CC BY 4.0 （除特别声明或转载文章外）
如果这篇博客帮助到你，可以请我喝一杯咖啡~
根据习题(1.2.3)已知下列非齐次两点边值问题(1.2.28)
\[\begin{cases} \mathbf{L}u=-\frac{d}{dx}(p\frac{du}{dx})+qu=f, x\in(a,b)\\ u(a)=\alpha, u'(b)=\beta \end{cases}\]与下列变微分方程等价：求$u\in H^1, u(a)=\alpha$ ，使
\[J(u_*)=\min_{u\in H^1,u(a)=\alpha}J(u)\]其中
\[J(u)=\frac{1}{2}a(u,u)-(f,u)-p(b)\beta u(b)\\ a(u,v)=\int_a^{b}(p\frac{du}{dx}\frac{dv}{dx})d\\ (g,u)=\int_a^b gudx\]设$[a,b]=[-1,1],p(x)\equiv-(\pi^2-1)^{-1},q(x)\equiv1,\alpha=0,\beta=-e\pi$，以及
\[f(x)=\frac{2\pi}{\pi^2-1}\cos(x\pi)e^x\]
- 分别取$h\in\lbrace 0.20,0.10,0.05,0.02\rbrace$，将求解域等分为长度为$h$的单元或子空间
- 根据上述剖分，就边值问题(1.2.28)和基函数(2.1.16)中$\varphi_i$而设计$\varphi_0(x)$，编程构建相应的 Ritz-Galekin 方程（即有限元方程）
- 分别使用高斯消元法和雅克比迭代法（迭代 30 次），求解上述有限元方程
- 计算得到有限元解 $u_h$ 并绘制其函数图像
- 已知$u(x)=\sin(x\pi)e^x$是上述边值问题的解析解，针对不同的步长$h$和线性方程组解法得到的数值解$u_h$，绘制误差函数$(u_h-u)$的函数图像，且进行观察分析。
首先构建第 2 问的有限元方程。仿照(2.1.16)设计$\varphi_0(x)$如下：
\[\phi_i(x)=\begin{cases} 1+\frac{1-x_i}{h}, x\in[x_{i-1},x_i]\\ 1-\frac{1-x_i}{h}, x\in[x_i,x_{i+1}]\\ 0, else \end{cases}\\ \phi_n(x)=\begin{cases} 1+\frac{x-x_n}{h}, x\in[x_{i-1},x_i]\\ 0, else \end{cases}\\\]试探函数
\[u_h(x)=\sum_{i=0}^n\sigma_i\varphi_i(x), \, \sigma_i=u_h(x_i)\]因为$\alpha=0,\beta\ne 0$，故右边值条件非齐次，根据(2.1.4)可得有限元方程为：
\[\sum_{i=1}^nu_i\int_a^b [p\varphi_i'\varphi_j'+q\varphi_i\varphi_j]\, dx=(f,\varphi_j)+p(b)\beta\varphi_j(b),\, j\in \lbrace 1,2,\dots,n\rbrace\]由于$\mid i-j\mid >1$时，$a(\varphi(i)-\varphi(j))=0，且$$p(b)\beta\varphi_j(b)=0$ 对于 $j\ne n$ 恒成立，故方程组形式如下:
\[\begin{bmatrix} a(\varphi_1,\varphi_1) & a(\varphi_2,\varphi_1) & 0 & 0 & \dots & 0 \\ a(\varphi_1,\varphi_2) & a(\varphi_2,\varphi_2) & a(\varphi_3,\varphi_2) & 0 & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & a(\varphi_{n-2},\varphi_{n-1}) & a(\varphi_{n-1},\varphi_{n-1}) & a(\varphi_{n},\varphi_{n-1}) \\ 0 & 0 & \dots & 0 & a(\varphi_{n-1},\varphi_{n}) & a(\varphi_{n},\varphi_{n}) \end{bmatrix} \begin{bmatrix} v_1\\ v_2\\ \dots\\ v_{n-1}\\ v_n \end{bmatrix}\\=\begin{bmatrix} (f,\varphi_{1})\\ (f,\varphi_{1})\\ (f,\varphi_{1})\\ (f,\varphi_{n})+p(b)\beta\varphi_n(b) \end{bmatrix}\]带入$a(u,v)=\int_a^b(p\frac{du}{dx}+quv)dx$有
\[a(\varphi_{i-1},\varphi_i) = \int_{x_{i-1}}^{x_i}[p\varphi_i'\varphi_j'+q\varphi_i\varphi_j]\, dx \\ = \int_{x_{i-1}}^{x_i} [-\frac{p(x)}{h^2}+q(x)\varphi_i(x)\varphi_{i-1}(x)]\, dx\]通过仿射变换到$[0,1]$上的标准山形函数
\[\varphi_i(x)=\begin{cases} \frac{x-x_{i-1}}{h}, \, x\in[x_{i-1},x_i]\\ 1-\frac{x-x_{i-1}}{h}, \, x\in[x_,x_{i+1}]\\ 0,\, else \end{cases} \\ \varphi_n(x)=\begin{cases} \frac{x-x_{i-1}}{h}, \, x\in[x_{i-1},x_i]\\ 0,\, else \end{cases} \\\]有
\[a(\varphi_{i-1},\varphi_i) = \int_0^1[-\frac{p(x_{i-1}+h\theta)}{h}+hq(x_{i-1}+h\theta)(1-\theta)\theta]\,d\theta\]带入$p(x)\equiv-(\pi^2-1)^{-1},q(x)\equiv1$有
\[a(\varphi_{i-1},\varphi_i)=\int_0^1\frac{1}{h(\pi^2-1)}+h(1-\theta)\, d\theta \\ =\frac{1}{h(\pi^2-1)}+\frac{h}{6}\]类似可得$a(\varphi_i,\varphi_i),a(\varphi_i,\varphi_{i+1})$，于是对$x\in[2,n-1]$总结有
\[a(\varphi_i,\varphi_j)= \begin{cases} \frac{h}{6}-\frac{p}{h} & i=j-1 \\ 2(\frac{p}{h}+\frac{h}{3}) & i=j \\ \frac{h}{6}-\frac{p}{h} & i=j+1 \\ 0 & \mathit{else} \end{cases}\]上式对$i=1$时仍满足条件，仍然成立；$i=n$时山形函数仅满足一半部分$x\in[x_{n-1},x_n]$，不含右半部分，于是其值应为上式的二分之一。
以下编写 python 程序求解各问题。
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
import altair as alt
import numpy as np
import scipy
import sys
def p(x):
return -1 / (np.pi * np.pi - 1)
def q(x):
return 0
def f(x):
return ((2 * np.pi) / (np.pi * np.pi - 1)) * np.cos(np.pi * x) * np.exp(x)
def phi(a, b, n, i, x):
h = (b - a) / n
xi = a + i * h
if xi - h <= x and x <= xi:
return 1 + (x - xi) / h
if xi <= x and x <= xi + h:
return 1 - (x - xi) / h
return 0
def dphi(a, b, n, i, x):
h = (b - a) / n
xi = a + i * h
if xi - h <= x and x <= xi:
return 1 / h
if xi <= x and x <= xi + h:
return -1 / h
return 0
def a_phi(a, b, n, i, j):
h = (b - a) / n
if i == j - 1:
return h / 6 - p(0) / h
if i == j:
return 2 * (p(0) / h + h / 3)
if i == j + 1:
return h / 6 - p(0) / h
return 0
def vec_b(a, b, n, j, beta):
h = (b - a) / n
xi = a + j * h
tmp = scipy.integrate.quad(
lambda x: f(x) * phi(a, b, n, j, x), max(a, xi - h), min(b, xi + h)
)[0]
return p(b) * beta * phi(a, b, n, j, b) + tmp
def get_A(a, b, n):
A = np.ndarray((n, n), dtype=np.float64)
for i in range(n):
for j in range(n):
if i == n - 1 and j == n - 1:
A[i][j] = a_phi(a, b, n, i + 1, j + 1) * 0.5
else:
A[i][j] = a_phi(a, b, n, i + 1, j + 1)
return A
def get_B(a, b, n, beta):
ret = np.ndarray((n,), dtype=np.float64)
for j in range(n):
ret[j] = vec_b(a, b, n, j + 1, beta)
return ret
def gaussian(A, b):
n = len(A)
for i in range(0, n - 1):
for j in range(i + 1, n):
tmp = A[j, i] / A[i, i]
A[j, i:n] = A[j, i:n] - tmp * A[i, i:n]
b[j] -= tmp * b[i]
for i in range(n - 1, -1, -1):
b[i] = (b[i] - np.dot(A[i, i + 1 : n], b[i + 1 : n])) / A[i, i]
return b
def jacobi(A, b, iter):
n = len(A)
L = np.array(np.tril(A, -1))
U = np.array(np.triu(A, 1))
D_inv = np.diag(1 / np.diag(A))
x = np.zeros(n)
for _it in range(iter):
x = D_inv.dot(b - L.dot(x) - U.dot(x))
return x.flatten()
def problem1():
print("Problem 1:")
a, b = (-1, 1)
for h in [0.20, 0.10, 0.05, 0.02]:
range = np.arange(a, b + h, h)
print(h)
print(np.around(range, 2))
def problem2():
print("Problem 2:")
a, b, alpha, beta = (-1, 1, 0, -np.pi * np.e)
for h in [0.20, 0.10, 0.05, 0.02]:
print(h)
n = int((b - a) / h)
A = get_A(a, b, n)
B = get_B(a, b, n, beta)
print(np.around(A, 2))
print(np.around(B, 2))
def problem3():
print("Problem 3:")
a, b, alpha, beta = (-1, 1, 0, -np.pi * np.e)
for h in [0.20, 0.10, 0.05, 0.02]:
print(h)
n = int((b - a) / h)
A = get_A(a, b, n)
B = get_B(a, b, n, beta)
x0 = gaussian(A.copy(), B.copy())
x1 = jacobi(A.copy(), B.copy(), 30)
print(np.around(x0, 2))
print(np.around(x1, 2))
def problem4():
print("Problem 4:")
chart = []
a, b, alpha, beta = (-1, 1, 0, -np.pi * np.e)
for h in [0.20, 0.10, 0.05, 0.02]:
print(h)
n = int((b - a) / h)
A = get_A(a, b, n)
B = get_B(a, b, n, beta)
x0 = gaussian(A.copy(), B.copy())
x1 = jacobi(A.copy(), B.copy(), 30)
for idx in range(n):
x = a + idx * h
chart.append({"h": h, "x": x, "y": x0[idx], "method": "gaussian"})
chart.append({"h": h, "x": x, "y": x1[idx], "method": "jacobi"})
idx = idx + 1
chart = (
alt.Chart(data=alt.InlineData(chart))
.encode(
x=alt.X("x:Q"),
y=alt.Y("y:Q", scale=alt.Scale(domain=(-5, 2))),
column=alt.Column("method:N"),
color=alt.Color("h:N"),
)
.mark_line(clip=True)
)
chart.save("problem4.html")
def problem5():
print("Problem 5:")
chart = []
