gctl/lib/math/gmath.cpp

/********************************************************
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 * Geophysical Computational Tools & Library (GCTL)
 *
 * Copyright (c) 2023  Yi Zhang (yizhang-geo@zju.edu.cn)
 *
 * GCTL is distributed under a dual licensing scheme. 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 
 * of the License, or (at your option) any later version. You should have 
 * received a copy of the GNU Lesser General Public License along with this 
 * program. If not, see <http://www.gnu.org/licenses/>.
 * 
 * If the terms and conditions of the LGPL v.2. would prevent you from using 
 * the GCTL, please consider the option to obtain a commercial license for a 
 * fee. These licenses are offered by the GCTL's original author. As a rule, 
 * licenses are provided "as-is", unlimited in time for a one time fee. Please 
 * send corresponding requests to: yizhang-geo@zju.edu.cn. Please do not forget 
 * to include some description of your company and the realm of its activities. 
 * Also add information on how to contact you by electronic and paper mail.
 ******************************************************/

#include "gmath.h"

int gctl::random(int low, int hig)
{
	return rand()%(hig - low + 1) + low;
}

double gctl::random(double low, double hig)
{
	double f = (double) rand()/RAND_MAX;
	return f*(hig-low) + low;
}

std::complex<double> gctl::random(std::complex<double> low, std::complex<double> hig)
{
	std::complex<double> c;
	double r = (double) rand()/RAND_MAX;
	double i = (double) rand()/RAND_MAX;
	double rh = std::max(hig.real(), low.real());
	double rl = std::min(hig.real(), low.real());
	double ih = std::max(hig.imag(), low.imag());
	double il = std::min(hig.imag(), low.imag());

	c.real(r*(rh - rl) + rl);
	c.imag(i*(ih - il) + il);
	return c;
}

bool gctl::isequal(double f, double v, double eps)
{
	return fabs(f - v) < eps;
}

double gctl::sign(double a)
{
	if (a > GCTL_ZERO) return 1.0;
	if (a < -1.0*GCTL_ZERO) return -1.0; 
	return 0.0;
}

//利用二分法求一个正数的n次方根 注意输入值小于1的情况
double gctl::sqrtn(double d,int n,double eps)
{
	double xmin,xmax,halfx;
	if (d == 1)
	{
		return d;
	}
	else if (d > 1)
	{
		xmin = 1;
		xmax = d;
		halfx = 0.5*(xmin+xmax);
		while (fabs(d - pow(halfx,n)) > eps)
		{
			if (pow(halfx,n) > d)
			{
				xmax = halfx;
				halfx = 0.5*(xmin+xmax);
			}
			else
			{
				xmin = halfx;
				halfx = 0.5*(xmin+xmax);
			}
		}
	}
	else
	{
		xmin = 0;
		xmax = 1;
		halfx = 0.5*(xmin+xmax);
		while (fabs(d - pow(halfx,n)) > eps)
		{
			if (pow(halfx,n) > d)
			{
				xmax = halfx;
				halfx = 0.5*(xmin+xmax);
			}
			else
			{
				xmin = halfx;
				halfx = 0.5*(xmin+xmax);
			}
		}
	}
	return halfx;
}

double gctl::geographic_area(double lon1, double lon2, double lat1, double lat2, double R)
{
	return fabs(R*R*(arc(lon2) - arc(lon1))*(sind(lat2) - sind(lat1)));
}

double gctl::geographic_distance(double lon1, double lon2, double lat1, double lat2, double R)
{
	double n1 = arc(lon1), n2 = arc(lon2), t1 = arc(lat1), t2 = arc(lat2);
	double a = sin(0.5*(t2 - t1)), b = sin(0.5*(n2 - n1));
	return 2*R*asin(sqrt(a*a + cos(t1)*cos(t2)*b*b));
}

double gctl::ellipse_radius_2d(double x_len, double y_len, double arc, double x_arc)
{
	if (fabs(x_len - y_len) < 1e-16) // 就是个圆 直接加
	{
		return 0.5*(x_len + y_len);
	}
	
	return sqrt(power2(x_len*cos(arc - x_arc)) + power2(y_len*sin(arc - x_arc)));
}

void gctl::ellipse_plus_elevation_2d(double x_len, double y_len, double arc, double elev,
	double &out_arc, double &out_rad, double x_arc)
{
	if (fabs(x_len - y_len) < 1e-8) // 就是个圆 直接加
	{
		out_arc = arc;
		out_rad = 0.5*(x_len + y_len) + elev;
		return;
	}
	
	if (fabs(arc) < 1e-8) // 弧度太小 直接加和
	{
		out_arc = arc;
		out_rad = x_len + elev;
		return;
	}

	if (fabs(fabs(arc) - 0.5*GCTL_Pi) < 1e-8) // 到了弧顶 直接加和
	{
		out_arc = arc;
		out_rad = y_len + elev;
		return;
	}
	
	double ellip_rad = ellipse_radius_2d(x_len, y_len, arc, x_arc);
	double alpha = atan(x_len*x_len*sin(arc)/(y_len*y_len*cos(arc)));
	double sin_alpha = sin(alpha);
	double lx = ellip_rad * sin(alpha - arc)/sin_alpha;
	double lr = ellip_rad * sin(arc)/sin_alpha + elev;
	out_rad = sqrt(lx*lx + lr*lr - 2.0*lx*lr*cos(GCTL_Pi - alpha));
	out_arc = acos((lx*lx + out_rad*out_rad - lr*lr)/(2.0*out_rad*lx));
	if (arc < 0.0) out_arc *= -1.0;

	return;
}

double gctl::ellipsoid_radius(double x_len, double y_len, double z_len, double phi, double theta)
{
	return x_len*y_len*z_len/sqrt(pow(y_len*z_len*sin(theta)*cos(phi),2) + 
		pow(x_len*z_len*sin(theta)*sin(phi),2) + pow(x_len*y_len*cos(theta),2));
}

