1.9 Numerical Methods

• What are numerical methods used for?

  Approximating solutions
• Numerical methods provide an approximate solution to mathematical problems.
• Non-iterative methods provide a direct, one-step solution without iteration.


  True
• Why are numerical methods useful in mathematics?

  Solve unsolvable problems
• Iterative methods require less computational effort than non-iterative methods

  False
• Steps of the bisection method algorithm

  1️⃣ Start with an interval [a, b] containing a root

  2️⃣ Calculate the midpoint c = (a + b) / 2

  3️⃣ If f(a) and f(c) have opposite signs, update b = c

  4️⃣ If f(b) and f(c) have opposite signs, update a = c

  5️⃣ Repeat steps 2-4 until the interval is sufficiently small
• The update formula for the Newton-Raphson method is x1 = x0 - f(x0) / f'(x0)
• Numerical methods use algorithms to approximate solutions to mathematical problems.

  True
• What is the trade-off when using numerical methods?

  Accuracy versus complexity
• Match the method with its type:

  Newton's method ↔️ Iterative

  Midpoint rule ↔️ Non-iterative
• Into how many main types can numerical methods be classified?

  Two
• What is an example of a non-iterative method for integration?

  Midpoint rule
• The bisection method is faster than other root-finding methods for all problems

  False
• The Newton-Raphson method requires the ability to calculate the derivative of the function


  True
• Analytical methods derive an exact, closed-form solution
• The bisection method starts with an interval where f(a) and f(b) have opposite signs
• What is one advantage of the bisection method?

  Simplicity
• The bisection method starts with an interval [a, b] where f(a) and f(b) have opposite signs
• The bisection method guarantees finding a root if one exists in the initial interval.

  True
• What is one disadvantage of the bisection method?

  Slow convergence
• What is the formula to calculate the midpoint in the bisection method?

  $c =$$(a + b) / 2$
• What is one advantage of the Newton-Raphson method?

  Faster convergence
• The Newton-Raphson method has faster convergence than the bisection
• What is the key requirement for the Newton-Raphson method to work effectively?

  Initial guess near the root
• The Newton-Raphson method is always guaranteed to converge regardless of the initial guess.

  False
• The fixed point iteration method requires the calculation of derivatives.

  False
• Order the methods from slowest to fastest convergence rate:

  1️⃣ Bisection

  2️⃣ Fixed-Point Iteration

  3️⃣ Newton-Raphson
• Root-finding is a key example of a numerical method used to find the roots or zeros of a function
• How are numerical methods classified based on their approach?

  Iterative and non-iterative
• What is the bisection method used for?

  Finding roots of a function
• What is the bisection method used for?

  Finding roots of functions
• In the bisection method, the midpoint `c` is calculated as `(a + b) / 2
• What is a key advantage of the bisection method?

  Simplicity
• For the function `f(x) = x^2 - 4`, the derivative `f'(x) = 2x`.


  True
• What is a major advantage of the Newton-Raphson method over the bisection method?

  Faster convergence
• In fixed point iteration, the formula for calculating the next estimate is `x_{n+1} = g(x_n)
• Match the numerical method with its convergence rate:

  Bisection ↔️ Linear

  Newton-Raphson ↔️ Quadratic
• In fixed-point iteration, the process stops when the absolute difference between successive estimates is less than a specified tolerance
• Steps of fixed-point iteration in the correct order:

  1️⃣ Choose an initial guess x0

  2️⃣ Iterate using x_{n+1} = g(x_n)

  3️⃣ Stop when |x_{n+1} - x_n| < tolerance
• The bisection method is guaranteed to converge regardless of the initial interval.


  True