Free Essay

Linear Progr

In: Other Topics

Submitted By cogni84
Words 3328
Pages 14
362

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.

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

} nrerror("Maximum number of iterations exceeded in zbrent"); return 0.0; Never get here. }

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 expansion of a function in the neighborhood of a point, f (x + δ) ≈ f (x) + f (x)δ + f (x) 2 δ + .... 2 (9.4.1)

For small enough values of δ, and for well-behaved functions, the terms beyond linear are unimportant, hence f (x + δ) = 0 implies δ=− f (x) . f (x) (9.4.2)

Newton-Raphson is not restricted to one dimension. The method readily generalizes to multiple dimensions, as we shall see in §9.6 and §9.7, below. Far from a root, where the higher-order terms in the series are important, the Newton-Raphson formula can give grossly inaccurate, meaningless corrections. For instance, the initial guess for the root might be so far from the true root as to let the search interval include a local maximum or minimum of the function. This can be death to the method (see Figure 9.4.2). If an iteration places a trial guess near such a local extremum, so that the first derivative nearly vanishes, then NewtonRaphson sends its solution off to limbo, with vanishingly small hope of recovery.

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

Figure 9.4.1. Newton’s method extrapolates the local derivative to find the next estimate of the root. In this example it works well and converges quadratically.

Figure 9.4.2. Unfortunate case where Newton’s method encounters a local extremum and shoots off to outer space. Here bracketing bounds, as in rtsafe, would save the day.

363

1

x

9.4 Newton-Raphson Method Using Derivative

2

3

f (x)

f(x)

3

2

1

x

364

Chapter 9.

Root Finding and Nonlinear Sets of Equations

f (x)

1

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

x

2

Figure 9.4.3. Unfortunate case where Newton’s method enters a nonconvergent cycle. This behavior is often encountered when the function f is obtained, in whole or in part, by table interpolation. With a better initial guess, the method would have succeeded.

Like most powerful tools, Newton-Raphson can be destructive used in inappropriate circumstances. Figure 9.4.3 demonstrates another possible pathology. Why do we call Newton-Raphson powerful? The answer lies in its rate of convergence: Within a small distance of x the function and its derivative are approximately: f (x + ) = f (x) + f (x) +
2f

f (x + ) = f (x) + f (x) + · · · By the Newton-Raphson formula, xi+1 = xi − so that i+1 (x) + ···, 2

(9.4.3)

f (xi ) , f (xi ) f (xi ) . f (xi )

(9.4.4)

=

i



(9.4.5)

When a trial solution xi differs from the true root by i , we can use (9.4.3) to express f (xi ), f (xi ) in (9.4.4) in terms of i and derivatives at the root itself. The result is a recurrence relation for the deviations of the trial solutions i+1 =−

2 i

f (x) . 2f (x)

(9.4.6)

9.4 Newton-Raphson Method Using Derivative

365

Equation (9.4.6) says that Newton-Raphson converges quadratically (cf. equation 9.2.3). Near a root, the number of significant digits approximately doubles with each step. This very strong convergence property makes Newton-Raphson the method of choice for any function whose derivative can be evaluated efficiently, and whose derivative is continuous and nonzero in the neighborhood of a root. Even where Newton-Raphson is rejected for the early stages of convergence (because of its poor global convergence properties), it is very common to “polish up” a root with one or two steps of Newton-Raphson, which can multiply by two or four its number of significant figures! For an efficient realization of Newton-Raphson the user provides a routine that evaluates both f (x) and its first derivative f (x) at the point x. The Newton-Raphson formula can also be applied using a numerical difference to approximate the true local derivative, f (x) ≈ f (x + dx) − f (x) . dx (9.4.7)

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

This is not, however, a recommended procedure for the following reasons: (i) You are doing two function evaluations per step, so at best the superlinear order of √ convergence will be only 2. (ii) If you take dx too small you will be wiped out by roundoff, while if you take it too large your order of convergence will be only linear, no better than using the initial evaluation f (x0 ) for all subsequent steps. Therefore, Newton-Raphson with numerical derivatives is (in one dimension) always dominated by the secant method of §9.2. (In multidimensions, where there is a paucity of available methods, Newton-Raphson with numerical derivatives must be taken more seriously. See §§9.6–9.7.) The following function calls a user supplied function funcd(x,fn,df) which supplies the function value as fn and the derivative as df. We have included input bounds on the root simply to be consistent with previous root-finding routines: Newton does not adjust bounds, and works only on local information at the point x. The bounds are used only to pick the midpoint as the first guess, and to reject the solution if it wanders outside of the bounds.
#include #define JMAX 20 Set to maximum number of iterations.

