In: Science

Submitted By photon88

Words 803

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Words 803

Pages 4

Solved Examples

By

Mahmoud SAYED AHMED

Ph.D. Candidate

Department of Civil Engineering, Ryerson University

Toronto, Ontario 2013

Table of Contents

Part I: Numerical Solution for Single Variable............................................................................................... 2

1.1.

Newton-Raphson Method ............................................................................................................ 2

1.2.

Secant Methods ............................................................................................................................ 4

Part Two: Numerical Solutions for Multiple Variables ................................................................................. 6

2.1.

Generalized Newton-Raphson Method for Two Variables ........................................................... 6

2.2.

Multi-dimensional case for Newton-Raphson Method ................................................................ 9

Appendix: Matrix ........................................................................................................................................ 10

Sayed-Ahmed, M.

Ryerson University

Jan. 2013

Part I: Numerical Solution for Single Variable

1.1.

Newton-Raphson Method

The Newton-Raphson method (NRM) is powerful numerical method based on the simple idea of linear approximation. NRM is usually home in on a root with devastating efficiency. It starts with initial guess, where the NRM is usually very good if

, and horrible if the guess are not close.

Question:

Answer:

Find the value of if using Newton-Raphson Method for three iterations?

Start with guess value of

The function equation should equal to zero; ( )

So the function equation; ( )

( )

( )

NRM:

The first iteration

( )

( )

then

( )

( )

( )

The absolute error,

|…...

...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). Numerical Recipes in C The Art of Scientiﬁc Computing Cambridge New York Port Chester Melbourne Sydney EXXON Research and Engineering Company Harvard-Smithsonian Center for Astrophysics Department of Physics, Cornell University CAMBRIDGE UNIVERSITY PRESS William T. Vetterling Saul A. Teukolsky Brian P. Flannery Second Edition William H. Press Polaroid Corporation Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC, 3207, Australia Copyright c Cambridge University Press 1988, 1992 except for §13.10 and Appendix B, which are placed into the public domain, and except for all other computer programs and procedures, which are Copyright c Numerical Recipes Software 1987, 1988,......

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...Powder Technology 217 (2012) 148–156 Contents lists available at SciVerse ScienceDirect Powder Technology journal homepage: www.elsevier.com/locate/powtec Numerical modeling of ﬂuid and particle behaviors in impact pulverizer Hirohisa Takeuchi, Hideya Nakamura, Tomohiro Iwasaki, Satoru Watano ⁎ Department of Chemical Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan a r t i c l e i n f o a b s t r a c t Numerical modeling of ﬂuid ﬂow and individual particle motion in an impact pulverizer was conducted using a Computational Fluid Dynamics (CFD)–Discrete Phase Model (DPM) coupling model. The impact pulverizer used is a dry system. Its grinding chamber consists of high-speed rotating hammers and a static concavo–convex stator. First of all, calculated results of ﬂuid pressure in the grinding chamber were compared with the experimental ones, showing the both results were in good agreement. The ﬂuid ﬂow in the grinding chamber indicated that the ﬂuid mainly swirled in the direction of the hammer rotation. The ﬂuid ﬂow pattern in the concaves of the stator was also swirling ﬂow, while its velocity was much lower than that in the outside of the concaves. Analyses of the particle motion suggested that the particles were accelerated by the ﬂuid drag force caused by the rotating hammers but not by the impact force from the hammers, resulting in impacts with the static stator at the velocity 1.4 times higher than the......

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...The Scientific Method Ashley Burrell SCI/230 07/28/2013 Dawn Gleason The Scientific Method This paper seeks to explain what the scientific method is and exactly how it is used or can be used on an everyday basis. The scientific method is a method often used to help a person or a group logically or rationally draw a conclusions about an in particular subject or topic. The scientific method is not just used by scientists, but is also used by many others such as historians, criminal investigators, doctors, engineers, and anyone else who is trying to intelligently solve some sort of problem. One thing to understand about the scientific method is that it can’t be used to answer every question or solve every problem a person may be having, but it can be used as a way to solve many problems such as how to get a car started or how to stop the roof from leaking when it rains. The objective of this paper is to define what the scientific method is and to explain what steps are used to problem solving. According to the American Heritage Dictionary of the English Language the scientific method is defined as “The principles and empirical processes of discovery and demonstration considered characteristic of or necessary for scientific investigation, generally involving the observation of phenomena, the formulation of a hypothesis concerning the phenomena, experimentation to demonstrate the truth or falseness of the hypothesis, and a conclusion that validates or modifies the......

