/*
Copyright © 1999 CERN - European Organization for Nuclear Research.
Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose 
is hereby granted without fee, provided that the above copyright notice appear in all copies and 
that both that copyright notice and this permission notice appear in supporting documentation. 
CERN makes no representations about the suitability of this software for any purpose. 
It is provided "as is" without expressed or implied warranty.
*/
package cern.colt.matrix.linalg;

import cern.colt.matrix.DoubleMatrix1D;
import cern.colt.matrix.DoubleMatrix2D;
import cern.colt.matrix.DoubleMatrix3D;
/**
Tests matrices for linear algebraic properties (equality, tridiagonality, symmetry, singularity, etc).
<p>
Except where explicitly indicated, all methods involving equality tests (<tt>==</tt>) allow for numerical instability, to a degree specified upon instance construction and returned by method {@link #tolerance()}.
The public static final variable <tt>DEFAULT</tt> represents a default Property object with a tolerance of <tt>1.0E-9</tt>.
The public static final variable <tt>ZERO</tt> represents a Property object with a tolerance of <tt>0.0</tt>.
The public static final variable <tt>TWELVE</tt> represents a Property object with a tolerance of <tt>1.0E-12</tt>.
As long as you are happy with these tolerances, there is no need to construct Property objects.
Simply use idioms like <tt>Property.DEFAULT.equals(A,B)</tt>, <tt>Property.ZERO.equals(A,B)</tt>, <tt>Property.TWELVE.equals(A,B)</tt>.
<p>
To work with a different tolerance (e.g. <tt>1.0E-15</tt> or <tt>1.0E-5</tt>) use the constructor and/or method {@link #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.
<p>
Note that this implementation is not synchronized.
<p>
Example: <tt>equals(DoubleMatrix2D A, DoubleMatrix2D B)</tt> is defined as follows
<table>
<td class="PRE"> 
<pre>
{ some other tests not related to tolerance go here }
double epsilon = tolerance();
for (int row=rows; --row >= 0;) {
&nbsp;&nbsp;&nbsp;for (int column=columns; --column >= 0;) {
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;//if (!(A.getQuick(row,column) == B.getQuick(row,column))) return false;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if (Math.abs(A.getQuick(row,column) - B.getQuick(row,column)) > epsilon) return false;
&nbsp;&nbsp;&nbsp;}
}
return true;
</pre>
</td>
</table>
Here are some example properties
<table border="1" cellspacing="0">
  <tr align="left" valign="top"> 
	<td valign="middle" align="left"><tt>matrix</tt></td>
	<td> <tt>4&nbsp;x&nbsp;4&nbsp;<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;0 </tt></td>
	<td><tt>4&nbsp;x&nbsp;4<br>
	  1&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;1 </tt></td>
	<td><tt>4&nbsp;x&nbsp;4<br>
	  1&nbsp;1&nbsp;0&nbsp;0<br>
	  1&nbsp;1&nbsp;1&nbsp;0<br>
	  0&nbsp;1&nbsp;1&nbsp;1<br>
	  0&nbsp;0&nbsp;1&nbsp;1 </tt></td>
	<td><tt> 4&nbsp;x&nbsp;4<br>
	  0&nbsp;1&nbsp;1&nbsp;1<br>
	  0&nbsp;1&nbsp;1&nbsp;1<br>
	  0&nbsp;0&nbsp;0&nbsp;1<br>
	  0&nbsp;0&nbsp;0&nbsp;1 </tt></td>
	<td><tt> 4&nbsp;x&nbsp;4<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  1&nbsp;1&nbsp;0&nbsp;0<br>
	  1&nbsp;1&nbsp;0&nbsp;0<br>
	  1&nbsp;1&nbsp;1&nbsp;1 </tt></td>
	<td><tt>4&nbsp;x&nbsp;4<br>
	  1&nbsp;1&nbsp;0&nbsp;0<br>
	  0&nbsp;1&nbsp;1&nbsp;0<br>
	  0&nbsp;1&nbsp;0&nbsp;1<br>
	  1&nbsp;0&nbsp;1&nbsp;1 </tt><tt> </tt> </td>
	<td><tt>4&nbsp;x&nbsp;4<br>
	  1&nbsp;1&nbsp;1&nbsp;0<br>
	  0&nbsp;1&nbsp;0&nbsp;0<br>
	  1&nbsp;1&nbsp;0&nbsp;1<br>
	  0&nbsp;0&nbsp;1&nbsp;1 </tt> </td>
  </tr>
  <tr align="center" valign="middle"> 
	<td><tt>upperBandwidth</tt></td>
	<td> 
	  <div align="center"><tt>0</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>0</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>1</tt></div>
	</td>
	<td><tt>3</tt></td>
	<td align="center" valign="middle"><tt>0</tt></td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>1</tt></div>
	</td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>2</tt></div>
	</td>
  </tr>
  <tr align="center" valign="middle"> 
	<td><tt>lowerBandwidth</tt></td>
	<td> 
	  <div align="center"><tt>0</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>0</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>1</tt></div>
	</td>
	<td><tt>0</tt></td>
	<td align="center" valign="middle"><tt>3</tt></td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>3</tt></div>
	</td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>2</tt></div>
	</td>
  </tr>
  <tr align="center" valign="middle"> 
	<td><tt>semiBandwidth</tt></td>
	<td> 
	  <div align="center"><tt>1</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>1</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>2</tt></div>
	</td>
	<td><tt>4</tt></td>
	<td align="center" valign="middle"><tt>4</tt></td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>4</tt></div>
	</td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>3</tt></div>
	</td>
  </tr>
  <tr align="center" valign="middle"> 
	<td><tt>description</tt></td>
	<td> 
	  <div align="center"><tt>zero</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>diagonal</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>tridiagonal</tt></div>
	</td>
	<td><tt>upper triangular</tt></td>
	<td align="center" valign="middle"><tt>lower triangular</tt></td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>unstructured</tt></div>
	</td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>unstructured</tt></div>
	</td>
  </tr>
</table>

