Example 1. Solve the equation $5 x-\cos x=0$.
(4)

Here it should be emphasized: When $x$ represents an angle, it is always calculated in radians.
Solving (4) can be transformed into $x=\frac{1}{5} \cos x$, yielding the iterative formula $x_{n+1}=\frac{1}{5} \cos x_{n}$. Taking $x_{0}=0.2$, after displaying $x_{0}$ on the keyboard, set the "mode key" to "RAD" (radians) position, and then press the above four keys in sequence, to find $x_{5}, \cdots$, the results are as follows:
$$
\begin{array}{l}
x_{1}=0.1861311, \\
x_{2}=0.19517616, \\
x_{3}=0.19616405, \\
x_{4}=0.19616429, \\
x_{5}=x_{8}=0.19616428 .
\end{array}
$$

Clearly, the sequence $\left\{x_{n}\right\}$ converges, so we can take $\mathbf{x}=\mathrm{x}_{0}=0.19616428$ as the approximate root of equation (4).
After completing Example 1, one cannot help but ask the following questions:
How can we construct an iterative format so that the approximate root sequence $\left\{x_{n}\right\}$ converges and converges quickly? How do we determine the initial value $x_{0}$? Can we estimate the error bound between the approximate root and the exact root?

Indeed, without solving these problems, the iterative method is difficult to implement. For this, we have the following convergence theorem:
Consider the equation
$$
\mathrm{x}=\varphi(\mathrm{x})
$$

which has a root $\mathrm{x}^{*}$ in $[\mathrm{a}, \mathrm{b}]$, and $\varphi(\mathrm{x})$ has a continuous first derivative in $[\mathrm{a}, \mathrm{b}]$. Let $\rho=\max_{a \leqslant x \leqslant b}\left|\varphi^{\prime}(x)\right|$, if $0<\rho<1$,
$$
\mathrm{a} \leqslant \mathrm{x} \leqslant \mathrm{b}
$$

then equation (2) has a unique solution in $[a, b]$. Furthermore, for any $x_{0} \in [a, b]$, the sequence $\left\{x_{n}\right\}$ determined by the iterative format
$$
x_{n+1}=\varphi \quad\left(x_{n}\right)
$$

converges to $x^{*}$ (at this time, we also say that format (3) converges), and the following error estimate holds:
$$
\left|x^{*}-x_{n}\right| \leqslant \frac{\rho}{1-\rho}\left|x_{n}-x_{n-1}\right| .
$$

The proof can be found in [3]. According to the above theorem, for a given equation (2), before establishing the corresponding iterative format, two things should be done:
1. Determine whether $\rho=\max_{a \leqslant x \leqslant b} \mid \varphi^{\prime}(x) \mid$ is less than 1. For those who have not studied calculus, we can take
$\varphi^{\prime}(x) \approx \frac{\varphi(x+h)-\varphi(x)}{h}$, where $h$ is a small positive number. At this point, it is only necessary to verify whether the following inequality holds:
$$
\max _{\mathrm{a} \leqslant \mathrm{b}}\left|\frac{\varphi(\mathrm{x}+\mathrm{h})-\varphi(\mathrm{x})}{\mathrm{h}}\right|<1 .
$$
2. Determine the root interval $[\mathrm{a}, \mathrm{b}]$ of $x=\varphi(x)$. Here, we introduce two simple methods:
i) Graphical method. Draw the line $y=x$ and the curve $y=\varphi(x)$. If the two lines intersect, then a small interval containing the x-coordinate of the intersection point is the root interval $[a, b]$;
ii) Take $f(x)=x-\varphi(x)$, and calculate the values of $f(x)$ at some points. Find two points $a, b$ (as close as possible), and it is known that the equation $x=\varphi(x)$ has at least one root in $(a, b)$.
To speed up convergence, the length of the interval $[\mathrm{a}, \mathrm{b}]$ should be as small as possible.