-
Notifications
You must be signed in to change notification settings - Fork 1
/
plotting lebesgue function
189 lines (137 loc) · 4.64 KB
/
plotting lebesgue function
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
### INSERT CODE HERE
## Plot the Lebesgue function for x∈[−1,1]x∈[−1,1] for N=5,10,20N=5,10,20 with equal distance nodes
#generate x_i's
data_5N= numpy.linspace(-1.0, 1.0, 6)
data_10N= numpy.linspace(-1.0, 1.0, 11)
data_20N= numpy.linspace(-1.0, 1.0, 21)
#===============
#for N=5
#generate lagrange basis
x = numpy.linspace(-1.0, 1.0, 100)
lagrange_basis_5 = numpy.ones((len(data_5N), x.shape[0]))
for i in xrange(len(data_5N)):
for j in xrange(len(data_5N)):
if i != j:
lagrange_basis_5 [i, :] *= (x - data_5N[j]) / (data_5N[i] -data_5N[j])
#lebesgue function values
lebesgue_5=numpy.zeros( x.shape[0])
for n in xrange(len(data_5N)):
lebesgue_5 += abs(lagrange_basis_5[n, :])
#===============
#for N=10
#generate lagrange basis
lagrange_basis_10 = numpy.ones((len(data_10N), x.shape[0]))
for i in xrange(len(data_10N)):
for j in xrange(len(data_10N)):
if i != j:
lagrange_basis_10 [i, :] *= (x - data_10N[j]) / (data_10N[i] -data_10N[j])
#lebesgue function values
lebesgue_10=numpy.zeros( x.shape[0])
for n in xrange(len(data_10N)):
lebesgue_10 += abs(lagrange_basis_10[n, :])
#=================
#for N=20
#generate lagrange basis
lagrange_basis_20 = numpy.ones((len(data_20N), x.shape[0]))
for i in xrange(len(data_20N)):
for j in xrange(len(data_20N)):
if i != j:
lagrange_basis_20 [i, :] *= (x - data_20N[j]) / (data_20N[i] -data_20N[j])
#lebesgue function values
lebesgue_20=numpy.zeros( x.shape[0])
for n in xrange(len(data_20N)):
lebesgue_20 += abs(lagrange_basis_20[n, :])
#graph
fig1 = plt.figure()
axes1 = fig1.add_subplot(1, 1, 1)
axes1.set_title("lebesgue function with N=5")
axes1.set_xlabel("x")
axes1.set_ylabel("$\lambda_N(x)$")
axes1.plot (x,lebesgue_5)
plt.show()
fig2 = plt.figure()
axes2 = fig2.add_subplot(1, 1, 1)
axes2.set_title("lebesgue function with N=5")
axes2.set_xlabel("x")
axes2.set_ylabel("$\lambda_N(x)$")
axes2.plot (x,lebesgue_10)
plt.show()
fig3 = plt.figure()
axes3 = fig3.add_subplot(1, 1, 1)
axes3.set_title("lebesgue function with N=5")
axes3.set_xlabel("x")
axes3.set_ylabel("$\lambda_N(x)$")
axes3.plot (x,lebesgue_20)
plt.show()
### INSERT CODE HERE
## Plot the Lebesgue function for x∈[−1,1]x∈[−1,1] for N=5,10,20N=5,10,20 with Chebyshev's nodes
#generate nodes
data_5N=numpy.zeros(6)
for n in xrange(1,len(data_5N)+1):
data_5N[n-1]=numpy.cos(((2*n-1)*numpy.pi)/(2*5))
data_10N=numpy.zeros(11)
for n in xrange(1,len(data_10N)+1):
data_10N[n-1]=numpy.cos(((2*n-1)*numpy.pi)/(2*10))
data_20N=numpy.zeros(21)
for n in xrange(1,len(data_20N)+1):
data_20N[n-1]=numpy.cos(((2*n-1)*numpy.pi)/(2*20))
#===============
#generate values for lebesgue funtion
#for N=5
#generate lagrange basis
x = numpy.linspace(-1.0, 1.0, 100)
lagrange_basis_5 = numpy.ones((len(data_5N), x.shape[0]))
for i in xrange(len(data_5N)):
for j in xrange(len(data_5N)):
if i != j:
lagrange_basis_5 [i, :] *= (x - data_5N[j]) / (data_5N[i] -data_5N[j])
#lebesgue function values
lebesgue_5=numpy.zeros( x.shape[0])
for n in xrange(len(data_5N)):
lebesgue_5 += abs(lagrange_basis_5[n, :])
#===============
#for N=10
#generate lagrange basis
lagrange_basis_10 = numpy.ones((len(data_10N), x.shape[0]))
for i in xrange(len(data_10N)):
for j in xrange(len(data_10N)):
if i != j:
lagrange_basis_10 [i, :] *= (x - data_10N[j]) / (data_10N[i] -data_10N[j])
#lebesgue function values
lebesgue_10=numpy.zeros( x.shape[0])
for n in xrange(len(data_10N)):
lebesgue_10 += abs(lagrange_basis_10[n, :])
#=================
#for N=20
#generate lagrange basis
lagrange_basis_20 = numpy.ones((len(data_20N), x.shape[0]))
for i in xrange(len(data_20N)):
for j in xrange(len(data_20N)):
if i != j:
lagrange_basis_20 [i, :] *= (x - data_20N[j]) / (data_20N[i] -data_20N[j])
#lebesgue function values
lebesgue_20=numpy.zeros( x.shape[0])
for n in xrange(len(data_20N)):
lebesgue_20 += abs(lagrange_basis_20[n, :])
#graph
fig1 = plt.figure()
axes1 = fig1.add_subplot(1, 1, 1)
axes1.set_title("lebesgue function with N=5")
axes1.set_xlabel("x")
axes1.set_ylabel("$\lambda_N(x)$")
axes1.plot (x,lebesgue_5)
plt.show()
fig2 = plt.figure()
axes2 = fig2.add_subplot(1, 1, 1)
axes2.set_title("lebesgue function with N=5")
axes2.set_xlabel("x")
axes2.set_ylabel("$\lambda_N(x)$")
axes2.plot (x,lebesgue_10)
plt.show()
fig3 = plt.figure()
axes3 = fig3.add_subplot(1, 1, 1)
axes3.set_title("lebesgue function with N=5")
axes3.set_xlabel("x")
axes3.set_ylabel("$\lambda_N(x)$")
axes3.plot (x,lebesgue_20)
plt.show()