Repository navigation
Expand file tree
/
Copy pathCubic_Spline_Interpolation.py
More file actions
150 lines (100 loc) · 4.85 KB
/
Copy pathCubic_Spline_Interpolation.py
File metadata and controls
150 lines (100 loc) · 4.85 KB
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
# Author: Jerome Richards
# Cubic Spline Interpolation
#
# Interpolates the data using Cubic Splines
##########################################
# Please reference textbook: #
# #
# Numerical Analysis Second Edition #
# By: #
# Timothy Sauer #
# #
# to understand the logic and reasoning #
# behind each declared variable and #
# array. #
# #
##########################################
import numpy as np
import matplotlib.pyplot as plt
# Description: Uses Cubic Spline interpolation on the given x and y data points
# Parameters: data on the horizontal axis, x_data, and on the vertical axis, y_data
# Returns: two arrays - x values and y values with calculated interpolation
def SplineFunction(x_data, data):
y_data = data; # Data on the y-axis
#################################################################
# #
# The cubic spline fomula function once all constants are found #
# #
#################################################################
def S(x, x_0, a, b, c, d):
return ( a + b*(x - x_0) + c*((x - x_0)**2) + d*((x - x_0)**3) );
def dS(x, x_0, a, b, c, d):
return ( b + 2*c*(x - x_0) + 3*d*((x - x_0)**2) );
delta_x = np.zeros((len(x_data) - 1, 1)); # The differences in length between x values
delta_y = np.zeros((len(x_data) - 1, 1)); # The differences in length between y values
D = np.zeros((len(x_data) - 1, 1)); # List of 'd' terms that will be used in the Spline Function
B = np.zeros((len(x_data) - 1, 1)); # List of 'b' terms that will be used in the Spline Function
M = np.zeros((len(x_data), len(x_data))); # The Spline Matrix
h = np.zeros((len(x_data), 1)); # The Spline Vector
############################################
# #
# Populates the delta_x and delta_y arrays #
# #
############################################
for i in range(0, len(x_data) - 1):
delta_x[i] = x_data[i + 1] - x_data[i];
delta_y[i] = y_data[i + 1] - y_data[i];
###############################
# #
# Populates the Spline Matrix #
# #
###############################
M[0][0] = 1;
for i in range(1, len(x_data) - 1):
M[i][i - 1] = delta_x[i - 1];
M[i][i] = 2*(delta_x[i - 1] + delta_x[i]);
M[i][i + 1] = delta_x[i];
M[len(x_data) - 1][len(x_data) - 1] = 1;
###############################
# #
# Populates the Spline Vector #
# #
###############################
h[0] = 0;
for i in range(0, len(h) - 2):
h[i + 1] = 3*( (delta_y[i + 1]/delta_x[i + 1]) - (delta_y[i]/delta_x[i]) );
h[len(h) - 1] = 0;
###################################################################################################
# #
# The Formula is M*C = h where C is a vector of unknowns. Solving for C, we take the inverse of M #
# and multiply it to both sides to attain C = M_inv*h #
# #
###################################################################################################
M_inv = np.linalg.inv(M); # inverse of matrix M
C = np.matmul(M_inv, h); # Solving for C
#####################################
# #
# Populates both the B and D arrays #
# #
#####################################
for i in range(0, len(D)):
D[i] = (C[i + 1] - C[i])/(3*delta_x[i]);
B[i] = (delta_y[i]/delta_x[i]) - (delta_x[i]/3)*(2*C[i] + C[i + 1]);
#######################################
# #
# Return new x and y points as arrays #
# #
#######################################
x = [];
y = [];
end = x_data[len(x_data) - 1];
I = 0;
i = x_data[0];
dx = 0.01;
while i < end:
x.append(i);
if i >= x_data[I + 1]:
I = I + 1;
y.append(S(i, x_data[I], y_data[I], B[I], C[I], D[I]));
i = i + dx;
return x, y;