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Words 3328

Pages 14

Chapter 9.

Root Finding and Nonlinear Sets of Equations

} a=b; fa=fb; if (fabs(d) > tol1) b += d; else b += SIGN(tol1,xm); fb=(*func)(b);

Move last best guess to a. Evaluate new trial root.

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CITED REFERENCES AND FURTHER READING: Brent, R.P. 1973, Algorithms for Minimization without Derivatives (Englewood Cliffs, NJ: PrenticeHall), Chapters 3, 4. [1] Forsythe, G.E., Malcolm, M.A., and Moler, C.B. 1977, Computer Methods for Mathematical Computations (Englewood Cliffs, NJ: Prentice-Hall), §7.2.

9.4 Newton-Raphson Method Using Derivative

Perhaps the most celebrated of all one-dimensional root-ﬁnding routines is Newton’s method, also called the Newton-Raphson method. This method is distinguished from the methods of previous sections by the fact that it requires the evaluation of both the function f (x), and the derivative f (x), at arbitrary points x. The Newton-Raphson formula consists geometrically of extending the tangent line at a current point x i until it crosses zero, then setting the next guess x i+1 to the abscissa of that zero-crossing (see Figure 9.4.1). Algebraically, the method derives from the familiar Taylor series…...

...DECISION MODELING DECISION WITH WITH MICROSOFT EXCEL MICROSOFT Linear Optimization Linear Optimization A constrained optimization model takes the form of a constrained performance measure to be optimized over a range of feasible values of the decision variables. The feasible values of the decision variables are determined by a set of inequality constraints. constraints Values of the decision variables must be chosen such that the inequality constraints are all satisfied while either maximizing or minimizing the desired performance variable. These models can contain tens, hundreds, or thousands of decision variables and constraints. Linear Optimization Very efficient search techniques exist to optimize constrained linear models. constrained These models are historically called linear programs linear (LP). In this chapter we will: 1. Develop techniques for formulating LP models 2. Give some recommended rules for expressing LP models in a spreadsheet that facilitates application of Excel’s Solver 3. Use Solver to optimize spreadsheet LP models Formulating LP Models Every linear programming model has two important features: Objective Function Constraints A single performance measure to be maximized or minimized (e.g., maximize profit, minimize cost) Constraints are limitations or requirements on the set of allowable decisions. Constraints may be further classified into physical, economic, or policy limitations......

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...LINEAR PROGRAMMING Definition. A mathematical technique for solving constrained maximization and minimization problems when there are many constraints and the objective function to be optimized, as well as the constraints faced, are linear (i.e., can be represented by straight lines) Assumptions. -LP is based on the assumption that the objective function that the organization seeks to optimize (maximize or minimize), as well as the constraints that it faces, are linear and can be represented GRAPHICALLY by straight lines. -Input and output prices are constant -Average and marginal costs are constant and equal (they are linear) -Profit per unit is constant; profit function is linear Applications of Linear Programming 1. Optimal process selection 2. Optimal product mix 3. Satisfying minimum product requirements 4. Long-run capacity planning 5. Other specific applications of linear programming Basic Linear Programming Concepts A. Production Process and Isoquants -where a production process or activity can be represented by a straight line ray from the origin in input space B. Optimal Mix of Production Process Procedure Used in Formulating and Solving Linear Programming Problems The steps followed in solving linear programming problem are: 1. Express the objective function of the problem as an equation and the constraints as inequalities. 2. Graph the inequality constraints and define the feasible region. 3. Graph......

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...Linear Regression I would like to know if people who enjoy thrill seeking have tattoos. I believe thrill seeking and tattoos go hand in hand. Most people I know are adventurous, risk takers, and daredevils and all of them have tattoos. I have a strong feeling that the correlation between the two will have a strong positive relationship. X= Tattoos Y= Thrill Seeking The scatter plot shows an extremely rough linear pattern but there is an upward sloping. Line of best fit: y = 0.9148x +25.505 Analysis: 1. r = .14 little or no correlation 2. R^2 = 2% 2% of the variance in thrill seeking is accounted by tattoos. 3. Slope = 0.0196(m) For every 1 tattoo people have there is an increase we expected of 0.9148 in thrill seeking. Conclusion: Between these two variables, there are no correlations between the two. It was shocking to see there is no relationship between the two. I truly believed people who are thrill seekers have tattoo. T-Test Independent 2 Sample My gym teacher believes that males are stronger than females and that is why males have more tattoos. The scale is determine by the number of tattoos both males and females have. Eighty-four males and one hundred and eleven females responded. The males average 39 (s.d. 1.42) while the females average 38 (s.d. 0.98). At the .10 significance level, test to see if there is a difference between males having more tattoos than females? Ho: Null Hypothesis Males equal Females Ha: Null......