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...Numerical Precision by John Nelems Professor Lu CIS512 Strayer University January 26th, 2014 Floating Point Format for storing floating numbers in memory Numeric precision and representation are very familiar discussions in computer science. The way that numbers are stored and represented in a computer system has to be represented correctly in order to avoid costly mistakes. Numbers are stored as binary numbers in computerized system. Floating point numbers are expressed as the product of two parts: the mantissa and a power of two. For example: ± mantissa x 2 exponent The mantissa represents the binary digits of the floating number. The power of two is represented by the exponent. The stored form of the exponent is an 8 –bit value ranging from 0-255. The mantissa is a 24-bit value (represents about seven decimal digits) that has a most significant bit (MSB) that is always 1 and not stored. Also, it has a sign bit that indicates if the floating point number is positive or negative. Floating point arithmetic is one of the most widely used ways of representing real numbers in today’s computer systems. The use of floating point arithmetic provides the following benefits: speed, accuracy, range, portability and ease of implementation and use for numerical applications. Floating point arithmetic uses IEEE 754 standard that is used by most computer manufacturers which makes it easy to use and manipulate floating numbers. Is the floating point......

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...Using examples discuss the strengths and weaknesses of interactionist research methods. Part B One method of research used by Interactionists is known as the un-structured interview. They are less strict than an ordinary interview and are similar to that of an everyday conversation. Although it will be based around a specific subject or topic it allows the interviewee to go deeper into the question and give a more in-depth opinion. Unstructured interviews are commonly, open-ended and flexible as they can go beyond the question. (reference) whilst having an specific topic for the overall direction of the interview’s structure the flexibleness of the style can enter areas which may not be or have seemed not to be important. Although unstructured interviews are helpful they do hold some weakness, firstly the process can be expensive and take a great amount of time for discussion with more than just one person to improve reliability. A great deal of time is also spent in collecting and analysing the answers given afterwards due to key words, progression of trends and so on. Furthermore it is known that some participants to put themselves in a favourable position to the reality. Lastly it can be difficult to separate bias from opinion if the interviewer becomes friendly with the interviewee. However in comparison to this, it allows the interviewer the opportunity to come across in a more friendly and open manner than an appearance which may make the interviewee feel......

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...corrective measures. This entire process by which we attempt to solve problems is called research. Thus research involves a series of well-thought-out and carefully executed activities that will enable the manager to know how organizational problems can be solved, or at least considerably minimized. Research thus encompass the process of inquiry, investigation, examination, and experimentation. These processes have to be carried out systematically, diligently, critically, objectively, and logically. The expected end results would be the discovery that will help the manager to deal with the problem situation. Definition of research 1. A voyage of discovery or a journey/movement from the known to unknown; An attitude; An experience; A method of critical thinking; A careful critical inquiry in seeking facts for principles. 2. An art of scientific investigation • Scientific and systematic search for pertinent information on a specific topic • Process of arriving at dependable solutions to problems through the planned and systematic collection, analysis and interpretation of data. 3. A systematized effort to gain new knowledge • Search for (new) knowledge/facts through objective, systematic and scientific method of finding solution to a problem. • Implicit question + Explicit answer + data to support the answer • Not synonymous with commonsense, but systematic, objective (purposeful), reproducible, relevant activity having control over some factors. We can......

