Question 12.1: Consider a function f(x) = x³ - 2x² + 3x - 5 = 0 Find the va......
Consider a function
f(x) = x³ – 2x² + 3x – 5 = 0
Find the value of x.
Step-by-Step
The derivative is
f^{\prime}(x) = 3x² – 4x + 3
Let the initial value of x = 3, then f(x) = 13 and f^{\prime}(x) = 18. For k = 1, from Equation 12.4,
0 = f(x_{0}) + f^{\prime}(x_{0})(x_{1} – x_{0})
x_{1} = x_{0} – \frac{f(x_{0})}{f^{\prime}(x_{0})} (12.4)
x_{1} = 3 – (13/18) = 2.278. Table 12.1 is compiled to k = 4 and gives x = 1.843. As a verification, substituting this value into the original equation, the identity is approximately satisfied.
| TABLE 12.1 Iterative Solution of a Function of One Variable (Example 12.1) |
|||
| k | x_{k} | f(x_{k}) | f^{\prime}(x_{k}) |
| 0 | 3 | 13 | 18 |
| 1 | 2.278 | 3.277 | 9.456 |
| 2 | 1.931 | 0.536 | 6.462 |
| 3 | 1.848 | 0.025 | 5.853 |
| 4 | 1.844 | ≈ 0 | |
Related Answered Questions
Question: 12.3
Verified Answer:
First, form a Y matrix as follows:
\overlin...
Question: 12.5
Verified Answer:
The only load on the transformer secondary is the ...
Question: 12.4
Verified Answer:
P–θ Network
First construct the P–θ network shown ...
Question: 12.2
Verified Answer:
\overline{J}(x)=\left|\begin{matrix} \frac{...