src/math/SquareMatrix.hpp

/*
 * Copyright (C) 2000-2004  Object Oriented Parallel Simulation Engine (OOPSE) project
 * 
 * Contact: oopse@oopse.org
 * 
 * This program is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public License
 * as published by the Free Software Foundation; either version 2.1
 * of the License, or (at your option) any later version.
 * All we ask is that proper credit is given for our work, which includes
 * - but is not limited to - adding the above copyright notice to the beginning
 * of your source code files, and to any copyright notice that you may distribute
 * with programs based on this work.
 * 
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU Lesser General Public License for more details.
 * 
 * You should have received a copy of the GNU Lesser General Public License
 * along with this program; if not, write to the Free Software
 * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA.
 *
 */

/**
 * @file SquareMatrix.hpp
 * @author Teng Lin
 * @date 10/11/2004
 * @version 1.0
 */
 #ifndef MATH_SQUAREMATRIX_HPP 
#define MATH_SQUAREMATRIX_HPP 

#include "math/RectMatrix.hpp"

namespace oopse {

    /**
     * @class SquareMatrix SquareMatrix.hpp "math/SquareMatrix.hpp"
     * @brief A square matrix class
     * @template Real the element type
     * @template Dim the dimension of the square matrix
     */
    template<typename Real, int Dim>
    class SquareMatrix : public RectMatrix<Real, Dim, Dim> {
        public:

        /** default constructor */
        SquareMatrix() {
            for (unsigned int i = 0; i < Dim; i++)
                for (unsigned int j = 0; j < Dim; j++)
                    data_[i][j] = 0.0;
         }

        /** copy constructor */
        SquareMatrix(const RectMatrix<Real, Dim, Dim>& m)  : RectMatrix<Real, Dim, Dim>(m) {
        }
        
        /** copy assignment operator */
        SquareMatrix<Real, Dim>& operator =(const RectMatrix<Real, Dim, Dim>& m) {
            RectMatrix<Real, Dim, Dim>::operator=(m);
            return *this;
        }
                               
        /** Retunrs  an identity matrix*/

       static SquareMatrix<Real, Dim> identity() {
            SquareMatrix<Real, Dim> m;
            
            for (unsigned int i = 0; i < Dim; i++) 
                for (unsigned int j = 0; j < Dim; j++) 
                    if (i == j)
                        m(i, j) = 1.0;
                    else
                        m(i, j) = 0.0;

            return m;
        }

        /** 
         * Retunrs  the inversion of this matrix. 
         * @todo need implementation
         */
         SquareMatrix<Real, Dim>  inverse() {
             SquareMatrix<Real, Dim> result;

             return result;
        }        

        /**
         * Returns the determinant of this matrix.
         * @todo need implementation
         */
        Real determinant() const {
            Real det;
            return det;
        }

        /** Returns the trace of this matrix. */
        Real trace() const {
           Real tmp = 0;
           
            for (unsigned int i = 0; i < Dim ; i++)
                tmp += data_[i][i];

            return tmp;
        }

        /** Tests if this matrix is symmetrix. */            
        bool isSymmetric() const {
            for (unsigned int i = 0; i < Dim - 1; i++)
                for (unsigned int j = i; j < Dim; j++)
                    if (fabs(data_[i][j] - data_[j][i]) > oopse::epsilon) 
                        return false;
                    
            return true;
        }

        /** Tests if this matrix is orthogonal. */            
        bool isOrthogonal() {
            SquareMatrix<Real, Dim> tmp;

            tmp = *this * transpose();

            return tmp.isDiagonal();
        }

        /** Tests if this matrix is diagonal. */
        bool isDiagonal() const {
            for (unsigned int i = 0; i < Dim ; i++)
                for (unsigned int j = 0; j < Dim; j++)
                    if (i !=j && fabs(data_[i][j]) > oopse::epsilon) 
                        return false;
                    
            return true;
        }

        /** Tests if this matrix is the unit matrix. */
        bool isUnitMatrix() const {
            if (!isDiagonal())
                return false;
            
            for (unsigned int i = 0; i < Dim ; i++)
                if (fabs(data_[i][i] - 1) > oopse::epsilon)
                    return false;
                
            return true;
        }         

        /** @todo need implementation */
        void diagonalize() {
            //jacobi(m, eigenValues, ortMat);
        }