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...Linear Accelerator July 16, 2013 HCS212 Health Care Vocabulary Ashley Fritz Being the Administrative Director here at Pediatric Cancer Specialist for over ten years, there has been notice of need of improvement to our growing facility and the radiation therapy given to the patients. Since our facility has opened in the year of 2000 we have been operating the same way every year and it is time to introduce some new technology in to the center. Research has been done for many days to find new machines that will help improve our radiation therapy process to our patients and also improve the way that this center operates. What has come about during the research is the newest and most advanced machine used in radiation therapy to treat cancer tumors and some benign diseases. This new machine we will be using is called a LINAC short abbreviation for the term Linear Accelerator. The LINAC is pretty expensive and is at a pricey one million dollars, but this machine will improve the radiation therapy given to our patients. Having this machine will improve radiation therapy here at Pediatric Cancer Specialist. Also there is belief that our satisfaction rates from our patients will improve with the new technology. Before bringing the machine into our building, everyone will be required to take a course and a test on how to operate this machine. We need everyone to know how to work this machine, just in case the specific technician scheduled will not be at work that......

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...Linear Regression deals with the numerical measures to express the relationship between two variables. Relationships between variables can either be strong or weak or even direct or inverse. A few examples may be the amount McDonald’s spends on advertising per month and the amount of total sales in a month. Additionally the amount of study time one puts toward this statistics in comparison to the grades they receive may be analyzed using the regression method. The formal definition of Regression Analysis is the equation that allows one to estimate the value of one variable based on the value of another. Key objectives in performing a regression analysis include estimating the dependent variable Y based on a selected value of the independent variable X. To explain, Nike could possibly measurer how much they spend on celebrity endorsements and the affect it has on sales in a month. When measuring, the amount spent celebrity endorsements would be the independent X variable. Without the X variable, Y would be impossible to estimate. The general from of the regression equation is Y hat "=a + bX" where Y hat is the estimated value of the estimated value of the Y variable for a selected X value. a represents the Y-Intercept, therefore, it is the estimated value of Y when X=0. Furthermore, b is the slope of the line or the average change in Y hat for each change of one unit in the independent variable X. Finally, X is any value of the independent variable that is......

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...The development of linear programming has been ranked among the most important scientific advances of the mid 20th century. Its impact since the 1950’s has been extraordinary. Today it is a standard tool used by some companies (around 56%) of even moderate size. Linear programming uses a mathematical model to describe the problem of concern. Linear programming involves the planning of activities to obtain an optimal result, i.e., a result that reaches the specified goal best (according to the mathematical model) among all feasible alternatives. Linear Programming as seen by various reports by many companies has saved them thousands to even millions of dollars. Since this is true why isn’t everyone using Linear Programming? Maybe the reason is because there has never been an in-depth experiment focusing on certain companies that do or do not use linear programming. My main argument is that linear programming is one of the most optimal ways of resource allocation and making the most money for any company today. I used (in conjunction with another field supporter – My Dad) the survey method to ask 28 companies that were in Delaware, New Jersey, and Pennsylvania whether they were linear programming users. In addition, I wanted to examine the effect of the use of linear programming across three different but key decision support areas of the participating companies to include (1) Planning (2) Forecasting and (3) Resource Allocation. The companies were selected randomly from the......

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...Executive Summary Entering the 4th quarter of Linear Technology’s fiscal year 2003 the market continues to show signs of improvement. The company has shown steady growth in the last year and revenues are estimated to increase 19% over FY 2002. Based on this estimate, FY 2003 net income will hit $222.7 million ($0.71 earnings per share); a 12.6% growth from the previous year. Operating cash flow; while lower than 2000 and 2001 has shown a modest increase since 2002 and continues to be positive due to the company’s variable cost structure. This is in-part is due to more efficient working capital investments and “other” adjustments to income, awarding the company a 10% increase in net cash flow year-over-year. Linear Technology has increased its cash holdings to excess of $1.5 billion through employing cost savings initiatives, though these holdings have only shown investors modest returns in the neighborhood of 4.25% ($0.10 earnings per share). While modest, investors have come to expect this form of conservativeness and there has been little outcry of agency issues. Looking ahead, based on an analog “fabs” life expectancy of 10 plus years, capital investments, for a new “fab”, will be required in the next one or two years in excess of $200 million; leaving more than sufficient cash holdings while requiring no leveraging. Based on these financials, Linear Technology should look to increase its dividend payout by $0.01 per share. This has become the expected trend over the......