// 使用牛顿迭代法计算一个矩阵的近似逆
double gctl::newton_inverse(const _2d_matrix &in_mat, _2d_matrix &inverse_mat, 
	double epsilon, int iter_times, bool initiate)
{
	if (in_mat.empty())
		throw runtime_error("The input matrix is empty. Thrown by gctl::newton_inverse(...)");

	if (in_mat.row_size() != in_mat.col_size())
		throw logic_error("The input matrix is square. Thrown by gctl::newton_inverse(...)");

	if (iter_times < 0)
		throw invalid_argument("Invalid iteration times. Thrown by gctl::newton_inverse(...)");

	int m_size = in_mat.row_size();

	if (initiate)
	{
		// 初始化逆矩阵 使用输入矩阵的对角元素构建初始矩阵
		if (!inverse_mat.empty()) inverse_mat.clear();
		inverse_mat.resize(m_size, m_size, 0.0);
		for (int i = 0; i < m_size; i++)
		{
			inverse_mat[i][i] = 1.0/in_mat[i][i];
		}
	}

	if (inverse_mat.row_size() != m_size || inverse_mat.col_size() != m_size)
		throw logic_error("Invalid output matrix size. From gctl::newton_inverse(...)");

	// 迭代的收敛条件 || E - in_mat*inverse_mat ||_inf < 1
	// 计算矩阵的无穷大范数
	double maxi_mod = 0.0, mod, ele;
	for (int i = 0; i < m_size; i++)
	{
		mod = 0.0;
		for (int j = 0; j < m_size; j++)
		{
			ele = 0.0;
			for (int k = 0; k < m_size; k++)
			{
				ele += in_mat[i][k] * inverse_mat[k][j];
			}

			if (i == j)
			{
				mod += GCTL_FABS(1.0 - ele);
			}
			else
			{
				mod += GCTL_FABS(ele);
			}
		}

		maxi_mod = GCTL_MAX(maxi_mod, mod);
	}

	if (maxi_mod >= 1.0)
	{
		GCTL_ShowWhatError("The iteration may not converge. From gctl::newton_inverse(...)", 
			GCTL_WARNING_ERROR, 0, 0, 0);
	}

	array<double> col_ax(m_size, 0.0); // A*X 的列向量
	_2d_matrix tmp_mat(m_size, m_size, 0.0);
	for (int t = 0; t < iter_times; t++)
	{
		if (maxi_mod <= epsilon)
			break;

		for (int i = 0; i < m_size; i++)
		{
			for (int j = 0; j < m_size; j++)
			{
				tmp_mat[i][j] = inverse_mat[i][j];
			}
		}

		for (int n = 0; n < m_size; n++)
		{
			for (int i = 0; i < m_size; i++)
			{
				for (int j = 0; j < m_size; j++)
				{
					col_ax[j] = 0.0;
					for (int k = 0; k < m_size; k++)
					{
						col_ax[j] += in_mat[j][k] * tmp_mat[k][i];
					}
					col_ax[j] *= -1.0;
				}
				col_ax[i] += 2.0;

				ele = 0.0;
				for (int j = 0; j < m_size; j++)
				{
					ele += tmp_mat[n][j] * col_ax[j];
				}

				inverse_mat[n][i] = ele;
			}
		}

		maxi_mod = 0.0;
		for (int i = 0; i < m_size; i++)
		{
			mod = 0.0;
			for (int j = 0; j < m_size; j++)
			{
				ele = 0.0;
				for (int k = 0; k < m_size; k++)
				{
					ele += in_mat[i][k] * inverse_mat[k][j];
				}

				if (i == j)
				{
					mod += GCTL_FABS(1.0 - ele);
				}
				else
				{
					mod += GCTL_FABS(ele);
				}
			}

			maxi_mod = GCTL_MAX(maxi_mod, mod);
		}
	}

	return maxi_mod;
}

// 使用牛顿迭代法计算一个矩阵的近似逆
double gctl::newton_inverse(const spmat<double> &in_mat, _2d_matrix &inverse_mat, 
	double epsilon, int iter_times, bool initiate)
{
	if (in_mat.empty())
		throw runtime_error("The input matrix is empty. Thrown by gctl::newton_inverse(...)");

	if (in_mat.row_size() != in_mat.col_size())
		throw logic_error("The input matrix is square. Thrown by gctl::newton_inverse(...)");

	if (iter_times < 0)
		throw invalid_argument("Invalid iteration times. Thrown by gctl::newton_inverse(...)");

	int m_size = in_mat.row_size();

	if (initiate)
	{
		// 初始化逆矩阵 使用输入矩阵的对角元素构建初始矩阵
		if (!inverse_mat.empty()) inverse_mat.clear();
		inverse_mat.resize(m_size, m_size, 0.0);
		for (int i = 0; i < m_size; i++)
		{
			inverse_mat[i][i] = 1.0/in_mat.at(i, i);
		}
	}

	if (inverse_mat.row_size() != m_size || inverse_mat.col_size() != m_size)
		throw logic_error("Invalid output matrix size. From gctl::newton_inverse(...)");

	// 迭代的收敛条件 || E - in_mat*inverse_mat ||_inf < 1
	// 计算矩阵的无穷大范数
	double maxi_mod = 0.0, mod, ele;
	array<double> tmp_arr(m_size, 0.0); // A*X 的列向量
	for (int i = 0; i < m_size; i++)
	{
		mod = 0.0;
		for (int j = 0; j < m_size; j++)
		{
			for (int k = 0; k < m_size; k++)
			{
				tmp_arr[k] = inverse_mat[k][j];
			}

			ele = in_mat.multiply_vector(tmp_arr, i);

			if (i == j)
			{
				mod += GCTL_FABS(1.0 - ele);
			}
			else
			{
				mod += GCTL_FABS(ele);
			}
		}

		maxi_mod = GCTL_MAX(maxi_mod, mod);
	}

	if (maxi_mod >= 1.0)
	{
		GCTL_ShowWhatError("The iteration may not converge. From gctl::newton_inverse(...)", 
			GCTL_WARNING_ERROR, 0, 0, 0);
	}

	array<double> col_ax(m_size, 0.0);
	_2d_matrix tmp_mat(m_size, m_size, 0.0);
	for (int t = 0; t < iter_times; t++)
	{
		if (maxi_mod <= epsilon)
			break;
		//else std::cout << "epsilon = " << maxi_mod << std::endl;

