C++实现矩阵求逆

  1 #ifndef __MATRIX_H__
  2 #define __MATRIX_H__
  3 
  4 #pragma once
  5 
  6 #include <iostream>
  7 #include <fstream>
  8 #include <sstream>
  9 #include <vector>
 10 #include <string>
 11 
 12 using std::vector;
 13 using std::string;
 14 using std::cout;
 15 using std::cin;
 16 using std::istream;
 17 using std::ostream;
 18 
 19 // 任意类型矩阵类
 20 template <typename Object>
 21 class MATRIX
 22 {
 23 public:
 24     explicit MATRIX() : array( 0 ) {}
 25 
 26     MATRIX( int rows, int cols):array( rows )
 27     {
 28         for( int i = 0; i < rows; ++i )
 29         {
 30             array[i].resize( cols );
 31         }
 32     }
 33 
 34     MATRIX( const MATRIX<Object>& m ){ *this = m;}
 35 
 36     void resize( int rows, int cols );           // 改变当前矩阵大小
 37     bool push_back( const vector<Object>& v );   // 在矩阵末尾添加一行数据
 38     void swap_row( int row1, int row2 );         // 将换两行的数据
 39 
 40     int  rows() const{ return array.size(); }
 41     int  cols() const { return rows() ? (array[0].size()) : 0; }
 42     bool empty() const { return rows() == 0; }        // 是否为空
 43     bool square() const { return (!(empty()) && rows() == cols()); }  // 是否为方阵
 44     
 45     
 46     const vector<Object>& operator[](int row) const { return array[row]; } //[]操作符重载 
 47     vector<Object>& operator[](int row){ return array[row]; }
 48     
 49 protected:
 50     vector< vector<Object> > array;
 51 };
 52 
 53 // 改变当前矩阵大小
 54 template <typename Object>
 55 void MATRIX<Object>::resize( int rows, int cols )
 56 {
 57     int rs = this->rows();
 58     int cs = this->cols();
 59 
 60     if ( rows == rs && cols == cs )
 61     {
 62         return;
 63     }
 64     else if ( rows == rs && cols != cs )
 65     {
 66         for ( int i = 0; i < rows; ++i )
 67         {
 68             array[i].resize( cols );
 69         }
 70     }
 71     else if ( rows != rs && cols == cs )
 72     {
 73         array.resize( rows );
 74         for ( int i = rs; i < rows; ++i )
 75         {
 76             array[i].resize( cols );
 77         }
 78     }
 79     else
 80     {
 81         array.resize( rows );
 82         for ( int i = 0; i < rows; ++i )
 83         {
 84             array[i].resize( cols );
 85         }
 86     }
 87 }
 88 
 89 // 在矩阵末尾添加一行
 90 template <typename Object>
 91 bool MATRIX<Object>::push_back( const vector<Object>& v )
 92 {
 93     if ( rows() == 0 || cols() == (int)v.size() )
 94     {
 95         array.push_back( v );
 96     }
 97     else
 98     {
 99         return false;
100     }
101 
102     return true;
103 }
104 
105 // 将换两行
106 template <typename Object>
107 void MATRIX<Object>::swap_row( int row1, int row2 )
108 {
109     if ( row1 != row2 && row1 >=0 &&
110         row1 < rows() && row2 >= 0 && row2 < rows() )
111     {
112         vector<Object>& v1 = array[row1];
113         vector<Object>& v2 = array[row2];
114         vector<Object> tmp = v1;
115         v1 = v2;
116         v2 = tmp;
117     }
118 }
119 
120 // 矩阵转置
121 template <typename Object>
122 const MATRIX<Object> trans( const MATRIX<Object>& m )
123 {
124     MATRIX<Object> ret;
125     if ( m.empty() ) return ret;
126 
127     int row = m.cols();
128     int col = m.rows();
129     ret.resize( row, col );
130 
131     for ( int i = 0; i < row; ++i )
132     {
