Iteration
Iteration
Introduction
- Iteration is a process that involves a methodical and repetitive approach to solving problems, particularly equations.
- Equations may not always be solvable through straightforward algebra, therefore requiring this iterative method.
The Process of Iteration
- A initial ‘guess’ or trial value, typically denoted as x₀, of the root (solution) of the equation is made.
- This trial value is then substituted into a rearranged version of the original equation to generate a new value; subsequently, this process is repeated.
- The process is continued until the root (solution) is found to the required degree of accuracy. However, some equations may not have solutions, or the solutions may not be found through iteration.
Iterative Formula
- To carry out the process of iteration, an iterative formula, often in the form of xₙ₊₁ = f(xₙ), is generated.
- In this formula, xₙ₊₁ is the next value, and xₙ is the current value.
- f(xₙ) is the function generated through rearranging the original equation, into which the current value is substituted.
Examples
- In the equation x² - x - 2 = 0, rewritten as x = √(x+2), if we use a trial value of x₀ = 2, the process would be: x₁ = √(2+2) = √4 = 2 x₂ = √(2+2) = √4 = 2 Since x₁ = x₂ = 2, then the root of the equation is x = 2.
Conclusion
- Iteration is a practical and effective method for solving complex equations that cannot be easily solved through traditional algebraic methods.
- Understanding iteration and being able to apply the iterative method can be critical in solving a wide array of mathematical problems. Regular practice of this method can enhance your problem-solving abilities in mathematics.