		for (int i = 0; i < m_size; i++)
		{
			for (int j = 0; j < m_size; j++)
			{
				tmp_mat[i][j] = inverse_mat[i][j];
			}
		}

		for (int n = 0; n < m_size; n++)
		{
			for (int i = 0; i < m_size; i++)
			{
				for (int j = 0; j < m_size; j++)
				{
					for (int k = 0; k < m_size; k++)
					{
						tmp_arr[k] = tmp_mat[k][i];
					}
					col_ax[j] = in_mat.multiply_vector(tmp_arr, j);
					col_ax[j] *= -1.0;
				}
				col_ax[i] += 2.0;

				ele = 0.0;
				for (int j = 0; j < m_size; j++)
				{
					ele += tmp_mat[n][j] * col_ax[j];
				}

				inverse_mat[n][i] = ele;
			}
		}

		maxi_mod = 0.0;
		for (int i = 0; i < m_size; i++)
		{
			mod = 0.0;
			for (int j = 0; j < m_size; j++)
			{
				for (int k = 0; k < m_size; k++)
				{
					tmp_arr[k] = inverse_mat[k][j];
				}

				ele = in_mat.multiply_vector(tmp_arr, i);

				if (i == j)
				{
					mod += GCTL_FABS(1.0 - ele);
				}
				else
				{
					mod += GCTL_FABS(ele);
				}
			}

			maxi_mod = GCTL_MAX(maxi_mod, mod);
		}
	}

	return maxi_mod;
}

void gctl::schmidt_orthogonal(const array<double> &a, array<double> &e, int a_s)
{
	if (a.empty())
		throw runtime_error("The input array is empty. Thrown by gctl::schmidt_orthogonal(...)");

	if (a_s <= 1) // a_s >= 2
		throw invalid_argument("vector size must be bigger than one. Thrown by gctl::schmidt_orthogonal(...)");

	int t_s = a.size();
	if (t_s%a_s != 0 || t_s <= 3) // t_s >= 4
		throw invalid_argument("incompatible total array size. Thrown by gctl::schmidt_orthogonal(...)");

	int len = t_s/a_s;
	if (len < a_s)
		throw invalid_argument("the vectors are over-determined. Thrown by gctl::schmidt_orthogonal(...)");

	e.resize(t_s, 0.0);

	double ae, ee;
	for (int i = 0; i < a_s; i++)
	{
		for (int l = 0; l < len; l++)
		{
			e[l + i*len] = a[l + i*len];
		}

		for (int m = 0; m < i; m++)
		{
			ae = ee = 0.0;
			for (int n = 0; n < len; n++)
			{
				ae += a[n + i*len] * e[n + m*len];
				ee += e[n + m*len] * e[n + m*len];
			}

			for (int n = 0; n < len; n++)
			{
				e[n + i*len] -= e[n + m*len] * ae/ee;
			}
		}
	}

	for (int i = 0; i < a_s; i++)
	{
		ee = 0.0;
		for (int l = 0; l < len; l++)
		{
			ee += e[l + i*len] * e[l + i*len];
		}
		ee = sqrt(ee);

		for (int l = 0; l < len; l++)
		{
			e[l + i*len] /= ee;
		}
	}
	return;
}

double gctl::dist_inverse_weight(std::vector<double> *dis_vec, std::vector<double> *val_vec, int order)
{
	if (dis_vec->size() != val_vec->size())
		throw runtime_error("The arrays have different sizes. Thrown by gctl::dist_inverse_weight(...)");

	double total_dist = 0;
	for (int i = 0; i < dis_vec->size(); i++)
	{
		dis_vec->at(i) = 1.0/(GCTL_ZERO + pow(dis_vec->at(i),order));
		total_dist += dis_vec->at(i);
	}

	double ret = 0.0;
	for (int i = 0; i < dis_vec->size(); i++)
	{
		ret += val_vec->at(i)*dis_vec->at(i)/total_dist;
	}
	return ret;
}

int gctl::find_index(const double *in_array, int array_size, double in_val, int &index)
{
	if (array_size <= 0)
	{
		index = -1;
		return -1;
	}
	else if (array_size == 1)
	{
		index = -1;
		return -1;
	}
	else if (in_val < in_array[0] || in_val > in_array[array_size-1])
	{
		index = -1;
		return -1;
	}
	else if (array_size == 2)
	{
		index = 0;
		return 0;
	}
	else
	{
		int low_range = 0;
		int high_range = array_size - 1;
		int test_index;
		bool found = false;
		while (high_range - low_range >= 1)
		{
			test_index = floor(0.5*(low_range + high_range));
			if (in_val >= in_array[test_index] && in_val <= in_array[test_index+1])
			{
				index = test_index;
				found = true;
				break;
			}
			else if (in_val < in_array[test_index])
			{
				high_range = test_index;
			}
			else if (in_val > in_array[test_index+1])
			{
				low_range = test_index+1;
			}
		}
		if (found) return 0;
		else return -1;
	}
}

int gctl::find_index(array<double> *in_array, double in_val, int &index)
{
	return find_index(in_array->get(), in_array->size(), in_val, index);
}

void gctl::fractal_model_1d(array<double> &out_arr, int out_size, double l_val, 
	double r_val, double maxi_range, double smoothness)
{
	if (out_size <= 0)
		throw invalid_argument("Negative output size. Thrown by gctl::fractal_model_1d(...)");
	if (maxi_range <= 0)
		throw invalid_argument("Negative maximal range. Thrown by gctl::fractal_model_1d(...)");
	if (smoothness <= 0)
		throw invalid_argument("Negative smoothness. Thrown by gctl::fractal_model_1d(...)");

	out_arr.resize(out_size);
	int bigger_num = (int) pow(2, ceil(log(out_arr.size()-1)/log(2))) + 1;
	array<double> tmp_arr(bigger_num); // 计算的长度必须为2的次方数

	int step_size = (int) (bigger_num-1)/2;

	tmp_arr[0] = l_val;
	tmp_arr[bigger_num-1] = r_val;

	srand(time(0));

