2.0 Iterative Solution

Question 1

Guess and Check vs Direct Substitution

Answer 1

• The method of guessing and checking (also known as “successive Approximation” or “Trial and Error”) will find all roots of all equations whichever form of the equation is used
  
  • This is possible since we can see the direction of change(using intervals is inappropriate) caused by a new guess and adjust the next guess accordingly to find the root
  • The method of Direct Substitution can be applied only to Form 2 of the equation and will find a root only if a certain condition is satisfied

Question 2

General Rule for “Direct Substitution” to find the root

Answer 2

• Based on these graphs, you can see that for the method of Direct Substitution to work, the ABSolute value of the slope of the RHS function must be less than the slope of the LHS(=1) at the root

Question 3

In order to check whether a root could be found using the Direct Substitution method…

Answer 3

We calculate the ABSolute value of the derivative of the RHS of the equation
If the result is smaller than 1, the direct substitution method can find that root

Question 4

How to access Excel’s “Goal Seek” and “Solver” ?

Answer 4

Data->What-if Analysis
Data->Solver

OR
Data>Analysis

Question 5

How to load “Goal Seek” and “Solver”?

Answer 5

Excel>File>Options>Add-Ins>GO>Select “Analysis ToolPak” and “Solver Add-Ins”>OK

Question 6

Excel’s Goal Seek

Answer 6

• The method of Excel’s Goal Seek will find a root automatically when you provide just the initial guess
  
  • This is because the Goal Seek technique contains the algorithm that can sense the direction of change caused by the subsequent guess
  • It will find all roots if you can give the initial guess in the appropriate range for each root (that is why plotting a graph could be very helpful!)

Question 7

Excel’s Solver

Answer 7

• Excel’s Solver can find all roots for similar reasons to Goal Seek
  
  • Most importantly, Solver can change multiple variables that define the function, within given constraints
  • This is useful in
    1. Nonlinear Regression: Finding the best-fitting function to a data set following a given criteria
    2. Linear Programming/Optimization: Finding the optimum solution to problems that involve multiple variables