133         for ( int j = 0; j < col; ++j )
134         {
135             ret[i][j] = m[j][i];
136         }
137     }
138 
139     return ret;
140 }
141 
142 //////////////////////////////////////////////////////////
143 // double类型矩阵类，用于科学计算
144 // 继承自MATRIX类
145 // 实现常用操作符重载，并实现计算矩阵的行列式、逆以及LU分解
146 class Matrix:public MATRIX<double>
147 {
148 public:
149     Matrix():MATRIX<double>(){}
150     Matrix( int c, int r ):MATRIX<double>(c,r){}
151     Matrix( const Matrix& m){ *this  = m; }
152 
153     const Matrix& operator+=( const Matrix& m );
154     const Matrix& operator-=( const Matrix& m );
155     const Matrix& operator*=( const Matrix& m );
156     const Matrix& operator/=( const Matrix& m );
157 };
158 
159 bool  operator==( const Matrix& lhs, const Matrix& rhs );        // 重载操作符==
160 bool  operator!=( const Matrix& lhs, const Matrix& rhs );        // 重载操作符!=
161 const Matrix operator+( const Matrix& lhs, const Matrix& rhs );  // 重载操作符+
162 const Matrix operator-( const Matrix& lhs, const Matrix& rhs );  // 重载操作符-
163 const Matrix operator*( const Matrix& lhs, const Matrix& rhs );  // 重载操作符*
164 const Matrix operator/( const Matrix& lhs, const Matrix& rhs );  // 重载操作符/
165 const double det( const Matrix& m );                             // 计算行列式
166 const double det( const Matrix& m, int start, int end );         // 计算子矩阵行列式
167 const Matrix abs( const Matrix& m );                             // 计算所有元素的绝对值
168 const double max( const Matrix& m );                             // 所有元素的最大值
169 const double max( const Matrix& m, int& row, int& col);          // 所有元素中的最大值及其下标
170 const double min( const Matrix& m );                             // 所有元素的最小值
171 const double min( const Matrix& m, int& row, int& col);          // 所有元素的最小值及其下标
172 const Matrix trans( const Matrix& m );                           // 返回转置矩阵
173 const Matrix submatrix(const Matrix& m,int rb,int re,int cb,int ce);  // 返回子矩阵
174 const Matrix inverse( const Matrix& m );                         // 计算逆矩阵
175 const Matrix LU( const Matrix& m );                              // 计算方阵的LU分解
176 const Matrix readMatrix( istream& in = std::cin );               // 从指定输入流读入矩阵
177 const Matrix readMatrix( string file );                          // 从文本文件读入矩阵
178 const Matrix loadMatrix( string file );                          // 从二进制文件读取矩阵
179 void  printMatrix( const Matrix& m, ostream& out = std::cout );  // 从指定输出流打印矩阵
180 void  printMatrix( const Matrix& m, string file);                // 将矩阵输出到文本文件
181 void  saveMatrix( const Matrix& m, string file);                 // 将矩阵保存为二进制文件
182 
183 
184 #endif 

  1 #include "Matrix.h"  
  2 #include <iomanip>  //用于设置输出格式  
  3   
  4 using std::ifstream;  
  5 using std::ofstream;  
  6 using std::istringstream;  
  7 using std::cerr;  
  8 using std::endl;  
  9   
 10 const Matrix& Matrix::operator+=( const Matrix& m )  
 11 {  
 12     if ( rows() != m.rows() || rows() != m.cols() )  
 13     {  
 14         return *this;  
 15     }  
 16   
 17     int r = rows();  
 18     int c = cols();  
 19   
 20     for ( int i = 0; i < r; ++i )  
 21     {  
 22         for ( int j = 0; j < c; ++j )  
 23         {  
 24             array[i][j] += m[i][j];  
 25         }  
 26     }  
 27   
 28     return *this;  
 29 }  
 30   
 31   
 32 const Matrix& Matrix::operator-=( const Matrix& m )  
