | 
| Field Summary | |
| static Property | DEFAULTThe default Property object; currently has tolerance()==1.0E-9. | 
| static Property | TWELVEA Property object with tolerance()==1.0E-12. | 
| static Property | ZEROA Property object with tolerance()==0.0. | 
| Fields inherited from class cern.colt.PersistentObject | 
| serialVersionUID | 
| Constructor Summary | |
| Property(double newTolerance)Constructs an instance with a tolerance of Math.abs(newTolerance). | |
| Method Summary | |
|  void | checkRectangular(DoubleMatrix2D A)Checks whether the given matrix A is rectangular. | 
|  void | checkSquare(DoubleMatrix2D A)Checks whether the given matrix A is square. | 
|  double | density(DoubleMatrix2D A)Returns the matrix's fraction of non-zero cells; A.cardinality() / A.size(). | 
|  boolean | equals(DoubleMatrix1D A,
       double value)Returns whether all cells of the given matrix A are equal to the given value. | 
|  boolean | equals(DoubleMatrix1D A,
       DoubleMatrix1D B)Returns whether both given matrices A and B are equal. | 
|  boolean | equals(DoubleMatrix2D A,
       double value)Returns whether all cells of the given matrix A are equal to the given value. | 
|  boolean | equals(DoubleMatrix2D A,
       DoubleMatrix2D B)Returns whether both given matrices A and B are equal. | 
|  boolean | equals(DoubleMatrix3D A,
       double value)Returns whether all cells of the given matrix A are equal to the given value. | 
|  boolean | equals(DoubleMatrix3D A,
       DoubleMatrix3D B)Returns whether both given matrices A and B are equal. | 
|  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. | 
|  boolean | isDiagonal(DoubleMatrix2D A)A matrix A is diagonal if A[i,j] == 0 whenever i != j. | 
|  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. | 
|  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. | 
|  boolean | isIdentity(DoubleMatrix2D A)A matrix A is an identity matrix if A[i,i] == 1 and all other cells are zero. | 
|  boolean | isLowerBidiagonal(DoubleMatrix2D A)A matrix A is lower bidiagonal if A[i,j]==0 unless i==j || i==j+1. | 
|  boolean | isLowerTriangular(DoubleMatrix2D A)A matrix A is lower triangular if A[i,j]==0 whenever i < j. | 
|  boolean | isNonNegative(DoubleMatrix2D A)A matrix A is non-negative if A[i,j] >= 0 holds for all cells. | 
|  boolean | isOrthogonal(DoubleMatrix2D A)A square matrix A is orthogonal if A*transpose(A) = I. | 
|  boolean | isPositive(DoubleMatrix2D A)A matrix A is positive if A[i,j] > 0 holds for all cells. | 
|  boolean | isSingular(DoubleMatrix2D A)A matrix A is singular if it has no inverse, that is, iff det(A)==0. | 
|  boolean | isSkewSymmetric(DoubleMatrix2D A)A square matrix A is skew-symmetric if A = -transpose(A), that is A[i,j] == -A[j,i]. | 
|  boolean | isSquare(DoubleMatrix2D A)A matrix A is square if it has the same number of rows and columns. | 
|  boolean | isStrictlyLowerTriangular(DoubleMatrix2D A)A matrix A is strictly lower triangular if A[i,j]==0 whenever i <= j. | 
|  boolean | isStrictlyTriangular(DoubleMatrix2D A)A matrix A is strictly triangular if it is triangular and its diagonal elements all equal 0. | 
|  boolean | isStrictlyUpperTriangular(DoubleMatrix2D A)A matrix A is strictly upper triangular if A[i,j]==0 whenever i >= j. | 
|  boolean | isSymmetric(DoubleMatrix2D A)A matrix A is symmetric if A = tranpose(A), that is A[i,j] == A[j,i]. | 
|  boolean | isTriangular(DoubleMatrix2D A)A matrix A is triangular iff it is either upper or lower triangular. | 
|  boolean | isTridiagonal(DoubleMatrix2D A)A matrix A is tridiagonal if A[i,j]==0 whenever Math.abs(i-j) > 1. | 
|  boolean | isUnitTriangular(DoubleMatrix2D A)A matrix A is unit triangular if it is triangular and its diagonal elements all equal 1. | 
|  boolean | isUpperBidiagonal(DoubleMatrix2D A)A matrix A is upper bidiagonal if A[i,j]==0 unless i==j || i==j-1. | 
|  boolean | isUpperTriangular(DoubleMatrix2D A)A matrix A is upper triangular if A[i,j]==0 whenever i > j. | 
|  boolean | isZero(DoubleMatrix2D A)A matrix A is zero if all its cells are zero. | 
|  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. | 
|  int | semiBandwidth(DoubleMatrix2D A)Returns the semi-bandwidth of the given square matrix A. | 
|  void | setTolerance(double newTolerance)Sets the tolerance to Math.abs(newTolerance). | 
|  double | tolerance()Returns the current tolerance. | 
|  String | toString(DoubleMatrix2D A)Returns summary information about the given matrix A. | 
|  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. | 
| 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 | 
public static final Property DEFAULT
public static final Property ZERO
public static final Property TWELVE
| Constructor Detail | 
public Property(double newTolerance)
| Method Detail | 
public void checkRectangular(DoubleMatrix2D A)
IllegalArgumentException - if A.rows() < A.columns().public void checkSquare(DoubleMatrix2D A)
IllegalArgumentException - if A.rows() != A.columns().public double density(DoubleMatrix2D A)
public boolean equals(DoubleMatrix1D A,
                      double value)
A - the first matrix to compare.value - the value to compare against.
