1. What is the Method of Successive Substitution?

The Method of Successive Substitution (also called Fixed-Point Iteration) is a numerical method for solving equations of the form x = g(x). It generates a sequence of approximations that (under certain conditions) converges to the solution.

2. How Does the Calculator Work?

The calculator implements the iterative process:

Where:

• \( x_n \) — Current approximation
• \( x_{n+1} \) — Next approximation
• \( g(x) \) — The function to iterate

Explanation: The method starts with an initial guess and repeatedly applies the function g(x) to generate successive approximations to the solution.

3. Convergence Criteria

Details: For the method to converge, the function g(x) must satisfy certain conditions (typically |g'(x)| < 1 in a neighborhood of the solution). The iterations stop when the difference between successive approximations is less than the specified tolerance.

4. Using the Calculator

Tips:

• Enter the function g(x) using standard mathematical notation (e.g., "cos(x)", "sqrt(x+1)", "(x^2+2)/3")
• Provide a reasonable initial guess close to the expected solution
• Smaller tolerance values require more iterations but give more precise results

5. Frequently Asked Questions (FAQ)

Q1: What if the method doesn't converge?
A: Try a different initial guess or reformulate the equation as x = g(x) with a different g(x) function.

Q2: How do I know if the method will converge?
A: The method converges if |g'(x)| < 1 near the solution. You can check this by calculating the derivative.

Q3: What's the difference between this and Newton's method?
A: Newton's method typically converges faster but requires calculating derivatives. Successive substitution is simpler but may require more iterations.

Q4: Can I use this for systems of equations?
A: This calculator handles single equations only. Systems require more advanced methods.

Q5: What functions can I enter?
A: Most standard mathematical functions: sin, cos, tan, exp, log, sqrt, etc. Use x as the variable.