 33 {  
 34     if ( rows() != m.rows() || cols() != m.cols() )  
 35     {  
 36         return *this;  
 37     }  
 38   
 39     int r = rows();  
 40     int c = cols();  
 41   
 42     for ( int i = 0; i < r; ++i )  
 43     {  
 44         for ( int j = 0; j < c; ++j )  
 45         {  
 46             array[i][j] -= m[i][j];  
 47         }  
 48     }  
 49   
 50     return *this;  
 51 }  
 52   
 53 const Matrix& Matrix::operator*=( const Matrix& m )  
 54 {  
 55     if ( cols() != m.rows() || !m.square() )  
 56     {  
 57         return *this;  
 58     }  
 59   
 60     Matrix ret( rows(), cols() );  
 61   
 62     int r = rows();  
 63     int c = cols();  
 64   
 65     for ( int i = 0; i < r; ++i )  
 66     {  
 67         for ( int j = 0; j < c; ++j )  
 68         {  
 69             double sum = 0.0;  
 70             for ( int k = 0; k < c; ++k )  
 71             {  
 72                 sum += array[i][k] * m[k][j];  
 73             }  
 74             ret[i][j] = sum;  
 75         }  
 76     }  
 77   
 78     *this = ret;  
 79     return *this;  
 80 }  
 81   
 82 const Matrix& Matrix::operator/=( const Matrix& m )  
 83 {  
 84     Matrix tmp = inverse( m );  
 85     return operator*=( tmp );  
 86 }  
 87   
 88   
 89 bool operator==( const Matrix& lhs, const Matrix& rhs )  
 90 {  
 91     if ( lhs.rows() != rhs.rows() || lhs.cols() != rhs.cols() )  
 92     {  
 93         return false;  
 94     }  
 95   
 96     for ( int i = 0; i < lhs.rows(); ++i )  
 97     {  
 98         for ( int j = 0; j < lhs.cols(); ++j )  
 99         {  
100             if ( rhs[i][j] != rhs[i][j] )  
101             {  
102                 return false;  
103             }  
104         }  
105     }  
106   
107     return true;  
108 }  
109   
110 bool operator!=( const Matrix& lhs, const Matrix& rhs )  
111 {  
112     return !( lhs == rhs );  
113 }  
114   
115 const Matrix operator+( const Matrix& lhs, const Matrix& rhs )  
116 {  
117     Matrix m;  
118     if ( lhs.rows() != rhs.rows() || lhs.cols() != rhs.cols() )  
119     {  
120         return m;  
121     }  
122   
123     m = lhs;  
124     m += rhs;  
125   
126     return m;  
127 }  
128   
129 const Matrix operator-( const Matrix& lhs, const Matrix& rhs )  
130 {  
131     Matrix m;  
132     if ( lhs.rows() != rhs.rows() || lhs.cols() != rhs.cols() )  
133     {  
134         return m;  
135     }  
136   
137     m = lhs;  
138     m -= rhs;  
139   
140     return m;  
141 }  
142   
143 const Matrix operator*( const Matrix& lhs, const Matrix& rhs )  
144 {  
145     Matrix m;  
146     if ( lhs.cols() != rhs.rows() )  
147     {  
148         return m;  
149     }  
150   
151     m.resize( lhs.rows(), rhs.cols() );  
152   
153     int r = m.rows();  
154     int c = m.cols();  
155     int K = lhs.cols();  
156   
157     for ( int i = 0; i < r; ++i )  
158     {  
159         for ( int j = 0; j < c; ++j )  
160         {  
161             double sum = 0.0;  
162             for ( int k = 0; k < K; ++k )  
163             {  
164                 sum += lhs[i][k] * rhs[k][j];  
165             }  
166             m[i][j] = sum;  
167         }  
168     }  
169   
170     return m;  
171 }  
172   
173 const Matrix operator/( const Matrix& lhs, const Matrix& rhs )  
174 {  
