CE 2020 Exam 2

What is the formula used to compute the golden ratio, ? What is the geometrical definition of the golden ratio? Provide a sketch to illustrate your answer as needed.

( 1+sqrt(5) ) / 2 If you divide a line into 2 segments, (segment1+segment2) / segment1 = segment1 / segment2

Produce a diagram showing the typical case of searching the root of a function using the False Position method for which the convergence to the solution will be slow.

(x^10) - 1

Briefly explain what the Brent method for solving root problems is. Produce a diagram showing schematically the implementation of the method. Highlight also the main differences with the Secant method.

Brent's method is an algorithm that applies a speedy open method whenever possible, but reverts to a reliable bracketing method if necessary. The bracketing technique is the bisection method, whereas the open methods are the secant method and the inverse quadratic interpolation.

With reference to an engineering model, what does the term optimization refer to? What is the difference between roots of a function and optima of a function?

From an engineering point of view, optimization is the process of creating something that is effective as possible. The optima includes the maxima and minima of a plot, which is usually on the y-axis.

In the context of optimization of one-dimensional functions, what do the terms global optimum, local optimum and multimodal cases refer to? Provide a sketch to illustrate your answer.

Global optimum: the absolute highest and lowest points on the plot Local optimum: maximas and minimas on the plot, but not the largest and smallest points of the plot Multimodal: cases that include local optima

The Incremental Search method for solving a root problem is an example of a bracketing method. Explain what the main drawback of the method is with regard to the number of subintervals chosen to divide the main search interval.

If the spacing between the points of an incremental search are too far apart, brackets may be missed due to capturing an even number of roots within two points. Incremental searches cannot find brackets containing even-multiplicity roots regardless of spacing.

What is the difference between one-dimensional and two-dimensional optimization? Provide a sketch (or diagrams) to illustrate your explanation.

One-dimensional optimization involves functions that depend on a single dependent variable (f(x)). Two-dimensional optimization involves functions that depend on 2 dependent variables (f(x,y)).

Explain the difference between the Bracketing methods and the Open methods for root problems, with regard to the number of points required to start the search process.

Open methods differ from bracketing methods, in that open methods require only a single starting value or two starting values that do not necessarily bracket a root.

Briefly explain what the Golden-Section Search method is and explain the advantage of using the golden ratio value in the implementation of the method. Provide a sketch to illustrate your answer.

The Golden-Section search method is used for finding a minimum on an interval, when there is only one minimum. The first step is to use the endpoints, and split that "segment" into three equal subsegments, where you create x1 and x2. Whichever value is the largest / highest, gets replaced.

Briefly explain what the Newton-Raphson method for solving root problems is. To illustrate your explanation, produce a diagram showing schematically the function to solve the roots for and the progression of the root search.

The Newton-Raphson method is based on computing the next best guess value from the tangent line to the curve f(x) at the initial guess value x. x(i+1) = x(i) - ( f(xi)/f')xi )

The Bisection method for solving a root problem is another example of a bracketing method. Explain briefly what the method is about and how it is implemented. Provide a diagram to illustrate your explanation (the diagram should show the different abscissas considered in the search process as the different iterations are performed).

The bisection method is a variation of the incremental search method. This method "brackets" the root, finds the midpoint of the brackets, and evaluates the relationship between them. If the midpoint and the left sided bracket are the same sign, then the midpoint replaces the left sided bracket for the next iteration. If they are different signs, then the midpoint replaces the right sided bracket.

What does the built-in MATLAB function contour() do? What are the arguments that the function receives?

The contour function creates a contour plot. In the command contour(x,y,z), x and y are coordinates in the value z.

The False Position method for solving a root problem is similar in nature to the Bisection method. Explain briefly the implementation of the method and the main difference with the Bisection method (provide a diagram to illustrate the explanation).

The false position is another bracketing method. This method determines the next guess, not by using the midpoint, bu connects the bracket endpoints with a straight line and determines the location of the x-intercept (xr). The value of xr replaces whichever of the 2 initial guess has the same sign as (f(xr)). The main difference between this and bisection is that the bisection doesn't take into account the shape of a function.