float rtnewt(void (*funcd)(float, float *, float *), float x1, float x2, float xacc) Using the Newton-Raphson method, find the root of a function known to lie in the interval [x1, x2]. The root rtnewt will be refined until its accuracy is known within ±xacc. funcd is a user-supplied routine that returns both the function value and the first derivative of the function at the point x. { void nrerror(char error_text[]); int j; float df,dx,f,rtn; rtn=0.5*(x1+x2); Initial guess. for (j=1;j 0.0 && fh > 0.0) || (fl < 0.0 && fh < 0.0)) nrerror("Root must be bracketed in rtsafe"); if (fl == 0.0) return x1; if (fh == 0.0) return x2; if (fl < 0.0) { Orient the search so that f (xl) < 0. xl=x1; xh=x2; } else { xh=x1; xl=x2; } rts=0.5*(x1+x2); Initialize the guess for root, dxold=fabs(x2-x1); the “stepsize before last,” dx=dxold; and the last step. (*funcd)(rts,&f,&df); for (j=1;j 0.0) Bisect if Newton out of range, || (fabs(2.0*f) > fabs(dxold*df))) { or not decreasing fast enough. dxold=dx; dx=0.5*(xh-xl); rts=xl+dx; Change in root is negligible. if (xl == rts) return rts; } else { Newton step acceptable. Take it. dxold=dx; dx=f/df; temp=rts; rts -= dx; if (temp == rts) return rts; } if (fabs(dx) < xacc) return rts; Convergence criterion. (*funcd)(rts,&f,&df); The one new function evaluation per iteration.

9.4 Newton-Raphson Method Using Derivative

367

if (f < 0.0) xl=rts; else xh=rts;

Maintain the bracket on the root.

} nrerror("Maximum number of iterations exceeded in rtsafe"); return 0.0; Never get here. }
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

For many functions the derivative f (x) often converges to machine accuracy before the function f (x) itself does. When that is the case one need not subsequently update f (x). This shortcut is recommended only when you confidently understand the generic behavior of your function, but it speeds computations when the derivative calculation is laborious. (Formally this makes the convergence only linear, but if the derivative isn’t changing anyway, you can do no better.)

Newton-Raphson and Fractals
An interesting sidelight to our repeated warnings about Newton-Raphson’s unpredictable global convergence properties — its very rapid local convergence notwithstanding — is to investigate, for some particular equation, the set of starting values from which the method does, or doesn’t converge to a root. Consider the simple equation z3 − 1 = 0 (9.4.8)

whose single real root is z = 1, but which also has complex roots at the other two cube roots of unity, exp(±2πi/3). Newton’s method gives the iteration zj+1 = zj −
3 zj − 1 2 3zj

(9.4.9)

Up to now, we have applied an iteration like equation (9.4.9) only for real starting values z0 , but in fact all of the equations in this section also apply in the complex plane. We can therefore map out the complex plane into regions from which a starting value z 0 , iterated in equation (9.4.9), will, or won’t, converge to z = 1. Naively, we might expect to find a “basin of convergence” somehow surrounding the root z = 1. We surely do not expect the basin of convergence to fill the whole plane, because the plane must also contain regions that converge to each of the two complex roots. In fact, by symmetry, the three regions must have identical shapes. Perhaps they will be three symmetric 120 ◦ wedges, with one root centered in each? Now take a look at Figure 9.4.4, which shows the result of a numerical exploration. The basin of convergence does indeed cover 1/3 the area of the complex plane, but its boundary is highly irregular — in fact, fractal. (A fractal, so called, has self-similar structure that repeats on all scales of magnification.) How does this fractal emerge from something as simple as Newton’s method, and an equation as simple as (9.4.8)? The answer is already implicit in Figure 9.4.2, which showed how, on the real line, a local extremum causes Newton’s method to shoot off to infinity. Suppose one is slightly removed from such a point. Then one might be shot off not to infinity, but — by luck — right into the basin of convergence of the desired

368

Chapter 9.

Root Finding and Nonlinear Sets of Equations

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

Figure 9.4.4. The complex z plane with real and imaginary components in the range (−2, 2). The black region is the set of points from which Newton’s method converges to the root z = 1 of the equation z 3 − 1 = 0. Its shape is fractal.

root. But that means that in the neighborhood of an extremum there must be a tiny, perhaps distorted, copy of the basin of convergence — a kind of “one-bounce away” copy. Similar logic shows that there can be “two-bounce” copies, “three-bounce” copies, and so on. A fractal thus emerges. Notice that, for equation (9.4.8), almost the whole real axis is in the domain of convergence for the root z = 1. We say “almost” because of the peculiar discrete points on the negative real axis whose convergence is indeterminate (see figure). What happens if you start Newton’s method from one of these points? (Try it.)