        /**
         * Jacobi iteration routines for computing eigenvalues/eigenvectors of 
         * real symmetric matrix
         *
         * @return true if success, otherwise return false
         * @param a symmetric matrix whose eigenvectors are to be computed. On return, the matrix is
         *     overwritten
         * @param w will contain the eigenvalues of the matrix On return of this function
         * @param v the columns of this matrix will contain the eigenvectors. The eigenvectors are 
         *    normalized and mutually orthogonal. 
         */
       
        static int jacobi(SquareMatrix<Real, Dim>& a, Vector<Real, Dim>& d, 
                              SquareMatrix<Real, Dim>& v);
    };//end SquareMatrix


/*=========================================================================

  Program:   Visualization Toolkit
  Module:    $RCSfile: SquareMatrix.hpp,v $

  Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen
  All rights reserved.
  See Copyright.txt or http://www.kitware.com/Copyright.htm for details.

     This software is distributed WITHOUT ANY WARRANTY; without even
     the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
     PURPOSE.  See the above copyright notice for more information.

=========================================================================*/

#define VTK_ROTATE(a,i,j,k,l) g=a(i, j);h=a(k, l);a(i, j)=g-s*(h+g*tau);\
        a(k, l)=h+s*(g-h*tau)

#define VTK_MAX_ROTATIONS 20

    // Jacobi iteration for the solution of eigenvectors/eigenvalues of a nxn
    // real symmetric matrix. Square nxn matrix a; size of matrix in n;
    // output eigenvalues in w; and output eigenvectors in v. Resulting
    // eigenvalues/vectors are sorted in decreasing order; eigenvectors are
    // normalized.
    template<typename Real, int Dim>
    int SquareMatrix<Real, Dim>::jacobi(SquareMatrix<Real, Dim>& a, Vector<Real, Dim>& w, 
                                  SquareMatrix<Real, Dim>& v) {
      const int n = Dim;  
      int i, j, k, iq, ip, numPos;
      Real tresh, theta, tau, t, sm, s, h, g, c, tmp;
      Real bspace[4], zspace[4];
      Real *b = bspace;
      Real *z = zspace;

      // only allocate memory if the matrix is large
      if (n > 4)
        {
        b = new Real[n];
        z = new Real[n]; 
        }

      // initialize
      for (ip=0; ip<n; ip++) 
        {
        for (iq=0; iq<n; iq++)
          {
          v(ip, iq) = 0.0;
          }
        v(ip, ip) = 1.0;
        }
      for (ip=0; ip<n; ip++) 
        {
        b[ip] = w[ip] = a(ip, ip);
        z[ip] = 0.0;
        }

      // begin rotation sequence
      for (i=0; i<VTK_MAX_ROTATIONS; i++) 
        {
        sm = 0.0;
        for (ip=0; ip<n-1; ip++) 
          {
          for (iq=ip+1; iq<n; iq++)
            {
            sm += fabs(a(ip, iq));
            }
          }
        if (sm == 0.0)
          {
          break;
          }

        if (i < 3)                                // first 3 sweeps
          {
          tresh = 0.2*sm/(n*n);
          }
        else
          {
          tresh = 0.0;
          }

        for (ip=0; ip<n-1; ip++) 
          {
          for (iq=ip+1; iq<n; iq++) 
            {
            g = 100.0*fabs(a(ip, iq));

            // after 4 sweeps
            if (i > 3 && (fabs(w[ip])+g) == fabs(w[ip])
            && (fabs(w[iq])+g) == fabs(w[iq]))
              {
              a(ip, iq) = 0.0;
              }
            else if (fabs(a(ip, iq)) > tresh) 
              {
              h = w[iq] - w[ip];
              if ( (fabs(h)+g) == fabs(h))
                {
                t = (a(ip, iq)) / h;
                }
              else 
                {
                theta = 0.5*h / (a(ip, iq));
                t = 1.0 / (fabs(theta)+sqrt(1.0+theta*theta));
                if (theta < 0.0)
                  {
                  t = -t;
                  }
                }
              c = 1.0 / sqrt(1+t*t);
              s = t*c;
              tau = s/(1.0+c);
              h = t*a(ip, iq);
              z[ip] -= h;
              z[iq] += h;
              w[ip] -= h;
              w[iq] += h;
              a(ip, iq)=0.0;