a, b, alpha, beta = (-1, 1, 0, -np.pi * np.e)
for h in [0.20, 0.10, 0.05, 0.02]:
print(h)
n = int((b - a) / h)
A = get_A(a, b, n)
B = get_B(a, b, n, beta)
x0 = gaussian(A.copy(), B.copy())
x1 = jacobi(A.copy(), B.copy(), 30)
for idx in range(n):
x = a + idx * h
fx = np.sin(np.pi * x) * np.e**x
chart.append({"h": h, "x": x, "diff": x0[idx] - fx, "method": "gaussian"})
chart.append({"h": h, "x": x, "diff": x1[idx] - fx, "method": "jacobi"})
idx = idx + 1
chart = (
alt.Chart(data=alt.InlineData(chart))
.encode(
x=alt.X("x:Q"),
y=alt.Y("diff:Q", scale=alt.Scale(domain=(-5, 2))),
column=alt.Column("method:N"),
color=alt.Color("h:N"),
)
.mark_line(clip=True)
)
chart.save("problem5.html")
def main(*argv):
problem1()
problem2()
problem3()
problem4()
problem5()
return 0
if __name__ == "__main__":
sys.exit(main(*sys.argv))
以下是该程序的屏幕输出，对应问题 1 ～ 3 的解答。注意为保持输出格式整齐，输出时仅保留了两位小数，实际计算结果和计算精度均使用六十四位浮点类型，可认为保持了高精度。
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
Problem 1:
0.2
[-1. -0.8 -0.6 -0.4 -0.2 -0. 0.2 0.4 0.6 0.8 1. ]
0.1
[-1. -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0. 0.1 0.2 0.3
0.4 0.5 0.6 0.7 0.8 0.9 1. ]
0.05
[-1. -0.95 -0.9 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45
-0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0. 0.05 0.1 0.15
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75
0.8 0.85 0.9 0.95 1. ]
0.02
[-1. -0.98 -0.96 -0.94 -0.92 -0.9 -0.88 -0.86 -0.84 -0.82 -0.8 -0.78
-0.76 -0.74 -0.72 -0.7 -0.68 -0.66 -0.64 -0.62 -0.6 -0.58 -0.56 -0.54
-0.52 -0.5 -0.48 -0.46 -0.44 -0.42 -0.4 -0.38 -0.36 -0.34 -0.32 -0.3
-0.28 -0.26 -0.24 -0.22 -0.2 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06
-0.04 -0.02 0. 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42
0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66
0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9
0.92 0.94 0.96 0.98 1. ]
Problem 2:
0.2
[[-0.99 0.6 0. 0. 0. 0. 0. 0. 0. 0. ]
[ 0.6 -0.99 0.6 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 0.6 -0.99 0.6 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0.6 -0.99 0.6 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0.6 -0.99 0.6 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0.6 -0.99 0.6 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0.6 -0.99 0.6 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 0.6 -0.99 0.6 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 0.6 -0.99 0.6 ]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0.6 -0.5 ]]
[-0.05 -0.02 0.03 0.09 0.14 0.13 0.06 -0.08 -0.25 0.79]
0.1
[[-2.19 1.14 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. ]
[ 1.14 -2.19 1.14 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 1.14 -2.19 1.14 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 1.14 -2.19 1.14 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 1.14 -2.19 1.14 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 1.14 -2.19 1.14 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 1.14 -2.19 1.14 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 1.14 -2.19 1.14 0. 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 1.14 -2.19 1.14 0. 0.
0. 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 0. 1.14 -2.19 1.14 0.
0. 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.14 -2.19 1.14
0. 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.14 -2.19
1.14 0. 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.14
-2.19 1.14 0. 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
1.14 -2.19 1.14 0. 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 1.14 -2.19 1.14 0. 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 1.14 -2.19 1.14 0. 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 1.14 -2.19 1.14 0. 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 1.14 -2.19 1.14 0. ]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 1.14 -2.19 1.14]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0. 1.14 -1.09]]
[-0.03 -0.03 -0.02 -0.01 0. 0.01 0.03 0.05 0.06 0.07 0.07 0.07
0.06 0.03 -0. -0.04 -0.08 -0.13 -0.16 0.87]
0.05
[[-4.48 2.26 0. ... 0. 0. 0. ]
[ 2.26 -4.48 2.26 ... 0. 0. 0. ]
[ 0. 2.26 -4.48 ... 0. 0. 0. ]
...
[ 0. 0. 0. ... -4.48 2.26 0. ]
[ 0. 0. 0. ... 2.26 -4.48 2.26]
[ 0. 0. 0. ... 0. 2.26 -2.24]]
[-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0. 0. 0. 0.01
0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.03
0.03 0.03 0.02 0.02 0.01 -0. -0.01 -0.02 -0.03 -0.04 -0.05 -0.06
-0.07 -0.08 -0.09 0.92]
0.02
[[-11.26 5.64 0. ... 0. 0. 0. ]
[ 5.64 -11.26 5.64 ... 0. 0. 0. ]
[ 0. 5.64 -11.26 ... 0. 0. 0. ]
...
[ 0. 0. 0. ... -11.26 5.64 0. ]
[ 0. 0. 0. ... 5.64 -11.26 5.64]
[ 0. 0. 0. ... 0. 5.64 -5.63]]
[-0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.
-0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0.
0. 0. 0. 0. 0. 0. 0. 0. 0. 0.01 0.01 0.01
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.
0. 0. -0. -0. -0. -0. -0.01 -0.01 -0.01 -0.01 -0.01 -0.01
-0.02 -0.02 -0.02 -0.02 -0.02 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03
-0.04 -0.04 -0.04 0.94]
Problem 3:
0.2
[-0.32 -0.62 -0.75 -0.57 -0.05 0.72 1.47 1.83 1.44 0.14]
[ -21.27 -34.61 -64. -68.79 -104.79 -99.25 -138.33 -121.45 -158.9
-131.79]
0.1
[-0.13 -0.28 -0.42 -0.55 -0.63 -0.66 -0.62 -0.5 -0.3 -0.01 0.34 0.72
1.1 1.43 1.67 1.76 1.66 1.34 0.79 0.04]
[-0. -0.04 -0.12 -0.23 -0.39 -0.56 -0.75 -0.91 -1.09 -1.17 -1.32 -1.3
-1.45 -1.4 -1.66 -1.7 -2.23 -2.51 -3.38 -3.94]
0.05
[-0.06 -0.13 -0.2 -0.27 -0.34 -0.41 -0.47 -0.53 -0.58 -0.61 -0.64 -0.64
-0.63 -0.61 -0.56 -0.49 -0.4 -0.28 -0.15 -0. 0.16 0.34 0.53 0.72
0.91 1.09 1.27 1.42 1.55 1.65 1.72 1.74 1.71 1.64 1.5 1.32
1.07 0.77 0.41 0.01]
[ 0.02 0.04 0.05 0.05 0.05 0.05 0.04 0.02 0. -0.02 -0.04 -0.07
-0.09 -0.12 -0.14 -0.17 -0.19 -0.21 -0.22 -0.23 -0.24 -0.24 -0.23 -0.21
-0.19 -0.16 -0.13 -0.1 -0.07 -0.03 -0.02 -0. -0.04 -0.07 -0.2 -0.31
-0.55 -0.78 -1.16 -1.51]
0.02
[-0.02 -0.05 -0.07 -0.1 -0.13 -0.15 -0.18 -0.21 -0.24 -0.26 -0.29 -0.32
-0.35 -0.38 -0.4 -0.43 -0.45 -0.48 -0.5 -0.52 -0.54 -0.56 -0.58 -0.59
-0.61 -0.62 -0.63 -0.63 -0.64 -0.64 -0.64 -0.63 -0.62 -0.61 -0.6 -0.58
-0.56 -0.54 -0.51 -0.48 -0.45 -0.41 -0.37 -0.33 -0.28 -0.23 -0.18 -0.12
-0.06 -0. 0.06 0.13 0.2 0.27 0.34 0.41 0.49 0.57 0.64 0.72
0.79 0.87 0.95 1.02 1.09 1.16 1.23 1.3 1.36 1.42 1.47 1.53
1.57 1.61 1.65 1.68 1.7 1.72 1.73 1.73 1.73 1.72 1.7 1.67
1.63 1.58 1.53 1.46 1.39 1.31 1.22 1.12 1.01 0.89 0.76 0.63
0.48 0.33 0.17 0. ]
[ 0. 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0. 0. 0. 0.