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...SUBJECT: NUMERICAL METHODS CODE: BUM2313 FACULTY OF INDUSTRIAL SCIENCES & TECHNOLOGY INSTRUCTION: Use MATHEMATICAL SOFTWARE such as EXCEL/ MATLAB/ MAPLE/ C to facilitate the computation. SUBMIT the solution in HARDCOPY & SOFTCOPY. Please save the solution in CD for softcopy. Do the assignment in group as allow by your lecturer. QUESTION 1 The nonlinear resistive circuit shown below is described by the nonlinear equation f ( x) g ( x) ( E x) 0 R TOPIC: CHAPTER 1, 2,3 & 4 DUE/DURATION: MARKS: ASSESSMENT: ASSIGNMENT 2nd May 2014 (before 5 P.M) WEEK 11 100 The function g ( x) gives the current through the nonlinear resistor as a function of the voltage x cross its terminals as shown in the following Figure 1. Figure 1 Assuming that g ( x) 9sin( x 5) 10 and consider the three following cases: Case 1: E 5, R 1, Case 2: E 15, R 3, Case 3: E 4, R 0.5. (a) (b) (c) By using an appropriate method that you have learned in this course, find all the solutions of the nonlinear resistive circuit equation for the all cases. Select suitable starting points for xl and xu by plotting f over the interval [0,4] for the all cases, and visually selecting a good starting point. Find the lowest root over the the interval [0,4] by using (i) Bisection method and (ii) False position method. Use the starting points xl and xu in (b) and terminate the computation if a 104. (For (a) and (b) use two decimal places, for (c) use eight decimal places)......

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...benefitting from rounding errors. Propose a plan on how the rounding error should be handled. Differentiate between the exponential format and the floating point format in terms of: o format o processing speed o accuracy While programming computers, floating point describes a method of representing an estimate or an approximation of real numbers in a way that can support a wide range of different values. Numbers are represented roughly to a fixed number of major digits and scaled using an exponent. The term floating point refers to the fact that the radix point or the decimal point can "float" from space to space. It can be placed anywhere relative to the significant digits of the number. Over the years, a variety of floating point representations has been used in computers. The advantage of floating point representation over fixed-point and integer representation is that it can support a much wider range of values. Some floating point hardware can only support the simplest of operations, like addition, subtraction, and multiplication. But even the most complex floating point hardware has a finite number of operations it can support. For example, none of them directly support arbitrary precision arithmetic. When a CPU is executing a program that calls for a floating point operation not directly supported by the hardware, the CPU uses a series of simpler floating point operations. In systems without any floating point hardware, the CPU......

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...Ch 12 – Numerical Problem 6 Jake Goldstone is working 30 hours per week. His marginal utility of income is 2, his marginal utility of leisure is 60, and his hourly wage is $20. Assume throughout this problem that the income effect is zero. Is Mr. Goldstone maximizing his utility? Would working more or less increase his utility? If his wage rose to $30 per hour, would he be maximizing his utility by working 30 hours per week? If not, should he work more or fewer hours? At a wage of $40 per hour, would he be maximizing his utility? If not, would working more or less than 30 hours per week increase his utility? 1. He’s not maximizing his utility since 2 * $20 = 40, which is less than his marginal utility of leisure of 60. 2. He would have to work more hours in order to maximizing his utility. 3. If his wage rose to $30, the total utility of his would be 30 *2 = 60. He then maximizes his utility. 4. At wage of $40, he would have to work less because his utility is excess his utility of leisure time (60). His total utility is 40 *2 = 80 Ch 13 – Numerical Problem 4 A new health club has just opened up in your town. Struggling to bring in money now, the club is offering 10-year memberships for a one-time payment now of $800. You cannot be sure that you will still be in town for the next 10 years, but you expect that you will be. You anticipate that your beneﬁt of belonging to the club will be $10 per month (think of this as an annual beneﬁt of $120). Decide whether you should......

