ChE 104

• What is a non-linear algebraic equation?

  An equation with at least one variable exponent other than 1
• What characterizes a non-linear algebraic equation?

  It includes a product of variables
• What is the standard form of a linear equation?

  Ax + By + C = 0
• What does the equation f(x) = 0 represent?

  It represents a non-linear algebraic equation
• What is a solution of the equation f(x) = 0?

  x = s such that f(s) = 0
• What are the values of x called in an equation?

  Roots or zeros of the equation
• What is a method for determining roots of non-linear equations?

  Numerical approximation
• What is the nature of the numerical approximation method?

  It is systematic trial and error
• What is required to terminate the iteration in numerical approximation?

  Evaluation of the error after nth iteration
• How is approximate percent relative error computed?

  $\epsilon_a =$$\frac{X_{new} - X_{old}}{X_{new}} \times 100$
• What are the methods for determining roots of non-linear equations?

  1. Bisection Method
  2. Newton-Raphson Method
  3. Successive Substitution Method
  4. Secant Method
• What is the Bisection Method also known as?

  Binary chopping or Bolzano’s method
• What does the Bisection Method involve?

  Dividing the interval in half
• What condition must be met for the Bisection Method to find a root?

  f(x1) and f(xn) must have opposite signs
• What indicates the presence of at least one real root in the Bisection Method?

  f(x1)f(xn) < 0
• What happens if f(x) does not change sign between two points?

  Roots may still exist between the two points
• What does it mean if f(x) changes sign between two points?

  More than one root may exist between the two points
• What is the process of the Incremental Search Method?

  • Locate an interval where the function changes sign
  • Identify the root by dividing the interval into subintervals
• What is the first step in the Bisection Method?

  Choose xl and xu where the function changes
• How is the root xr estimated in the Bisection Method?

  $x_r =$$\frac{x_l + x_u}{2}$
• What do you check after estimating xr in the Bisection Method?

  If f(xl)f(xr) < 0
• What happens if f(xl)f(xr) < 0?

  The root lies in the lower subinterval
• What happens if f(xl)f(xr) > 0?

  The root lies in the upper subinterval
• What do you do if f(xl)f(xr) = 0?

  The root is xr; terminate the computation
• What is checked after the iterations in the Bisection Method?

  Approximate percent relative error
• What indicates the stopping criteria in the Bisection Method?

  Absolute relative error less than tolerance
• What is the sample problem involving the Bisection Method about?

  Finding height 'h' in a spherical tank
• What is the equation for height 'h' in the sample problem?

  $f(h) =$$h^3 - 9h^2 +$$3.8197$
• What is the stopping criterion for the sample problem?

  0.09 unit absolute error
• What is the Successive Substitution Method also known as?

  Fixed point iteration
• What does the Successive Substitution Method use to predict the root?

  A formula rearranging f(x) = 0
• What does the equation x = g(x) represent in the Successive Substitution Method?

  A new value of x as a function of old x
• What is the Newton-Raphson Method known for?

  It is widely used for root-locating
• How does the Newton-Raphson Method improve the estimate of the root?

  By extending a tangent from the point [xi, f(xi)]
• What is the principle behind Newton's method?

  The tangent line is close to the curve
• What does the Secant Method define?

  The derivative by a backward finite difference
• What is required for the Secant Method?

  Two initial estimates of x
• What is the concentration of pollutant bacteria model in the assignment?

  $C =$$75e^{-1.5t} +$$20e^{-0.075t}$
• What is the stopping criterion for the assignment problem?

  0.5% reduction in concentration
• What are the learning objectives of the study material?

  • Define non-linear algebraic equations (NLAE)
  • Familiarize with methods of determining roots of NLAE
  • Solve problems involving NLAE