-0. -0. -0. -0. -0.01 -0.01 -0.01 -0.01 -0.01 -0.02 -0.02 -0.02
-0.02 -0.02 -0.02 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.04 -0.04
-0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04
-0.04 -0.03 -0.03 -0.03 -0.03 -0.03 -0.02 -0.02 -0.02 -0.02 -0.01 -0.01
-0.01 -0. 0. 0.01 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04
0.04 0.05 0.05 0.05 0.05 0.05 0.03 0.02 -0.01 -0.04 -0.1 -0.17
-0.27 -0.36 -0.51 -0.66]
Problem 4:
0.2
0.1
0.05
0.02
Problem 5:
0.2
0.1
0.05
0.02
以下是问题 4 的输出图像。注意到 jacobi 方法在 $h=0.20$ 时极值远远超过了坐标轴范围的十倍，因此并没有在图中显示出来。
{
"config": {"view": {"continuousWidth": 400, "continuousHeight": 300}},
"data": {"name": "data-8179d865bf4dea9647cb306c153f1a05"},
"mark": {"type": "line", "clip": true},
"encoding": {
"color": {"field": "h", "type": "nominal"},
"column": {"field": "method", "type": "nominal"},
"x": {"field": "x", "type": "quantitative"},
"y": {"field": "y", "scale": {"domain": [-5, 2]}, "type": "quantitative"}
},
"$schema": "https://vega.github.io/schema/vega-lite/v4.17.0.json",
"datasets": {
"data-8179d865bf4dea9647cb306c153f1a05": [
{"h": 0.2, "x": -1, "y": -0.32492686134931326, "method": "gaussian"},
{"h": 0.2, "x": -1, "y": -21.269118150127948, "method": "jacobi"},
{"h": 0.2, "x": -0.8, "y": -0.6234543499346311, "method": "gaussian"},
{"h": 0.2, "x": -0.8, "y": -34.611601079696825, "method": "jacobi"},
{"h": 0.2, "x": -0.6, "y": -0.7496666984401675, "method": "gaussian"},
{"h": 0.2, "x": -0.6, "y": -64.00110253299722, "method": "jacobi"},
{
"h": 0.2,
"x": -0.3999999999999999,
"y": -0.5739551643156318,
"method": "gaussian"
},
{
"h": 0.2,
"x": -0.3999999999999999,
"y": -68.79492267237313,
"method": "jacobi"
},
{
"h": 0.2,
"x": -0.19999999999999996,
"y": -0.051088818275673256,
"method": "gaussian"
},
{
"h": 0.2,
"x": -0.19999999999999996,
"y": -104.78629143087304,
"method": "jacobi"
},
{"h": 0.2, "x": 0, "y": 0.7192128915169901, "method": "gaussian"},
{"h": 0.2, "x": 0, "y": -99.24690233482727, "method": "jacobi"},
{
"h": 0.2,
"x": 0.20000000000000018,
"y": 1.472699868658632,
"method": "gaussian"
},
{
"h": 0.2,
"x": 0.20000000000000018,
"y": -138.33198586681883,
"method": "jacobi"
},
{
"h": 0.2,
"x": 0.40000000000000013,
"y": 1.8321588685316437,
"method": "gaussian"
},
{
"h": 0.2,
"x": 0.40000000000000013,
"y": -121.44801198936979,
"method": "jacobi"
},
{
"h": 0.2,
"x": 0.6000000000000001,
"y": 1.4397904627284976,
"method": "gaussian"
},
{
"h": 0.2,
"x": 0.6000000000000001,
"y": -158.8965880582449,
"method": "jacobi"
},
{"h": 0.2, "x": 0.8, "y": 0.14408242248524275, "method": "gaussian"},
{"h": 0.2, "x": 0.8, "y": -131.7928923698842, "method": "jacobi"},
{"h": 0.1, "x": -1, "y": -0.13351119121079627, "method": "gaussian"},
{"h": 0.1, "x": -1, "y": -0.0032387810032757087, "method": "jacobi"},
{"h": 0.1, "x": -0.9, "y": -0.27907710665125157, "method": "gaussian"},
{"h": 0.1, "x": -0.9, "y": -0.039577232851867254, "method": "jacobi"},
{"h": 0.1, "x": -0.8, "y": -0.4225069235510638, "method": "gaussian"},
{"h": 0.1, "x": -0.8, "y": -0.11781675194164949, "method": "jacobi"},
{"h": 0.1, "x": -0.7, "y": -0.5468176430082995, "method": "gaussian"},
{"h": 0.1, "x": -0.7, "y": -0.2324682346123158, "method": "jacobi"},
{"h": 0.1, "x": -0.6, "y": -0.6335891717352686, "method": "gaussian"},
{"h": 0.1, "x": -0.6, "y": -0.38792816366859095, "method": "jacobi"},
{"h": 0.1, "x": -0.5, "y": -0.664787012095379, "method": "gaussian"},
{"h": 0.1, "x": -0.5, "y": -0.5555859517046723, "method": "jacobi"},
{
"h": 0.1,
"x": -0.3999999999999999,
"y": -0.6249436069248552,
"method": "gaussian"
},
{
"h": 0.1,
"x": -0.3999999999999999,
"y": -0.7502269707052441,
"method": "jacobi"
},
{
"h": 0.1,
"x": -0.29999999999999993,
"y": -0.5035224011511343,
"method": "gaussian"
},
{
"h": 0.1,
"x": -0.29999999999999993,
"y": -0.9088646313831512,
"method": "jacobi"
},
{
"h": 0.1,
"x": -0.19999999999999996,
"y": -0.2972274300898505,
"method": "gaussian"
},
{
"h": 0.1,
"x": -0.19999999999999996,
"y": -1.0898625326157396,
"method": "jacobi"
},
{
"h": 0.1,
"x": -0.09999999999999998,
"y": -0.011974342247126558,
"method": "gaussian"
},
{
"h": 0.1,
"x": -0.09999999999999998,
"y": -1.174382554612051,
"method": "jacobi"
},
{"h": 0.1, "x": 0, "y": 0.33578494046759966, "method": "gaussian"},
{"h": 0.1, "x": 0, "y": -1.3164696393156858, "method": "jacobi"},
{
"h": 0.1,
"x": 0.10000000000000009,
"y": 0.7186841645233705,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.10000000000000009,
"y": -1.3047108056410144,
"method": "jacobi"
},
{
"h": 0.1,
"x": 0.20000000000000018,
"y": 1.0992665675430395,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.20000000000000018,
"y": -1.4496759205170122,
"method": "jacobi"
},
{
"h": 0.1,
"x": 0.30000000000000004,
"y": 1.432180452120568,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.30000000000000004,
"y": -1.3979371891210561,
"method": "jacobi"
},
{
"h": 0.1,
"x": 0.40000000000000013,
"y": 1.6677916309686578,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.40000000000000013,
"y": -1.6627308729463497,
"method": "jacobi"
},
{"h": 0.1, "x": 0.5, "y": 1.7570998325304716, "method": "gaussian"},
{"h": 0.1, "x": 0.5, "y": -1.701665119679169, "method": "jacobi"},
{
"h": 0.1,
"x": 0.6000000000000001,
"y": 1.6576662377681086,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.6000000000000001,
"y": -2.2327607680459596,
"method": "jacobi"
},
{
"h": 0.1,
"x": 0.7000000000000002,
"y": 1.3400740715573132,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.7000000000000002,
"y": -2.5056258600952437,
"method": "jacobi"
},
{"h": 0.1, "x": 0.8, "y": 0.7942744919092701, "method": "gaussian"},
{"h": 0.1, "x": 0.8, "y": -3.383367759649291, "method": "jacobi"},
{
"h": 0.1,
"x": 0.9000000000000001,
"y": 0.03503961041951632,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.9000000000000001,
"y": -3.939668346220933,
"method": "jacobi"
},
{"h": 0.05, "x": -1, "y": -0.061496106137998066, "method": "gaussian"},
{"h": 0.05, "x": -1, "y": 0.02193574334870496, "method": "jacobi"},
{"h": 0.05, "x": -0.95, "y": -0.12759936666608956, "method": "gaussian"},
{"h": 0.05, "x": -0.95, "y": 0.03795348334341902, "method": "jacobi"},
{"h": 0.05, "x": -0.9, "y": -0.1969213528756319, "method": "gaussian"},
{"h": 0.05, "x": -0.9, "y": 0.04805539614548926, "method": "jacobi"},
{"h": 0.05, "x": -0.85, "y": -0.2678379317242652, "method": "gaussian"},
{"h": 0.05, "x": -0.85, "y": 0.052477664888660916, "method": "jacobi"},
{"h": 0.05, "x": -0.8, "y": -0.33851047792831396, "method": "gaussian"},
{"h": 0.05, "x": -0.8, "y": 0.05148094714265911, "method": "jacobi"},
{"h": 0.05, "x": -0.75, "y": -0.40691539358247386, "method": "gaussian"},
{"h": 0.05, "x": -0.75, "y": 0.04552726373514087, "method": "jacobi"},
{"h": 0.05, "x": -0.7, "y": -0.47088197500156853, "method": "gaussian"},
{"h": 0.05, "x": -0.7, "y": 0.03501183385432144, "method": "jacobi"},
{
"h": 0.05,
"x": -0.6499999999999999,
"y": -0.5281384333392113,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.6499999999999999,
"y": 0.02049842025568367,
"method": "jacobi"
},
{"h": 0.05, "x": -0.6, "y": -0.5763656232085925, "method": "gaussian"},
{"h": 0.05, "x": -0.6, "y": 0.0024641145679928996, "method": "jacobi"},
{"h": 0.05, "x": -0.55, "y": -0.6132577671768001, "method": "gaussian"},
{"h": 0.05, "x": -0.55, "y": -0.01847311556069981, "method": "jacobi"},
{"h": 0.05, "x": -0.5, "y": -0.6365891897390382, "method": "gaussian"},
{"h": 0.05, "x": -0.5, "y": -0.041752120998741496, "method": "jacobi"},
{
"h": 0.05,
"x": -0.44999999999999996,
"y": -0.644285799188812,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.44999999999999996,
"y": -0.06669442360204814,
"method": "jacobi"
},
{
"h": 0.05,
"x": -0.3999999999999999,
"y": -0.6344997874362689,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.3999999999999999,
"y": -0.09263471941717062,
"method": "jacobi"
},
{"h": 0.05, "x": -0.35, "y": -0.6056857646662013, "method": "gaussian"},
{"h": 0.05, "x": -0.35, "y": -0.11881436251295618, "method": "jacobi"},
{
"h": 0.05,
"x": -0.29999999999999993,
"y": -0.5566763165983929,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.29999999999999993,
"y": -0.1444596114735799,
"method": "jacobi"
},
{"h": 0.05, "x": -0.25, "y": -0.48675477609207635, "method": "gaussian"},
{"h": 0.05, "x": -0.25, "y": -0.1687320438964767, "method": "jacobi"},
{
"h": 0.05,
"x": -0.19999999999999996,
"y": -0.3957228470136154,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.19999999999999996,
"y": -0.19078126846809612,
"method": "jacobi"
},
{
"h": 0.05,
"x": -0.1499999999999999,
"y": -0.2839606155086455,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.1499999999999999,
"y": -0.20972239795690428,
"method": "jacobi"
},
{
"h": 0.05,
"x": -0.09999999999999998,
"y": -0.1524764404110915,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.09999999999999998,
"y": -0.22471348984087391,
"method": "jacobi"
},
{
"h": 0.05,
"x": -0.04999999999999993,
"y": -0.0029442379909527076,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.04999999999999993,
"y": -0.23487348888943999,
"method": "jacobi"
},
{"h": 0.05, "x": 0, "y": 0.1622742270157864, "method": "gaussian"},
{"h": 0.05, "x": 0, "y": -0.23956147740464964, "method": "jacobi"},
{
"h": 0.05,
"x": 0.050000000000000044,
"y": 0.3401242571603884,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.050000000000000044,
"y": -0.237895049902178,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.10000000000000009,
"y": 0.5268723013711385,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.10000000000000009,
"y": -0.2299127892938506,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.15000000000000013,
"y": 0.7181408146109048,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.15000000000000013,
"y": -0.21446167823976284,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.20000000000000018,
"y": 0.9089650941380875,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.20000000000000018,