175     Matrix tmp = inverse( rhs );  
176     Matrix m;  
177   
178     if ( tmp.empty() )  
179     {  
180         return m;  
181     }  
182   
183     return m = lhs * tmp;  
184 }  
185   
186 inline static double LxAbs( double d )  
187 {  
188     return (d>=0)?(d):(-d);  
189 }  
190   
191 inline   
192 static bool isSignRev( const vector<double>& v )  
193 {  
194     int p = 0;  
195     int sum = 0;  
196     int n = (int)v.size();  
197   
198     for ( int i = 0; i < n; ++i )  
199     {  
200         p = (int)v[i];  
201         if ( p >= 0 )  
202         {  
203             sum += p + i;  
204         }  
205     }  
206   
207     if ( sum % 2 == 0 ) // 如果是偶数，说明不变号  
208     {  
209         return false;  
210     }  
211     return true;  
212 }  
213   
214 // 计算方阵行列式  
215 const double det( const Matrix& m )  
216 {  
217     double ret = 0.0;  
218   
219     if ( m.empty() || !m.square() ) return ret;  
220   
221     Matrix N = LU( m );  
222   
223     if ( N.empty() ) return ret;  
224   
225     ret = 1.0;  
226     for ( int i = 0; i < N.cols(); ++ i )  
227     {  
228         ret *= N[i][i];  
229     }  
230   
231     if ( isSignRev( N[N.rows()-1] ))  
232     {  
233         return -ret;  
234     }  
235   
236     return ret;  
237 }  
238   
239 // 计算矩阵指定子方阵的行列式   
240 const double det( const Matrix& m, int start, int end )  
241 {  
242     return det( submatrix(m, start, end, start, end) );  
243 }  
244   
245   
246 // 计算矩阵转置  
247 const Matrix trans( const Matrix& m )  
248 {  
249     Matrix ret;  
250     if ( m.empty() ) return ret;  
251   
252     int r = m.cols();  
253     int c = m.rows();  
254   
255     ret.resize(r, c);  
256     for ( int i = 0; i < r; ++i )  
257     {  
258         for ( int j = 0; j < c; ++j )  
259         {  
260             ret[i][j] = m[j][i];  
261         }  
262     }  
263   
264     return ret;  
265 }  
266   
267 // 计算逆矩阵  
268 const Matrix  inverse( const Matrix& m )   
269 {  
270     Matrix ret;  
271   
272     if ( m.empty() || !m.square() )  
273     {  
274         return ret;  
275     }  
276   
277     int n = m.rows();  
278   
279     ret.resize( n, n );  
280     Matrix A(m);  
281   
282     for ( int i = 0; i < n; ++i ) ret[i][i] = 1.0;  
283   
284     for ( int j = 0; j < n; ++j )  //每一列  
285     {  
286         int p = j;  
287         double maxV = LxAbs(A[j][j]);  
288         for ( int i = j+1; i < n; ++i )  // 找到第j列中元素绝对值最大行  
289         {  
290             if ( maxV < LxAbs(A[i][j]) )  
291             {  
292                 p = i;  
293                 maxV = LxAbs(A[i][j]);  
294             }  
295         }  
296   
297         if ( maxV < 1e-20 )   
298         {  
299             ret.resize(0,0);  
300             return ret;  
301         }  
302   
303         if ( j!= p )  
304         {  
305             A.swap_row( j, p );  
306             ret.swap_row( j, p );  
307         }  
308           
309         double d = A[j][j];  
310         for ( int i = j; i < n; ++i ) A[j][i] /= d;  
311         for ( int i = 0; i < n; ++i ) ret[j][i] /= d;  
312   
313         for ( int i = 0; i < n; ++i )  
314         {  
315             if ( i != j )  
316             {  
317                 double q = A[i][j];  
318                 for ( int k = j; k < n; ++k )  
319                 {  