              // ip already shifted left by 1 unit
              for (j = 0;j <= ip-1;j++) 
                {
                VTK_ROTATE(a,j,ip,j,iq);
                }
              // ip and iq already shifted left by 1 unit
              for (j = ip+1;j <= iq-1;j++) 
                {
                VTK_ROTATE(a,ip,j,j,iq);
                }
              // iq already shifted left by 1 unit
              for (j=iq+1; j<n; j++) 
                {
                VTK_ROTATE(a,ip,j,iq,j);
                }
              for (j=0; j<n; j++) 
                {
                VTK_ROTATE(v,j,ip,j,iq);
                }
              }
            }
          }

        for (ip=0; ip<n; ip++) 
          {
          b[ip] += z[ip];
          w[ip] = b[ip];
          z[ip] = 0.0;
          }
        }

      //// this is NEVER called
      if ( i >= VTK_MAX_ROTATIONS )
        {
           std::cout << "vtkMath::Jacobi: Error extracting eigenfunctions" << std::endl;
           return 0;
        }

      // sort eigenfunctions                 these changes do not affect accuracy 
      for (j=0; j<n-1; j++)                  // boundary incorrect
        {
        k = j;
        tmp = w[k];
        for (i=j+1; i<n; i++)                // boundary incorrect, shifted already
          {
          if (w[i] >= tmp)                   // why exchage if same?
            {
            k = i;
            tmp = w[k];
            }
          }
        if (k != j) 
          {
          w[k] = w[j];
          w[j] = tmp;
          for (i=0; i<n; i++) 
            {
            tmp = v(i, j);
            v(i, j) = v(i, k);
            v(i, k) = tmp;
            }
          }
        }
      // insure eigenvector consistency (i.e., Jacobi can compute vectors that
      // are negative of one another (.707,.707,0) and (-.707,-.707,0). This can
      // reek havoc in hyperstreamline/other stuff. We will select the most
      // positive eigenvector.
      int ceil_half_n = (n >> 1) + (n & 1);
      for (j=0; j<n; j++)
        {
        for (numPos=0, i=0; i<n; i++)
          {
          if ( v(i, j) >= 0.0 )
            {
            numPos++;
            }
          }
    //    if ( numPos < ceil(double(n)/double(2.0)) )
        if ( numPos < ceil_half_n)
          {
          for(i=0; i<n; i++)
            {
            v(i, j) *= -1.0;
            }
          }
        }

      if (n > 4)
        {
        delete [] b;
        delete [] z;
        }
      return 1;
    }