	while (step_size >= 1)
	{
		for (int i = step_size; i < bigger_num; i += 2*step_size)
		{
			tmp_arr[i] = random(-1.0*maxi_range, maxi_range) + 
				0.5*(tmp_arr[i-step_size] + tmp_arr[i+step_size]);
		}
		maxi_range = pow(2.0, -1.0*smoothness)*maxi_range;
		step_size /= 2;
	}

	for (int i = 0; i < out_arr.size(); i++)
	{
		out_arr[i] = tmp_arr[i];
	}
	return;
}

void gctl::fractal_model_2d(_2d_matrix &out_arr, int r_size, int c_size, double dl_val, 
	double dr_val, double ul_val, double ur_val, double maxi_range, double smoothness, 
	unsigned int seed)
{
	if (r_size <= 0 || c_size <= 0)
		throw invalid_argument("Negative output size. Thrown by gctl::fractal_model_2d(...)");
	if (maxi_range <= 0)
		throw invalid_argument("Negative maximal range. Thrown by gctl::fractal_model_2d(...)");
	if (smoothness <= 0)
		throw invalid_argument("Negative smoothness. Thrown by gctl::fractal_model_2d(...)");

	int i, j, m, n, R, jmax;
	double random_d; 

	out_arr.resize(r_size, c_size);
	int xnum = out_arr.col_size();
	int ynum = out_arr.row_size();

	int order_x = ceil(log(xnum-1)/log(2));
	int order_y = ceil(log(ynum-1)/log(2));
	int imax = GCTL_MAX(order_x,order_y);
	int ntotal = (int) pow(2.0, imax) + 1; //总数据点数为2的imax次方加1

	_2d_matrix topo(ntotal, ntotal, 0.0);
	for (i=0; i<ntotal; i++)//设定地形数据初始值，角点数据必须给出
	{
		for (j=0; j<ntotal; j++)
		{
			if (i == 0 && j == 0)
			{
				topo[i][j] = ul_val; //角点初始值；
			} 
			else if (i == ntotal-1 && j==0)
			{
				topo[i][j] = dl_val; //角点初始值；
			} 
			else if (i==0 && j==ntotal-1)
			{
				topo[i][j] = ur_val; //角点初始值；
			} 
			else if (i==ntotal-1 && j==ntotal-1)
			{
				topo[i][j] = dr_val; //角点初始值；
			} 
			else
			{
				topo[i][j] = GCTL_BDL_MAX;
			}
		}
	}

	if (seed == 0) srand(time(0));
	else srand(seed);

	for (i = 1; i <= imax; i++)//开始迭代，生成正方形区域随机地形
	{
		R = int(double(ntotal-1)/pow(2.0,i));

		jmax=int(pow(4.0,i-1));

		for (j=1; j<=jmax; j++)
		{
			random_d = random(-1.0*maxi_range, maxi_range);

			m=2*R*(j-(ceil((double)j/pow(2.0,i-1))-1) * pow(2.0,i-1))-R;
			n=2*R*(ceil((double)j/pow(2.0,i-1)))-R;
			topo[m][n]=(topo[m-R][n-R]+topo[m+R][n-R]+topo[m-R][n+R]+topo[m+R][n+R])/4+random_d;
		}

		for (j=1; j<=jmax; j++)
		{
			m=2*R*(j-(ceil((double)j/pow(2.0,i-1))-1)* pow(2.0,i-1))-R;
			n=2*R*(ceil((double)j/pow(2.0,i-1)))-R;

			if (topo[m][n-R] == GCTL_BDL_MAX)
			{
				random_d = random(-1.0*maxi_range, maxi_range);

				if ((n-R)!=0)
				{
					topo[m][n-R]=(topo[m][n-2*R]+topo[m-R][n-R]+topo[m+R][n-R]+topo[m][n])/4+random_d;
				} 
				else
				{
					topo[m][n-R]=(topo[m-R][n-R]+topo[m+R][n-R]+topo[m][n])/3+random_d;
				}
			}

			if (topo[m-R][n] == GCTL_BDL_MAX)
			{
				random_d = random(-1.0*maxi_range, maxi_range);

				if ((m-R)!=0)
				{
					topo[m-R][n]=(topo[m-R][n-R]+topo[m-2*R][n]+topo[m][n]+topo[m-R][n+R])/4+random_d;
				} 
				else
				{
					topo[m-R][n]=(topo[m-R][n-R]+topo[m][n]+topo[m-R][n+R])/3+random_d;
				}
			}

			if (topo[m+R][n] == GCTL_BDL_MAX)
			{
				random_d = random(-1.0*maxi_range, maxi_range);

				if ((m+R)!=(ntotal-1))
				{
					topo[m+R][n]=(topo[m+R][n-R]+topo[m][n]+topo[m+2*R][n]+topo[m+R][n+R])/4+random_d;
				} 
				else
				{
					topo[m+R][n]=(topo[m+R][n-R]+topo[m][n]+topo[m+R][n+R])/3+random_d;
				}
			}

			if (topo[m][n+R] == GCTL_BDL_MAX)
			{
				random_d = random(-1.0*maxi_range, maxi_range);

				if ((n+R)!=(ntotal-1))
				{
					topo[m][n+R]=(topo[m][n]+topo[m-R][n+R]+topo[m+R][n+R]+topo[m][n+2*R])/4+random_d;
				} 
				else
				{
					topo[m][n+R]=(topo[m][n]+topo[m-R][n+R]+topo[m+R][n+R])/3+random_d;
				}
			}
		}

		maxi_range = pow(2.0, -1.0*smoothness)*maxi_range;
	}

	for (int j = 0; j < ynum; j++)//按预设区域剪裁数组
	{
		for (int i=0; i < xnum; i++)
		{
			out_arr[i][j] = topo[i][j];
		}
	}

	return;
}


void gctl::difference_1d(const array<double> &in, array<double> &diff, double spacing, int order)
{
	if (order < 1 || order > 4)
		throw invalid_argument("The input order can only be 1 to 4. Thrown by gctl::difference_1d(...)");

	if (spacing <= 0.0)
		throw invalid_argument("The input spacing can't be negative or zero. Thrown by void gctl::difference_1d(...)");

	if (order == 1 && in.size() < 3)
		throw runtime_error("The input array size must be equal to or bigger than three for the first order derivative. Thrown by gctl::difference_1d(...)");

	if (order == 2 && in.size() < 4)
		throw runtime_error("The input array size must be equal to or bigger than four for the second order derivative. Thrown by gctl::difference_1d(...)");

	if (order == 3 && in.size() < 6)
		throw runtime_error("The input array size must be equal to or bigger than six for the third order derivative. Thrown by gctl::difference_1d(...)");

	if (order == 4 && in.size() < 7)
		throw runtime_error("The input array size must be equal to or bigger than seven for the fourth order derivative. Thrown by gctl::difference_1d(...)");

	int t_size = in.size();
	diff.resize(t_size);