"y": -0.19346075219252035,
"method": "jacobi"
},
{"h": 0.05, "x": 0.25, "y": 1.093872788708935, "method": "gaussian"},
{"h": 0.05, "x": 0.25, "y": -0.16447230552900055, "method": "jacobi"},
{
"h": 0.05,
"x": 0.30000000000000004,
"y": 1.266986204826347,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.30000000000000004,
"y": -0.1340005421886404,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.3500000000000001,
"y": 1.4221468980375234,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.3500000000000001,
"y": -0.09571542058648858,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.40000000000000013,
"y": 1.5530613487113973,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.40000000000000013,
"y": -0.06584529437642239,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.4500000000000002,
"y": 1.653465794786797,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.4500000000000002,
"y": -0.028937790295414937,
"method": "jacobi"
},
{"h": 0.05, "x": 0.5, "y": 1.717307545354916, "method": "gaussian"},
{"h": 0.05, "x": 0.5, "y": -0.018887642852077162, "method": "jacobi"},
{"h": 0.05, "x": 0.55, "y": 1.738939347683239, "method": "gaussian"},
{"h": 0.05, "x": 0.55, "y": -0.0027796607727244233, "method": "jacobi"},
{
"h": 0.05,
"x": 0.6000000000000001,
"y": 1.7133226475447099,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.6000000000000001,
"y": -0.041658512722785666,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.6500000000000001,
"y": 1.6362348913280287,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.6500000000000001,
"y": -0.07486024607984344,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.7000000000000002,
"y": 1.5044753924425571,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.7000000000000002,
"y": -0.19811742961984427,
"method": "jacobi"
},
{"h": 0.05, "x": 0.75, "y": 1.3160637487347284, "method": "gaussian"},
{"h": 0.05, "x": 0.75, "y": -0.3142729375101187, "method": "jacobi"},
{"h": 0.05, "x": 0.8, "y": 1.0704243767775534, "method": "gaussian"},
{"h": 0.05, "x": 0.8, "y": -0.5545953409486208, "method": "jacobi"},
{
"h": 0.05,
"x": 0.8500000000000001,
"y": 0.7685504470882278,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.8500000000000001,
"y": -0.7831436326635387,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.9000000000000001,
"y": 0.4131403842523758,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.9000000000000001,
"y": -1.1575636893387267,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.9500000000000002,
"y": 0.008700158045234203,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.9500000000000002,
"y": -1.5109546792036979,
"method": "jacobi"
},
{"h": 0.02, "x": -1, "y": -0.02363008841603029, "method": "gaussian"},
{"h": 0.02, "x": -1, "y": 0.0037974489539095145, "method": "jacobi"},
{"h": 0.02, "x": -0.98, "y": -0.04811694205860219, "method": "gaussian"},
{"h": 0.02, "x": -0.98, "y": 0.006782896741906355, "method": "jacobi"},
{"h": 0.02, "x": -0.96, "y": -0.07338704040322307, "method": "gaussian"},
{"h": 0.02, "x": -0.96, "y": 0.009064996079378057, "method": "jacobi"},
{"h": 0.02, "x": -0.94, "y": -0.09936039893066095, "method": "gaussian"},
{"h": 0.02, "x": -0.94, "y": 0.010757260831655496, "method": "jacobi"},
{"h": 0.02, "x": -0.92, "y": -0.12595065962572963, "method": "gaussian"},
{"h": 0.02, "x": -0.92, "y": 0.01195393228458958, "method": "jacobi"},
{"h": 0.02, "x": -0.9, "y": -0.15306521339901186, "method": "gaussian"},
{"h": 0.02, "x": -0.9, "y": 0.012752564078187398, "method": "jacobi"},
{"h": 0.02, "x": -0.88, "y": -0.1806053551929255, "method": "gaussian"},
{"h": 0.02, "x": -0.88, "y": 0.013223122815767301, "method": "jacobi"},
{"h": 0.02, "x": -0.86, "y": -0.20846647241978702, "method": "gaussian"},
{"h": 0.02, "x": -0.86, "y": 0.013437661451266465, "method": "jacobi"},
{"h": 0.02, "x": -0.84, "y": -0.236538267258258, "method": "gaussian"},
{"h": 0.02, "x": -0.84, "y": 0.013440885443561704, "method": "jacobi"},
{
"h": 0.02,
"x": -0.8200000000000001,
"y": -0.26470501320600126,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.8200000000000001,
"y": 0.013278745430638561,
"method": "jacobi"
},
{"h": 0.02, "x": -0.8, "y": -0.2928458461507974, "method": "gaussian"},
{"h": 0.02, "x": -0.8, "y": 0.012975806249490612, "method": "jacobi"},
{"h": 0.02, "x": -0.78, "y": -0.320835090080095, "method": "gaussian"},
{"h": 0.02, "x": -0.78, "y": 0.012557380715703713, "method": "jacobi"},
{"h": 0.02, "x": -0.76, "y": -0.3485426174003425, "method": "gaussian"},
{"h": 0.02, "x": -0.76, "y": 0.012035075889917264, "method": "jacobi"},
{"h": 0.02, "x": -0.74, "y": -0.37583424368288315, "method": "gaussian"},
{"h": 0.02, "x": -0.74, "y": 0.011420998548343199, "method": "jacobi"},
{"h": 0.02, "x": -0.72, "y": -0.402572156493119, "method": "gaussian"},
{"h": 0.02, "x": -0.72, "y": 0.01072000814958333, "method": "jacobi"},
{"h": 0.02, "x": -0.7, "y": -0.4286153777945611, "method": "gaussian"},
{"h": 0.02, "x": -0.7, "y": 0.00993736498700762, "method": "jacobi"},
{
"h": 0.02,
"x": -0.6799999999999999,
"y": -0.4538202592498017,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.6799999999999999,
"y": 0.009075242612075853,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.6599999999999999,
"y": -0.4780410095669383,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.6599999999999999,
"y": 0.008136190708276208,
"method": "jacobi"
},
{"h": 0.02, "x": -0.64, "y": -0.5011302528631649, "method": "gaussian"},
{"h": 0.02, "x": -0.64, "y": 0.007121824853688932, "method": "jacobi"},
{"h": 0.02, "x": -0.62, "y": -0.5229396168377601, "method": "gaussian"},
{"h": 0.02, "x": -0.62, "y": 0.00603414123955603, "method": "jacobi"},
{"h": 0.02, "x": -0.6, "y": -0.5433203493652452, "method": "gaussian"},
{"h": 0.02, "x": -0.6, "y": 0.004875112204429027, "method": "jacobi"},
{
"h": 0.02,
"x": -0.5800000000000001,
"y": -0.56212396193677,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.5800000000000001,
"y": 0.0036471022595456278,
"method": "jacobi"
},
{"h": 0.02, "x": -0.56, "y": -0.579202898194577, "method": "gaussian"},
{"h": 0.02, "x": -0.56, "y": 0.002352768765618362, "method": "jacobi"},
{"h": 0.02, "x": -0.54, "y": -0.5944112256214849, "method": "gaussian"},
{"h": 0.02, "x": -0.54, "y": 0.0009951707260344913, "method": "jacobi"},
{"h": 0.02, "x": -0.52, "y": -0.6076053482655387, "method": "gaussian"},
{"h": 0.02, "x": -0.52, "y": -0.00042224965346264763, "method": "jacobi"},
{"h": 0.02, "x": -0.5, "y": -0.6186447382162596, "method": "gaussian"},
{"h": 0.02, "x": -0.5, "y": -0.0018956445619300601, "method": "jacobi"},
{"h": 0.02, "x": -0.48, "y": -0.6273926832753282, "method": "gaussian"},
{"h": 0.02, "x": -0.48, "y": -0.003420764353956768, "method": "jacobi"},
{
"h": 0.02,
"x": -0.45999999999999996,
"y": -0.6337170483329724,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.45999999999999996,
"y": -0.004992956612570121,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.43999999999999995,
"y": -0.6374910474705574,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.43999999999999995,
"y": -0.0066071707037712165,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.42000000000000004,
"y": -0.638594023888589,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.42000000000000004,
"y": -0.008257963529967974,
"method": "jacobi"
},
{"h": 0.02, "x": -0.4, "y": -0.6369122345063548, "method": "gaussian"},
{"h": 0.02, "x": -0.4, "y": -0.00993950768444344, "method": "jacobi"},
{"h": 0.02, "x": -0.38, "y": -0.6323396359466785, "method": "gaussian"},
{"h": 0.02, "x": -0.38, "y": -0.011645601573653597, "method": "jacobi"},
{"h": 0.02, "x": -0.36, "y": -0.6247786684789163, "method": "gaussian"},
{"h": 0.02, "x": -0.36, "y": -0.01336968165352573, "method": "jacobi"},
{
"h": 0.02,
"x": -0.33999999999999997,
"y": -0.6141410343624337,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.33999999999999997,
"y": -0.015104836757091851,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.31999999999999995,
"y": -0.6003484669123438,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.31999999999999995,
"y": -0.016843824558398907,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.29999999999999993,
"y": -0.5833334865002168,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.29999999999999993,
"y": -0.018579090169618978,
"method": "jacobi"
},
{"h": 0.02, "x": -0.28, "y": -0.563040139605736, "method": "gaussian"},
{"h": 0.02, "x": -0.28, "y": -0.020302786888613058, "method": "jacobi"},
{"h": 0.02, "x": -0.26, "y": -0.5394247169518103, "method": "gaussian"},
{"h": 0.02, "x": -0.26, "y": -0.022006799083197127, "method": "jacobi"},
{"h": 0.02, "x": -0.24, "y": -0.5124564466863732, "method": "gaussian"},
{"h": 0.02, "x": -0.24, "y": -0.023682767204737804, "method": "jacobi"},
{
"h": 0.02,
"x": -0.21999999999999997,
"y": -0.48211815851988316,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.21999999999999997,
"y": -0.02532211490221476,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.19999999999999996,
"y": -0.4484069146892496,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.19999999999999996,
"y": -0.026916078205515677,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.17999999999999994,
"y": -0.4113346035973662,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.17999999999999994,
"y": -0.02845573673086105,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.16000000000000003,
"y": -0.370928491973407,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.16000000000000003,
"y": -0.02993204685385301,
"method": "jacobi"
},
{"h": 0.02, "x": -0.14, "y": -0.32723173141328105, "method": "gaussian"},
{"h": 0.02, "x": -0.14, "y": -0.03133587678253279, "method": "jacobi"},
{"h": 0.02, "x": -0.12, "y": -0.28030381519281206, "method": "gaussian"},
{"h": 0.02, "x": -0.12, "y": -0.03265804345293442, "method": "jacobi"},
{
"h": 0.02,
"x": -0.09999999999999998,
"y": -0.230220981298952,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.09999999999999998,
