def g_u(x):
"""u的梯度(解析值)
为尽量减少计算与误差，直接代入梯度解析式计算梯度.
Args:
x: (theta,phi)^T.
Returns:
g: 梯度矢量.
"""
assert isinstance(x, Matrix), 'x不是Matrix'
n = int(x.row / 2)
g = Matrix(zero_mat(2 * n, 1))
for i in range(n):
a = 0
b = 0
for j in range(n):
if j != i:
a -= (sin(x[i][0] - x[j][0]) + cos(x[i][0]) * sin(x[j][0]) *
(1 - cos(x[i + n][0] - x[j + n][0]))) / (rij(
x[i][0], x[j][0], x[i + n][0], x[j + n][0])**3)
b -= (sin(x[i][0]) * sin(x[j][0]) *
sin(x[i + n][0] - x[j + n][0])) / (rij(
x[i][0], x[j][0], x[i + n][0], x[j + n][0])**3)
g[i][0] = a
g[i + n][0] = b
return g
def fij(n, h, x0, y0):
"""
线性方程组右侧列向量
input：剖分数n，步长h，x0，y0
output：Matrix fij
"""
n0 = (n - 1) * (n - 1)
f = Matrix(zero_mat(n0, 1))
for i in range(1, n):
for j in range(1, n):
k = (i - 1) * (n - 1) + j - 1
f[k][0] = fxy(x0 + i * h, y0 + j * h) * h * h
return f
def get_G(n):
'''
获取六边形电阻网络的导纳矩阵
进行星->三角变换后变为等边长三角形网络
取边上节点数为n，节点标号方式为从一个顶点开始，算作第一层，一次向上取第二层，第三层……
return Matrix
'''
g = Matrix(zero_mat(int((n**2 + n) / 2), int((n**2 + n) / 2)))
g[0][0] = 1
g[0][1] = -1 / 2
g[0][2] = -1 / 2
i = 1
for k in range(1, n - 1):
g[i][i - k] = -1 / 2
g[i][i + 1] = -1 / 3
g[i][i + k + 1] = -1 / 2
g[i][i + k + 2] = -1 / 3
g[i][i] = 5 / 3
for j in range(1, k):
s = i + j
g[s][s - k - 1] = -1 / 3
g[s][s - k] = -1 / 3
g[s][s - 1] = -1 / 3
g[s][s + 1] = -1 / 3
g[s][s + k + 1] = -1 / 3
g[s][s + k + 2] = -1 / 3
g[s][s] = 2
s = i + k
g[s][s - k - 1] = -1 / 2
g[s][s - 1] = -1 / 3
g[s][s + k + 1] = -1 / 3
g[s][s + k + 2] = -1 / 2
g[s][s] = 5 / 3
i = s + 1
g[i][i - n + 1] = -1 / 2
g[i][i + 1] = -1 / 2
g[i][i] = 1
for j in range(1, n - 1):
s = i + j
g[s][s - n] = -1 / 3
g[s][s - n + 1] = -1 / 3
g[s][s - 1] = -1 / 2
g[s][s + 1] = -1 / 2
g[s][s] = 5 / 3
s = i + n - 1
g[s][s - n] = -1 / 2
g[s][s - 1] = -1 / 2
g[s][s] = 1
# 带状矩阵，半宽为n
return g
def get_G(n):
'''
获取三角形电阻网络的导纳矩阵
取边上节点数为n，节点标号方式为从一个顶点开始，算作第一层，一次向上取第二层，第三层……
return Matrix
'''
g = Matrix(zero_mat(int((n**2 + n) / 2), int((n**2 + n) / 2)))
g[0][0] = 2
g[0][1] = -1
g[0][2] = -1
i = 1
for k in range(1, n - 1):
g[i][i - k] = -1
g[i][i + 1] = -1
g[i][i + k + 1] = -1
g[i][i + k + 2] = -1
g[i][i] = 4
for j in range(1, k):
s = i + j
g[s][s - k - 1] = -1
g[s][s - k] = -1
g[s][s - 1] = -1
g[s][s + 1] = -1
g[s][s + k + 1] = -1
g[s][s + k + 2] = -1
g[s][s] = 6
s = i + k
g[s][s - k - 1] = -1
g[s][s - 1] = -1
g[s][s + k + 1] = -1
g[s][s + k + 2] = -1
g[s][s] = 4
i = s + 1
g[i][i - n + 1] = -1
g[i][i + 1] = -1
g[i][i] = 2
for j in range(1, n - 1):
s = i + j
g[s][s - n] = -1
g[s][s - n + 1] = -1
g[s][s - 1] = -1
g[s][s + 1] = -1
g[s][s] = 4
s = i + n - 1
g[s][s - n] = -1
g[s][s - 1] = -1
g[s][s] = 2
# 带状矩阵，半宽为n
return g
def get_G(n, w):
'''
获取三角形交流网络的导纳矩阵
取边上节点数为n，节点标号方式为从右下一个顶点开始，算作第一层，一次向上取第二层，第三层……
return Matrix
'''
g = Matrix(zero_mat(int((n**2 + n) / 2), int((n**2 + n) / 2)))
