Solving polynomial equations by iteration

Solving polynomial equations by iteration

Lessons

In this lesson, we will learn:

  • Solving Equations Using Direct Iteration
  • Evaluating equations Using Iteration by Bisection
  • Analyzing Equations Using Newton-Raphson Method
• Iteration means to repeatedly solving an equation to obtain a result using the result from the previous calculation.
• Direct iteration:
1. Rearrange the original equation such that the term in which the variable with the highest exponent is isolated.
2. Leave the variable on its own on the LHS by performing inverse operation.
3. The LHS becomes xn+1x_{n+1}.
4. The RHS becomes xnx_n.
• Iteration by bisection:
1. Shrink the interval where the roots lies within 2 equal parts.
2. Decide in which part the solution resides.
3. Repeat the steps until a consistent answer is achieved.
• Newton-Raphson method:
xn+1=xnf(xn)f(xn)x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}
  • Introduction
    Introduction to solving polynomial equations by iteration
    a)
    Direct/Fixed point iteration

    b)
    Iteration by bisection

    c)
    Newton-Raphson method


  • 1.
    Solving Equations Using Direct Iteration
    a)
    Show that x25x8=0x^2-5x-8=0 can be written in the form x=8+5xx=\sqrt{8+5x}.

    b)
    Use the iteration formula xn+1=8+5xnx_{n+1}=\sqrt{8+5x_n} to find x3x_3 to 22 decimal places. Start with x0=2x_0=2.


  • 2.
    Solving Equations Using Direct Iteration
    a)
    Show that x3x8=0x^3-x-8=0 can be written in the form x=3x+8x={^3}\sqrt{x+8}.

    b)
    Use the iteration formula xn+1=3xn+8x_{n+1}={^3}\sqrt{x_n+8} to find x4x_4 to 22 decimal places. Start with x1=0x_1=0.


  • 3.
    Evaluating equations Using Iteration by Bisection
    The equation x3+5x7=91x^3+5x-7=91 has a solution between 4 and 5. Use bisection iteration to find the solution and give the answer to 1 decimal place.

  • 4.
    Use bisection iteration to solve x3x2=39x^3-x^2=39. Give your answer to 1 decimal place.

  • 5.
    Analyzing Equations Using Newton-Raphson Method
    Given x26x+5=0x^2-6x+5=0.
    a)
    Find the iteration formula.

    b)
    Use the iteration formula found in (a) to approximate the solution. Start with x1=2x_1=2.