320                     A [i][k] -= q * A[j][k];  
321                 }  
322                 for ( int k = 0; k < n; ++k )  
323                 {  
324                     ret[i][k] -= q * ret[j][k];  
325                 }  
326             }  
327         }  
328     }  
329   
330     return ret;  
331 }  
332   
333 // 计算绝对值  
334 const Matrix abs( const Matrix& m )  
335 {  
336     Matrix ret;  
337   
338     if( m.empty() )  
339     {  
340         return ret;  
341     }  
342   
343     int r = m.rows();  
344     int c = m.cols();  
345     ret.resize( r, c );  
346   
347     for ( int i = 0; i < r; ++i )  
348     {  
349         for ( int j = 0; j < c; ++j )  
350         {  
351             double t = m[i][j];  
352             if ( t < 0 ) ret[i][j] = -t;  
353             else ret[i][j] = t;  
354         }  
355     }  
356   
357     return ret;  
358 }  
359   
360 // 返回矩阵所有元素的最大值  
361 const double max( const Matrix& m )  
362 {  
363     if ( m.empty() ) return 0.;  
364       
365     double ret = m[0][0];  
366     int r = m.rows();  
367     int c = m.cols();  
368   
369     for ( int i = 0; i < r; ++i )  
370     {  
371         for ( int j = 0; j < c; ++j )  
372         {  
373             if ( m[i][j] > ret ) ret = m[i][j];  
374         }  
375     }  
376     return ret;  
377 }  
378   
379 // 计算矩阵最大值，并返回该元素的引用  
380 const double max( const Matrix& m, int& row, int& col )  
381 {  
382     if ( m.empty() ) return 0.;  
383   
384     double ret = m[0][0];  
385     row = 0;  
386     col = 0;  
387   
388     int r = m.rows();  
389     int c = m.cols();  
390   
391     for ( int i = 0; i < r; ++i )  
392     {  
393         for ( int j = 0; j < c; ++j )  
394         {  
395             if ( m[i][j] > ret )  
396             {  
397                 ret = m[i][j];  
398                 row = i;  
399                 col = j;  
400             }  
401         }  
402     }  
403     return ret;  
404 }  
405   
406 // 计算矩阵所有元素最小值  
407 const double min( const Matrix& m )  
408 {  
409     if ( m.empty() ) return 0.;  
410   
411     double ret = m[0][0];  
412     int r = m.rows();  
413     int c = m.cols();  
414   
415     for ( int i = 0; i < r; ++i )  
416     {  
417         for ( int j = 0; j < c; ++j )  
418         {  
419             if ( m[i][j] > ret ) ret = m[i][j];  
420         }  
421     }  
422   
423     return ret;  
424 }  
425   
426 // 计算矩阵最小值，并返回该元素的引用  
427 const double min( const Matrix& m, int& row, int& col)  
428 {  
429     if ( m.empty() ) return 0.;  
430   
431     double ret = m[0][0];  
432     row = 0;  
433     col = 0;  
434     int r = m.rows();  
435     int c = m.cols();  
436   
437     for ( int i = 0; i < r; ++i )  
438     {  
439         for ( int j = 0; j < c; ++j )  
440         {  
441             if ( m[i][j] > ret )  
442             {  
443                 ret = m[i][j];  
444                 row = i;  
445                 col = j;  
446             }  
447         }  
448     }  
449   
450     return ret;  
451 }  
452   
453 // 取矩阵中指定位置的子矩阵   
454 const Matrix submatrix(const Matrix& m,int rb,int re,int cb,int ce)  
455 {  
456     Matrix ret;  
457     if ( m.empty() ) return ret;  
458   
459     if ( rb < 0 || re >= m.rows() || rb > re ) return ret;  
460     if ( cb < 0 || ce >= m.cols() || cb > ce ) return ret;  