public boolean equals(DoubleMatrix1D A,
                      DoubleMatrix1D B)
A - the first matrix to compare.B - the second matrix to compare.
public boolean equals(DoubleMatrix2D A,
                      double value)
A - the first matrix to compare.value - the value to compare against.
public boolean equals(DoubleMatrix2D A,
                      DoubleMatrix2D B)
A - the first matrix to compare.B - the second matrix to compare.
public boolean equals(DoubleMatrix3D A,
                      double value)
A - the first matrix to compare.value - the value to compare against.
public boolean equals(DoubleMatrix3D A,
                      DoubleMatrix3D B)
A - the first matrix to compare.B - the second matrix to compare.
public void generateNonSingular(DoubleMatrix2D A)
A - the square matrix to modify.
IllegalArgumentException - if !isSquare(A).public boolean isDiagonal(DoubleMatrix2D A)
public boolean isDiagonallyDominantByColumn(DoubleMatrix2D A)
Note: Ignores tolerance.
public boolean isDiagonallyDominantByRow(DoubleMatrix2D A)
Note: Ignores tolerance.
public boolean isIdentity(DoubleMatrix2D A)
public boolean isLowerBidiagonal(DoubleMatrix2D A)
public boolean isLowerTriangular(DoubleMatrix2D A)
public boolean isNonNegative(DoubleMatrix2D A)
Note: Ignores tolerance.
public boolean isOrthogonal(DoubleMatrix2D A)
IllegalArgumentException - if !isSquare(A).public boolean isPositive(DoubleMatrix2D A)
Note: Ignores tolerance.
public boolean isSingular(DoubleMatrix2D A)
public boolean isSkewSymmetric(DoubleMatrix2D A)
IllegalArgumentException - if !isSquare(A).public boolean isSquare(DoubleMatrix2D A)
public boolean isStrictlyLowerTriangular(DoubleMatrix2D A)
public boolean isStrictlyTriangular(DoubleMatrix2D A)
public boolean isStrictlyUpperTriangular(DoubleMatrix2D A)
public boolean isSymmetric(DoubleMatrix2D A)
IllegalArgumentException - if !isSquare(A).public boolean isTriangular(DoubleMatrix2D A)
public boolean isTridiagonal(DoubleMatrix2D A)
public boolean isUnitTriangular(DoubleMatrix2D A)
public boolean isUpperBidiagonal(DoubleMatrix2D A)
public boolean isUpperTriangular(DoubleMatrix2D A)
public boolean isZero(DoubleMatrix2D A)
public int lowerBandwidth(DoubleMatrix2D A)
A - the square matrix to analyze.
IllegalArgumentException - if !isSquare(A).semiBandwidth(DoubleMatrix2D), 
upperBandwidth(DoubleMatrix2D)public int semiBandwidth(DoubleMatrix2D A)
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 | 
A - the square matrix to analyze.
IllegalArgumentException - if !isSquare(A).lowerBandwidth(DoubleMatrix2D), 
upperBandwidth(DoubleMatrix2D)public void setTolerance(double newTolerance)
UnsupportedOperationException - if this==DEFAULT || this==ZERO || this==TWELVE.public double tolerance()
public String toString(DoubleMatrix2D A)
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.
public int upperBandwidth(DoubleMatrix2D A)
A - the square matrix to analyze.
IllegalArgumentException - if !isSquare(A).semiBandwidth(DoubleMatrix2D), 
lowerBandwidth(DoubleMatrix2D)| 
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