lequations.cpp

科学与工程数值算法求解方程组的类

// 4. CMatrix& mtxResultImag - CMatrix引用对象，返回方程组解矩阵的虚部矩阵
//
// 返回值：BOOL 型，方程组求解是否成功
//////////////////////////////////////////////////////////////////////
BOOL CLEquations::GetRootsetGaussJordan(const CMatrix& mtxCoefImag, const CMatrix& mtxConstImag, CMatrix& mtxResult, CMatrix& mtxResultImag)
{
	int *pnJs,l,k,i,j,nIs,u,v;
    double p,q,s,d;

	// 方程组的属性，将常数矩阵赋给解矩阵
	mtxResult = m_mtxConst;
	mtxResultImag = mtxConstImag;
	double *pDataCoef = m_mtxCoef.GetData();
	double *pDataConst = mtxResult.GetData();
	double *pDataCoefImag = mtxCoefImag.GetData();
	double *pDataConstImag = mtxResultImag.GetData();
	int n = GetNumUnknowns();
	int m = m_mtxConst.GetNumColumns();

	// 临时缓冲区，存放变换的列数
    pnJs = new int[n];

	// 消元
    for (k=0;k<=n-1;k++)
    { 
		d=0.0;
        for (i=k;i<=n-1;i++)
        {
			for (j=k;j<=n-1;j++)
			{ 
				u=i*n+j;
				p=pDataCoef[u]*pDataCoef[u]+pDataCoefImag[u]*pDataCoefImag[u];
				if (p>d) 
				{
					d=p;
					pnJs[k]=j;
					nIs=i;
				}
			}
        }
        
		// 求解失败
		if (d == 0.0)
        {
			delete[] pnJs;
            return FALSE;
        }
        
		if (nIs!=k)
        { 
			for (j=k;j<=n-1;j++)
            { 
				u=k*n+j; 
				v=nIs*n+j;
                p=pDataCoef[u]; 
				pDataCoef[u]=pDataCoef[v]; 
				pDataCoef[v]=p;
                p=pDataCoefImag[u]; 
				pDataCoefImag[u]=pDataCoefImag[v]; 
				pDataCoefImag[v]=p;
            }
            
			for (j=0;j<=m-1;j++)
            { 
				u=k*m+j; 
				v=nIs*m+j;
                p=pDataConst[u]; 
				pDataConst[u]=pDataConst[v]; 
				pDataConst[v]=p;
                p=pDataConstImag[u]; 
				pDataConstImag[u]=pDataConstImag[v]; 
				pDataConstImag[v]=p;
            }
        }
        
		if (pnJs[k]!=k)
        {
			for (i=0;i<=n-1;i++)
            { 
				u=i*n+k; 
				v=i*n+pnJs[k];
				p=pDataCoef[u]; 
				pDataCoef[u]=pDataCoef[v]; 
				pDataCoef[v]=p;
				p=pDataCoefImag[u]; 
				pDataCoefImag[u]=pDataCoefImag[v]; 
				pDataCoefImag[v]=p;
            }
		}

        v=k*n+k;
        for (j=k+1;j<=n-1;j++)
        { 
			u=k*n+j;
            p=pDataCoef[u]*pDataCoef[v]; 
			q=-pDataCoefImag[u]*pDataCoefImag[v];
            s=(pDataCoef[v]-pDataCoefImag[v])*(pDataCoef[u]+pDataCoefImag[u]);
            pDataCoef[u]=(p-q)/d; 
			pDataCoefImag[u]=(s-p-q)/d;
        }
        
		for (j=0;j<=m-1;j++)
        { 
			u=k*m+j;
            p=pDataConst[u]*pDataCoef[v]; 
			q=-pDataConstImag[u]*pDataCoefImag[v];
            s=(pDataCoef[v]-pDataCoefImag[v])*(pDataConst[u]+pDataConstImag[u]);
            pDataConst[u]=(p-q)/d; 
			pDataConstImag[u]=(s-p-q)/d;
        }
        
		for (i=0;i<=n-1;i++)
        {
			if (i!=k)
            { 
				u=i*n+k;
				for (j=k+1;j<=n-1;j++)
                { 
					v=k*n+j; 
					l=i*n+j;
					p=pDataCoef[u]*pDataCoef[v]; 
					q=pDataCoefImag[u]*pDataCoefImag[v];
					s=(pDataCoef[u]+pDataCoefImag[u])*(pDataCoef[v]+pDataCoefImag[v]);
					pDataCoef[l]=pDataCoef[l]-p+q;
					pDataCoefImag[l]=pDataCoefImag[l]-s+p+q;
                }
            
				for (j=0;j<=m-1;j++)
				{ 
					l=i*m+j; 
					v=k*m+j;
					p=pDataCoef[u]*pDataConst[v]; q=pDataCoefImag[u]*pDataConstImag[v];
					s=(pDataCoef[u]+pDataCoefImag[u])*(pDataConst[v]+pDataConstImag[v]);
					pDataConst[l]=pDataConst[l]-p+q; 
					pDataConstImag[l]=pDataConstImag[l]-s+p+q;
				}
            }
		}
    }