@author wolfgang.hoschek@cern.ch
@version 1.1, 28/May/2000 (fixed strange bugs involving NaN, -inf, inf)
*/
public class Property extends cern.colt.PersistentObject {
	/**
	 * The default Property object; currently has <tt>tolerance()==1.0E-9</tt>.
	 */
	public static final Property DEFAULT = new Property(1.0E-9);
	
	/**
	 * A Property object with <tt>tolerance()==0.0</tt>.
	 */
	public static final Property ZERO = new Property(0.0);
	
	/**
	 * A Property object with <tt>tolerance()==1.0E-12</tt>.
	 */
	public static final Property TWELVE = new Property(1.0E-12);
	
	protected double tolerance;
/**
 * Not instantiable by no-arg constructor.
 */
private Property() {
	this(1.0E-9); // just to be on the safe side
}
/**
 * Constructs an instance with a tolerance of <tt>Math.abs(newTolerance)</tt>.
 */
public Property(double newTolerance) {
	tolerance = Math.abs(newTolerance);
}
/**
 * Returns a String with <tt>length</tt> blanks.
 */
protected static String blanks(int length) {
	if (length <0 ) length = 0;
	StringBuffer buf = new StringBuffer(length);
	for (int k = 0; k < length; k++) {
		buf.append(' ');
	}
	return buf.toString();
}
/**
 * Checks whether the given matrix <tt>A</tt> is <i>rectangular</i>.
 * @throws IllegalArgumentException if <tt>A.rows() < A.columns()</tt>.
 */
public void checkRectangular(DoubleMatrix2D A) {
	if (A.rows() < A.columns()) {
		throw new IllegalArgumentException("Matrix must be rectangular: "+cern.colt.matrix.doublealgo.Formatter.shape(A));
	}
}
/**
 * Checks whether the given matrix <tt>A</tt> is <i>square</i>.
 * @throws IllegalArgumentException if <tt>A.rows() != A.columns()</tt>.
 */
public void checkSquare(DoubleMatrix2D A) {
	if (A.rows() != A.columns()) throw new IllegalArgumentException("Matrix must be square: "+cern.colt.matrix.doublealgo.Formatter.shape(A));
}
/**
 * Returns the matrix's fraction of non-zero cells; <tt>A.cardinality() / A.size()</tt>.
 */
public double density(DoubleMatrix2D A) {
	return A.cardinality() / (double) A.size();
}
/**
 * Returns whether all cells of the given matrix <tt>A</tt> are equal to the given value.
 * The result is <tt>true</tt> if and only if <tt>A != null</tt> and
 * <tt>! (Math.abs(value - A[i]) > tolerance())</tt> holds for all coordinates.
 * @param   A   the first matrix to compare.
 * @param   value   the value to compare against.
 * @return  <tt>true</tt> if the matrix is equal to the value;
 *          <tt>false</tt> otherwise.
 */
public boolean equals(DoubleMatrix1D A, double value) {
	if (A==null) return false;
	double epsilon = tolerance();
	for (int i = A.size(); --i >= 0;) {
		//if (!(A.getQuick(i) == value)) return false;
		//if (Math.abs(value - A.getQuick(i)) > epsilon) return false;
		double x = A.getQuick(i);
		double diff = Math.abs(value - x);
		if ((diff!=diff) && ((value!=value && x!=x) || value==x)) diff = 0;
		if (!(diff <= epsilon)) return false;
	}
	return true;
}
/**
 * Returns whether both given matrices <tt>A</tt> and <tt>B</tt> are equal.
 * The result is <tt>true</tt> if <tt>A==B</tt>. 
 * Otherwise, the result is <tt>true</tt> if and only if both arguments are <tt>!= null</tt>, 
 * have the same size and 
 * <tt>! (Math.abs(A[i] - B[i]) > tolerance())</tt> holds for all indexes.
 * @param   A   the first matrix to compare.
 * @param   B   the second matrix to compare.
 * @return  <tt>true</tt> if both matrices are equal;
 *          <tt>false</tt> otherwise.
 */
public boolean equals(DoubleMatrix1D A, DoubleMatrix1D B) {
	if (A==B) return true;
	if (! (A != null && B != null)) return false;
	int size = A.size();
	if (size != B.size()) return false;