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...zero at the start of the algorithm. Step 1: Start Step 2: Initialize sum, count and maximum to zero Step 3: Examine top card. Step 4: If it says “end of data” proceed to step 9; otherwise, proceed to next step. Step 5: Add value from top card to sum. Step 6: Increase count by 1. Step 7: If value is greater than maximum, set maximum to value. Step 7: Discard top card Step 8: Return to Step 3. Step 9: Is the count greater than zero? If yes, proceed to step 10. If no, proceed to step 11. Step 10: Calculate average = sum/count Step 11: End 2.4 Flowchart: [pic] 2.5 Students could implement the subprogram in any number of languages. The following Fortran 90 program is one example. It should be noted that the availability of complex variables in Fortran 90 would allow this subroutine to be made even more concise. However, we did not exploit this feature, in order to make the code more compatible with languages such as Visual BASIC or C. PROGRAM Rootfind IMPLICIT NONE INTEGER::ier REAL::a, b, c, r1, i1, r2, i2 DATA a,b,c/1.,6.,2./ CALL Roots(a, b, c, ier, r1, i1, r2, i2) IF (ier == 0) THEN PRINT *, r1,i1," i" PRINT *, r2,i2," i" ELSE PRINT *, "No roots" END IF END SUBROUTINE Roots(a, b, c, ier, r1, i1, r2, i2) IMPLICIT NONE INTEGER::ier REAL::a, b, c, d, r1, i1, r2, i2 r1=0. r2=0. i1=0. i2=0. IF (a == 0.)......

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...time to waste. The atmosphere in this group was definitely neither efficient nor positive. After that meeting, I talked to John and Allen about the missed communication and mistakes in the group, and how can we do to pull the team back on track. Although, we were all frustrated, we all knew that the deadline was coming soon and there was not much time to do it. We had to submit our report by third week. With limited time pressure and some talking, we stopped complaining and pointing fingers at each other. This time we clearly divided responsibility, and made sure people got their tasks right. We send all of our work to John, and he did the report with Allen. Rests of us were assigned to do more work on presentation. The problems were all solved. Finally we handed in our group report on time, and the presentation was finished in the class. Our performance and report was not reviewed strictly after we completed. Not to my surprised, we did not achieve the goal we set at the beginning. It was an unforgettable experience working in a group since all the conflicts. Case Analysis In my opinion, working in this group was an unpleasant and frustrating experience, but I learned some lessons from it. When I reflected all the conflicts and problems back to theories from the textbook, I have gained some knowledge. The main reasons that caused conflicts were personality differences, self-serving bias, and miscommunication. First of all, people’s personalities can be......

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... Costing Methods Paper ACC /561 Maurice Guliford Costing Methods Paper ACC /561 Maurice Guliford 2015 2015 After working this problem several times, I feel in this case, it would be better to use the absorption method because this method incorporates only the overhead that is allocated to the 81,300 units sold. The variable method counts fixed overhead as a period expense, meaning that the fixed overhead for this period is calculated on the basis of the 95,400 units produced, if the absorption method is used. The variable method only calculates fixed overhead on the basis of the 81,300 units that were sold. This provides management with a more accurate picture of the profitability of the fishing lures. Thus, the variable costing method is optimal. The ability to provide accurate information to management about product costs is the main benefit of the absorption method. The variable method is beneficial because it provides an output (net income) that is closer to the cash flow of the business. This is useful in particular for businesses that might be short on cash flow. Further, the variable costing method provides management with a clearer picture of the effect that fixed costs have on the total profitability of the company. Absorption costing is particularly useful for firms that do not sell all of their output during the manufacturing period, as is the case with Polk. Under absorption costing, the cost of a good is not shown until the good has been sold. This can be a......

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...ME 2173 MATLAB Project 1 Numerical Methods Using MATLAB Click Link Below To Buy: http://hwcampus.com/shop/2173-matlab-project-1/ MATLAB Project 1 1.) Save Table 1 below in an excel file called 'Superheat' and complete the instructions that follow: Table 1: Properties of Superheated Steam at three different Pressures (1MPa =10116 N/m^2) T em p °C p1=0.20 MPa (120.2 C) p2=0.30 MPa (133.5 C) p3=0.40 MPa (143.6 C) volume v1(m^3/kg) energy u1(k)/kg) enthalpy h1(k)/kg) volume v2(m^3/kg) energy u2(k)/kg) enthalpy h2(k)/kg) volume v3(m^3/kg) energy u3(k)/kg) enthalpy h3(k)/kg) 150 0.960 2577.1 0.634 2571.0 0.471 2564.4 2752.8 200 1.081 2654.6 0.716 2651.0 0.534 2647.2 2860.8 250 1.199 2731.4 0.796 2728.9 0.595 2726.4 2964.4 300 1.316 2808.8 0.875 2807.0 0.655 2805.1 3067.1 350 1.433 2887.3 0.954 2885.9 0.714 2884.4 3170 400 1.549 2967.1 1.032 2966.0 0.773 2964.9 3274.1 450 1.666 3048.5 1.109 3047.5 0.831 3046.6 3379 500 1.781 3131.4 1.187 3130.6 0.889 3129.8 3485.4 600 2.013 3302.2 1.341 3301.6 1.006 3301.0 3703.4 700 2.244 3479.9 1.496 3479.5 1.122 3479.0 3927.8 800 2.476 3664.7 1.650 3664.3 1.237 3663.9 4158.7 900 2.707 3856.3 1.804 3856.0 1.353 3855.7 4396.9 1000 2.938 4054.8 1.958 4054.5 1.469 4054.3 4641.9 a. Use a MATLAB command to import the data from an excel file, as a (13x10) matrix 'SteamProps' b. Given that h=u+pv, use the column vectors of the 'SteamProps' matrix with operations to......

