Chapter 3

• Analytical Solution


  A solution to the equation (also called a root of the equation) is a numerical value of 𝒙 that satisfies the equation.
• Graphical Solution


  the solution is the point where the function 𝑓(𝑥) crosses or touches the 𝑥-axis. An equation might have no solution or can have one or several (possibly many) roots.
• Numerical Solution


  a solution of an equation 𝑓 𝑥 = 0 is a value of 𝑥 that satisfies the equation approximately. This means that when 𝑥 is substituted in the equation, the value of 𝑓(𝑥) is close to zero, but not exactly zero.
• bracketing methods and open methods


  The methods used for solving equations numerically
• Bracketing Method


  the endpoints of the interval are the upper bound and lower bound of the solution. Then, by using a numerical scheme, the size of the interval is successively reduced until the distance between the endpoints is less than the desired accuracy of the solution.
• Open Method


  an initial estimate (one point) for the solution is assumed. The value of this initial guess for the solution should be close to the actual solution. Then, by using a numerical scheme, better (more accurate) values for the solution are calculated
• $X_{TS}$


  the true (exact) solution such that 𝑓 𝑥𝑇𝑆 = 0
• $X_{NS}$


  a numerically approximated solution such that 𝑓 𝑥𝑁𝑆 = $\epsilon$
• Four measures can be considered for estimating the error


  1. True error
  2. Tolerance in f(x)
  3. Tolerance in the solution
  4. Relative Error
• True Error


  the difference between the true solution and a numerical solution
• Tolerance in f(x)


  defined as the absolute value of the difference between 𝑓(𝑥��𝑆) and 𝑓(𝑥𝑁𝑆)
• Tolerance in the solution


  is the maximum amount by which the true solution can deviate from an approximate numerical solution
• Bracketing Methods


  1. Bisection Method
  2. Regula-Falsi Method
• Regula-Falsi Method


  (also called false position and linear interpolation methods) is a bracketing method for finding a numerical solution of an equation of the form 𝑓(𝑥) = 0 when it is known that, within a given interval [𝑎, 𝑏], 𝑓(��) is continuous and the equation has a solution
• Open Methods

  1. Newton's Method
  2. Secant Method
  3. Fixed-Point Iteration Method
• Newton's Method


  (also called the Newton-Raphson method) is a scheme for finding a numerical solution of an equation of the form f(x) = 0 where f(x) is continuous and differentiable, and the equation is known to have a solution near a given point
• Slope of the Curve


  is a slope of a tangent line for a curve at one point
• Rate of Change


  The second derivative is the rate of change of the slope.
• y''


  rate of change of slope
• Second Method


  is a scheme for finding a numerical solution of an equation of the form 𝑓 𝑥 = 0. The method uses two points in the neighborhood of the solution to determine a new estimate for the solution.
• Fixed-point Iteration Method


  is a method for solving an equation of the form �� 𝑥 = 0 .The method is carried out by rewriting the equation in the form 𝒙 = 𝒈(𝒙).
• Lipschitz continuous


  The fixed-point iteration method converges if, in the neighborhood of the fixed point, the derivative of g(x) has an absolute value that is smaller than 1 (also called