	// 利用向前或向后差分计算边缘元素的导数
	if (order == 1)
	{
		diff[0] = (-3.0*in[0]+4.0*in[1]-in[2])/(2.0*spacing);
		diff[t_size-1] = (3.0*in[t_size-1]-4.0*in[t_size-2]+in[t_size-3])/(2.0*spacing);

		for (int i = 1; i < t_size-1; i++)
		{
			diff[i] = (in[i+1] - in[i-1])/(2.0*spacing);
		}
	}
	else if (order == 2)
	{
		diff[0] = (2.0*in[0]-5.0*in[1]+4.0*in[2]-in[3])/(spacing*spacing);
		diff[t_size-1] = (2.0*in[t_size-1]-5.0*in[t_size-2]+4.0*in[t_size-3]-in[t_size-4])/(spacing*spacing);

		for (int i = 1; i < t_size-1; i++)
		{
			diff[i] = (in[i-1]-2.0*in[i]+in[i+1])/(spacing*spacing);
		}
	}
	else if (order == 3)
	{
		diff[0] = (-5.0*in[0]+18.0*in[1]-24.0*in[2]+14.0*in[3]-3.0*in[4])/(2.0*spacing*spacing*spacing);
		diff[1] = (-5.0*in[1]+18.0*in[2]-24.0*in[3]+14.0*in[4]-3.0*in[5])/(2.0*spacing*spacing*spacing);
		diff[t_size-1] = (5.0*in[t_size-1]-18.0*in[t_size-2]+24.0*in[t_size-3]-14.0*in[t_size-4]+3.0*in[t_size-5])/(2.0*spacing*spacing*spacing);
		diff[t_size-2] = (5.0*in[t_size-2]-18.0*in[t_size-3]+24.0*in[t_size-4]-14.0*in[t_size-5]+3.0*in[t_size-6])/(2.0*spacing*spacing*spacing);

		for (int i = 2; i < t_size-2; i++)
		{
			diff[i] = (-1.0*in[i-2]+2.0*in[i-1]-2.0*in[i+1]+in[i+2])/(2.0*spacing*spacing*spacing);
		}
	}
	else
	{
		diff[0] = (3.0*in[0]-14.0*in[1]+26.0*in[2]-24.0*in[3]+11.0*in[4]-2.0*in[5])/(spacing*spacing*spacing*spacing);
		diff[1] = (3.0*in[1]-14.0*in[2]+26.0*in[3]-24.0*in[4]+11.0*in[5]-2.0*in[6])/(spacing*spacing*spacing*spacing);
		diff[t_size-1] = (3.0*in[t_size-1]-14.0*in[t_size-2]+26.0*in[t_size-3]-24.0*in[t_size-4]+11.0*in[t_size-5]-2.0*in[t_size-6])/(spacing*spacing*spacing*spacing);
		diff[t_size-2] = (3.0*in[t_size-2]-14.0*in[t_size-3]+26.0*in[t_size-4]-24.0*in[t_size-5]+11.0*in[t_size-6]-2.0*in[t_size-7])/(spacing*spacing*spacing*spacing);

		for (int i = 2; i < t_size-2; i++)
		{
			diff[i] = (in[i-2]-4.0*in[i-1]+6.0*in[i]-4.0*in[i+1]+in[i+2])/(spacing*spacing*spacing*spacing);
		}
	}
	return;
}

void gctl::difference_2d(const _2d_matrix &in, _2d_matrix &diff, double spacing, gradient_type_e d_type, int order)
{
	std::string err_str;
	if (order < 1 || order > 4)
		throw invalid_argument("The input order can only be 1 to 4. Thrown by void gctl::difference_2d(...)");

	if (spacing <= 0.0)
		throw invalid_argument("The input spacing can't be negative or zero. Thrown by void gctl::difference_2d(...)");

	int t_size;
	if (d_type == Dx)
	{
		t_size = in.col_size();
	}
	else if (d_type == Dy)
	{
		t_size = in.row_size();
	}
	else
	{
		throw logic_error("The calculation type must be Dx or Dy. Thrown by gctl::difference_2d(...)");
	}

	if (order == 1 && t_size < 3)
		throw runtime_error("The input array size must be equal to or bigger than 3x3 for the first order derivative. Thrown by void gctl::difference_2d(...)");

	if (order == 2 && t_size < 4)
		throw runtime_error("The input array size must be equal to or bigger than 4x4 for the second order derivative. Thrown by gctl::difference_2d(...)");

	if (order == 3 && t_size < 6)
		throw runtime_error("The input array size must be equal to or bigger than 6x6 for the third order derivative. Thrown by gctl::difference_2d(...)");

	if (order == 4 && t_size < 7)
		throw runtime_error("The input array size must be equal to or bigger than 7x7 for the fourth order derivative. Thrown by void gctl::difference_2d(...)");

	int tr_size = in.row_size(), tl_size = in.col_size();
	diff.resize(tr_size, tl_size);

	if (d_type == Dx)
	{
		if (order == 1)
		{
			for (int i = 0; i < tr_size; i++)
			{
				diff[i][0] = (-3.0*in[i][0]+4.0*in[i][1]-in[i][2])/(2.0*spacing);
				diff[i][tl_size-1] = (3.0*in[i][tl_size-1]-4.0*in[i][tl_size-2]+in[i][tl_size-3])/(2.0*spacing);

