Numerical methods

Study Notes

Numerical methods involve calculating approximate solutions to equations. These methods help locate roots of equations using graphical approaches or by analyzing changes in sign. Fixed point iteration is a technique to refine approximations of roots using iterative formulas.

• Locating roots — Finding where a function equals zero by graphical methods or sign changes. Example: If $f(x)$ changes sign between $x = -1$ and $x = 0$, there is a root in that interval.
• Graphical approach — Using graphs to determine the number of real roots by observing intersections. Example: Sketching graphs to find intersections that indicate roots.
• Change of sign approach — Identifying roots by checking for sign changes in a continuous interval. Example: $f(-1)$ and $f(0)$ have opposite signs, indicating a root.
• Fixed point iteration — Using an iterative formula to approximate a root by refining initial guesses. Example: Rearranging $x^3 + 3x = 21$ to $x = F(x)$ and iterating to find a solution.

Exam Tips

Key Definitions to Remember

• Locating roots involves finding where a function equals zero.
• Fixed point iteration is a method to refine root approximations using iterative formulas.

Common Confusions

• Confusing the graphical approach with the change of sign approach.
• Misunderstanding the iterative process and stopping too early before convergence.

Typical Exam Questions

• How do you determine the number of real roots using a graph? By counting the intersections of the graph with the x-axis.
• What indicates a root in the change of sign method? A change in sign of $f(x)$ over an interval.
• How is a fixed point iteration performed? By rearranging the equation and iterating to refine the root approximation.

What Examiners Usually Test

• Ability to locate roots using graphical and sign change methods.
• Understanding and application of fixed point iteration to find accurate root approximations.