Iterative methods are often used to find the zeros of transcendental equations, which are equations involving transcendental functions such as trigonometric functions, exponential functions, or logarithmic functions. One commonly used iterative method for finding zeros is the Newton-Raphson method, also known as the Newton’s method.

Newton-Raphson Method:

1. Initial Guess: Start with an initial guess x0​ close to the actual root of the equation.
2. Iteration: Repeat the following steps until a desired level of accuracy is achieved:
   • Compute the function value

     $$

     f(xn​) and its derivative
     $f′(xn​) at the current iterate $

     
     ${\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord"><span\; class="mord\; mathnormal">x</span><span\; class="msupsub"><span\; class="vlist-t\; vlist-t2"><span\; class="vlist-r"><span\; class="vlist"><span\; class="pstrut"></span><span\; class="sizing\; reset-size6\; size3\; mtight"><span\; class="mord\; mathnormal\; mtight">n</span></span></span><span\; class="vlist-s"></span></span></span></span></span></span></span></span>.</span>}}_{}$

   • Compute the next iterate using the formula:
     

     xn+1​=xn​−f′(xn​)f(xn​)​

3. Convergence Criterion: Check for convergence by evaluating the absolute difference between successive iterates. If this difference is below a specified tolerance level, the iteration is terminated, and the current iterate is accepted as the approximate root.
4. Output: The final iterate xn​ is an approximation to the root of the transcendental equation.

Example:

Consider the equation

f(x)=sin(x)−x2=0, and we want to find its root.

1. Initial Guess: Start with an initial guess, say x0​=1.
2. Iteration:
   • At

     ${}_{}$

     x0​=1,
     $\mathrm{<br>}(1)=\sin (1)-1^2\approx -0.1585$

     f(1)=sin(1)−12≈−0.1585 and 

     
     ${\mathrm{<br>}}^{}(1)=\cos (1)-2\approx -1.5836$

     f′(1)=cos(1)−2≈−1.5836.

   • Compute the next iterate using the formula:

     ${\mathrm{<br>}}_{}=1-\frac{-0.1585}{-1.5836}\approx 1.1009$

     x1​=1−−1.5836−0.1585​≈1.1009

   • Repeat this process until convergence is achieved.
3. Convergence Criterion: Check if the absolute difference between successive iterates is below a specified tolerance level (e.g., ∣xn+1​−xn​∣<ϵ).
4. Output: The final iterate

   ${}_{}$

   xn​ is an approximation to the root of the transcendental equation.

Iterative methods like Newton-Raphson are powerful tools for finding zeros of transcendental equations but may require careful selection of initial guesses and convergence criteria to ensure accuracy and convergence. Additionally, they may not converge for all types of functions or under certain conditions, so robustness and convergence analysis are essential considerations when using these methods.