				for (int j = 1; j < tl_size-1; j++)
				{
					diff[i][j] = (in[i][j+1] - in[i][j-1])/(2.0*spacing);
				}
			}
		}
		else if (order == 2)
		{
			for (int i = 0; i < tr_size; i++)
			{
				diff[i][0] = (2.0*in[i][0]-5.0*in[i][1]+4.0*in[i][2]-in[i][3])/(spacing*spacing);
				diff[i][tl_size-1] = (2.0*in[i][tl_size-1]-5.0*in[i][tl_size-2]+4.0*in[i][tl_size-3]-in[i][tl_size-4])/(spacing*spacing);

				for (int j = 1; j < tl_size-1; j++)
				{
					diff[i][j] = (in[i][j-1]-2.0*in[i][j]+in[i][j+1])/(spacing*spacing);
				}
			}
		}
		else if (order == 3)
		{
			for (int i = 0; i < tr_size; i++)
			{
				diff[i][0] = (-5.0*in[i][0]+18.0*in[i][1]-24.0*in[i][2]+14.0*in[i][3]-3.0*in[i][4])/(2.0*spacing*spacing*spacing);
				diff[i][1] = (-5.0*in[i][1]+18.0*in[i][2]-24.0*in[i][3]+14.0*in[i][4]-3.0*in[i][5])/(2.0*spacing*spacing*spacing);
				diff[i][tl_size-1] = (5.0*in[i][tl_size-1]-18.0*in[i][tl_size-2]+24.0*in[i][tl_size-3]-14.0*in[i][tl_size-4]+3.0*in[i][tl_size-5])/(2.0*spacing*spacing*spacing);
				diff[i][tl_size-2] = (5.0*in[i][tl_size-2]-18.0*in[i][tl_size-3]+24.0*in[i][tl_size-4]-14.0*in[i][tl_size-5]+3.0*in[i][tl_size-6])/(2.0*spacing*spacing*spacing);

				for (int j = 2; j < tl_size-2; j++)
				{
					diff[i][j] = (-1.0*in[i][j-2]+2.0*in[i][j-1]-2.0*in[i][j+1]+in[i][j+2])/(2.0*spacing*spacing*spacing);
				}
			}
		}
		else
		{
			for (int i = 0; i < tr_size; i++)
			{
				diff[i][0] = (3.0*in[i][0]-14.0*in[i][1]+26.0*in[i][2]-24.0*in[i][3]+11.0*in[i][4]-2.0*in[i][5])/(spacing*spacing*spacing*spacing);
				diff[i][1] = (3.0*in[i][1]-14.0*in[i][2]+26.0*in[i][3]-24.0*in[i][4]+11.0*in[i][5]-2.0*in[i][6])/(spacing*spacing*spacing*spacing);
				diff[i][tl_size-1] = (3.0*in[i][tl_size-1]-14.0*in[i][tl_size-2]+26.0*in[i][tl_size-3]-24.0*in[i][tl_size-4]+11.0*in[i][tl_size-5]-2.0*in[i][tl_size-6])/(spacing*spacing*spacing*spacing);
				diff[i][tl_size-2] = (3.0*in[i][tl_size-2]-14.0*in[i][tl_size-3]+26.0*in[i][tl_size-4]-24.0*in[i][tl_size-5]+11.0*in[i][tl_size-6]-2.0*in[i][tl_size-7])/(spacing*spacing*spacing*spacing);

				for (int j = 2; j < tl_size-2; j++)
				{
					diff[i][j] = (in[i][j-2]-4.0*in[i][j-1]+6.0*in[i][j]-4.0*in[i][j+1]+in[i][j+2])/(spacing*spacing*spacing*spacing);
				}
			}
		}
	}
	else
	{
		if (order == 1)
		{
			for (int j = 0; j < tl_size; j++)
			{
				diff[0][j] = (-3.0*in[0][j]+4.0*in[1][j]-in[2][j])/(2.0*spacing);
				diff[tr_size-1][j] = (3.0*in[tr_size-1][j]-4.0*in[tr_size-2][j]+in[tr_size-3][j])/(2.0*spacing);

				for (int i = 1; i < tr_size-1; i++)
				{
					diff[i][j] = (in[i+1][j] - in[i-1][j])/(2.0*spacing);
				}
			}
		}
		else if (order == 2)
		{
			for (int j = 0; j < tl_size; j++)
			{
				diff[0][j] = (2.0*in[0][j]-5.0*in[1][j]+4.0*in[2][j]-in[3][j])/(spacing*spacing);
				diff[tr_size-1][j] = (2.0*in[tr_size-1][j]-5.0*in[tr_size-2][j]+4.0*in[tr_size-3][j]-in[tr_size-4][j])/(spacing*spacing);

				for (int i = 1; i < tr_size-1; i++)
				{
					diff[i][j] = (in[i-1][j]-2.0*in[i][j]+in[i+1][j])/(spacing*spacing);
				}
			}
		}
		else if (order == 3)
		{
			for (int j = 0; j < tl_size; j++)
			{
				diff[0][j] = (-5.0*in[0][j]+18.0*in[1][j]-24.0*in[2][j]+14.0*in[3][j]-3.0*in[4][j])/(2.0*spacing*spacing*spacing);
				diff[1][j] = (-5.0*in[1][j]+18.0*in[2][j]-24.0*in[3][j]+14.0*in[4][j]-3.0*in[5][j])/(2.0*spacing*spacing*spacing);
				diff[tr_size-1][j] = (5.0*in[tr_size-1][j]-18.0*in[tr_size-2][j]+24.0*in[tr_size-3][j]-14.0*in[tr_size-4][j]+3.0*in[tr_size-5][j])/(2.0*spacing*spacing*spacing);
				diff[tr_size-2][j] = (5.0*in[tr_size-2][j]-18.0*in[tr_size-3][j]+24.0*in[tr_size-4][j]-14.0*in[tr_size-5][j]+3.0*in[tr_size-6][j])/(2.0*spacing*spacing*spacing);