	double epsilon = tolerance();
	for (int i=size; --i >= 0;) {
		//if (!(getQuick(i) == B.getQuick(i))) return false;
		//if (Math.abs(A.getQuick(i) - B.getQuick(i)) > epsilon) return false;
		double x = A.getQuick(i);
		double value = B.getQuick(i);
		double diff = Math.abs(value - x);
		if ((diff!=diff) && ((value!=value && x!=x) || value==x)) diff = 0;
		if (!(diff <= epsilon)) return false;
	}
	return true;
}
/**
 * Returns whether all cells of the given matrix <tt>A</tt> are equal to the given value.
 * The result is <tt>true</tt> if and only if <tt>A != null</tt> and
 * <tt>! (Math.abs(value - A[row,col]) > tolerance())</tt> holds for all coordinates.
 * @param   A   the first matrix to compare.
 * @param   value   the value to compare against.
 * @return  <tt>true</tt> if the matrix is equal to the value;
 *          <tt>false</tt> otherwise.
 */
public boolean equals(DoubleMatrix2D A, double value) {
	if (A==null) return false;
	int rows = A.rows();
	int columns = A.columns();

	double epsilon = tolerance();
	for (int row=rows; --row >= 0;) {
		for (int column=columns; --column >= 0;) {
			//if (!(A.getQuick(row,column) == value)) return false;
			//if (Math.abs(value - A.getQuick(row,column)) > epsilon) return false;
			double x = A.getQuick(row,column);
			double diff = Math.abs(value - x);
			if ((diff!=diff) && ((value!=value && x!=x) || value==x)) diff = 0;
			if (!(diff <= epsilon)) return false;
		}
	}
	return true;
}
/**
 * Returns whether both given matrices <tt>A</tt> and <tt>B</tt> are equal.
 * The result is <tt>true</tt> if <tt>A==B</tt>. 
 * Otherwise, the result is <tt>true</tt> if and only if both arguments are <tt>!= null</tt>, 
 * have the same number of columns and rows and 
 * <tt>! (Math.abs(A[row,col] - B[row,col]) > tolerance())</tt> holds for all coordinates.
 * @param   A   the first matrix to compare.
 * @param   B   the second matrix to compare.
 * @return  <tt>true</tt> if both matrices are equal;
 *          <tt>false</tt> otherwise.
 */
public boolean equals(DoubleMatrix2D A, DoubleMatrix2D B) {
	if (A==B) return true;
	if (! (A != null && B != null)) return false;
	int rows = A.rows();
	int columns = A.columns();
	if (columns != B.columns() || rows != B.rows()) return false;

	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;
			double x = A.getQuick(row,column);
			double value = B.getQuick(row,column);
			double diff = Math.abs(value - x);
			if ((diff!=diff) && ((value!=value && x!=x) || value==x)) diff = 0;
			if (!(diff <= epsilon)) return false;
		}
	}
	return true;
}
/**
 * Returns whether all cells of the given matrix <tt>A</tt> are equal to the given value.
 * The result is <tt>true</tt> if and only if <tt>A != null</tt> and
 * <tt>! (Math.abs(value - A[slice,row,col]) > tolerance())</tt> holds for all coordinates.
 * @param   A   the first matrix to compare.
 * @param   value   the value to compare against.
 * @return  <tt>true</tt> if the matrix is equal to the value;
 *          <tt>false</tt> otherwise.
 */
public boolean equals(DoubleMatrix3D A, double value) {
	if (A==null) return false;
	int rows = A.rows();
	int columns = A.columns();

	double epsilon = tolerance();
	for (int slice=A.slices(); --slice >= 0;) {
		for (int row=rows; --row >= 0;) {
			for (int column=columns; --column >= 0;) {
				//if (!(A.getQuick(slice,row,column) == value)) return false;
				//if (Math.abs(value - A.getQuick(slice,row,column)) > epsilon) return false;
				double x = A.getQuick(slice,row,column);
				double diff = Math.abs(value - x);
				if ((diff!=diff) && ((value!=value && x!=x) || value==x)) diff = 0;
				if (!(diff <= epsilon)) return false;
			}
		}
	}
	return true;
}
/**
 * Returns whether both given matrices <tt>A</tt> and <tt>B</tt> are equal.
 * The result is <tt>true</tt> if <tt>A==B</tt>. 
 * Otherwise, the result is <tt>true</tt> if and only if both arguments are <tt>!= null</tt>, 
 * have the same number of columns, rows and slices, and
 * <tt>! (Math.abs(A[slice,row,col] - B[slice,row,col]) > tolerance())</tt> holds for all coordinates.
 * @param   A   the first matrix to compare.
 * @param   B   the second matrix to compare.
 * @return  <tt>true</tt> if both matrices are equal;
 *          <tt>false</tt> otherwise.
 */
public boolean equals(DoubleMatrix3D A, DoubleMatrix3D B) {
	if (A==B) return true;
	if (! (A != null && B != null)) return false;
	int slices = A.slices();
	int rows = A.rows();
	int columns = A.columns();
	if (columns != B.columns() || rows != B.rows() || slices != B.slices()) return false;