	// 求解调整
	for (k=n-1;k>=0;k--)
    {
		if (pnJs[k]!=k)
		{
			for (j=0;j<=m-1;j++)
			{ 
				u=k*m+j;
				v=pnJs[k]*m+j;
				p=pDataConst[u]; 
				pDataConst[u]=pDataConst[v]; 
				pDataConst[v]=p;
				p=pDataConstImag[u]; 
				pDataConstImag[u]=pDataConstImag[v]; 
				pDataConstImag[v]=p;
			}
		}
	}

	// 清理内存
	delete[] pnJs;

	return TRUE;
}

//////////////////////////////////////////////////////////////////////
// 求解三对角线方程组的追赶法
//
// 参数：
// 1. CMatrix& mtxResult - CMatrix引用对象，返回方程组解矩阵
//
// 返回值：BOOL 型，方程组求解是否成功
//////////////////////////////////////////////////////////////////////
BOOL CLEquations::GetRootsetTriDiagonal(CMatrix& mtxResult)
{ 
	int k,j;
    double s;
    
	// 将常数矩阵赋给解矩阵
	mtxResult = m_mtxConst;
	double *pDataConst = mtxResult.GetData();

	int n = GetNumUnknowns();
	ASSERT(m_mtxCoef.GetNumRows() == n);

	// 为系数矩阵对角线数组分配内存
	double* pDiagData = new double[3*n-2];

	// 构造系数矩阵对角线元素数组
	k = j = 0;
	if (k == 0)
	{
		pDiagData[j++] = m_mtxCoef.GetElement(k, k);
		pDiagData[j++] = m_mtxCoef.GetElement(k, k+1);
	}
    for (k=1; k<n-1; ++k)
	{
		pDiagData[j++] = m_mtxCoef.GetElement(k, k-1);
		pDiagData[j++] = m_mtxCoef.GetElement(k, k);
		pDiagData[j++] = m_mtxCoef.GetElement(k, k+1);
	}
	if (k == n-1)
	{
		pDiagData[j++] = m_mtxCoef.GetElement(k, k-1);
		pDiagData[j++] = m_mtxCoef.GetElement(k, k);
	}

	// 求解
	for (k=0;k<=n-2;k++)
    { 
		j=3*k; 
		s=pDiagData[j];

		// 求解失败
        if (fabs(s)+1.0==1.0)
		{
			delete[] pDiagData;
			return FALSE;
		}

        pDiagData[j+1]=pDiagData[j+1]/s;
        pDataConst[k]=pDataConst[k]/s;
        pDiagData[j+3]=pDiagData[j+3]-pDiagData[j+2]*pDiagData[j+1];
        pDataConst[k+1]=pDataConst[k+1]-pDiagData[j+2]*pDataConst[k];
    }
    
	s=pDiagData[3*n-3];
    if (s == 0.0)
	{
		delete[] pDiagData;
		return FALSE;
	}
    
	// 调整
	pDataConst[n-1]=pDataConst[n-1]/s;
    for (k=n-2;k>=0;k--)
		pDataConst[k]=pDataConst[k]-pDiagData[3*k+1]*pDataConst[k+1];
    
	// 释放内存
	delete[] pDiagData;

	return TRUE;
}

//////////////////////////////////////////////////////////////////////
// 一般带型方程组的求解
//
// 参数：
// 1. int nBandWidth - 带宽
// 2. CMatrix& mtxResult - CMatrix引用对象，返回方程组解矩阵
//
// 返回值：BOOL 型，方程组求解是否成功
//////////////////////////////////////////////////////////////////////
BOOL CLEquations::GetRootsetBand(int nBandWidth, CMatrix& mtxResult)
{ 
	int ls,k,i,j,is,u,v;
    double p,t;
    
	// 带宽必须为奇数
	if ((nBandWidth-1)%2 != 0)
		return FALSE;

	// 将常数矩阵赋给解矩阵
	mtxResult = m_mtxConst;
	double *pDataConst = mtxResult.GetData();

	// 方程组属性
	int m = m_mtxConst.GetNumColumns();
	int n = GetNumUnknowns();
	ASSERT(m_mtxCoef.GetNumRows() == n);