461   
462     ret.resize( re-rb+1, ce-cb+1 );  
463   
464     for ( int i = rb; i <= re; ++i )  
465     {  
466         for ( int j = cb; j <= ce; ++j )  
467         {  
468             ret[i-rb][j-cb] = m[i][j];  
469         }  
470     }  
471   
472     return ret;  
473 }  
474   
475   
476 inline static  
477 int max_idx( const Matrix& m, int k, int n )  
478 {  
479     int p = k;  
480     for ( int i = k+1; i < n; ++i )  
481     {  
482         if ( LxAbs(m[p][k]) < LxAbs(m[i][k]) )  
483         {  
484             p = i;  
485         }  
486     }  
487     return p;  
488 }  
489   
490 // 计算方阵 M 的 LU 分解  
491 // 其中L为对角线元素全为1的下三角阵，U为对角元素依赖M的上三角阵  
492 // 使得 M = LU  
493 // 返回矩阵下三角部分存储L(对角元素除外)，上三角部分存储U(包括对角线元素)  
494 const Matrix LU( const Matrix& m )  
495 {  
496     Matrix ret;  
497   
498     if ( m.empty() || !m.square() ) return ret;  
499   
500     int n = m.rows();  
501     ret.resize( n+1, n );  
502   
503     for ( int i = 0; i < n; ++i )  
504     {  
505         ret[n][i] = -1.0;  
506     }  
507   
508     for ( int i = 0; i < n; ++i )  
509     {  
510         for ( int j = 0; j < n; ++j )  
511         {  
512             ret[i][j] = m[i][j];  
513         }  
514     }  
515   
516     for ( int k = 0; k < n-1; ++k )  
517     {  
518         int p = max_idx( ret, k, n );  
519         if ( p != k )              // 进行行交换  
520         {  
521             ret.swap_row( k, p );  
522             ret[n][k] = (double)p; // 记录将换信息  
523         }  
524   
525         if ( ret[k][k] == 0.0 )  
526         {  
527             cout << "ERROR: " << endl;  
528             ret.resize(0,0);  
529             return ret;  
530         }  
531   
532         for ( int i = k+1; i < n; ++i )  
533         {  
534             ret[i][k] /= ret[k][k];  
535             for ( int j = k+1; j < n; ++j )  
536             {  
537                 ret[i][j] -= ret[i][k] * ret[k][j];  
538             }  
539         }  
540     }  
541   
542     return ret;  
543 }  
544   
545 //---------------------------------------------------  
546 //                      读取和打印  
547 //---------------------------------------------------  
548 // 从输入流读取矩阵  
549 const Matrix readMatrix( istream& in )  
550 {  
551     Matrix M;  
552     string str;  
553     double b;  
554     vector<double> v;  
555   
556     while( getline( in, str )  )  
557     {  
558         for ( string::size_type i = 0; i < str.size(); ++i )  
559         {  
560             if ( str[i] == ‘,‘ || str[i] == ‘;‘)  
561             {  
562                 str[i] = ‘ ‘;  
563             }  
564             else if ( str[i] != ‘.‘ && (str[i] < ‘0‘ || str[i] > ‘9‘)   
565                 && str[i] != ‘ ‘ && str[i] != ‘\t‘ && str[i] != ‘-‘)  
566             {  
567                 M.resize(0,0);  
568                 return M;  
569             }  
570         }  
571   
572         istringstream sstream(str);  
573         v.resize(0);  
574   
575         while ( sstream >> b )  
576         {  
577             v.push_back(b);  
578         }  
579         if ( v.size() == 0 )  
580         {  
581             continue;  
582         }  
583         if ( !M.push_back( v ) )  
584         {  
585             M.resize( 0, 0 );  
586             return M;  
587         }  
588     }  