	double epsilon = tolerance();
	for (int slice=slices; --slice >= 0;) {
		for (int row=rows; --row >= 0;) {
			for (int column=columns; --column >= 0;) {
				//if (!(A.getQuick(slice,row,column) == B.getQuick(slice,row,column))) return false;
				//if (Math.abs(A.getQuick(slice,row,column) - B.getQuick(slice,row,column)) > epsilon) return false;
				double x = A.getQuick(slice,row,column);
				double value = B.getQuick(slice,row,column);
				double diff = Math.abs(value - x);
				if ((diff!=diff) && ((value!=value && x!=x) || value==x)) diff = 0;
				if (!(diff <= epsilon)) return false;
			}
		}
	}
	return true;
}
/**
Modifies the given matrix square matrix <tt>A</tt> such that it is diagonally dominant by row and column, hence non-singular, hence invertible.
For testing purposes only.
@param A the square matrix to modify.
@throws IllegalArgumentException if <tt>!isSquare(A)</tt>.
*/
public void generateNonSingular(DoubleMatrix2D A) {
	checkSquare(A);
	cern.jet.math.Functions F = cern.jet.math.Functions.functions;
	int min = Math.min(A.rows(), A.columns());
	for (int i = min; --i >= 0; ) {
		A.setQuick(i,i, 0);
	}
	for (int i = min; --i >= 0; ) {
		double rowSum = A.viewRow(i).aggregate(F.plus,F.abs);
		double colSum = A.viewColumn(i).aggregate(F.plus,F.abs);
		A.setQuick(i,i, Math.max(rowSum,colSum) + i+1);
	}
}
/**
 */
protected static String get(cern.colt.list.ObjectArrayList list, int index) {
	return ((String) list.get(index));
}
/**
 * A matrix <tt>A</tt> is <i>diagonal</i> if <tt>A[i,j] == 0</tt> whenever <tt>i != j</tt>.
 * Matrix may but need not be square.
 */
public boolean isDiagonal(DoubleMatrix2D A) {
	double epsilon = tolerance();
	int rows = A.rows();
	int columns = A.columns();
	for (int row = rows; --row >=0; ) {
		for (int column = columns; --column >= 0; ) {
			//if (row!=column && A.getQuick(row,column) != 0) return false;
			if (row!=column && !(Math.abs(A.getQuick(row,column)) <= epsilon)) return false;
		}
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>diagonally dominant by column</i> 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.
 * <tt>returns true if for all i: abs(A[i,i]) &gt; Sum(abs(A[j,i])); j != i.</tt>
 * Matrix may but need not be square.
 * <p>
 * Note: Ignores tolerance.
 */
public boolean isDiagonallyDominantByColumn(DoubleMatrix2D A) {
	cern.jet.math.Functions F = cern.jet.math.Functions.functions;
	double epsilon = tolerance();
	int min = Math.min(A.rows(), A.columns());
	for (int i = min; --i >= 0; ) {
		double diag = Math.abs(A.getQuick(i,i));
		diag += diag;
		if (diag <= A.viewColumn(i).aggregate(F.plus,F.abs)) return false;
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>diagonally dominant by row</i> 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.
 * <tt>returns true if for all i: abs(A[i,i]) &gt; Sum(abs(A[i,j])); j != i.</tt> 
 * Matrix may but need not be square.
 * <p>
 * Note: Ignores tolerance.
 */
public boolean isDiagonallyDominantByRow(DoubleMatrix2D A) {
	cern.jet.math.Functions F = cern.jet.math.Functions.functions;
	double epsilon = tolerance();
	int min = Math.min(A.rows(), A.columns());
	for (int i = min; --i >= 0; ) {
		double diag = Math.abs(A.getQuick(i,i));
		diag += diag;
		if (diag <= A.viewRow(i).aggregate(F.plus,F.abs)) return false;
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is an <i>identity</i> matrix if <tt>A[i,i] == 1</tt> and all other cells are zero.
 * Matrix may but need not be square.
 */
public boolean isIdentity(DoubleMatrix2D A) {
	double epsilon = tolerance();
	int rows = A.rows();
	int columns = A.columns();
	for (int row = rows; --row >=0; ) {
		for (int column = columns; --column >= 0; ) {
			double v = A.getQuick(row,column);
			if (row==column) {
				if (!(Math.abs(1-v) < epsilon)) return false;
			}
			else if (!(Math.abs(v) <= epsilon)) return false;
		}
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>lower bidiagonal</i> if <tt>A[i,j]==0</tt> unless <tt>i==j || i==j+1</tt>.
 * Matrix may but need not be square.
 */
public boolean isLowerBidiagonal(DoubleMatrix2D A) {
	double epsilon = tolerance();
	int rows = A.rows();
	int columns = A.columns();
	for (int row = rows; --row >=0; ) {
		for (int column = columns; --column >= 0; ) {
			if (!(row==column || row==column+1)) {
				//if (A.getQuick(row,column) != 0) return false;
				if (!(Math.abs(A.getQuick(row,column)) <= epsilon)) return false;
			}
		}
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>lower triangular</i> if <tt>A[i,j]==0</tt> whenever <tt>i &lt; j</tt>.
 * Matrix may but need not be square.
 */
public boolean isLowerTriangular(DoubleMatrix2D A) {
	double epsilon = tolerance();
	int rows = A.rows();
	int columns = A.columns();
	for (int column = columns; --column >= 0; ) {
		for (int row = Math.min(column,rows); --row >= 0; ) {
			//if (A.getQuick(row,column) != 0) return false;
			if (!(Math.abs(A.getQuick(row,column)) <= epsilon)) return false;
		}
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>non-negative</i> if <tt>A[i,j] &gt;= 0</tt> holds for all cells.
 * <p>
 * Note: Ignores tolerance.
 */
public boolean isNonNegative(DoubleMatrix2D A) {
	int rows = A.rows();
	int columns = A.columns();
	for (int row = rows; --row >=0; ) {
		for (int column = columns; --column >= 0; ) {
			if (! (A.getQuick(row,column) >= 0)) return false;
		}
	}
	return true;
}
/**
 * A square matrix <tt>A</tt> is <i>orthogonal</i> if <tt>A*transpose(A) = I</tt>.
 * @throws IllegalArgumentException if <tt>!isSquare(A)</tt>.
 */
public boolean isOrthogonal(DoubleMatrix2D A) {
	checkSquare(A);
	return equals(A.zMult(A,null,1,0,false,true), cern.colt.matrix.DoubleFactory2D.dense.identity(A.rows()));
}
/**
 * A matrix <tt>A</tt> is <i>positive</i> if <tt>A[i,j] &gt; 0</tt> holds for all cells.
 * <p>
 * Note: Ignores tolerance.
 */
public boolean isPositive(DoubleMatrix2D A) {
	int rows = A.rows();
	int columns = A.columns();
	for (int row = rows; --row >=0; ) {
		for (int column = columns; --column >= 0; ) {
			if (!(A.getQuick(row,column) > 0)) return false;
		}
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>singular</i> if it has no inverse, that is, iff <tt>det(A)==0</tt>.
 */
public boolean isSingular(DoubleMatrix2D A) {
	return !(Math.abs(Algebra.DEFAULT.det(A)) >= tolerance());
}
/**
 * A square matrix <tt>A</tt> is <i>skew-symmetric</i> if <tt>A = -transpose(A)</tt>, that is <tt>A[i,j] == -A[j,i]</tt>.
 * @throws IllegalArgumentException if <tt>!isSquare(A)</tt>.
 */
public boolean isSkewSymmetric(DoubleMatrix2D A) {
	checkSquare(A);
	double epsilon = tolerance();
	int rows = A.rows();
	int columns = A.columns();
	for (int row = rows; --row >=0; ) {
		for (int column = rows; --column >= 0; ) {
			//if (A.getQuick(row,column) != -A.getQuick(column,row)) return false;
			if (!(Math.abs(A.getQuick(row,column) + A.getQuick(column,row)) <= epsilon)) return false;
		}
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>square</i> if it has the same number of rows and columns.
 */
public boolean isSquare(DoubleMatrix2D A) {
	return A.rows() == A.columns();
}
/**
 * A matrix <tt>A</tt> is <i>strictly lower triangular</i> if <tt>A[i,j]==0</tt> whenever <tt>i &lt;= j</tt>.
 * Matrix may but need not be square.
 */
public boolean isStrictlyLowerTriangular(DoubleMatrix2D A) {
	double epsilon = tolerance();
	int rows = A.rows();
	int columns = A.columns();
	for (int column = columns; --column >= 0; ) {
		for (int row = Math.min(rows,column+1); --row >= 0; ) {
			//if (A.getQuick(row,column) != 0) return false;
			if (!(Math.abs(A.getQuick(row,column)) <= epsilon)) return false;
		}
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>strictly triangular</i> if it is triangular and its diagonal elements all equal 0.
 * Matrix may but need not be square.
 */
public boolean isStrictlyTriangular(DoubleMatrix2D A) {
	if (!isTriangular(A)) return false;

