Hugging Face's logo

Spaces:
r3hab
/
numerical-analysis
Running

App Files Community
r3hab commited on
Commit
f5ba7f3
·
verified ·
1 Parent(s): 9fee162

Create 1.html

Browse files
Files changed (1) hide show
  1. 1.html +159 -0
1.html ADDED
@@ -0,0 +1,159 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ <!DOCTYPE html>
2
+ <html>
3
+ <head>
4
+ <title>Newton's Divided Difference Interpolation</title>
5
+ <script src="https://polyfill.io/v3/polyfill.min.js?presets=full"></script>
6
+ <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
7
+ <style>
8
+ body {
9
+ font-family: Arial, sans-serif;
10
+ }
11
+ .equation {
12
+ display: block;
13
+ margin-left: auto;
14
+ margin-right: auto;
15
+ text-align: center;
16
+ margin-top: 1em;
17
+ margin-bottom: 1em;
18
+ }
19
+ .table-container {
20
+ display: flex;
21
+ justify-content: center;
22
+ margin-top: 1em;
23
+ margin-bottom: 1em;
24
+ }
25
+ .data-table {
26
+ border-collapse: collapse;
27
+ }
28
+ .data-table th, .data-table td {
29
+ border: 1px solid black;
30
+ padding: 8px;
31
+ text-align: center;
32
+ }
33
+ .boxed-answer {
34
+ border: 1px solid black;
35
+ padding: 10px;
36
+ display: inline-block;
37
+ margin-top: 1em;
38
+ }
39
+ ol li {
40
+ margin-bottom: 0.5em;
41
+ }
42
+ </style>
43
+ </head>
44
+ <body>
45
+
46
+ <p>Given the values:</p>
47
+
48
+ <div class="table-container">
49
+ <table class="data-table">
50
+ <thead>
51
+ <tr>
52
+ <th>\(x\)</th>
53
+ <th>\(f(x)\)</th>
54
+ </tr>
55
+ </thead>
56
+ <tbody>
57
+ <tr>
58
+ <td>5</td>
59
+ <td>150</td>
60
+ </tr>
61
+ <tr>
62
+ <td>7</td>
63
+ <td>392</td>
64
+ </tr>
65
+ <tr>
66
+ <td>11</td>
67
+ <td>1452</td>
68
+ </tr>
69
+ <tr>
70
+ <td>13</td>
71
+ <td>2366</td>
72
+ </tr>
73
+ <tr>
74
+ <td>17</td>
75
+ <td>5202</td>
76
+ </tr>
77
+ </tbody>
78
+ </table>
79
+ </div>
80
+
81
+ <p>We need to evaluate \( f(9) \) using Newton’s divided difference formula.</p>
82
+
83
+ <p>First, we compute the divided differences:</p>
84
+
85
+ <ol>
86
+ <li>
87
+ <p><b>First divided differences:</b></p>
88
+ <div class="equation">
89
+ \[
90
+ \begin{aligned}
91
+ f[5, 7] &= \frac{392 - 150}{7 - 5} = 121, \\
92
+ f[7, 11] &= \frac{1452 - 392}{11 - 7} = 265, \\
93
+ f[11, 13] &= \frac{2366 - 1452}{13 - 11} = 457, \\
94
+ f[13, 17] &= \frac{5202 - 2366}{17 - 13} = 709.
95
+ \end{aligned}
96
+ \]
97
+ </div>
98
+ </li>
99
+ <li>
100
+ <p><b>Second divided differences:</b></p>
101
+ <div class="equation">
102
+ \[
103
+ \begin{aligned}
104
+ f[5, 7, 11] &= \frac{265 - 121}{11 - 5} = 24, \\
105
+ f[7, 11, 13] &= \frac{457 - 265}{13 - 7} = 32, \\
106
+ f[11, 13, 17] &= \frac{709 - 457}{17 - 11} = 42.
107
+ \end{aligned}
108
+ \]
109
+ </div>
110
+ </li>
111
+ <li>
112
+ <p><b>Third divided differences:</b></p>
113
+ <div class="equation">
114
+ \[
115
+ \begin{aligned}
116
+ f[5, 7, 11, 13] &= \frac{32 - 24}{13 - 5} = 1, \\
117
+ f[7, 11, 13, 17] &= \frac{42 - 32}{17 - 7} = 1.
118
+ \end{aligned}
119
+ \]
120
+ </div>
121
+ </li>
122
+ <li>
123
+ <p><b>Fourth divided difference:</b></p>
124
+ <div class="equation">
125
+ \[
126
+ f[5, 7, 11, 13, 17] &= \frac{1 - 1}{17 - 5} = 0.
127
+ \]
128
+ </div>
129
+ </li>
130
+ </ol>
131
+
132
+ <p>Using these divided differences, the interpolating polynomial is constructed as:</p>
133
+ <div class="equation">
134
+ \[
135
+ P(x) = 150 + 121(x - 5) + 24(x - 5)(x - 7) + 1(x - 5)(x - 7)(x - 11)
136
+ \]
137
+ </div>
138
+
139
+ <p>Evaluating this polynomial at \( x = 9 \):</p>
140
+ <div class="equation">
141
+ \[
142
+ \begin{aligned}
143
+ P(9) &= 150 + 121(9 - 5) + 24(9 - 5)(9 - 7) + 1(9 - 5)(9 - 7)(9 - 11) \\
144
+ &= 150 + 121 \cdot 4 + 24 \cdot 4 \cdot 2 + 1 \cdot 4 \cdot 2 \cdot (-2) \\
145
+ &= 150 + 484 + 192 - 16 \\
146
+ &= 810.
147
+ \end{aligned}
148
+ \]
149
+ </div>
150
+
151
+ <p>Thus, the value of \( f(9) \) is:</p>
152
+ <div class="boxed-answer">
153
+ \[
154
+ \boxed{810}
155
+ \]
156
+ </div>
157
+
158
+ </body>
159
+ </html>