g[0][0] = 1j * (w - 1 / w)
g[0][1] = -1j * w
g[0][2] = 1j / w
i = 1
for k in range(1, n - 1):
g[i][i - k] = -1j * w
g[i][i + 1] = -1
g[i][i + k + 1] = -1j * w
g[i][i + k + 2] = 1j / w
g[i][i] = 1 + 1j * (2 * w - 1 / w)
for j in range(1, k):
s = i + j
g[s][s - k - 1] = 1j / w
g[s][s - k] = -1j * w
g[s][s - 1] = -1
g[s][s + 1] = -1
g[s][s + k + 1] = -1j * w
g[s][s + k + 2] = 1j / w
g[s][s] = 2 + 2j * (w - 1 / w)
s = i + k
g[s][s - k - 1] = 1j / w
g[s][s - 1] = -1
g[s][s + k + 1] = -1j * w
g[s][s + k + 2] = 1j / w
g[s][s] = 1 + 1j * (w - 2 / w)
i = s + 1
g[i][i - n + 1] = -1j * w
g[i][i + 1] = -1
g[i][i] = 1 + 1j * w
for j in range(1, n - 1):
s = i + j
g[s][s - n] = 1j / w
g[s][s - n + 1] = -1j * w
g[s][s - 1] = -1
g[s][s + 1] = -1
g[s][s] = 2 + 1j * (w - 1 / w)
s = i + n - 1
g[s][s - n] = 1j / w
g[s][s - 1] = -1
g[s][s] = 1 - 1j / w
# 带状矩阵，半宽为n
return g
def initialize_x0(n):
"""初始化电子位置.
cos(theta)和phi坐标在各自范围内随机取值.
Args:
n: 电子数.
Returns:
x0: 初始化的位置分布.
"""
x0 = Matrix(zero_mat(2 * n, 1))
for i in range(n):
x0[i][0] = acos(random.uniform(-1, 1))
x0[i + n][0] = random.uniform(0, 2 * pi)
return x0
def solve_R(g, a, b, n):
'''
直接法求解标号为a,b节点间的等效电阻R
return R
'''
assert a < b, "b>a不符合规范"
# 删去g的第a行a列，使其成为非奇异矩阵
g.pop_col(a)
g.pop_row(a)
m = g.row
# 对称正定矩阵Cholesky分解改进算法，G=LDL^T,带状矩阵优化
for i in range(m):
if i < n:
for j in range(i + 1):
for k in range(j):
g[i][j] = g[i][j] - g[i][k] * g[j][k] / g[k][k]
else:
for j in range(i - n, i + 1):
for k in range(j):
g[i][j] = g[i][j] - g[i][k] * g[j][k] / g[k][k]
# 对电流I进行处理，I=LDI'
# 求解 LDI'=I 得到 I'，下三角前代算法
I = Matrix(zero_mat(m, 1))
I[b - 1][0] = 1
for i in range(m):
assert g[i][i] != 0, "系数矩阵主元存在‘0’项"
for j in range(i):
I[i][0] = I[i][0] - g[i][j] / g[j][j] * I[j][0]
I[i][0] = I[i][0]
"""
# 利用(TD)^(-1)特性方法失败
I = Matrix(zero_mat(m, 1))
I[b-1][0] = 1
for i in range(b, m):
I[i][0] = -g[i][b - 1] / g[b - 1][b - 1]
"""
g = g.T()
# 上三角矩阵回代得解
# 直接调用wheels通法
x = up_tri_msolve(g, I)
return x[b - 1][0]
"""
def get_G(n):
'''
获取正方形电阻网络的导纳矩阵
节点标号方式为从左下起，由左向右，依次取完每行
return Matrix
'''
g = Matrix(zero_mat(n**2, n**2))
for i in range(n**2):
for j in range(n**2):
if (abs(j - i) == 1) | (abs(j - i) == n):
g[i][j] = -1
g[i][i] += 1
for i in range(1, n):
g[i * n][i * n - 1] = 0
g[i * n][i * n] -= 1
g[i * n - 1][i * n] = 0
g[i * n - 1][i * n - 1] -= 1
return g
def coefficientmatrix(n):
"""
系数矩阵生成, 已包含边界条件‘u=0’
input： 剖分数 n
output：系数矩阵A
"""
n0 = (n - 1) * (n - 1) # 系数矩阵的维数
A = Matrix(zero_mat(n0, n0))
for i in range(0, n0): # 设置中央对角线
A[i][i] = 4
for j in range(0, n - 1): # 设置里面的两条斜对角线
for k in range(0, n - 2):
kk = j * (n - 1) + k
A[kk][kk + 1] = -1
A[kk + 1][kk] = -1
for k in range(0, n0 - n + 1): # 设置外面的两条斜对角线
A[k][k + n - 1] = -1
A[k + n - 1][k] = -1
return A
def initialize_x0(n):
"""初始化电子位置.
cos(theta)和phi坐标在各自范围内均匀取值
Args:
n: 电子数.
Returns:
x0: 初始化的位置分布.