	double epsilon = tolerance();
	for (int i = Math.min(A.rows(), A.columns()); --i >= 0; ) {
		//if (A.getQuick(i,i) != 0) return false;
		if (!(Math.abs(A.getQuick(i,i)) <= epsilon)) return false;
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>strictly upper triangular</i> if <tt>A[i,j]==0</tt> whenever <tt>i &gt;= j</tt>.
 * Matrix may but need not be square.
 */
public boolean isStrictlyUpperTriangular(DoubleMatrix2D A) {
	double epsilon = tolerance();
	int rows = A.rows();
	int columns = A.columns();
	for (int column = columns; --column >= 0; ) {
		for (int row = rows; --row >= column; ) {
			//if (A.getQuick(row,column) != 0) return false;
			if (!(Math.abs(A.getQuick(row,column)) <= epsilon)) return false;
		}
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>symmetric</i> if <tt>A = tranpose(A)</tt>, that is <tt>A[i,j] == A[j,i]</tt>.
 * @throws IllegalArgumentException if <tt>!isSquare(A)</tt>.
 */
public boolean isSymmetric(DoubleMatrix2D A) {
	checkSquare(A);
	return equals(A,A.viewDice());
}
/**
 * A matrix <tt>A</tt> is <i>triangular</i> iff it is either upper or lower triangular.
 * Matrix may but need not be square.
 */
public boolean isTriangular(DoubleMatrix2D A) {
	return isLowerTriangular(A) || isUpperTriangular(A);
}
/**
 * A matrix <tt>A</tt> is <i>tridiagonal</i> if <tt>A[i,j]==0</tt> whenever <tt>Math.abs(i-j) > 1</tt>.
 * Matrix may but need not be square.
 */
public boolean isTridiagonal(DoubleMatrix2D A) {
	double epsilon = tolerance();
	int rows = A.rows();
	int columns = A.columns();
	for (int row = rows; --row >=0; ) {
		for (int column = columns; --column >= 0; ) {
			if (Math.abs(row-column) > 1) {
				//if (A.getQuick(row,column) != 0) return false;
				if (!(Math.abs(A.getQuick(row,column)) <= epsilon)) return false;
			}
		}
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>unit triangular</i> if it is triangular and its diagonal elements all equal 1.
 * Matrix may but need not be square.
 */
public boolean isUnitTriangular(DoubleMatrix2D A) {
	if (!isTriangular(A)) return false;
	