"y": -0.03388935115763988,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.07999999999999996,
"y": -0.1770765576972089,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.07999999999999996,
"y": -0.03502063180690909,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.05999999999999994,
"y": -0.12098124594697045,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.05999999999999994,
"y": -0.03604278671040685,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.040000000000000036,
"y": -0.06206333939096437,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.040000000000000036,
"y": -0.036946829756343454,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.020000000000000018,
"y": -0.00046887228106461907,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.020000000000000018,
"y": -0.03772393185357008,
"method": "jacobi"
},
{"h": 0.02, "x": 0, "y": 0.06363830363969693, "method": "gaussian"},
{"h": 0.02, "x": 0, "y": -0.038365466491106344, "method": "jacobi"},
{
"h": 0.02,
"x": 0.020000000000000018,
"y": 0.13007651839953746,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.020000000000000018,
"y": -0.03886305625867239,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.040000000000000036,
"y": 0.19864636200389205,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.040000000000000036,
"y": -0.039208620161157455,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.06000000000000005,
"y": 0.26913084446063906,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.06000000000000005,
"y": -0.039394421549616074,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.08000000000000007,
"y": 0.34129564025495546,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.08000000000000007,
"y": -0.039413116481409445,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.10000000000000009,
"y": 0.4148894196371738,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.10000000000000009,
"y": -0.039257802312559494,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.1200000000000001,
"y": 0.48964426879863193,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.1200000000000001,
"y": -0.03892206631631616,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.14000000000000012,
"y": 0.5652762007011524,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.14000000000000012,
"y": -0.03840003411341529,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.15999999999999992,
"y": 0.6414857579959719,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.15999999999999992,
"y": -0.03768641769158492,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.17999999999999994,
"y": 0.7179587091182982,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.17999999999999994,
"y": -0.03677656278460075,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.19999999999999996,
"y": 0.7943668382749308,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.19999999999999996,
"y": -0.03566649537465776,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.21999999999999997,
"y": 0.8703688296554135,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.21999999999999997,
"y": -0.034352967076071286,
"method": "jacobi"
},
{"h": 0.02, "x": 0.24, "y": 0.9456112457929456, "method": "gaussian"},
{"h": 0.02, "x": 0.24, "y": -0.032833499153406144, "method": "jacobi"},
{"h": 0.02, "x": 0.26, "y": 1.0197295995808695, "method": "gaussian"},
{"h": 0.02, "x": 0.26, "y": -0.031106424923111175, "method": "jacobi"},
{"h": 0.02, "x": 0.28, "y": 1.0923495190151624, "method": "gaussian"},
{"h": 0.02, "x": 0.28, "y": -0.02917093028466481, "method": "jacobi"},
{
"h": 0.02,
"x": 0.30000000000000004,
"y": 1.1630880032843232,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.30000000000000004,
"y": -0.02702709212516553,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.32000000000000006,
"y": 1.2315547683667858,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.32000000000000006,
"y": -0.02467591434028124,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.3400000000000001,
"y": 1.2973536798240577,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.3400000000000001,
"y": -0.022119361214549032,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.3600000000000001,
"y": 1.3600842699968305,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.3600000000000001,
"y": -0.019360387905238335,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.3800000000000001,
"y": 1.4193433363231158,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.3800000000000001,
"y": -0.01640296777639499,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.40000000000000013,
"y": 1.4747266170038693,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.40000000000000013,
"y": -0.013252116668680808,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.41999999999999993,
"y": 1.525830539744572,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.41999999999999993,
"y": -0.00991391218236238,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.43999999999999995,
"y": 1.5722540388028883,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.43999999999999995,
"y": -0.0063955211523194725,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.45999999999999996,
"y": 1.613600435074978,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.45999999999999996,
"y": -0.0027051813941842665,
"method": "jacobi"
},
{"h": 0.02, "x": 0.48, "y": 1.649479373458544, "method": "gaussian"},
{"h": 0.02, "x": 0.48, "y": 0.0011476142429211423, "method": "jacobi"},
{"h": 0.02, "x": 0.5, "y": 1.679508811241554, "method": "gaussian"},
{"h": 0.02, "x": 0.5, "y": 0.005152562590014439, "method": "jacobi"},
{"h": 0.02, "x": 0.52, "y": 1.7033170507841913, "method": "gaussian"},
{"h": 0.02, "x": 0.52, "y": 0.009296751838868416, "method": "jacobi"},
{"h": 0.02, "x": 0.54, "y": 1.7205448092903988, "method": "gaussian"},
{"h": 0.02, "x": 0.54, "y": 0.013568902260879432, "method": "jacobi"},
{"h": 0.02, "x": 0.56, "y": 1.730847318006897, "method": "gaussian"},
{"h": 0.02, "x": 0.56, "y": 0.017946098223747663, "method": "jacobi"},
{
"h": 0.02,
"x": 0.5800000000000001,
"y": 1.733896442744332,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.5800000000000001,
"y": 0.02242276418135323,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.6000000000000001,
"y": 1.7293828171898242,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.6000000000000001,
"y": 0.02693529673549128,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.6200000000000001,
"y": 1.7170179800752787,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.6200000000000001,
"y": 0.0315147912311649,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.6400000000000001,
"y": 1.6965365068839735,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.6400000000000001,
"y": 0.03594443229574648,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.6600000000000001,
"y": 1.667698126421855,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.6600000000000001,
"y": 0.04040527576336838,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.6799999999999999,
"y": 1.6302898122522107,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.6799999999999999,
"y": 0.04419559569334923,
"method": "jacobi"
},
{"h": 0.02, "x": 0.7, "y": 1.5841278386956172, "method": "gaussian"},
{"h": 0.02, "x": 0.7, "y": 0.047980186007372146, "method": "jacobi"},
{"h": 0.02, "x": 0.72, "y": 1.5290597908338257, "method": "gaussian"},
{"h": 0.02, "x": 0.72, "y": 0.049777402339108975, "method": "jacobi"},
{"h": 0.02, "x": 0.74, "y": 1.4649665177290938, "method": "gaussian"},
{"h": 0.02, "x": 0.74, "y": 0.051536056087096174, "method": "jacobi"},
{"h": 0.02, "x": 0.76, "y": 1.391764017881896, "method": "gaussian"},
{"h": 0.02, "x": 0.76, "y": 0.04844320166624886, "method": "jacobi"},
{"h": 0.02, "x": 0.78, "y": 1.309405245802324, "method": "gaussian"},
{"h": 0.02, "x": 0.78, "y": 0.04529487017123339, "method": "jacobi"},
{"h": 0.02, "x": 0.8, "y": 1.2178818284661848, "method": "gaussian"},
{"h": 0.02, "x": 0.8, "y": 0.03201217273211242, "method": "jacobi"},
{
"h": 0.02,
"x": 0.8200000000000001,
"y": 1.1172256803680203,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.8200000000000001,
"y": 0.018695878714153718,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.8400000000000001,
"y": 1.0075105058721499,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.8400000000000001,
"y": -0.012950187628817802,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.8600000000000001,
"y": 0.8888531776013501,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.8600000000000001,
"y": -0.04453139526616195,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.8800000000000001,
"y": 0.7614149796928037,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.8800000000000001,
"y": -0.10498095984893495,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.9000000000000001,
"y": 0.6254027048941774,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.9000000000000001,
"y": -0.1651380101967946,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.9199999999999999,
"y": 0.48106959467064253,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.9199999999999999,
"y": -0.2650129204885398,
"method": "jacobi"
},
{"h": 0.02, "x": 0.94, "y": 0.3287161117477186, "method": "gaussian"},
{"h": 0.02, "x": 0.94, "y": -0.3641779437333099, "method": "jacobi"},
{"h": 0.02, "x": 0.96, "y": 0.16869053482613744, "method": "gaussian"},
{"h": 0.02, "x": 0.96, "y": -0.5111397029898572, "method": "jacobi"},
{"h": 0.02, "x": 0.98, "y": 0.0013893655744645797, "method": "gaussian"},
{"h": 0.02, "x": 0.98, "y": -0.6567299340209907, "method": "jacobi"}
]
}
}
以下是问题 5 的输出图像。与问题 4 相同地，jacobi 方法在 $h=0.20$ 时没有在图中显示出来。由图像可以看出，gaussian 法随着 h 的减小逐渐收敛，但 jacobi 法波动不稳定，总体趋势略有减少。
{
"config": {"view": {"continuousWidth": 400, "continuousHeight": 300}},
"data": {"name": "data-cb6cd9b7596cdf937fb83fc34af5ad01"},
"mark": {"type": "line", "clip": true},
"encoding": {
"color": {"field": "h", "type": "nominal"},