	// 带宽数组：带型矩阵的有效数据
	double* pBandData = new double[nBandWidth*n];

	// 半带宽
    ls = (nBandWidth-1)/2;

	// 构造带宽数组
    for (i=0; i<n; ++i)
	{
		j = 0;
		for (k=max(0, i-ls); k<max(0, i-ls)+nBandWidth; ++k)
		{
			if (k < n)
				pBandData[i*nBandWidth+j++] = m_mtxCoef.GetElement(i, k);
			else
				pBandData[i*nBandWidth+j++] = 0;
		}
	}

	// 求解
    for (k=0;k<=n-2;k++)
    { 
		p=0.0;
        for (i=k;i<=ls;i++)
        { 
			t=fabs(pBandData[i*nBandWidth]);
            if (t>p) 
			{
				p=t; 
				is=i;
			}
        }
        
		if (p == 0.0)
		{
			delete[] pBandData;
			return FALSE;
		}

        for (j=0;j<=m-1;j++)
        { 
			u=k*m+j; 
			v=is*m+j;
            t=pDataConst[u]; 
			pDataConst[u]=pDataConst[v]; 
			pDataConst[v]=t;
        }
        
		for (j=0;j<=nBandWidth-1;j++)
        { 
			u=k*nBandWidth+j; 
			v=is*nBandWidth+j;
            t=pBandData[u]; 
			pBandData[u]=pBandData[v]; 
			pBandData[v]=t;
        }
        
		for (j=0;j<=m-1;j++)
        { 
			u=k*m+j; 
			pDataConst[u]=pDataConst[u]/pBandData[k*nBandWidth];
		}
        
		for (j=1;j<=nBandWidth-1;j++)
        { 
			u=k*nBandWidth+j; 
			pBandData[u]=pBandData[u]/pBandData[k*nBandWidth];
		}
        
		for (i=k+1;i<=ls;i++)
        { 
			t=pBandData[i*nBandWidth];
            for (j=0;j<=m-1;j++)
            { 
				u=i*m+j; 
				v=k*m+j;
                pDataConst[u]=pDataConst[u]-t*pDataConst[v];
            }
            
			for (j=1;j<=nBandWidth-1;j++)
            { 
				u=i*nBandWidth+j; 
				v=k*nBandWidth+j;
                pBandData[u-1]=pBandData[u]-t*pBandData[v];
            }
            
			u=i*nBandWidth+nBandWidth-1; pBandData[u]=0.0;
        }
        
		if (ls!=(n-1)) 
			ls=ls+1;
    }
    
	p=pBandData[(n-1)*nBandWidth];
    if (p == 0.0)
	{
		delete[] pBandData;
		return FALSE;
	}

    for (j=0;j<=m-1;j++)
    { 
		u=(n-1)*m+j; 
		pDataConst[u]=pDataConst[u]/p;
	}
    
	ls=1;
    for (i=n-2;i>=0;i--)
    { 
		for (k=0;k<=m-1;k++)
        { 
			u=i*m+k;
            for (j=1;j<=ls;j++)
            { 
				v=i*nBandWidth+j; 
				is=(i+j)*m+k;
                pDataConst[u]=pDataConst[u]-pBandData[v]*pDataConst[is];
            }
        }
        
		if (ls!=(nBandWidth-1)) 
			ls=ls+1;
    }
    
	// 释放内存
	delete[] pBandData;

	return TRUE;
}

//////////////////////////////////////////////////////////////////////
// 求解对称方程组的分解法
//
// 参数：
// 1. CMatrix& mtxResult - CMatrix引用对象，返回方程组解矩阵
//
// 返回值：BOOL 型，方程组求解是否成功
//////////////////////////////////////////////////////////////////////
BOOL CLEquations::GetRootsetDjn(CMatrix& mtxResult)
{ 
	int i,j,l,k,u,v,w,k1,k2,k3;
    double p;

	// 方程组属性，将常数矩阵赋给解矩阵
	CMatrix mtxCoef = m_mtxCoef;
	mtxResult = m_mtxConst;
	int n = mtxCoef.GetNumColumns();
	int m = mtxResult.GetNumColumns();
	double* pDataCoef = mtxCoef.GetData();
	double* pDataConst = mtxResult.GetData();