				for (int i = 2; i < tr_size-2; i++)
				{
					diff[i][j] = (-1.0*in[i-2][j]+2.0*in[i-1][j]-2.0*in[i+1][j]+in[i+2][j])/(2.0*spacing*spacing*spacing);
				}
			}
		}
		else
		{
			for (int j = 0; j < tl_size; j++)
			{
				diff[0][j] = (3.0*in[0][j]-14.0*in[1][j]+26.0*in[2][j]-24.0*in[3][j]+11.0*in[4][j]-2.0*in[5][j])/(spacing*spacing*spacing*spacing);
				diff[1][j] = (3.0*in[1][j]-14.0*in[2][j]+26.0*in[3][j]-24.0*in[4][j]+11.0*in[5][j]-2.0*in[6][j])/(spacing*spacing*spacing*spacing);
				diff[tr_size-1][j] = (3.0*in[tr_size-1][j]-14.0*in[tr_size-2][j]+26.0*in[tr_size-3][j]-24.0*in[tr_size-4][j]+11.0*in[tr_size-5][j]-2.0*in[tr_size-6][j])/(spacing*spacing*spacing*spacing);
				diff[tr_size-2][j] = (3.0*in[tr_size-2][j]-14.0*in[tr_size-3][j]+26.0*in[tr_size-4][j]-24.0*in[tr_size-5][j]+11.0*in[tr_size-6][j]-2.0*in[tr_size-7][j])/(spacing*spacing*spacing*spacing);

				for (int i = 2; i < tr_size-2; i++)
				{
					diff[i][j] = (in[i-2][j]-4.0*in[i-1][j]+6.0*in[i][j]-4.0*in[i+1][j]+in[i+2][j])/(spacing*spacing*spacing*spacing);
				}
			}
		}
	}
	return;
}

void gctl::difference_2d(const array<double> &in, array<double> &diff, int row_size, int col_size, 
	double spacing, gradient_type_e d_type, int order)
{
	std::string err_str;
	if (order < 1 || order > 4)
		throw invalid_argument("The input order can only be 1 to 4. Thrown by void gctl::difference_2d(...)");

	if (spacing <= 0.0)
		throw invalid_argument("The input spacing can't be negative or zero. Thrown by void gctl::difference_2d(...)");

	if (row_size*col_size != in.size())
		throw invalid_argument("The input array size does not match. Thrown by void gctl::difference_2d(...)");

	int t_size;
	if (d_type == Dx)
	{
		t_size = col_size;
	}
	else if (d_type == Dy)
	{
		t_size = row_size;
	}
	else
	{
		throw logic_error("The calculation type must be Dx or Dy. Thrown by gctl::difference_2d(...)");
	}

	if (order == 1 && t_size < 3)
		throw runtime_error("The input array size must be equal to or bigger than 3x3 for the first order derivative. Thrown by gctl::difference_2d(...)");

	if (order == 2 && t_size < 4)
		throw runtime_error("The input array size must be equal to or bigger than 4x4 for the second order derivative. Thrown by gctl::difference_2d(...)");

	if (order == 3 && t_size < 6)
		throw runtime_error("The input array size must be equal to or bigger than 6x6 for the third order derivative. Thrown by gctl::difference_2d(...)");

	if (order == 4 && t_size < 7)
		throw runtime_error("The input array size must be equal to or bigger than 7x7 for the fourth order derivative. Thrown by gctl::difference_2d(...)");

	int tr_size = row_size, tl_size = col_size;
	diff.resize(tr_size*tl_size);

	int idx;
	if (d_type == Dx)
	{
		if (order == 1)
		{
			for (int i = 0; i < tr_size; i++)
			{
				idx = i*tl_size;
				diff[idx] = (-3.0*in[idx]+4.0*in[idx+1]-in[idx+2])/(2.0*spacing);

				idx = i*tl_size + tl_size-1;
				diff[idx] = (3.0*in[idx]-4.0*in[idx-1]+in[idx-2])/(2.0*spacing);

				for (int j = 1; j < tl_size-1; j++)
				{
					idx = i*tl_size + j;
					diff[idx] = (in[idx+1] - in[idx-1])/(2.0*spacing);
				}
			}
		}
		else if (order == 2)
		{
			for (int i = 0; i < tr_size; i++)
			{
				idx = i*tl_size;
				diff[idx] = (2.0*in[idx]-5.0*in[idx+1]+4.0*in[idx+2]-in[idx+3])/(spacing*spacing);

				idx = i*tl_size + tl_size-1;
				diff[idx] = (2.0*in[idx]-5.0*in[idx-1]+4.0*in[idx-2]-in[idx-3])/(spacing*spacing);

				for (int j = 1; j < tl_size-1; j++)
				{
					idx = i*tl_size + j;
					diff[idx] = (in[idx-1]-2.0*in[idx]+in[idx+1])/(spacing*spacing);
				}
			}
		}
		else if (order == 3)
		{
			for (int i = 0; i < tr_size; i++)
			{
				idx = i*tl_size;
				diff[idx] = (-5.0*in[idx]+18.0*in[idx+1]-24.0*in[idx+2]+14.0*in[idx+3]-3.0*in[idx+4])/(2.0*spacing*spacing*spacing);

				idx = i*tl_size+1;
				diff[idx] = (-5.0*in[idx]+18.0*in[idx+1]-24.0*in[idx+2]+14.0*in[idx+3]-3.0*in[idx+4])/(2.0*spacing*spacing*spacing);

				idx = i*tl_size + tl_size-1;
				diff[idx] = (5.0*in[idx]-18.0*in[idx-1]+24.0*in[idx-2]-14.0*in[idx-3]+3.0*in[idx-4])/(2.0*spacing*spacing*spacing);

				idx = i*tl_size + tl_size-2;
				diff[idx] = (5.0*in[idx]-18.0*in[idx-1]+24.0*in[idx-2]-14.0*in[idx-3]+3.0*in[idx-4])/(2.0*spacing*spacing*spacing);

				for (int j = 2; j < tl_size-2; j++)
				{
					idx = i*tl_size + j;
					diff[idx] = (-1.0*in[idx-2]+2.0*in[idx-1]-2.0*in[idx+1]+in[idx+2])/(2.0*spacing*spacing*spacing);
				}
			}
		}
		else
		{
			for (int i = 0; i < tr_size; i++)
			{
				idx = i*tl_size;
				diff[idx] = (3.0*in[idx]-14.0*in[idx+1]+26.0*in[idx+2]-24.0*in[idx+3]+11.0*in[idx+4]-2.0*in[idx+5])/(spacing*spacing*spacing*spacing);