}
#endif //MATH_SQUAREMATRIX_HPP 

Revision:	1616
Committed:	Wed Oct 20 18:07:08 2004 UTC (19 years, 8 months ago) by tim
File size:	11587 byte(s)
Log Message:	Math library pass the unit test
#	User	Rev	Content
1	tim	1563	/*
2			* Copyright (C) 2000-2004 Object Oriented Parallel Simulation Engine (OOPSE) project
3			*
4			* Contact: oopse@oopse.org
5			*
6			* This program is free software; you can redistribute it and/or
7			* modify it under the terms of the GNU Lesser General Public License
8			* as published by the Free Software Foundation; either version 2.1
9			* of the License, or (at your option) any later version.
10			* All we ask is that proper credit is given for our work, which includes
11			* - but is not limited to - adding the above copyright notice to the beginning
12			* of your source code files, and to any copyright notice that you may distribute
13			* with programs based on this work.
14			*
15			* This program is distributed in the hope that it will be useful,
16			* but WITHOUT ANY WARRANTY; without even the implied warranty of
17			* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
18			* GNU Lesser General Public License for more details.
19			*
20			* You should have received a copy of the GNU Lesser General Public License
21			* along with this program; if not, write to the Free Software
22			* Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
23			*
24			*/
25
26			/**
27			* @file SquareMatrix.hpp
28			* @author Teng Lin
29			* @date 10/11/2004
30			* @version 1.0
31			*/
32	tim	1616	#ifndef MATH_SQUAREMATRIX_HPP
33	tim	1563	#define MATH_SQUAREMATRIX_HPP
34
35	tim	1567	#include "math/RectMatrix.hpp"
36	tim	1563
37			namespace oopse {
38
39			/**
40			* @class SquareMatrix SquareMatrix.hpp "math/SquareMatrix.hpp"
41			* @brief A square matrix class
42			* @template Real the element type
43			* @template Dim the dimension of the square matrix
44			*/
45			template<typename Real, int Dim>
46	tim	1567	class SquareMatrix : public RectMatrix<Real, Dim, Dim> {
47	tim	1563	public:
48
49			/** default constructor */
50			SquareMatrix() {
51			for (unsigned int i = 0; i < Dim; i++)
52			for (unsigned int j = 0; j < Dim; j++)
53			data_[i][j] = 0.0;
54			}
55
56			/** copy constructor */
57	tim	1567	SquareMatrix(const RectMatrix<Real, Dim, Dim>& m) : RectMatrix<Real, Dim, Dim>(m) {
58	tim	1563	}
59
60			/** copy assignment operator */
61	tim	1567	SquareMatrix<Real, Dim>& operator =(const RectMatrix<Real, Dim, Dim>& m) {
62			RectMatrix<Real, Dim, Dim>::operator=(m);
63			return *this;
64	tim	1563	}
65	tim	1567
66			/** Retunrs an identity matrix*/
67	tim	1563
68	tim	1567	static SquareMatrix<Real, Dim> identity() {
69			SquareMatrix<Real, Dim> m;
70	tim	1563
71			for (unsigned int i = 0; i < Dim; i++)
72	tim	1567	for (unsigned int j = 0; j < Dim; j++)
73	tim	1563	if (i == j)
74	tim	1567	m(i, j) = 1.0;
75	tim	1563	else
76	tim	1567	m(i, j) = 0.0;
77
78			return m;
79	tim	1563	}
80
81	tim	1594	/**
82			* Retunrs the inversion of this matrix.
83	tim	1603	* @todo need implementation
84	tim	1594	*/
85	tim	1567	SquareMatrix<Real, Dim> inverse() {
86			SquareMatrix<Real, Dim> result;
87
88			return result;
89	tim	1569	}
90	tim	1563
91	tim	1594	/**
92			* Returns the determinant of this matrix.
93	tim	1603	* @todo need implementation
94	tim	1594	*/
95	tim	1606	Real determinant() const {
96			Real det;
97	tim	1567	return det;
98	tim	1563	}
99
100			/** Returns the trace of this matrix. */
101	tim	1606	Real trace() const {
102			Real tmp = 0;
103	tim	1563
104			for (unsigned int i = 0; i < Dim ; i++)
105			tmp += data_[i][i];
106
107			return tmp;
108			}
109
110			/** Tests if this matrix is symmetrix. */
111			bool isSymmetric() const {