CITED REFERENCES AND FURTHER READING: Acton, F.S. 1970, Numerical Methods That Work; 1990, corrected edition (Washington: Mathematical Association of America), Chapter 2. Ralston, A., and Rabinowitz, P. 1978, A First Course in Numerical Analysis, 2nd ed. (New York: McGraw-Hill), §8.4. Ortega, J., and Rheinboldt, W. 1970, Iterative Solution of Nonlinear Equations in Several Variables (New York: Academic Press). Mandelbrot, B.B. 1983, The Fractal Geometry of Nature (San Francisco: W.H. Freeman). Peitgen, H.-O., and Saupe, D. (eds.) 1988, The Science of Fractal Images (New York: SpringerVerlag).

9.5 Roots of Polynomials

369

9.5 Roots of Polynomials
Here we present a few methods for finding roots of polynomials. These will serve for most practical problems involving polynomials of low-to-moderate degree or for well-conditioned polynomials of higher degree. Not as well appreciated as it ought to be is the fact that some polynomials are exceedingly ill-conditioned. The tiniest changes in a polynomial’s coefficients can, in the worst case, send its roots sprawling all over the complex plane. (An infamous example due to Wilkinson is detailed by Acton [1].) Recall that a polynomial of degree n will have n roots. The roots can be real or complex, and they might not be distinct. If the coefficients of the polynomial are real, then complex roots will occur in pairs that are conjugate, i.e., if x 1 = a + bi is a root then x2 = a − bi will also be a root. When the coefficients are complex, the complex roots need not be related. Multiple roots, or closely spaced roots, produce the most difficulty for numerical algorithms (see Figure 9.5.1). For example, P (x) = (x − a) 2 has a double real root at x = a. However, we cannot bracket the root by the usual technique of identifying neighborhoods where the function changes sign, nor will slope-following methods such as Newton-Raphson work well, because both the function and its derivative vanish at a multiple root. Newton-Raphson may work, but slowly, since large roundoff errors can occur. When a root is known in advance to be multiple, then special methods of attack are readily devised. Problems arise when (as is generally the case) we do not know in advance what pathology a root will display.

Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).

Deflation of Polynomials
When seeking several or all roots of a polynomial, the total effort can be significantly reduced by the use of deflation. As each root r is found, the polynomial is factored into a product involving the root and a reduced polynomial of degree one less than the original, i.e., P (x) = (x − r)Q(x). Since the roots of Q are exactly the remaining roots of P , the effort of finding additional roots decreases, because we work with polynomials of lower and lower degree as we find successive roots. Even more important, with deflation we can avoid the blunder of having our iterative method converge twice to the same (nonmultiple) root instead of separately to two different roots. Deflation, which amounts to synthetic division, is a simple operation that acts on the array of polynomial coefficients. The concise code for synthetic division by a monomial factor was given in §5.3 above. You can deflate complex roots either by converting that code to complex data type, or else — in the case of a polynomial with real coefficients but possibly complex roots — by deflating by a quadratic factor, [x − (a + ib)] [x − (a − ib)] = x2 − 2ax + (a2 + b2 ) (9.5.1)

The routine poldiv in §5.3 can be used to divide the polynomial by this factor. Deflation must, however, be utilized with care. Because each new root is known with only finite accuracy, errors creep into the determination of the coefficients of the successively deflated polynomial. Consequently, the roots can become more and more inaccurate. It matters a lot whether the inaccuracy creeps in stably (plus or…...

Similar Documents

Premium Essay

Linear Optimization

...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......

Words: 4037 - Pages: 17

Premium Essay

Linear Programming

...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......

Words: 277 - Pages: 2

Premium Essay

Linear Regression

...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......

Words: 478 - Pages: 2

Free Essay

Linear Accelerator

...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......

Words: 729 - Pages: 3

Premium Essay

Linear Regression

...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......

Words: 1324 - Pages: 6

Premium Essay

Linear Programming

...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......

Words: 326 - Pages: 2

Free Essay

Linear Algebra

... SCHAUM’S outlines SCHAUM’S outlines Linear Algebra Fourth Edition Seymour Lipschutz, Ph.D. Temple University Marc Lars Lipson, Ph.D. University of Virginia Schaum’s Outline Series New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2009, 2001, 1991, 1968 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-0-07-154353-8 MHID: 0-07-154353-8 The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-154352-1, MHID: 0-07-154352-X. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at bulksales@mcgraw-hill.com. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies...

Words: 229129 - Pages: 917

Premium Essay

Linear Tech

...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......

Words: 1555 - Pages: 7

Free Essay

A Linear Equation

...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...

Words: 304 - Pages: 2

Premium Essay

Linear

...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.......

Words: 848 - Pages: 4

Premium Essay

Linear Equations

...      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)......

Words: 404 - Pages: 2

Premium Essay

Linear Programming

...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......

Words: 2007 - Pages: 9

Free Essay

Linear Correlation

...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......

Words: 539 - Pages: 3

Free Essay

Linear

...(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......

Words: 586 - Pages: 3

Premium Essay

Linear Dividend

...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......

Words: 8046 - Pages: 33