				idx = i*tl_size+1;
				diff[idx] = (3.0*in[idx]-14.0*in[idx+1]+26.0*in[idx+2]-24.0*in[idx+3]+11.0*in[idx+4]-2.0*in[idx+5])/(spacing*spacing*spacing*spacing);

				idx = i*tl_size + tl_size-1;
				diff[idx] = (3.0*in[idx]-14.0*in[idx-1]+26.0*in[idx-2]-24.0*in[idx-3]+11.0*in[idx-4]-2.0*in[idx-5])/(spacing*spacing*spacing*spacing);

				idx = i*tl_size + tl_size-2;
				diff[idx] = (3.0*in[idx]-14.0*in[idx-1]+26.0*in[idx-2]-24.0*in[idx-3]+11.0*in[idx-4]-2.0*in[idx-5])/(spacing*spacing*spacing*spacing);

				for (int j = 2; j < tl_size-2; j++)
				{
					idx = i*tl_size + j;
					diff[idx] = (in[idx-2]-4.0*in[idx-1]+6.0*in[idx]-4.0*in[idx+1]+in[idx+2])/(spacing*spacing*spacing*spacing);
				}
			}
		}
	}
	else
	{
		if (order == 1)
		{
			for (int j = 0; j < tl_size; j++)
			{
				idx = j;
				diff[idx] = (-3.0*in[idx]+4.0*in[idx+tl_size]-in[idx+2*tl_size])/(2.0*spacing);

				idx = (tr_size-1)*tl_size + j;
				diff[idx] = (3.0*in[idx]-4.0*in[idx-tl_size]+in[idx-2*tl_size])/(2.0*spacing);

				for (int i = 1; i < tr_size-1; i++)
				{
					idx = i*tl_size + j;
					diff[idx] = (in[idx+tl_size] - in[idx-tl_size])/(2.0*spacing);
				}
			}
		}
		else if (order == 2)
		{
			for (int j = 0; j < tl_size; j++)
			{
				idx = j;
				diff[idx] = (2.0*in[idx]-5.0*in[idx+tl_size]+4.0*in[idx+2*tl_size]-in[idx+3*tl_size])/(spacing*spacing);

				idx = (tr_size-1)*tl_size + j;
				diff[idx] = (2.0*in[idx]-5.0*in[idx-tl_size]+4.0*in[idx-2*tl_size]-in[idx-3*tl_size])/(spacing*spacing);

				for (int i = 1; i < tr_size-1; i++)
				{
					idx = i*tl_size + j;
					diff[idx] = (in[idx-tl_size]-2.0*in[idx]+in[idx+tl_size])/(spacing*spacing);
				}
			}
		}
		else if (order == 3)
		{
			for (int j = 0; j < tl_size; j++)
			{
				idx = j;
				diff[idx] = (-5.0*in[idx]+18.0*in[idx+tl_size]-24.0*in[idx+2*tl_size]+14.0*in[idx+3*tl_size]-3.0*in[idx+4*tl_size])/(2.0*spacing*spacing*spacing);

				idx = tl_size+j;
				diff[idx] = (-5.0*in[idx]+18.0*in[idx+tl_size]-24.0*in[idx+2*tl_size]+14.0*in[idx+3*tl_size]-3.0*in[idx+4*tl_size])/(2.0*spacing*spacing*spacing);

				idx = (tr_size-1)*tl_size + j;
				diff[idx] = (5.0*in[idx]-18.0*in[idx-tl_size]+24.0*in[idx-2*tl_size]-14.0*in[idx-3*tl_size]+3.0*in[idx-4*tl_size])/(2.0*spacing*spacing*spacing);

				idx = (tr_size-2)*tl_size + j;
				diff[idx] = (5.0*in[idx]-18.0*in[idx-tl_size]+24.0*in[idx-2*tl_size]-14.0*in[idx-3*tl_size]+3.0*in[idx-4*tl_size])/(2.0*spacing*spacing*spacing);

				for (int i = 2; i < tr_size-2; i++)
				{
					idx = i*tl_size + j;
					diff[idx] = (-1.0*in[idx-2*tl_size]+2.0*in[idx-tl_size]-2.0*in[idx+tl_size]+in[idx+2*tl_size])/(2.0*spacing*spacing*spacing);
				}
			}
		}
		else
		{
			for (int j = 0; j < tl_size; j++)
			{
				idx = j;
				diff[idx] = (3.0*in[idx]-14.0*in[idx+tl_size]+26.0*in[idx+2*tl_size]-24.0*in[idx+3*tl_size]+11.0*in[idx+4*tl_size]-2.0*in[idx+5*tl_size])/(spacing*spacing*spacing*spacing);

				idx = tl_size+j;
				diff[idx] = (3.0*in[idx]-14.0*in[idx+tl_size]+26.0*in[idx+2*tl_size]-24.0*in[idx+3*tl_size]+11.0*in[idx+4*tl_size]-2.0*in[idx+5*tl_size])/(spacing*spacing*spacing*spacing);

				idx = (tr_size-1)*tl_size + j;
				diff[idx] = (3.0*in[idx]-14.0*in[idx-tl_size]+26.0*in[idx-2*tl_size]-24.0*in[idx-3*tl_size]+11.0*in[idx-4*tl_size]-2.0*in[idx-5*tl_size])/(spacing*spacing*spacing*spacing);

				idx = (tr_size-2)*tl_size + j;
				diff[idx] = (3.0*in[idx]-14.0*in[idx-tl_size]+26.0*in[idx-2*tl_size]-24.0*in[idx-3*tl_size]+11.0*in[idx-4*tl_size]-2.0*in[idx-5*tl_size])/(spacing*spacing*spacing*spacing);

				for (int i = 2; i < tr_size-2; i++)
				{
					idx = i*tl_size + j;
					diff[idx] = (in[idx-2*tl_size]-4.0*in[idx-tl_size]+6.0*in[idx]-4.0*in[idx+tl_size]+in[idx+2*tl_size])/(spacing*spacing*spacing*spacing);
				}
			}
		}
	}
	return;
}