112			for (unsigned int i = 0; i < Dim - 1; i++)
113			for (unsigned int j = i; j < Dim; j++)
114	tim	1567	if (fabs(data_[i][j] - data_[j][i]) > oopse::epsilon)
115	tim	1563	return false;
116
117			return true;
118			}
119
120	tim	1569	/** Tests if this matrix is orthogonal. */
121	tim	1567	bool isOrthogonal() {
122			SquareMatrix<Real, Dim> tmp;
123	tim	1563
124	tim	1567	tmp = this transpose();
125	tim	1563
126	tim	1569	return tmp.isDiagonal();
127	tim	1563	}
128
129			/** Tests if this matrix is diagonal. */
130			bool isDiagonal() const {
131			for (unsigned int i = 0; i < Dim ; i++)
132			for (unsigned int j = 0; j < Dim; j++)
133	tim	1567	if (i !=j && fabs(data_[i][j]) > oopse::epsilon)
134	tim	1563	return false;
135
136			return true;
137			}
138
139			/** Tests if this matrix is the unit matrix. */
140			bool isUnitMatrix() const {
141			if (!isDiagonal())
142			return false;
143
144			for (unsigned int i = 0; i < Dim ; i++)
145	tim	1567	if (fabs(data_[i][i] - 1) > oopse::epsilon)
146	tim	1563	return false;
147
148			return true;
149	tim	1567	}
150	tim	1563
151	tim	1603	/** @todo need implementation */
152	tim	1569	void diagonalize() {
153	tim	1594	//jacobi(m, eigenValues, ortMat);
154	tim	1569	}
155
156			/**
157			* Jacobi iteration routines for computing eigenvalues/eigenvectors of
158			* real symmetric matrix
159			*
160			* @return true if success, otherwise return false
161	tim	1616	* @param a symmetric matrix whose eigenvectors are to be computed. On return, the matrix is
162			* overwritten
163			* @param w will contain the eigenvalues of the matrix On return of this function
164			* @param v the columns of this matrix will contain the eigenvectors. The eigenvectors are
165			* normalized and mutually orthogonal.
166	tim	1569	*/
167	tim	1616
168			static int jacobi(SquareMatrix<Real, Dim>& a, Vector<Real, Dim>& d,
169	tim	1569	SquareMatrix<Real, Dim>& v);
170	tim	1563	};//end SquareMatrix
171
172	tim	1569
173	tim	1616	/*=========================================================================
174	tim	1569
175	tim	1616	Program: Visualization Toolkit
176			Module: $RCSfile: SquareMatrix.hpp,v $
177	tim	1569
178	tim	1616	Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen
179			All rights reserved.
180			See Copyright.txt or http://www.kitware.com/Copyright.htm for details.
181
182			This software is distributed WITHOUT ANY WARRANTY; without even
183			the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
184			PURPOSE. See the above copyright notice for more information.
185
186			=========================================================================*/
187
188			#define VTK_ROTATE(a,i,j,k,l) g=a(i, j);h=a(k, l);a(i, j)=g-s(h+gtau);\
189			a(k, l)=h+s(g-htau)
190
191			#define VTK_MAX_ROTATIONS 20
192
193			// Jacobi iteration for the solution of eigenvectors/eigenvalues of a nxn
194			// real symmetric matrix. Square nxn matrix a; size of matrix in n;
195			// output eigenvalues in w; and output eigenvectors in v. Resulting
196			// eigenvalues/vectors are sorted in decreasing order; eigenvectors are
197			// normalized.
198			template<typename Real, int Dim>
199			int SquareMatrix<Real, Dim>::jacobi(SquareMatrix<Real, Dim>& a, Vector<Real, Dim>& w,
200			SquareMatrix<Real, Dim>& v) {
201			const int n = Dim;
202			int i, j, k, iq, ip, numPos;
203			Real tresh, theta, tau, t, sm, s, h, g, c, tmp;
204			Real bspace[4], zspace[4];
205			Real *b = bspace;
206			Real *z = zspace;
207
208			// only allocate memory if the matrix is large
209			if (n > 4)
210			{
211			b = new Real[n];
212			z = new Real[n];
213			}
214
215			// initialize
216			for (ip=0; ip<n; ip++)
217			{
218			for (iq=0; iq<n; iq++)
219			{
220			v(ip, iq) = 0.0;
221			}