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...A linear equation In this lesson you can learn how to solve a simplest equation with one unknown variable. I will start with the following example. Solve an equation 5x - 8 + 2x - 2 = 7x - 1 - 3x - 3 for the unknown variable x. The left side of the equation is an expression, which is to the left of the equal sign. The right side of the equation is an expression, which is to the right of the equal sign. In our case the left side of the equation is 5x - 8 + 2x - 2, while the right side is 7x - 1 - 3x - 3. Terms containing variable x are called variable terms; terms containing the numbers only are called constant terms, or simply constants. The equation under consideration is called a linear equation, because its both sides are linear polynomials. The solution of an equation is such a value of the variable x that turns the equation into a valid equality when this value is substituted to both sides. I am explaining below how to solve this linear equation, in other words, how to find the unknown value of the variable x. The first step you should do is to simplify both sides of the equation by collecting the common terms containing variable x and the common constant terms separately at each side of the equation. Let us do it. By collecting common terms with the variable x at the left side, you will get 5x + 2x = 7x. By collecting common constant terms at the left side you will get -8 - 2 = -10. Thus, now the left side is 7x - 10. Making similar calculations...

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...ingredients with money she has made from the previous game. Julia has talked to some students and vendors who have sold food at previous football games at Tech as well as at other universities. From this she has discovered that she can expect to sell at least as many slices of pizza as hot dogs and barbecue sandwiches combined. She also anticipates that she will probably sell at least twice as many hot dogs as barbecue sandwiches. She believes that she will sell everything she can stock and develop a customer base for the season if she follows these general guidelines for demand. If Julia clears at least $1,000 in profit for each game after paying all her expenses, she believes it will be worth leasing the booth. A. Formulate and solve a linear programming model for Julia that will help you advise her if she should lease the booth. B. If Julia were to borrow some more money from a friend before the first game to purchase more ingredients, could she increase her profit? If so, how much should she borrow and how much additional profit would she make? What factor constrains her from borrowing even more money than this amount (indicated in your answer to the previous question)? C. When Julia looked at the solution in (A), she realized that it would be physically difficult for her to prepare all the hot dogs and barbecue sandwiches indicated in this solution. She believes she can hire a friend of hers to help her for $100 per game. Based on the results in (A) and (B), is this......

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...It may also attract investors that prefer companies that pay dividends. For example, a cash dividend is to be paid at specified times (usually quarterly), however a stock repurchase is not. For some investors, the dependability of the dividend may be more important. Also, I think that dividends help to avoid wasting firms’ cash on not necessarily needed or unproductive projects or acquisitions. 3. Linear Technology is considering increasing dividends in the 4th quarter of 2003. Is Linear Technology in a financial position to consider a dividend increase (particularly in light of the sharp decline in sales and profits in fiscal year 2002)? It seems to me that Linear Technology is not in an appropriate financial position to consider a dividend increase. The company`s performance was good, but numbers were far below the levels of fiscal year 2001. Case study claims that “management did not see a clear path of reaching levels of 2001 in the next years” (Baker, 2004). Moreover, there were some geopolitical factors (war in Iraq) that challenged the US economy. 4. Suppose Linear Technology determines it wants to return more cash to shareholders. One way to do this is through increasing its regular quarterly dividend payment to shareholders. Name another way to distribute cash to shareholders other than dividends. What are the advantages and disadvantages of this alternative way of distributing cash? Another way to distribute cash to shareholders is stock repurchase.......

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... One of the most important concepts that we learned in this course is formulating linear equations from everyday life problems that need solutions. With this concept under your belt you will always be able to find solutions in everday life. For example what if you have a new house interior to paint and you need to figure out how much paint you should purchase. Through this course we have attained the ability to determine the exact amount of paint needed to be purchased. What many fail to realize is that math is in our lives daily on multiple occasions. This course provided the comfort of being able to handle this daily math without worries. Out of all the concepts explained inequalities seem to be the least important to everyday life. Although they seem the least important does not mean they are useless. Someone somewhere is using these equations for a significant task. How do you think you will use the information you learned in this course in the future? Which concepts will be most important to you? Which will be least important? Explain your answers. I will continue to use basic mathematical (algebraic) calculations such as expenses versus income in applicable situations, estimation of materials based upon linear measurements and the calculation of expenses based upon the cost factor of those materials as I progress in my personal life. I do see myself for personal reasons, using simple graphs to get ideas or guidance on how my personal venture(s)......