"column": {"field": "method", "type": "nominal"},
"x": {"field": "x", "type": "quantitative"},
"y": {"field": "diff", "scale": {"domain": [-5, 2]}, "type": "quantitative"}
},
"$schema": "https://vega.github.io/schema/vega-lite/v4.17.0.json",
"datasets": {
"data-cb6cd9b7596cdf937fb83fc34af5ad01": [
{"h": 0.2, "x": -1, "diff": -0.3249268613493132, "method": "gaussian"},
{"h": 0.2, "x": -1, "diff": -21.269118150127948, "method": "jacobi"},
{"h": 0.2, "x": -0.8, "diff": -0.3593454113986744, "method": "gaussian"},
{"h": 0.2, "x": -0.8, "diff": -34.34749214116087, "method": "jacobi"},
{"h": 0.2, "x": -0.6, "diff": -0.22771581571433908, "method": "gaussian"},
{"h": 0.2, "x": -0.6, "diff": -63.4791516502714, "method": "jacobi"},
{
"h": 0.2,
"x": -0.3999999999999999,
"diff": 0.06355708346983036,
"method": "gaussian"
},
{
"h": 0.2,
"x": -0.3999999999999999,
"diff": -68.15741042458767,
"method": "jacobi"
},
{
"h": 0.2,
"x": -0.19999999999999996,
"diff": 0.4301490439818748,
"method": "gaussian"
},
{
"h": 0.2,
"x": -0.19999999999999996,
"diff": -104.30505356861549,
"method": "jacobi"
},
{"h": 0.2, "x": 0, "diff": 0.7192128915169901, "method": "gaussian"},
{"h": 0.2, "x": 0, "diff": -99.24690233482727, "method": "jacobi"},
{
"h": 0.2,
"x": 0.20000000000000018,
"diff": 0.7547773403027332,
"method": "gaussian"
},
{
"h": 0.2,
"x": 0.20000000000000018,
"diff": -139.04990839517473,
"method": "jacobi"
},
{
"h": 0.2,
"x": 0.40000000000000013,
"diff": 0.4133492686698659,
"method": "gaussian"
},
{
"h": 0.2,
"x": 0.40000000000000013,
"diff": -122.86682158923156,
"method": "jacobi"
},
{
"h": 0.2,
"x": 0.6000000000000001,
"diff": -0.2931474958468041,
"method": "gaussian"
},
{
"h": 0.2,
"x": 0.6000000000000001,
"diff": -160.6295260168202,
"method": "jacobi"
},
{"h": 0.2, "x": 0.8, "diff": -1.1640577136559274, "method": "gaussian"},
{"h": 0.2, "x": 0.8, "diff": -133.10103250602538, "method": "jacobi"},
{"h": 0.1, "x": -1, "diff": -0.13351119121079622, "method": "gaussian"},
{"h": 0.1, "x": -1, "diff": -0.0032387810032756636, "method": "jacobi"},
{"h": 0.1, "x": -0.9, "diff": -0.15344017239416655, "method": "gaussian"},
{"h": 0.1, "x": -0.9, "diff": 0.08605970140521776, "method": "jacobi"},
{"h": 0.1, "x": -0.8, "diff": -0.15839798501510705, "method": "gaussian"},
{"h": 0.1, "x": -0.8, "diff": 0.14629218659430726, "method": "jacobi"},
{"h": 0.1, "x": -0.7, "diff": -0.1450716930842031, "method": "gaussian"},
{"h": 0.1, "x": -0.7, "diff": 0.1692777153117806, "method": "jacobi"},
{"h": 0.1, "x": -0.6, "diff": -0.11163828900944017, "method": "gaussian"},
{"h": 0.1, "x": -0.6, "diff": 0.1340227190572375, "method": "jacobi"},
{
"h": 0.1,
"x": -0.5,
"diff": -0.058256352382745624,
"method": "gaussian"
},
{"h": 0.1, "x": -0.5, "diff": 0.05094470800796114, "method": "jacobi"},
{
"h": 0.1,
"x": -0.3999999999999999,
"diff": 0.012568640860606939,
"method": "gaussian"
},
{
"h": 0.1,
"x": -0.3999999999999999,
"diff": -0.11271472291978202,
"method": "jacobi"
},
{
"h": 0.1,
"x": -0.29999999999999993,
"diff": 0.09581212912298542,
"method": "gaussian"
},
{
"h": 0.1,
"x": -0.29999999999999993,
"diff": -0.30953010110903145,
"method": "jacobi"
},
{
"h": 0.1,
"x": -0.19999999999999996,
"diff": 0.18401043216769758,
"method": "gaussian"
},
{
"h": 0.1,
"x": -0.19999999999999996,
"diff": -0.6086246703581916,
"method": "jacobi"
},
{
"h": 0.1,
"x": -0.09999999999999998,
"diff": 0.2676357970723334,
"method": "gaussian"
},
{
"h": 0.1,
"x": -0.09999999999999998,
"diff": -0.894772415292591,
"method": "jacobi"
},
{"h": 0.1, "x": 0, "diff": 0.33578494046759966, "method": "gaussian"},
{"h": 0.1, "x": 0, "diff": -1.3164696393156858, "method": "jacobi"},
{
"h": 0.1,
"x": 0.10000000000000009,
"diff": 0.37716756914903227,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.10000000000000009,
"diff": -1.6462274010153526,
"method": "jacobi"
},
{
"h": 0.1,
"x": 0.20000000000000018,
"diff": 0.3813440391871409,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.20000000000000018,
"diff": -2.1675984488729108,
"method": "jacobi"
},
{
"h": 0.1,
"x": 0.30000000000000004,
"diff": 0.34012173678487945,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.30000000000000004,
"diff": -2.4899959044567446,
"method": "jacobi"
},
{
"h": 0.1,
"x": 0.40000000000000013,
"diff": 0.24898203110688,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.40000000000000013,
"diff": -3.0815404728081273,
"method": "jacobi"
},
{"h": 0.1, "x": 0.5, "diff": 0.10837856183034345, "method": "gaussian"},
{"h": 0.1, "x": 0.5, "diff": -3.350386390379297, "method": "jacobi"},
{
"h": 0.1,
"x": 0.6000000000000001,
"diff": -0.07527172080719313,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.6000000000000001,
"diff": -3.9656987266212615,
"method": "jacobi"
},
{
"h": 0.1,
"x": 0.7000000000000002,
"diff": -0.28908609125486406,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.7000000000000002,
"diff": -4.134786022907421,
"method": "jacobi"
},
{"h": 0.1, "x": 0.8, "diff": -0.5138656442319001, "method": "gaussian"},
{"h": 0.1, "x": 0.8, "diff": -4.691507895790461, "method": "jacobi"},
{
"h": 0.1,
"x": 0.9000000000000001,
"diff": -0.7250195503454733,
"method": "gaussian"
},
{
"h": 0.1,
"x": 0.9000000000000001,
"diff": -4.699727506985923,
"method": "jacobi"
},
{"h": 0.05, "x": -1, "diff": -0.061496106137998024, "method": "gaussian"},
{"h": 0.05, "x": -1, "diff": 0.021935743348705006, "method": "jacobi"},
{
"h": 0.05,
"x": -0.95,
"diff": -0.06709974155287324,
"method": "gaussian"
},
{"h": 0.05, "x": -0.95, "diff": 0.09845310845663534, "method": "jacobi"},
{
"h": 0.05,
"x": -0.9,
"diff": -0.07128441861854687,
"method": "gaussian"
},
{"h": 0.05, "x": -0.9, "diff": 0.17369233040257429, "method": "jacobi"},
{
"h": 0.05,
"x": -0.85,
"diff": -0.07379561317271832,
"method": "gaussian"
},
{"h": 0.05, "x": -0.85, "diff": 0.24651998344020776, "method": "jacobi"},
{
"h": 0.05,
"x": -0.8,
"diff": -0.07440153939235722,
"method": "gaussian"
},
{"h": 0.05, "x": -0.8, "diff": 0.31558988567861584, "method": "jacobi"},
{
"h": 0.05,
"x": -0.75,
"diff": -0.07290180093358939,
"method": "gaussian"
},
{"h": 0.05, "x": -0.75, "diff": 0.37954085638402535, "method": "jacobi"},
{
"h": 0.05,
"x": -0.7,
"diff": -0.06913602507747213,
"method": "gaussian"
},
{"h": 0.05, "x": -0.7, "diff": 0.43675778377841784, "method": "jacobi"},
{
"h": 0.05,
"x": -0.6499999999999999,
"diff": -0.06299224032016154,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.6499999999999999,
"diff": 0.4856446132747334,
"method": "jacobi"
},
{
"h": 0.05,
"x": -0.6,
"diff": -0.05441474048276407,
"method": "gaussian"
},
{"h": 0.05, "x": -0.6, "diff": 0.5244149972938213, "method": "jacobi"},
{
"h": 0.05,
"x": -0.55,
"diff": -0.043411166355417974,
"method": "gaussian"
},
{"h": 0.05, "x": -0.55, "diff": 0.5513734852606823, "method": "jacobi"},
{
"h": 0.05,
"x": -0.5,
"diff": -0.030058530026404795,
"method": "gaussian"
},
{"h": 0.05, "x": -0.5, "diff": 0.5647785387138919, "method": "jacobi"},
{
"h": 0.05,
"x": -0.44999999999999996,
"diff": -0.014507908196757846,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.44999999999999996,
"diff": 0.563083467390006,
"method": "jacobi"
},
{
"h": 0.05,
"x": -0.3999999999999999,
"diff": 0.0030124603491932245,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.3999999999999999,
"diff": 0.5448775283682915,
"method": "jacobi"
},
{
"h": 0.05,
"x": -0.35,
"diff": 0.022195920791010293,
"method": "gaussian"
},
{"h": 0.05, "x": -0.35, "diff": 0.5090673229442554, "method": "jacobi"},
{
"h": 0.05,
"x": -0.29999999999999993,
"diff": 0.042658213675726864,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.29999999999999993,
"diff": 0.4548749188005399,
"method": "jacobi"
},
{
"h": 0.05,
"x": -0.25,
"diff": 0.06394053881110734,
"method": "gaussian"
},
{"h": 0.05, "x": -0.25, "diff": 0.381963271006707, "method": "jacobi"},
{
"h": 0.05,
"x": -0.19999999999999996,
"diff": 0.08551501524393268,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.19999999999999996,
"diff": 0.29045659378945193,
"method": "jacobi"
},
{
"h": 0.05,
"x": -0.1499999999999999,
"diff": 0.10679262883838037,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.1499999999999999,
"diff": 0.18103084639012157,
"method": "jacobi"
},
{
"h": 0.05,
"x": -0.09999999999999998,
"diff": 0.12713369890836848,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.09999999999999998,
"diff": 0.05489664947858608,
"method": "jacobi"
},
{
"h": 0.05,
"x": -0.04999999999999993,
"diff": 0.145860828161343,
"method": "gaussian"
},
{
"h": 0.05,
"x": -0.04999999999999993,
"diff": -0.0860684227371443,
"method": "jacobi"
},
{"h": 0.05, "x": 0, "diff": 0.1622742270157864, "method": "gaussian"},
{"h": 0.05, "x": 0, "diff": -0.23956147740464964, "method": "jacobi"},
{
"h": 0.05,
"x": 0.050000000000000044,
"diff": 0.17566922558654793,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.050000000000000044,
"diff": -0.4023500814760185,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.10000000000000009,
"diff": 0.1853557059968003,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.10000000000000009,
"diff": -0.5714293846681888,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.15000000000000013,
"diff": 0.19067910614017314,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.15000000000000013,
"diff": -0.7419233867104945,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.20000000000000018,
"diff": 0.1910425657821888,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.20000000000000018,
"diff": -0.911383280548419,
"method": "jacobi"
},
{"h": 0.05, "x": 0.25, "diff": 0.18592970935315078, "method": "gaussian"},
{"h": 0.05, "x": 0.25, "diff": -1.0724153848847848, "method": "jacobi"},
{
"h": 0.05,
"x": 0.30000000000000004,
"diff": 0.17492748949065828,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.30000000000000004,
"diff": -1.226059257524329,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.3500000000000001,