What does the built-in MATLAB function fminbnd() do? What are the arguments that the function receives? What does the function return? How does the function work (i.e., what method of optimization does it use)?

The fminbnd function combines the golden-section search and the parabolic interpolation search.

What does the built-in MATLAB function fminsearch() do? What are the arguments that the function receives? What does the function return? How does the function work (i.e., what method of optimization does it use)?

The function fminsearch finds the minimum of a scalar function of multiple variables, starting at an initial estimate. In the command fminsearch(func,x0), x0 is the stating point and func is the function.

What does the built-in MATLAB function fzero(function, x0) (or fzero(function, [x0 x1])) do? What are the parameters x0 and x1 passed to the function fzero()? How does the function work (i.e., what root search method does the function use)?

The fzero function finds how many roots and where they are in a function. The "function" part is a function handle that's being evaluated, the "x0" is the initial guess, and "x1" is the second part of a bracket guess." This function uses both single initial guesses and initial search brackets.

Given a root problem of the form f (x) = 0, where x is (or are) the root (or roots) of the model, briefly explain what the Graphical Approach for finding the root(s) is and how it is implemented. Provide a diagram to illustrate the explanation.

The graphical approach for finding the roots of the function is to: 1. choose two points- one on the left side of the root and one on the right side 2. determine if the selected points have the same or different sign 3. if they are the same sign, then there is most likely an even number of roots 4. if they are different signs, then there is most likely an odd number of roots

Briefly explain what the Incremental Search method for root finding is and how it is implemented. Provide a diagram to illustrate the explanation.

The incremental search method takes the graph of the function, and then "brackets" it between two points. After this, it splits up this section into equal subsections, then determines how many times the graph changes sign in these subsections.

You should understand how to find the optima (i.e., minima and maxima) of a given simple function analytically, using calculus.

The maxima or minima will have a derivative of 0. It is the maxima if the second derivative is less than zero, and it is the minima if the second derivative is greater than zero.

Briefly explain what the Parabolic Interpolation method for finding an optimum is. Provide a sketch to illustrate your answer.

The parabolic interpolation method uses parabolic interpolation of 3 points to estimate the next location.

What does the built-in MATLAB function roots(c) do? What is the argument passed to the function? What is the format of the parameter c passed to the function? How does the function work (what method of root search does it use)?

The roots function determines all of the roots of a polynomial, including imaginary number. The "c" is a row vector that contains the polynomial coefficients.

Consider a simple engineering model that satisfies the following equation: Dependent variable = function (independent variable, parameters) Assuming the value of the dependent variable and the parameters are known for a certain practical application of the model, explain what the 'roots' of the problem are. Write down how the generic equation above has to be re- written to compute the roots.

The roots of a problem are when the function is equal to zero. For this problem, the generic equation would be: f(independent) = function(independent, parameters) - dependent = 0

Briefly explain what the Secant method for solving root problems is. Discuss the differences between the Newton- Raphson and the Secant methods.

The secant method uses a specific formula to find the derivative of a function, then substitutes that formula into the Newton-Raphson formula to give: x(i+1) = x(i) - ( f(xi)*(xi-1 - xi) / f(xi-1) - f(xi) ). The modified secant method is a little different, but has the same focus.

Briefly explain what the Simple Fixed-Point Iteration method (also called One-Point Iteration or Substitution method) for solving root problems is. To illustrate your explanation, produce a diagram showing schematically the function to solve the roots for and the two functions in which the original function is decomposed.

The simple fixed-point iteration method first requires that you solve for x (x = g(x)). Then, the new function g(x) is used to predict a new value of x.

There are two major classes of methods available to solve root problems. Which are these? What is the main difference between them?

The two major methods are graphical and bracketing. While graphical method merely use the plot of a function to evaluate the roots, bracketing methods are based on making 2 initial guess that "bracket" the root and evaluate to get as close as possible to the exact solution.

If you have developed an algorithm that searches for the minimum of a function, f (x), how could you use the very same algorithm to search for the maximum of a function?

change the values to negatives