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...Merton Trucks Case Note Abstract We discuss Merton Trucks [Dhe90a] as a case to introduce linear programming in the MBA program. This case adapted from Sherman Motor Company case, was used to introduce Linear Programming formulations as well as duality. Refer to the teaching note [Dhe90b]. Our approach differs from the approach suggested by Dhebar [Dhe90b]. First, our audience consists pre-dominantly of engineers with not too much work experience. As a result, handling math and algebra is relatively easy. Explaining the algebraic formulation, graphical approach and using the Excel solver do not consume that much time. Second, because this case is used during the ﬁrst week of the MBA program, students are still unfamiliar with the case methodology and we spend signiﬁcant time in understanding case facts. The circular logic used in allocating ﬁxed costs based on the product mix that in turn is used in deciding the product mix takes some time to understand. Third, because of the participant background, they have difﬁculty in translating the model to the speciﬁc business situation and interpreting the trade-offs involved in various what-if analyses that are prompted by the case questions. We return to the case when we teach duality. After explaining duality, we analyze the case to show how some of the questions and what-if analyses can be simpliﬁed using duality. This note is based on our experiences with teaching three large batches of students in our MBA programs. 1 1......

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...MODULE 6 EXERCISE Linear Correlation IRINA QUENGA EG 381 STATISTICS 02/22/2015 ITT TECHNICAL INSTITUTE Task 1: Listed below are baseball team statistics, consisting of the proportions of wins and the result of this difference: Difference (number of runs scored) - (number of runs allowed). The statistics are from a recent year, and the teams are NY—Yankees, Toronto, Boston, Cleveland, Texas, Houston, San Francisco, and Kansas City. Difference 163 55 –5 88 51 16 –214 Wins 0.599 0.537 0.531 0.481 0.494 0.506 0.383 A) Construct a scatter plot, find the value of the linear correlation coefficient r, and find the critical values of r from Table VI, Appendix A, p. A-14, of your textbook Elementary Statistics. Use α = 0.05. B) Is there sufficient evidence to conclude that there is a linear correlation between the proportion of wins and the above difference? Task 2: A classic application of correlation involves the association between temperature and the number of times a cricket chirps in a minute. Listed below are the numbers of chirps in 1 minute and the corresponding temperatures in °F: Chirps in 1 Min 882 1188 1104 864 1200 1032 960 900 Temperature (°F) 69.7 93.3 84.3 76.3 88.6 82.6 71.6 79.6 A) Construct a scatter plot, find the value of the linear correlation coefficient r, and find the critical values of r from Table VI, Appendix A, p. A-14, of your textbook Elementary Statistics. Use α = 0.05. B) Is there a linear correlation between the number......

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...(in connection with the planning activities of the military), linear programming and its many extensions have come into wide use. In academic circles decision scientists (operations researchers and management scientists), as well as numerical analysts, mathematicians, and economists have written hundreds of books and an uncountable number of articles on the subject. Curiously, in spite of its wide applicability today to everyday problems, it was unknown prior to 1947. This is not quite correct; there were some isolated exceptions. Fourier (of Fourier series fame) in 1823 and the wellknown Belgian mathematician de la Vallée Poussin in 1911 each wrote a paper about it, but that was about it. Their work had as much inﬂuence on Post-1947 developments as would ﬁnding in an Egyptian tomb an electronic computer built in 3000 BC. Leonid Kantorovich’s remarkable 1939 monograph on the subject was also neglected for ideological reasons in the USSR. It was resurrected two decades later after the major developments had already taken place in the West. An excellent paper by Hitchcock in 1941 on the transportation problem was also overlooked until after others in the late 1940’s and early 1950’s had independently rediscovered its properties. What seems to characterize the pre-1947 era was lack of any interest in trying to optimize. T. Motzkin in his scholarly thesis written in 1936 cites only 42 papers on linear inequality systems, none of which mentioned an......

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...ALISON BERKLEY WAGONFELD Dividend Policy at Linear Technology It was April 2003 and Paul Coghlan was pulling together his notes for Linear Technology’s board meeting the following day. As chief financial officer of the Silicon Valley semiconductor company, Coghlan was responsible for making a recommendation about whether or not Linear should increase its dividend this quarter. Coghlan and Linear’s CEO Robert Swanson were pleased with the company’s third-quarter financials for fiscal year 2003, but sales and net income still remained substantially below Linear’s record levels set in 2001. In addition, the technology industry was still emerging from a recessionary environment and it was unclear how strong business would be for the remainder of the year. Linear Technology Corporation Headquartered in Milpitas, California, Linear was founded in 1981 by Robert Swanson. Under his leadership, the company focused on designing, manufacturing, and marketing integrated circuits (semiconductors) that were used in various electronic applications such as cellular telephones, digital cameras, complex medical devices, and navigation systems. Linear’s customers spanned numerous industries and no single customer accounted for more than 5% of its business. In 2002, the communications industry accounted for 33% of Linear’s business, computers 27%, automotive 6%, and the remaining 34% was spread across many different applications. Linear focused on the analog segment within......

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