589   
590     return M;  
591 }  
592   
593 // 从文本文件读入矩阵  
594 const Matrix readMatrix( string file )  
595 {  
596     ifstream fin( file.c_str() );  
597     Matrix M;  
598   
599     if ( !fin )  
600     {  
601         cerr << "Error: open file " << file << " failed." << endl;  
602         return M;  
603     }  
604   
605     M = readMatrix( fin );  
606     fin.close();  
607   
608     return M;  
609 }  
610   
611 // 将矩阵输出到指定输出流  
612 void printMatrix( const Matrix& m, ostream& out )  
613 {  
614     if ( m.empty() )  
615     {  
616         return;  
617     }  
618   
619     int r = m.rows();  
620     int c = m.cols();  
621   
622     int n = 0;              // 数据小数点前最大位数  
623     double maxV = max(abs(m));  
624     while( maxV >= 1.0 )  
625     {  
626         maxV /= 10;  
627         ++n;  
628     }  
629     if ( n == 0 ) n = 1;    // 如果最大数绝对值小于1，这小数点前位数为1，为数字0  
630     int pre = 4;            // 小数点后数据位数  
631     int wid = n + pre + 3;  // 控制字符宽度=n+pre+符号位+小数点位  
632   
633     out<<std::showpoint;  
634     out<<std::setiosflags(std::ios::fixed);  
635     out<<std::setprecision( pre );  
636     for ( int i = 0; i < r; ++i )  
637     {  
638         for ( int j = 0; j < c; ++j )  
639         {  
640             out<<std::setw(wid) << m[i][j];  
641         }  
642         out << endl;  
643     }  
644   
645     out<<std::setprecision(6);  
646     out<<std::noshowpoint;  
647 }  
648   
649 // 将矩阵打印到指定文件   
650 void printMatrix( const Matrix& m, string file )  
651 {  
652     ofstream fout( file.c_str() );  
653     if ( !fout ) return;  
654   
655     printMatrix( m, fout );  
656     fout.close();  
657 }  
658   
659 // 将矩阵数据存为二进制文件   
660 void saveMatrix( const Matrix& m, string file )  
661 {  
662     if ( m.empty() ) return;  
663   
664     ofstream fout(file.c_str(), std::ios_base::out|std::ios::binary );  
665     if ( !fout ) return;  
666   
667     int r = m.rows();  
668     int c = m.cols();  
669     char Flag[12] = "MATRIX_DATA";  
670     fout.write( (char*)&Flag, sizeof(Flag) );  
671     fout.write( (char*)&r, sizeof(r) );  
672     fout.write( (char*)&c, sizeof(c) );  
673   
674     for ( int i = 0; i < r; ++i )  
675     {  
676         for ( int j = 0; j < c; ++j )  
677         {  
678             double t = m[i][j];  
679             fout.write( (char*)&t, sizeof(t) );  
680         }  
681     }  
682   
683     fout.close();  
684 }  
685   
686 // 从二进制文件load矩阵  
687 const Matrix loadMatrix( string file )  
688 {  
689     Matrix m;  
690   
691     ifstream fin( file.c_str(), std::ios_base::in|std::ios::binary );  
692     if ( !fin ) return m;  
693   
694     char Flag[12];  
695     fin.read((char*)&Flag, sizeof(Flag));  
696   
697     string str( Flag );  
698     if ( str != "MATRIX_DATA" )  
699     {  
700         return m;  
701     }  
702   
703     int r, c;  
704     fin.read( (char*)&r, sizeof(r) );  
705     fin.read( (char*)&c, sizeof(c) );  
706   
707     if ( r <=0 || c <=0 ) return m;  
708   
709     m.resize( r, c );  
710     double t;  
711   
712     for ( int i = 0; i < r; ++i )  
713     {  
714         for ( int j = 0; j < c; ++j )  
715         {  
716             fin.read( (char*)&t, sizeof(t) );  
717             m[i][j] = t;  
718         }  
719     }  
720   
721     return m;  
722 }