	double epsilon = tolerance();
	for (int i = Math.min(A.rows(), A.columns()); --i >= 0; ) {
		//if (A.getQuick(i,i) != 1) return false;
		if (!(Math.abs(1 - A.getQuick(i,i)) <= epsilon)) return false;
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>upper bidiagonal</i> if <tt>A[i,j]==0</tt> unless <tt>i==j || i==j-1</tt>.
 * Matrix may but need not be square.
 */
public boolean isUpperBidiagonal(DoubleMatrix2D A) {
	double epsilon = tolerance();
	int rows = A.rows();
	int columns = A.columns();
	for (int row = rows; --row >=0; ) {
		for (int column = columns; --column >= 0; ) {
			if (!(row==column || row==column-1)) {
				//if (A.getQuick(row,column) != 0) return false;
				if (!(Math.abs(A.getQuick(row,column)) <= epsilon)) return false;
			}
		}
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>upper triangular</i> if <tt>A[i,j]==0</tt> whenever <tt>i &gt; j</tt>.
 * Matrix may but need not be square.
 */
public boolean isUpperTriangular(DoubleMatrix2D A) {
	double epsilon = tolerance();
	int rows = A.rows();
	int columns = A.columns();
	for (int column = columns; --column >= 0; ) {
		for (int row = rows; --row > column; ) {
			//if (A.getQuick(row,column) != 0) return false;
			if (!(Math.abs(A.getQuick(row,column)) <= epsilon)) return false;
		}
	}
	return true;
}
/**
 * A matrix <tt>A</tt> is <i>zero</i> if all its cells are zero.
 */
public boolean isZero(DoubleMatrix2D A) {
	return equals(A,0);
}
/**
The <i>lower bandwidth</i> of a square matrix <tt>A</tt> is the maximum <tt>i-j</tt> for which <tt>A[i,j]</tt> is nonzero and <tt>i &gt; j</tt>.
A <i>banded</i> matrix has a "band" about the diagonal.
Diagonal, tridiagonal and triangular matrices are special cases.

@param A the square matrix to analyze.
@return the lower bandwith.
@throws IllegalArgumentException if <tt>!isSquare(A)</tt>.
@see #semiBandwidth(DoubleMatrix2D)
@see #upperBandwidth(DoubleMatrix2D)
*/
public int lowerBandwidth(DoubleMatrix2D A) {
	checkSquare(A);
	double epsilon = tolerance();
	int rows = A.rows();