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...ME 2173 MATLAB Project 5 Numerical Methods Using MATLAB Click Link Below To Buy: http://hwcampus.com/shop/matlab-project-5/ The properties of Superheated Steam at pressure 200 kPa are shown in the table below: Table 1 Temp °C p=200 kPa (120.2 C) volume v(m^3/kg) energy u(k)/kg) enthalpy h(k)/kg) entropy s(k)/kg.K) 150 0.960 2577.1 2706.2 7.127 200 1.081 2654.6 2769.1 7.281 250 1.199 2731.4 2870.7 7.508 300 1.316 2808.8 2971.2 7.710 350 1.433 2887.3 3072.1 7.894 400 1.549 2967.1 3173.9 8.064 450 1.666 3048.5 3277.0 8.224 500 1.781 3131.4 3381.6 8.373 600 2.013 3302.2 3487.7 8.515 700 2.244 3479.9 3704.8 8.779 800 2.476 3664.7 3928.8 9.022 900 2.707 3856.3 4159.8 9.248 1000 2.938 4054.8 4397.6 9.460 The Ideal-gas specific heat at constant pressure cp in kJ/kmol • K of water vapor as a function of temperature (in Kelvin, °K) is given by: cp(T) = a + bT + cT2 + dT3 where a = 32.24, b = 0.1923 x 10-2, c = 1.055 x 10-5, d = —3.595 x 10-9. cp = c,, + R, and the gas constant, R= 0.4615 kJ/kg • K. For the computations below, convert the temperature to Kelvin: K=273+°C a. Use spline interpolation to increase the data points in table 1 for T, v, u, h, s, by creating a temperature vector in K Tnew = [150: 50:1000] + 273 and using it in the function 'interpl' to form the new vectors vnew, anew, hnew, Snew; e.g. vnew= interpl(T+273, v, Tnew,'spline'). b. Create a 1x2 subplot: subplot (1, 2, 1) has anew , hnew vs Tnew and the data plots......

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...Expository Method Expository method of teaching is almost similar to showing or telling method. It is most used for lower grades, however, it can also be used in high school and collegiate level when there’s a difficult topic that needs further explanation. Steps of the Expository Method. Subject: History of Math Topic: Leibniz-Newton Controversy regarding the origin of Calculus 1. Approach. Reviewing past lessons regarding the basics of Calculus, its fundamentals and concepts. 2. Presentation. The teacher explains Leibniz and Newton’s controversy regarding calculus by: a. Showing a table regarding the comparisons of the contributions of the two mathematicians. b. Showing a timeline as to which when did the contributions took place. c. Show relevant yet reliable sources and clippings about the issue. 3. Application. To further test whether the students understand the topic or not, the teacher will conduct two activities. a. The teacher will set up a debate as to who really invented Calculus. b. An essay quiz regarding the topic. Integration Method Integration is the process of unifying the child’s mental, physical, socio-emotional and spiritual energies through learning experience. Emphasis is placed on the development of integrated personality of the child. Learning activities are organized in terms of larger units. The teacher must also know her pupil’s like and dislikes. The child’s nature and experience must be made the......

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