"diff": 0.15774845397693738,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.3500000000000001,
"diff": -1.3601138646470745,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.40000000000000013,
"diff": 0.13425174884961955,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.40000000000000013,
"diff": -1.4846548942382003,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.4500000000000002,
"diff": 0.10446213476487798,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.4500000000000002,
"diff": -1.577941450317334,
"method": "jacobi"
},
{"h": 0.05, "x": 0.5, "diff": 0.06858627465478784, "method": "gaussian"},
{"h": 0.05, "x": 0.5, "diff": -1.6676089135522054, "method": "jacobi"},
{
"h": 0.05,
"x": 0.55,
"diff": 0.027025550634276918,
"method": "gaussian"
},
{"h": 0.05, "x": 0.55, "diff": -1.7146934578216866, "method": "jacobi"},
{
"h": 0.05,
"x": 0.6000000000000001,
"diff": -0.019615311030591887,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.6000000000000001,
"diff": -1.7745964712980875,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.6500000000000001,
"diff": -0.07052448467254724,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.6500000000000001,
"diff": -1.7816196220804195,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.7000000000000002,
"diff": -0.12468477036962011,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.7000000000000002,
"diff": -1.8272775924320215,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.75,
"diff": -0.18088131878412783,
"method": "gaussian"
},
{"h": 0.05, "x": 0.75, "diff": -1.811218005028975, "method": "jacobi"},
{"h": 0.05, "x": 0.8, "diff": -0.23771575936361677, "method": "gaussian"},
{"h": 0.05, "x": 0.8, "diff": -1.862735477089791, "method": "jacobi"},
{
"h": 0.05,
"x": 0.8500000000000001,
"diff": -0.29362699643170953,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.8500000000000001,
"diff": -1.8453210761834762,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.9000000000000001,
"diff": -0.34691877651261377,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.9000000000000001,
"diff": -1.9176228501037162,
"method": "jacobi"
},
{
"h": 0.05,
"x": 0.9500000000000002,
"diff": -0.395793949259197,
"method": "gaussian"
},
{
"h": 0.05,
"x": 0.9500000000000002,
"diff": -1.915448786508129,
"method": "jacobi"
},
{"h": 0.02, "x": -1, "diff": -0.023630088416030245, "method": "gaussian"},
{"h": 0.02, "x": -1, "diff": 0.0037974489539095596, "method": "jacobi"},
{
"h": 0.02,
"x": -0.98,
"diff": -0.024550963176605243,
"method": "gaussian"
},
{"h": 0.02, "x": -0.98, "diff": 0.0303488756239033, "method": "jacobi"},
{
"h": 0.02,
"x": -0.96,
"diff": -0.025397836895193725,
"method": "gaussian"
},
{"h": 0.02, "x": -0.96, "diff": 0.0570541995874074, "method": "jacobi"},
{
"h": 0.02,
"x": -0.94,
"diff": -0.026164041627405113,
"method": "gaussian"
},
{"h": 0.02, "x": -0.94, "diff": 0.08395361813491134, "method": "jacobi"},
{
"h": 0.02,
"x": -0.92,
"diff": -0.026843004265375656,
"method": "gaussian"
},
{"h": 0.02, "x": -0.92, "diff": 0.11106158764494355, "method": "jacobi"},
{
"h": 0.02,
"x": -0.9,
"diff": -0.02742827914192683,
"method": "gaussian"
},
{"h": 0.02, "x": -0.9, "diff": 0.13838949833527242, "method": "jacobi"},
{
"h": 0.02,
"x": -0.88,
"diff": -0.02791358136889499,
"method": "gaussian"
},
{"h": 0.02, "x": -0.88, "diff": 0.1659148966397978, "method": "jacobi"},
{
"h": 0.02,
"x": -0.86,
"diff": -0.028292820793334866,
"method": "gaussian"
},
{"h": 0.02, "x": -0.86, "diff": 0.19361131307771862, "method": "jacobi"},
{
"h": 0.02,
"x": -0.84,
"diff": -0.028560136447924073,
"method": "gaussian"
},
{"h": 0.02, "x": -0.84, "diff": 0.22141901625389562, "method": "jacobi"},
{
"h": 0.02,
"x": -0.8200000000000001,
"diff": -0.028709931364754915,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.8200000000000001,
"diff": 0.2492738272718849,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.8,
"diff": -0.028736907614840657,
"method": "gaussian"
},
{"h": 0.02, "x": -0.8, "diff": 0.27708474478544737, "method": "jacobi"},
{
"h": 0.02,
"x": -0.78,
"diff": -0.028636101429136362,
"method": "gaussian"
},
{"h": 0.02, "x": -0.78, "diff": 0.30475636936666234, "method": "jacobi"},
{
"h": 0.02,
"x": -0.76,
"diff": -0.02840291825069874,
"method": "gaussian"
},
{"h": 0.02, "x": -0.76, "diff": 0.332174775039561, "method": "jacobi"},
{
"h": 0.02,
"x": -0.74,
"diff": -0.028033167561859418,
"method": "gaussian"
},
{"h": 0.02, "x": -0.74, "diff": 0.35922207466936695, "method": "jacobi"},
{
"h": 0.02,
"x": -0.72,
"diff": -0.027523097324970214,
"method": "gaussian"
},
{"h": 0.02, "x": -0.72, "diff": 0.38576906731773214, "method": "jacobi"},
{"h": 0.02, "x": -0.7, "diff": -0.0268694278704647, "method": "gaussian"},
{"h": 0.02, "x": -0.7, "diff": 0.41168331491110405, "method": "jacobi"},
{
"h": 0.02,
"x": -0.6799999999999999,
"diff": -0.026069385061693096,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.6799999999999999,
"diff": 0.43682611680018446,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.6599999999999999,
"diff": -0.025120732562276815,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.6599999999999999,
"diff": 0.4610564677129377,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.64,
"diff": -0.024021803028630828,
"method": "gaussian"
},
{"h": 0.02, "x": -0.64, "diff": 0.48423027468822305, "method": "jacobi"},
{
"h": 0.02,
"x": -0.62,
"diff": -0.022771528047851985,
"method": "gaussian"
},
{"h": 0.02, "x": -0.62, "diff": 0.5062022300294642, "method": "jacobi"},
{
"h": 0.02,
"x": -0.6,
"diff": -0.021369466639416745,
"method": "gaussian"
},
{"h": 0.02, "x": -0.6, "diff": 0.5268259949302575, "method": "jacobi"},
{
"h": 0.02,
"x": -0.5800000000000001,
"diff": -0.01981583213809579,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.5800000000000001,
"diff": 0.5459552320582198,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.56,
"diff": -0.01811151727521676,
"method": "gaussian"
},
{"h": 0.02, "x": -0.56, "diff": 0.5634441496849787, "method": "jacobi"},
{
"h": 0.02,
"x": -0.54,
"diff": -0.016258117275930184,
"method": "gaussian"
},
{"h": 0.02, "x": -0.54, "diff": 0.5791482790715892, "method": "jacobi"},
{
"h": 0.02,
"x": -0.52,
"diff": -0.014257950791462304,
"method": "gaussian"
},
{"h": 0.02, "x": -0.52, "diff": 0.5929251478206138, "method": "jacobi"},
{
"h": 0.02,
"x": -0.5,
"diff": -0.012114078503626224,
"method": "gaussian"
},
{"h": 0.02, "x": -0.5, "diff": 0.6046350151507034, "method": "jacobi"},
{
"h": 0.02,
"x": -0.48,
"diff": -0.009830319145296085,
"method": "gaussian"
},
{"h": 0.02, "x": -0.48, "diff": 0.6141415997760753, "method": "jacobi"},
{
"h": 0.02,
"x": -0.45999999999999996,
"diff": -0.007411262926153794,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.45999999999999996,
"diff": 0.6213128287942484,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.43999999999999995,
"diff": -0.004862282035654419,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.43999999999999995,
"diff": 0.6260215947311317,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.42000000000000004,
"diff": -0.00218953814260725,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.42000000000000004,
"diff": 0.6281465222160137,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.4,
"diff": 0.0006000132791073032,
"method": "gaussian"
},
{"h": 0.02, "x": -0.4, "diff": 0.6275727401010187, "method": "jacobi"},
{
"h": 0.02,
"x": -0.38,
"diff": 0.0034986219453730927,
"method": "gaussian"
},
{"h": 0.02, "x": -0.38, "diff": 0.624192656318398, "method": "jacobi"},
{
"h": 0.02,
"x": -0.36,
"diff": 0.0064977452152562565,
"method": "gaussian"
},
{"h": 0.02, "x": -0.36, "diff": 0.6179067320406468, "method": "jacobi"},
{
"h": 0.02,
"x": -0.33999999999999997,
"diff": 0.009588054131418056,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.33999999999999997,
"diff": 0.6086242517367599,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.31999999999999995,
"diff": 0.012759443165371476,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.31999999999999995,
"diff": 0.5962640855193163,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.29999999999999993,
"diff": 0.01600104377390299,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.29999999999999993,
"diff": 0.5807554401045008,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.28,
"diff": 0.019301241860533724,
"method": "gaussian"
},
{"h": 0.02, "x": -0.28, "diff": 0.5620385945776566, "method": "jacobi"},
{
"h": 0.02,
"x": -0.26,
"diff": 0.02264769922251819,
"method": "gaussian"
},
{"h": 0.02, "x": -0.26, "diff": 0.5400656170911313, "method": "jacobi"},
{"h": 0.02, "x": -0.24, "diff": 0.0260273790496105, "method": "gaussian"},
{"h": 0.02, "x": -0.24, "diff": 0.5148010585312459, "method": "jacobi"},
{
"h": 0.02,
"x": -0.21999999999999997,
"diff": 0.029426575525682508,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.21999999999999997,
"diff": 0.4862226191433509,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.19999999999999996,
"diff": 0.03283094756829846,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.19999999999999996,
"diff": 0.4543217840520324,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.17999999999999994,
"diff": 0.03622555672456451,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.17999999999999994,
"diff": 0.41910442359106964,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.16000000000000003,
"diff": 0.03959490922402209,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.16000000000000003,
"diff": 0.38059135434357605,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.14,
"diff": 0.042923002171084246,
"method": "gaussian"
},
{"h": 0.02, "x": -0.14, "diff": 0.3388188568018325, "method": "jacobi"},
{
"h": 0.02,
"x": -0.12,
"diff": 0.046193373840590746,
"method": "gaussian"
},
{"h": 0.02, "x": -0.12, "diff": 0.29383914558046836, "method": "jacobi"},
{
"h": 0.02,
"x": -0.09999999999999998,
"diff": 0.049389158020508,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.09999999999999998,