	for (int k=rows; --k >= 0; ) {
		for (int i = rows-k; --i >= 0; ) {
			int j = i+k;
			//if (A.getQuick(j,i) != 0) return k;
			if (!(Math.abs(A.getQuick(j,i)) <= epsilon)) return k;
		}
	}
	return 0;
}
/**
Returns the <i>semi-bandwidth</i> of the given square matrix <tt>A</tt>.
A <i>banded</i> 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 <tt>k</tt> diagonal lines above the diagonal, and the <tt>k</tt> diagonal lines below the diagonal.
The <i>semi-bandwith l</i> is the number <tt>k+1</tt>.
The <i>bandwidth p</i> is the number <tt>2*k + 1</tt>.
For example, a tridiagonal matrix corresponds to <tt>k=1, l=2, p=3</tt>, 
a diagonal or zero matrix corresponds to <tt>k=0, l=1, p=1</tt>, 
<p>
The <i>upper bandwidth</i> is the maximum <tt>j-i</tt> for which <tt>A[i,j]</tt> is nonzero and <tt>j &gt; i</tt>.
The <i>lower bandwidth</i> is the maximum <tt>i-j</tt> for which <tt>A[i,j]</tt> is nonzero and <tt>i &gt; j</tt>. 
Diagonal, tridiagonal and triangular matrices are special cases.
<p>
Examples:
<table border="1" cellspacing="0">
  <tr align="left" valign="top"> 
	<td valign="middle" align="left"><tt>matrix</tt></td>
	<td> <tt>4&nbsp;x&nbsp;4&nbsp;<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;0 </tt></td>
	<td><tt>4&nbsp;x&nbsp;4<br>
	  1&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  0&nbsp;0&nbsp;0&nbsp;1 </tt></td>
	<td><tt>4&nbsp;x&nbsp;4<br>
	  1&nbsp;1&nbsp;0&nbsp;0<br>
	  1&nbsp;1&nbsp;1&nbsp;0<br>
	  0&nbsp;1&nbsp;1&nbsp;1<br>
	  0&nbsp;0&nbsp;1&nbsp;1 </tt></td>
	<td><tt> 4&nbsp;x&nbsp;4<br>
	  0&nbsp;1&nbsp;1&nbsp;1<br>
	  0&nbsp;1&nbsp;1&nbsp;1<br>
	  0&nbsp;0&nbsp;0&nbsp;1<br>
	  0&nbsp;0&nbsp;0&nbsp;1 </tt></td>
	<td><tt> 4&nbsp;x&nbsp;4<br>
	  0&nbsp;0&nbsp;0&nbsp;0<br>
	  1&nbsp;1&nbsp;0&nbsp;0<br>
	  1&nbsp;1&nbsp;0&nbsp;0<br>
	  1&nbsp;1&nbsp;1&nbsp;1 </tt></td>
	<td><tt>4&nbsp;x&nbsp;4<br>
	  1&nbsp;1&nbsp;0&nbsp;0<br>
	  0&nbsp;1&nbsp;1&nbsp;0<br>
	  0&nbsp;1&nbsp;0&nbsp;1<br>
	  1&nbsp;0&nbsp;1&nbsp;1 </tt><tt> </tt> </td>
	<td><tt>4&nbsp;x&nbsp;4<br>
	  1&nbsp;1&nbsp;1&nbsp;0<br>
	  0&nbsp;1&nbsp;0&nbsp;0<br>
	  1&nbsp;1&nbsp;0&nbsp;1<br>
	  0&nbsp;0&nbsp;1&nbsp;1 </tt> </td>
  </tr>
  <tr align="center" valign="middle"> 
	<td><tt>upperBandwidth</tt></td>
	<td> 
	  <div align="center"><tt>0</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>0</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>1</tt></div>
	</td>
	<td><tt>3</tt></td>
	<td align="center" valign="middle"><tt>0</tt></td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>1</tt></div>
	</td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>2</tt></div>
	</td>
  </tr>
  <tr align="center" valign="middle"> 
	<td><tt>lowerBandwidth</tt></td>
	<td> 
	  <div align="center"><tt>0</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>0</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>1</tt></div>
	</td>
	<td><tt>0</tt></td>
	<td align="center" valign="middle"><tt>3</tt></td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>3</tt></div>
	</td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>2</tt></div>
	</td>
  </tr>
  <tr align="center" valign="middle"> 
	<td><tt>semiBandwidth</tt></td>
	<td> 
	  <div align="center"><tt>1</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>1</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>2</tt></div>
	</td>
	<td><tt>4</tt></td>
	<td align="center" valign="middle"><tt>4</tt></td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>4</tt></div>
	</td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>3</tt></div>
	</td>
  </tr>
  <tr align="center" valign="middle"> 
	<td><tt>description</tt></td>
	<td> 
	  <div align="center"><tt>zero</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>diagonal</tt></div>
	</td>
	<td> 
	  <div align="center"><tt>tridiagonal</tt></div>
	</td>
	<td><tt>upper triangular</tt></td>
	<td align="center" valign="middle"><tt>lower triangular</tt></td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>unstructured</tt></div>
	</td>
	<td align="center" valign="middle"> 
	  <div align="center"><tt>unstructured</tt></div>
	</td>
  </tr>
</table>

@param A the square matrix to analyze.
@return the semi-bandwith <tt>l</tt>.
@throws IllegalArgumentException if <tt>!isSquare(A)</tt>.
@see #lowerBandwidth(DoubleMatrix2D)
@see #upperBandwidth(DoubleMatrix2D)
*/
public int semiBandwidth(DoubleMatrix2D A) {
	checkSquare(A);
	double epsilon = tolerance();
	int rows = A.rows();

	for (int k=rows; --k >= 0; ) {
		for (int i = rows-k; --i >= 0; ) {
			int j = i+k;
			//if (A.getQuick(j,i) != 0) return k+1;
			//if (A.getQuick(i,j) != 0) return k+1;
			if (!(Math.abs(A.getQuick(j,i)) <= epsilon)) return k+1;
			if (!(Math.abs(A.getQuick(i,j)) <= epsilon)) return k+1;
		}
	}
	return 1;
}
/**
 * Sets the tolerance to <tt>Math.abs(newTolerance)</tt>.
 * @throws UnsupportedOperationException if <tt>this==DEFAULT || this==ZERO || this==TWELVE</tt>.
 */
public void setTolerance(double newTolerance) {
	if (this==DEFAULT || this==ZERO || this==TWELVE) {
		throw new IllegalArgumentException("Attempted to modify immutable object."); 
		//throw new UnsupportedOperationException("Attempted to modify object."); // since JDK1.2
	}
	tolerance = Math.abs(newTolerance);
}
/**
 * Returns the current tolerance.
 */
public double tolerance() {
	return tolerance;
}
/**
Returns summary information about the given matrix <tt>A</tt>.
That is a String with (propertyName, propertyValue) pairs.
Useful for debugging or to quickly get the rough picture of a matrix.
For example,
<pre>
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.
</pre>
*/
public String toString(DoubleMatrix2D A) {
	final cern.colt.list.ObjectArrayList names = new cern.colt.list.ObjectArrayList();
	final cern.colt.list.ObjectArrayList values = new cern.colt.list.ObjectArrayList();
	String unknown = "Illegal operation or error: ";

