Posted on 2006-06-04 20:22
Tauruser 阅读(3893)
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数值计算
3.2 Gauss消去法
3.2.1 顺序消去法
Gauss消去法就是将方程组(3.1.1)通过(n-1)步消元,将(3.1.1)转化为上三角方程组
(3.2.1)
再回代求此方程组的解.
下面记增广矩阵
,即

第1步 设
,计算
l
,记为
,若用
乘
第一行加到第i行,可消去
,用Gauss变换矩阵表示

令 
其中
一般地,假定已完成了(k-1)步消元,即已将
转化为以下形式:

第k步,假定
,计算
(3.2.2)
记
,
,则

其中
(3.2.3).
当k=1,2,…,n-1则可得到
,即方程组(3.2.1).
直接回代解(3.2.1)得,
(3.2.4)
并且有
,由以上顺序消去过程可得如下定理.
定理2.1 设
非奇异,则通过两行互换总可使
,k=1,2,…,n-1.可将方程组(3.1.1)转化为(3.2.1)并求得方程组(3.1.1)的解为(3.2.4),且有
.
如果不做行交换,则使
的条件如下.
定理2.2
非奇异,且各阶顺序主子式
, 则
,k=1,2,…,n-1.
证明 用归纳法,当
,故
.现假设(k-1)成立,即
,对i=1,2,…,k-1已推出
,故Gauss消去法能进行(k-1)步消元,A已约化为
,即

故
对k=1,2,…,n均成立,证毕.
在整个消去法消元过程中,k从1到(n-1)共需乘除法运算次数为

加减法次数为

回代过程中由公式(3.2.4)可知乘除法次数为
,加减法次数为
,于是Gauss消去法的乘除法总次数为
,加减法次数为
例3.4 用Gauss消去法解方程组

并求detA.
解 消元得

再由(3.2.4)回代,得解
讲解:
Gauss 消去法是将方程组AX=b,通过消元转化为上三角方程组(3,2,1)求解,消元第一步做完后有

用矩阵表示
第K-1步完成后得到
当
,可做K步,得到
得到
,对应的方程组就是(3.2.1),利用公式(3.2.4)就可求得解。
定理2.2给出了进行顺序消去法的条件,即A的所有顺序生子式
,而方程(3.1.1)解存在唯一的条件是
好了,原理讲完了,贴我的例程。
#include
<
iostream
>
#include
<
vector
>
#include
<
cmath
>
using
namespace
std;
class
CGAUSSSOLVEEQU

{
private
:
vector
<
vector
<
double
>>
m_equset;
vector
<
double
>
m_answer;
unsigned
int
m_n;
public
:
void
inputEquSet();
void
solveEquSet();
void
outputAnswer();
}
;
void
CGAUSSSOLVEEQU::inputEquSet()

{
double
dtemp;
vector
<
double
>
vtemp;
cout
<<
"
请输入你的方程个数:
"
;
cin
>>
m_n;
cout
<<
"
请按照向量的形式输入各变量的系数。最后一位为b。每个方程一行:
"
<<
endl;
for
(unsigned
int
i(
0
);i
<
m_n;
++
i)

{
m_equset.push_back(vtemp);
for
(unsigned
int
j(
0
);j
<=
m_n;
++
j)

{
cin
>>
dtemp;
m_equset[i].push_back(dtemp);
}
if
(m_equset[i].size()
!=
m_n
+
1
)

{
cout
<<
"
输入有误,请重新输入上一个方程。
"
<<
endl;
--
i;
}
}
}
void
CGAUSSSOLVEEQU::solveEquSet()

{
vector
<
vector
<
double
>>
::iterator iter;
iter
=
m_equset.begin();
for
(unsigned
int
m(
0
);m
<
m_n
-
1
;
++
m)

{
//
将绝对值最大的主元素移上去。此举是为了减少误差
for
(vector
<
vector
<
double
>>
::iterator iter2
=
iter
+
1
;iter2
!=
m_equset.end();
++
iter2)

{
if
(fabsl(iter
->
front())
<
fabsl(iter2
->
front()))

{
swap(
*
iter,
*
iter2);
}
}
//
进行消元
for
(unsigned
int
i
=
m
+
1
;i
<
m_n;
++
i)

{
double
dm;
dm
=
m_equset[i][m]
/
m_equset[m][m];
for
(unsigned
int
j
=
m;j
<
m_n
+
1
;
++
j)

{
m_equset[i][j]
-=
dm
*
m_equset[m][j];
}
}
++
iter;
}
//
初始化m_answer向量
for
(unsigned
int
i(
0
);i
<
m_n;
++
i) m_answer.push_back(
0
);
//
求解答案
m_answer[m_n
-
1
]
=
m_equset[m_n
-
1
][m_n]
/
m_equset[m_n
-
1
][m_n
-
1
];

for
(
int
i
=
m_n
-
2
;i
>=
0
;
--
i)

{
m_answer[i]
=
m_equset[i][m_n];
for
(
int
j
=
m_n
-
1
;j
>
i;
--
j)
m_answer[i]
-=
m_answer[j]
*
m_equset[i][j];
m_answer[i]
/=
m_equset[i][i];
}
}
void
CGAUSSSOLVEEQU::outputAnswer()

{
for
(unsigned
int
i(
1
);i
<=
m_n;
++
i)

{
cout
<<
"
x(
"
<<
i
<<
"
)=
"
<<
m_answer[i
-
1
]
<<
endl;
}
}
int
main()

{
CGAUSSSOLVEEQU myEqu;
myEqu.inputEquSet();
myEqu.solveEquSet();
myEqu.outputAnswer();
return
0
;
}
//
Power By Tauruser 2006.6.4