	// 非对称系数矩阵，不能用本方法求解
    if (pDataCoef[0] == 0.0)
		return FALSE;

    for (i=1; i<=n-1; i++)
    { 
		u=i*n; 
		pDataCoef[u]=pDataCoef[u]/pDataCoef[0];
	}
    
	for (i=1; i<=n-2; i++)
    { 
		u=i*n+i;
        for (j=1; j<=i; j++)
        { 
			v=i*n+j-1; 
			l=(j-1)*n+j-1;
            pDataCoef[u]=pDataCoef[u]-pDataCoef[v]*pDataCoef[v]*pDataCoef[l];
        }
        
		p=pDataCoef[u];
        if (p == 0.0)
			return FALSE;

        for (k=i+1; k<=n-1; k++)
        { 
			u=k*n+i;
            for (j=1; j<=i; j++)
            { 
				v=k*n+j-1; 
				l=i*n+j-1; 
				w=(j-1)*n+j-1;
				pDataCoef[u]=pDataCoef[u]-pDataCoef[v]*pDataCoef[l]*pDataCoef[w];
            }
            
			pDataCoef[u]=pDataCoef[u]/p;
        }
    }
    
	u=n*n-1;
    for (j=1; j<=n-1; j++)
    { 
		v=(n-1)*n+j-1; 
		w=(j-1)*n+j-1;
        pDataCoef[u]=pDataCoef[u]-pDataCoef[v]*pDataCoef[v]*pDataCoef[w];
    }
    
	p=pDataCoef[u];
    if (p == 0.0)
		return FALSE;

    for (j=0; j<=m-1; j++)
	{
	    for (i=1; i<=n-1; i++)
		{ 
			u=i*m+j;
			for (k=1; k<=i; k++)
			{ 
				v=i*n+k-1; 
				w=(k-1)*m+j;
				pDataConst[u]=pDataConst[u]-pDataCoef[v]*pDataConst[w];
			}
		}
	}

    for (i=1; i<=n-1; i++)
    { 
		u=(i-1)*n+i-1;
        for (j=i; j<=n-1; j++)
        { 
			v=(i-1)*n+j; 
			w=j*n+i-1;
            pDataCoef[v]=pDataCoef[u]*pDataCoef[w];
        }
    }
    
	for (j=0; j<=m-1; j++)
    { 
		u=(n-1)*m+j;
        pDataConst[u]=pDataConst[u]/p;
        for (k=1; k<=n-1; k++)
        { 
			k1=n-k; 
			k3=k1-1; 
			u=k3*m+j;
            for (k2=k1; k2<=n-1; k2++)
            { 
				v=k3*n+k2; 
				w=k2*m+j;
                pDataConst[u]=pDataConst[u]-pDataCoef[v]*pDataConst[w];
            }
            
			pDataConst[u]=pDataConst[u]/pDataCoef[k3*n+k3];
        }
    }
    
	return TRUE;
}

//////////////////////////////////////////////////////////////////////
// 求解对称正定方程组的平方根法
//
// 参数：
// 1. CMatrix& mtxResult - CMatrix引用对象，返回方程组解矩阵
//
// 返回值：BOOL 型，方程组求解是否成功
//////////////////////////////////////////////////////////////////////
BOOL CLEquations::GetRootsetCholesky(CMatrix& mtxResult)
{ 
	int i,j,k,u,v;
    
	// 方程组属性，将常数矩阵赋给解矩阵
	CMatrix mtxCoef = m_mtxCoef;
	mtxResult = m_mtxConst;
	int n = mtxCoef.GetNumColumns();
	int m = mtxResult.GetNumColumns();
	double* pDataCoef = mtxCoef.GetData();
	double* pDataConst = mtxResult.GetData();
    
	// 非对称正定系数矩阵，不能用本方法求解
	if (pDataCoef[0] <= 0.0)
		return FALSE;

	pDataCoef[0]=sqrt(pDataCoef[0]);
    for (j=1; j<=n-1; j++) 
		pDataCoef[j]=pDataCoef[j]/pDataCoef[0];
    
	for (i=1; i<=n-1; i++)
    { 
		u=i*n+i;
        for (j=1; j<=i; j++)
        { 
			v=(j-1)*n+i;
            pDataCoef[u]=pDataCoef[u]-pDataCoef[v]*pDataCoef[v];
        }
        
		if (pDataCoef[u] <= 0.0)
			return FALSE;

        pDataCoef[u]=sqrt(pDataCoef[u]);
        if (i!=(n-1))
        { 
			for (j=i+1; j<=n-1; j++)
            { 
				v=i*n+j;
                for (k=1; k<=i; k++)
					pDataCoef[v]=pDataCoef[v]-pDataCoef[(k-1)*n+i]*pDataCoef[(k-1)*n+j];
                pDataCoef[v]=pDataCoef[v]/pDataCoef[u];
            }
        }
    }
    
	for (j=0; j<=m-1; j++)
    { 
		pDataConst[j]=pDataConst[j]/pDataCoef[0];
        for (i=1; i<=n-1; i++)
        { 
			u=i*n+i; 
			v=i*m+j;
            for (k=1; k<=i; k++)