	// determine properties
	names.add("density");
	try { values.add(String.valueOf(density(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	// determine properties
	names.add("isDiagonal");
	try { values.add(String.valueOf(isDiagonal(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	// determine properties
	names.add("isDiagonallyDominantByRow");
	try { values.add(String.valueOf(isDiagonallyDominantByRow(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	// determine properties
	names.add("isDiagonallyDominantByColumn");
	try { values.add(String.valueOf(isDiagonallyDominantByColumn(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isIdentity");
	try { values.add(String.valueOf(isIdentity(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isLowerBidiagonal");
	try { values.add(String.valueOf(isLowerBidiagonal(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isLowerTriangular");
	try { values.add(String.valueOf(isLowerTriangular(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isNonNegative");
	try { values.add(String.valueOf(isNonNegative(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isOrthogonal");
	try { values.add(String.valueOf(isOrthogonal(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isPositive");
	try { values.add(String.valueOf(isPositive(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isSingular");
	try { values.add(String.valueOf(isSingular(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isSkewSymmetric");
	try { values.add(String.valueOf(isSkewSymmetric(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isSquare");
	try { values.add(String.valueOf(isSquare(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isStrictlyLowerTriangular");
	try { values.add(String.valueOf(isStrictlyLowerTriangular(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isStrictlyTriangular");
	try { values.add(String.valueOf(isStrictlyTriangular(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isStrictlyUpperTriangular");
	try { values.add(String.valueOf(isStrictlyUpperTriangular(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isSymmetric");
	try { values.add(String.valueOf(isSymmetric(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isTriangular");
	try { values.add(String.valueOf(isTriangular(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isTridiagonal");
	try { values.add(String.valueOf(isTridiagonal(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isUnitTriangular");
	try { values.add(String.valueOf(isUnitTriangular(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isUpperBidiagonal");
	try { values.add(String.valueOf(isUpperBidiagonal(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
	
	names.add("isUpperTriangular");
	try { values.add(String.valueOf(isUpperTriangular(A)));} 
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }
		
	names.add("isZero");
	try { values.add(String.valueOf(isZero(A))); }
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }

	names.add("lowerBandwidth");
	try { values.add(String.valueOf(lowerBandwidth(A))); }
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }

	names.add("semiBandwidth");
	try { values.add(String.valueOf(semiBandwidth(A))); }
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }

	names.add("upperBandwidth");
	try { values.add(String.valueOf(upperBandwidth(A))); }
	catch (IllegalArgumentException exc) { values.add(unknown+exc.getMessage()); }

	
	// sort ascending by property name
	cern.colt.function.IntComparator comp = new cern.colt.function.IntComparator() {
		public int compare(int a, int b) {
			return get(names,a).compareTo(get(names,b));
		}
	};
	cern.colt.Swapper swapper = new cern.colt.Swapper() {
		public void swap(int a, int b) {
			Object tmp;
			tmp = names.get(a); names.set(a,names.get(b)); names.set(b,tmp);
			tmp = values.get(a); values.set(a,values.get(b)); values.set(b,tmp);
		}
	};	
	cern.colt.GenericSorting.quickSort(0,names.size(),comp,swapper);
	
	// determine padding for nice formatting
	int maxLength = 0;
	for (int i = 0; i < names.size(); i++) {
		int length = ((String) names.get(i)).length();
		maxLength = Math.max(length, maxLength);
	}

	// finally, format properties
	StringBuffer buf = new StringBuffer();
	for (int i = 0; i < names.size(); i++) {
		String name = ((String) names.get(i));
		buf.append(name);
		buf.append(blanks(maxLength - name.length()));
		buf.append(" : ");
		buf.append(values.get(i));
		if (i < names.size() - 1)
			buf.append('\n');
	}
	
	return buf.toString();
}
/**
The <i>upper bandwidth</i> of a square matrix <tt>A</tt> is the 
maximum <tt>j-i</tt> for which <tt>A[i,j]</tt> is nonzero and <tt>j &gt; i</tt>.
A <i>banded</i> matrix has a "band" about the diagonal. 
Diagonal, tridiagonal and triangular matrices are special cases.

@param A the square matrix to analyze.
@return the upper bandwith.
@throws IllegalArgumentException if <tt>!isSquare(A)</tt>.
@see #semiBandwidth(DoubleMatrix2D)
@see #lowerBandwidth(DoubleMatrix2D)
*/
public int upperBandwidth(DoubleMatrix2D A) {
	checkSquare(A);
	double epsilon = tolerance();
	int rows = A.rows();

	for (int k=rows; --k >= 0; ) {
		for (int i = rows-k; --i >= 0; ) {
			int j = i+k;
			//if (A.getQuick(i,j) != 0) return k;
			if (!(Math.abs(A.getQuick(i,j)) <= epsilon)) return k;
		}
	}
	return 0;
}
}
