有理数系的线性代数运算程序(2) 线性方程组
要解一个线性方程组： \( AX=B \) ,其中
\[A=\begin{bmatrix}a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{bmatrix} ,B = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \]
首先将A化为上三角矩阵，然后逐行将A化成单位阵，在这个变换过程中，B随A的变换同时等价变换，当A是E时，就可以得出运算结果。
private static void swapLine(ref List<RationalNumber> Mat, ref List<RationalNumber> Vect,int N,int k1,int k2)
{
RationalNumber tmp;
tmp = Vect[k1];
Vect[k1] = Vect[k2];
Vect[k2] = tmp;
for (int i=0;i<N;i++)
{
tmp = Mat[k1 * N + i];
Mat[k1 * N + i] = Mat[k2 * N + i];
Mat[k2 * N + i] = tmp;
}
}
private static void upperTriangularWithAugmented_Line(ref List<RationalNumber> Mat, ref List<RationalNumber> Vect, int N, int i)
{
int k1, k2;
for (k1 = i + 1; k1 < N; k1++)
{
RationalNumber rate = Mat[k1 * N + i] / Mat[i * N + i];
Vect[k1] = Vect[k1] - rate * Vect[i];
for (k2 = i; k2 < N; k2++)
{
Mat[k1 * N + k2] = Mat[k1 * N + k2] - rate * Mat[i * N + k2];
}
}
}
private static bool upperTriangularWithAugmented(ref List<RationalNumber> Mat, ref List<RationalNumber> Vect,int N)
{
bool allLineZero = true;
for(int i=0;i<N;i++)
{
for(int k1=i;k1<N;k1++)
{
if (!(Mat[k1 * N + i] == 0))
{
swapLine(ref Mat, ref Vect, N, k1, i);
allLineZero = false;
break;
}
}
if (allLineZero)
{
break;
}
else
{
upperTriangularWithAugmented_Line(ref Mat,ref Vect, N, i);
}
}
return allLineZero;
}
private static string Mat2LaTeX(ref List<RationalNumber> Mat, ref List<RationalNumber> Vect,int N)
{
string msg = @"\pmatrix{";
for(int i=0;i<N;i++)
{
if(i>0)
{
msg += @"\\";
}
for(int j=0;j<N;j++)
{
if(j>0)
{
msg += @"&";
}
msg += Mat[i * N + j].ToLaTeX();
}
}
msg += "}X=";
msg += @"\pmatrix{";
for(int i=0;i<N;i++)
{
if (i > 0)
msg += @"\\";
msg += Vect[i].ToLaTeX();
}
msg += "}";
return msg;
}
private static string Vect2LaTeX(ref List<RationalNumber> Vect,int N)
{
string msg = @"\pmatrix{";
for(int i=0;i<N;i++)
{
if(i>0)
{
msg += @"\\";
}
msg += Vect[i].ToLaTeX();
}
msg += "}";
return msg;
}
private static void identityMatrix(ref List<RationalNumber> Mat, ref List<RationalNumber> Vect,int N)
{
int i, j;
for(i=N-1;i>=0;i--)
{
RationalNumber rate = 1 / Mat[i * N + i];
Vect[i] = rate * Vect[i];
Mat[i * N + i] = new RationalNumber(1);
for( j=0;j<i;j++)
{
Vect[j] = Vect[j] - Vect[i] * Mat[j * N + i];
Mat[j * N + i] = new RationalNumber(0);
}
}
}
public static string linearEquation(ref List<RationalNumber> Mat,ref List<RationalNumber> Vect)
{
int N = 1;
while(Mat.Count>=(N+1)*(N+1))
{
N++;
}
if (Vect.Count >= N)
{
upperTriangularWithAugmented(ref Mat, ref Vect,N);
string latex_line1 = Mat2LaTeX(ref Mat, ref Vect, N);
identityMatrix(ref Mat, ref Vect, N);
Vect2LaTeX(ref Vect, N);
return latex_line1 +",X="+ Vect2LaTeX(ref Vect, N); ;
}
else
{
return @"SyntexError";
}
}计算结果返回的是LaTex形式。