"diff": 0.2457207881618201,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.07999999999999996,
"diff": 0.052493142325716446,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.07999999999999996,
"diff": 0.19454906821601625,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.05999999999999994,
"diff": 0.05548783038625772,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.05999999999999994,
"diff": 0.14042628962282133,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.040000000000000036,
"diff": 0.05835550779243076,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.040000000000000036,
"diff": 0.08347201742705168,
"method": "jacobi"
},
{
"h": 0.02,
"x": -0.020000000000000018,
"diff": 0.06107831165781031,
"method": "gaussian"
},
{
"h": 0.02,
"x": -0.020000000000000018,
"diff": 0.02382325208530485,
"method": "jacobi"
},
{"h": 0.02, "x": 0, "diff": 0.06363830363969693, "method": "gaussian"},
{"h": 0.02, "x": 0, "diff": -0.038365466491106344, "method": "jacobi"},
{
"h": 0.02,
"x": 0.020000000000000018,
"diff": 0.06601754623475573,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.020000000000000018,
"diff": -0.10292202842345413,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.040000000000000036,
"diff": 0.06819818214579301,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.040000000000000036,
"diff": -0.16965680001925648,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.06000000000000005,
"diff": 0.07016251649380345,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.06000000000000005,
"diff": -0.2383627495164517,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.08000000000000007,
"diff": 0.07189310162772333,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.08000000000000007,
"diff": -0.30881565510864156,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.10000000000000009,
"diff": 0.07337282426283559,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.10000000000000009,
"diff": -0.3807743976868977,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.1200000000000001,
"diff": 0.07458499465759116,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.1200000000000001,
"diff": -0.45398134045735694,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.14000000000000012,
"diff": 0.07551343751785683,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.14000000000000012,
"diff": -0.5281627972967108,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.15999999999999992,
"diff": 0.07614258429736365,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.15999999999999992,
"diff": -0.6030295913901931,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.17999999999999994,
"diff": 0.0764575665435333,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.17999999999999994,
"diff": -0.6782777053593656,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.19999999999999996,
"diff": 0.07644430991903295,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.19999999999999996,
"diff": -0.7535890237305556,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.21999999999999997,
"diff": 0.07608962851140222,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.21999999999999997,
"diff": -0.8286321682200826,
"method": "jacobi"
},
{"h": 0.02, "x": 0.24, "diff": 0.07538131902612355, "method": "gaussian"},
{"h": 0.02, "x": 0.24, "diff": -0.9030634259202283, "method": "jacobi"},
{"h": 0.02, "x": 0.26, "diff": 0.07430825444258948, "method": "gaussian"},
{"h": 0.02, "x": 0.26, "diff": -0.9765277700613912, "method": "jacobi"},
{"h": 0.02, "x": 0.28, "diff": 0.07286047669772255, "method": "gaussian"},
{"h": 0.02, "x": 0.28, "diff": -1.0486599726021046, "method": "jacobi"},
{
"h": 0.02,
"x": 0.30000000000000004,
"diff": 0.0710292879486345,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.30000000000000004,
"diff": -1.1190858074608543,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.32000000000000006,
"diff": 0.06880733995377919,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.32000000000000006,
"diff": -1.1874233427532879,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.3400000000000001,
"diff": 0.0661887211016492,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.3400000000000001,
"diff": -1.2532843199369574,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.3600000000000001,
"diff": 0.0631690406073524,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.3600000000000001,
"diff": -1.3162756172947165,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.3800000000000001,
"diff": 0.059745509390417784,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.3800000000000001,
"diff": -1.376000794709093,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.40000000000000013,
"diff": 0.05591701714209152,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.40000000000000013,
"diff": -1.4320617165304586,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.41999999999999993,
"diff": 0.05168420508722682,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.41999999999999993,
"diff": -1.4840602468397075,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.43999999999999995,
"diff": 0.04704953394479894,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.43999999999999995,
"diff": -1.5316000260104088,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.45999999999999996,
"diff": 0.0420173465921434,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.45999999999999996,
"diff": -1.5742882698770189,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.48,
"diff": 0.036593924941296097,
"method": "gaussian"
},
{"h": 0.02, "x": 0.48, "diff": -1.6117378342743267, "method": "jacobi"},
{"h": 0.02, "x": 0.5, "diff": 0.03078754054142574, "method": "gaussian"},
{"h": 0.02, "x": 0.5, "diff": -1.6435687081101138, "method": "jacobi"},
{
"h": 0.02,
"x": 0.52,
"diff": 0.024608498429316894,
"method": "gaussian"
},
{"h": 0.02, "x": 0.52, "diff": -1.669411800516006, "method": "jacobi"},
{
"h": 0.02,
"x": 0.54,
"diff": 0.018069173760273616,
"method": "gaussian"
},
{"h": 0.02, "x": 0.54, "diff": -1.6889067332692458, "method": "jacobi"},
{"h": 0.02, "x": 0.56, "diff": 0.01118404076472057, "method": "gaussian"},
{"h": 0.02, "x": 0.56, "diff": -1.701717179018429, "method": "jacobi"},
{
"h": 0.02,
"x": 0.5800000000000001,
"diff": 0.003969693591206136,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.5800000000000001,
"diff": -1.7075039849717726,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.6000000000000001,
"diff": -0.0035551413854775227,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.6000000000000001,
"diff": -1.7060026618398105,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.6200000000000001,
"diff": -0.011369602191741546,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.6200000000000001,
"diff": -1.6968727910358554,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.6400000000000001,
"diff": -0.01945069719569048,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.6400000000000001,
"diff": -1.6800427717839175,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.6600000000000001,
"diff": -0.02777332071230587,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.6600000000000001,
"diff": -1.6550661713707926,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.6799999999999999,
"diff": -0.03631027859654523,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.6799999999999999,
"diff": -1.6224044951554066,
"method": "jacobi"
},
{"h": 0.02, "x": 0.7, "diff": -0.04503232411656022, "method": "gaussian"},
{"h": 0.02, "x": 0.7, "diff": -1.5811799768048054, "method": "jacobi"},
{
"h": 0.02,
"x": 0.72,
"diff": -0.05390820436567223,
"method": "gaussian"
},
{"h": 0.02, "x": 0.72, "diff": -1.533190592860389, "method": "jacobi"},
{
"h": 0.02,
"x": 0.74,
"diff": -0.06290471743565273,
"method": "gaussian"
},
{"h": 0.02, "x": 0.74, "diff": -1.4763351790776504, "method": "jacobi"},
{
"h": 0.02,
"x": 0.76,
"diff": -0.07198678053533603,
"method": "gaussian"
},
{"h": 0.02, "x": 0.76, "diff": -1.4153075967509832, "method": "jacobi"},
{
"h": 0.02,
"x": 0.78,
"diff": -0.08111750919765259,
"method": "gaussian"
},
{"h": 0.02, "x": 0.78, "diff": -1.3452278848287431, "method": "jacobi"},
{"h": 0.02, "x": 0.8, "diff": -0.09025830767498544, "method": "gaussian"},
{"h": 0.02, "x": 0.8, "diff": -1.2761279634090579, "method": "jacobi"},
{
"h": 0.02,
"x": 0.8200000000000001,
"diff": -0.09936897057730132,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.8200000000000001,
"diff": -1.1978987722311678,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.8400000000000001,
"diff": -0.10840779576002357,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.8400000000000001,
"diff": -1.1288684892609913,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.8600000000000001,
"diff": -0.11733170841912866,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.8600000000000001,
"diff": -1.0507162812866406,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.8800000000000001,
"diff": -0.12609639629965275,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.8800000000000001,
"diff": -0.9924923358413914,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.9000000000000001,
"diff": -0.1346564558708122,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.9000000000000001,
"diff": -0.9251971709617842,
"method": "jacobi"
},
{
"h": 0.02,
"x": 0.9199999999999999,
"diff": -0.14296554926647093,
"method": "gaussian"
},
{
"h": 0.02,
"x": 0.9199999999999999,
"diff": -0.8890480644256533,
"method": "jacobi"
},
{"h": 0.02, "x": 0.94, "diff": -0.1509765717338491, "method": "gaussian"},
{"h": 0.02, "x": 0.94, "diff": -0.8438706272148776, "method": "jacobi"},
{
"h": 0.02,
"x": 0.96,
"diff": -0.15864182927647258,
"method": "gaussian"
},
{"h": 0.02, "x": 0.96, "diff": -0.8384720670924672, "method": "jacobi"},
{
"h": 0.02,
"x": 0.98,
"diff": -0.16591322611940593,
"method": "gaussian"
},
{"h": 0.02, "x": 0.98, "diff": -0.8240325257148612, "method": "jacobi"}
]
}
}