222	tim	1586	v(ip, ip) = 1.0;
223	tim	1616	}
224			for (ip=0; ip<n; ip++)
225			{
226			b[ip] = w[ip] = a(ip, ip);
227			z[ip] = 0.0;
228			}
229	tim	1569
230	tim	1616	// begin rotation sequence
231			for (i=0; i<VTK_MAX_ROTATIONS; i++)
232			{
233	tim	1586	sm = 0.0;
234	tim	1616	for (ip=0; ip<n-1; ip++)
235			{
236			for (iq=ip+1; iq<n; iq++)
237			{
238			sm += fabs(a(ip, iq));
239			}
240			}
241	tim	1586	if (sm == 0.0)
242	tim	1616	{
243			break;
244			}
245	tim	1569
246	tim	1616	if (i < 3) // first 3 sweeps
247			{
248			tresh = 0.2sm/(nn);
249			}
250	tim	1586	else
251	tim	1616	{
252			tresh = 0.0;
253			}
254	tim	1569
255	tim	1616	for (ip=0; ip<n-1; ip++)
256			{
257			for (iq=ip+1; iq<n; iq++)
258			{
259			g = 100.0*fabs(a(ip, iq));
260	tim	1569
261	tim	1616	// after 4 sweeps
262			if (i > 3 && (fabs(w[ip])+g) == fabs(w[ip])
263			&& (fabs(w[iq])+g) == fabs(w[iq]))
264			{
265			a(ip, iq) = 0.0;
266			}
267			else if (fabs(a(ip, iq)) > tresh)
268			{
269			h = w[iq] - w[ip];
270			if ( (fabs(h)+g) == fabs(h))
271			{
272			t = (a(ip, iq)) / h;
273			}
274			else
275			{
276			theta = 0.5*h / (a(ip, iq));
277			t = 1.0 / (fabs(theta)+sqrt(1.0+theta*theta));
278			if (theta < 0.0)
279			{
280			t = -t;
281			}
282			}
283			c = 1.0 / sqrt(1+t*t);
284			s = t*c;
285			tau = s/(1.0+c);
286			h = t*a(ip, iq);
287			z[ip] -= h;
288			z[iq] += h;
289			w[ip] -= h;
290			w[iq] += h;
291			a(ip, iq)=0.0;
292	tim	1569
293	tim	1616	// ip already shifted left by 1 unit
294			for (j = 0;j <= ip-1;j++)
295			{
296			VTK_ROTATE(a,j,ip,j,iq);
297	tim	1586	}
298	tim	1616	// ip and iq already shifted left by 1 unit
299			for (j = ip+1;j <= iq-1;j++)
300			{
301			VTK_ROTATE(a,ip,j,j,iq);
302			}
303			// iq already shifted left by 1 unit
304			for (j=iq+1; j<n; j++)
305			{
306			VTK_ROTATE(a,ip,j,iq,j);
307			}
308			for (j=0; j<n; j++)
309			{
310			VTK_ROTATE(v,j,ip,j,iq);
311			}
312			}
313	tim	1586	}
314	tim	1616	}
315	tim	1586
316	tim	1616	for (ip=0; ip<n; ip++)
317			{
318			b[ip] += z[ip];
319			w[ip] = b[ip];
320			z[ip] = 0.0;
321			}
322	tim	1586	}
323
324	tim	1616	//// this is NEVER called
325			if ( i >= VTK_MAX_ROTATIONS )
326			{
327			std::cout << "vtkMath::Jacobi: Error extracting eigenfunctions" << std::endl;
328			return 0;
329			}
330	tim	1569
331	tim	1616	// sort eigenfunctions these changes do not affect accuracy
332			for (j=0; j<n-1; j++) // boundary incorrect
333			{
334	tim	1586	k = j;
335	tim	1616	tmp = w[k];
336			for (i=j+1; i<n; i++) // boundary incorrect, shifted already
337			{
338			if (w[i] >= tmp) // why exchage if same?
339			{
340	tim	1586	k = i;
341	tim	1616	tmp = w[k];
342	tim	1586	}
343	tim	1616	}
344			if (k != j)
345			{
346			w[k] = w[j];
347			w[j] = tmp;
348			for (i=0; i<n; i++)
349			{
350			tmp = v(i, j);
351			v(i, j) = v(i, k);
352			v(i, k) = tmp;
353			}
354			}
355	tim	1586	}
356	tim	1616	// insure eigenvector consistency (i.e., Jacobi can compute vectors that
357			// are negative of one another (.707,.707,0) and (-.707,-.707,0). This can
358			// reek havoc in hyperstreamline/other stuff. We will select the most
359			// positive eigenvector.
360			int ceil_half_n = (n >> 1) + (n & 1);
361			for (j=0; j<n; j++)
362			{
363			for (numPos=0, i=0; i<n; i++)
364			{
365			if ( v(i, j) >= 0.0 )
366			{
367			numPos++;
368	tim	1586	}
369	tim	1616	}
370			// if ( numPos < ceil(double(n)/double(2.0)) )
371			if ( numPos < ceil_half_n)
372			{
373			for(i=0; i<n; i++)
374			{
375			v(i, j) *= -1.0;
376			}
377			}
378	tim	1586	}
379	tim	1569
380	tim	1616	if (n > 4)
381			{
382			delete [] b;
383			delete [] z;
384			}
385			return 1;
386	tim	1569	}
387
388
389			}
390	tim	1616	#endif //MATH_SQUAREMATRIX_HPP
391	tim	1569