"""
x0 = Matrix(zero_mat(2 * n, 1))
costheta = [1 - i * 2 / (n - 1) for i in range(n)]
phi = [i * 2 * pi / n for i in range(n)]
random.shuffle(costheta)
random.shuffle(phi)
for i in range(n):
x0[i][0] = acos(costheta[i])
x0[i + n][0] = phi[i]
return x0
def initialize_x0(n):
"""初始化电子位置.
cos(theta)和phi坐标在各自范围内随机取值.
Args:
n: 电子数.
Returns:
x0: 初始化的位置分布.
"""
x0 = Matrix(zero_mat(2 * n, 1))
i = 0
for point in splot(n):
if i >= n:
break
x0[i][0] = point[0]
x0[i + n][0] = point[1]
i += 1
return x0
def solve_R(g, a, b, n, M, e):
'''
迭代法求解标号为a,b节点间的等效电阻R
return R
'''
assert a < b, "b>a不符合规范"
# 删去g的第a行a列，使其成为非奇异矩阵
g.pop_col(a)
g.pop_row(a)
m = g.row
# 对电流I进行处理
I = Matrix(zero_mat(m, 1))
I[b - 1][0] = 1
"""
# Jacobi迭代法收敛速度太慢
# Jacobi迭代法求解 gx=I
x_0 = Matrix(zero_mat(m, 1))
for i in range(m):
x_0[i][0] = 1 # 内置初始解向量
x = Matrix(zero_mat(m, 1)) # 另一组解向量初始化
M = 1000 # 内置最大迭代次数100
e = 1e-10 # 内置判停标准1e-10
for k in range(M):
for i in range(m):
x[i][0] = I[i][0]
for j in range(m):
x[i][0] = x[i][0] - g[i][j] * x_0[j][0]
x[i][0] = x[i][0] / g[i][i] + x_0[i][0]
if col_abs(x - x_0) < e:
print("迭代次数: ", k+1)
break
elif k == M - 1:
print("Not converged at given M=", M, " and e=", e)
break
x_0 = copy.deepcopy(x)
"""
# Gauss-Seidel迭代法 + 带状对称矩阵优化
x = Matrix(zero_mat(m, 1))
for i in range(m):
x[i][0] = 1 # 内置初始解向量
x_0 = Matrix(zero_mat(m, 1)) # 另一组解向量初始化
for k in range(M):
x_0 = copy.deepcopy(x)
if m > 2 * n + 1:
for i in range(m):
x[i][0] = I[i][0]
if i < n:
for j in range(i):
x[i][0] = x[i][0] - g[i][j] * x[j][0]
for j in range(i + 1, i + n + 1):
x[i][0] = x[i][0] - g[i][j] * x[j][0]
x[i][0] = x[i][0] / g[i][i]
elif i > m - n - 1:
for j in range(i - n, i):
x[i][0] = x[i][0] - g[i][j] * x[j][0]
for j in range(i + 1, m):
x[i][0] = x[i][0] - g[i][j] * x[j][0]
x[i][0] = x[i][0] / g[i][i]
else:
for j in range(i - n, i):
x[i][0] = x[i][0] - g[i][j] * x[j][0]
for j in range(i + 1, i + n + 1):
x[i][0] = x[i][0] - g[i][j] * x[j][0]
x[i][0] = x[i][0] / g[i][i]
# 逐次超松弛迭代法
x[i][0] = -0.9 * x_0[i][0] + 1.9 * x[i][0]
else:
for i in range(m):
x[i][0] = I[i][0]
for j in range(i):
x[i][0] = x[i][0] - g[i][j] * x[j][0]
for j in range(i + 1, m):
x[i][0] = x[i][0] - g[i][j] * x[j][0]
x[i][0] = x[i][0] / g[i][i]
# 逐次超松弛迭代法
x[i][0] = -0.9 * x_0[i][0] + 1.9 * x[i][0]
if col_abs(x - x_0) < e:
print("迭代次数: ", k + 1)
break
elif k == M - 1:
print("Not converged at given M=", M, " and e=", e)
break
return x[b - 1][0]
def uxy(x, y):
"""泊松方程真解"""
return sin(pi * x) * sin(pi * y)
a_x = 0. # x方向边界
b_x = 1.
c_y = 0. # y方向边界
d_y = 1.
for m in range(4, 8):
n = 2**m # 剖分数
h = (b_x - a_x) / n # 步长
A0 = coefficientmatrix(n) # 计算系数矩阵
f_1d = Matrix(zero_mat((n - 1) * (n - 1), 1)) # 代求向量，初始化
b_matrix = fij(n, h, a_x, c_y) # 计算右端矩阵b
f_1d = cg(A0, b_matrix, f_1d) # 共轭梯度法求解
uij = [] # 转换成二维矩阵
fT = f_1d.T().matrix[0]
zero = [0 for i in range(n + 1)]
uij.append(zero)
for i in range(n - 1):
uij.append([0] + fT[i * (n - 1):(i + 1) * (n - 1):] + [0])
uij.append(zero)
El2 = e_l2(uij, n, h, a_x, c_y) # 计算误差
print('n=', n, ' 误差： ', El2)
plt.figure()
