7 The value of a function, \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), and its gradient function, \(\frac { \mathrm { dy } } { \mathrm { dx } }\), when \(x = 2\), is given in Table 7.1.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Table 7.1}
| \(x\) | \(\mathrm { f } ( x )\) | \(\frac { \mathrm { dy } } { \mathrm { dx } }\) |
| 2 | 6 | - 2.8 |
\end{table}
- Determine the approximate value of the error when \(f ( 2 )\) is used to estimate \(f ( 2.03 )\).
The Newton-Raphson method is used to find a sequence of approximations to a root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\). The spreadsheet output showing the iterates, together with some further analysis, is shown in Table 7.2.
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Table 7.2}
| A | B | C | D |
| 1 | r | Xr | difference | ratio |
| 2 | 0 | 12 | | |
| 3 | 1 | -13.1165572 | -25.1165572 | |
| 4 | 2 | 1.76283279 | 14.87939004 | -0.5924136 |
| 5 | 3 | 2.18052157 | 0.41768878 | 0.02807163 |
| 6 | 4 | 2.182419024 | 0.001897454 | 0.00454275 |
| 7 | 5 | 2.182419066 | \(4.13985 \mathrm { E } - 08\) | \(2.1818 \mathrm { E } - 05\) |
- Explain what the values in column D tell you about the order of convergence of this sequence of approximations.
- Without doing any further calculation, state the value of \(\alpha\) as accurately as you can, justifying the precision quoted.