Comparing Convergence Of False Position And Bisection Methods Engineering Essay

Comparing Convergence Of False Position And Bisection Methods Engineering Essay Explain with example that rate of convergence of false position method is faster than that of the bisection method. Introduction False position method In numerical analysis, the false position method or regula falsi method is a root-finding algorithm that combines features from the bisection method and the secant method. The method: The first two iterations of the false position method. The red curve shows the function f and the blue lines are the secants. Like the bisection method, the false position method starts with two points a0 and b0 such that f(a0) and f(b0) are of opposite signs, which implies by the intermediate value theorem that the function f has a root in the interval [a0, b0], assuming continuity of the function f. The method proceeds by producing a sequence of shrinking intervals [ak, bk] that all contain a root of f. At iteration number k, the number is computed. As explained below, ck is the root of the secant line through (ak, f(ak)) and (bk, f(bk)). If f(ak) and f(ck) have the same sign, then we set ak+1 = ck and bk+1 = bk, otherwise we set ak+1 = ak and bk+1 = ck. This process is repeated until the root is approximated sufficiently well. The above formula is also used in the secant method, but the secant method always retains the last two computed points, while the false position method retains two points which certainly bracket a root. On the other hand, the only difference between the false position method and the bisection method is that the latter uses ck = (ak + bk) / 2. Bisection method In mathematics, the bisection method is a root-finding algorithm which repeatedly bisects an interval then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow. The method is applicable when we wish to solve the equation for the scalar variable x, where f is a continuous function. The bisection method requires two initial points a and b such that f(a) and f(b) have opposite signs. This is called a bracket of a root, for by the intermediate value theorem the continuous function f must have at least one root in the interval (a, b). The method now divides the interval in two by computing the midpoint c = (a+b) / 2 of the interval. Unless c is itself a rootwhich is very unlikely, but possiblethere are now two possibilities: either f(a) and f(c) have opposite signs and bracket a root, or f(c) and f(b) have opposite signs and bracket a root. We select the subinterval that is a bracket, and apply the same bisection step to it. In this way the interval that might contain a zero of f is reduced in width by 50% at each step. We continue until we have a bracket sufficiently small for our purposes. This is similar to the computer science Binary Search, where the range of possible solutions is halved each iteration. Explicitly, if f(a) f(c) Advantages and drawbacks of the bisection method Advantages of Bisection Method The bisection method is always convergent. Since the method brackets the root, the method is guaranteed to converge. As iterations are conducted, the interval gets halved. So one can guarantee the decrease in the error in the solution of the equation. Drawbacks of Bisection Method The convergence of bisection method is slow as it is simply based on halving the interval. If one of the initial guesses is closer to the root, it will take larger number of iterations to reach the root. If a function is such that it just touches the x-axis (Figure 3.8) such as it will be unable to find the lower guess, , and upper guess, , such that For functions where there is a singularity and it reverses sign at the singularity, bisection method may converge on the singularity (Figure 3.9). An example include and, are valid initial guesses which satisfy . However, the function is not continuous and the theorem that a root exists is also not applicable. Figure.3.8. Function has a single root at that cannot be bracketed. Figure.3.9. Function has no root but changes sign. Explanation Source code for False position method: Example code of False-position method C code was written for clarity instead of efficiency. It was designed to solve the same problem as solved by the Newtons method and secant method code: to find the positive number x where cos(x) = x3. This problem is transformed into a root-finding problem of the form f(x) = cos(x) x3 = 0. #include #include double f(double x) { return cos(x) x*x*x; } double FalsiMethod(double s, double t, double e, int m) { int n,side=0; double r,fr,fs = f(s),ft = f(t); for (n = 1; n { r = (fs*t ft*s) / (fs ft); if (fabs(t-s) fr = f(r); if (fr * ft > 0) { t = r; ft = fr; if (side==-1) fs /= 2; side = -1; } else if (fs * fr > 0) { s = r; fs = fr; if (side==+1) ft /= 2; side = +1; } else break; } return r; } int main(void) { printf(%0.15fn, FalsiMethod(0, 1, 5E-15, 100)); return 0; } After running this code, the final answer is approximately 0.865474033101614 Example 1 Consider finding the root of f(x) = x2 3. Let ÃŽÂ µstep = 0.01, ÃŽÂ µabs = 0.01 and start with the interval [1, 2]. Table 1. False-position method applied to f(x)  =  x2 3. a b f(a) f(b) c f(c) Update Step Size 1.0 2.0 -2.00 1.00 1.6667 -0.2221 a = c 0.6667 1.6667 2.0 -0.2221 1.0 1.7273 -0.0164 a = c 0.0606 1.7273 2.0 -0.0164 1.0 1.7317 0.0012 a = c 0.0044 Thus, with the third iteration, we note that the last step 1.7273 à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ 1.7317 is less than 0.01 and |f(1.7317)| Note that after three iterations of the false-position method, we have an acceptable answer (1.7317 where f(1.7317) = -0.0044) whereas with the bisection method, it took seven iterations to find a (notable less accurate) acceptable answer (1.71344 where f(1.73144) = 0.0082) Example 2 Consider finding the root of f(x) = e-x(3.2 sin(x) 0.5 cos(x)) on the interval [3, 4], this time with ÃŽÂ µstep = 0.001, ÃŽÂ µabs = 0.001. Table 2. False-position method applied to f(x)  = e-x(3.2 sin(x) 0.5 cos(x)). a b f(a) f(b) c f(c) Update Step Size 3.0 4.0 0.047127 -0.038372 3.5513 -0.023411 b = c 0.4487 3.0 3.5513 0.047127 -0.023411 3.3683 -0.0079940 b = c 0.1830 3.0 3.3683 0.047127 -0.0079940 3.3149 -0.0021548 b = c 0.0534 3.0 3.3149 0.047127 -0.0021548 3.3010 -0.00052616 b = c 0.0139 3.0 3.3010 0.047127 -0.00052616 3.2978 -0.00014453 b = c 0.0032 3.0 3.2978 0.047127 -0.00014453 3.2969 -0.000036998 b = c 0.0009 Thus, after the sixth iteration, we note that the final step, 3.2978 à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ 3.2969 has a size less than 0.001 and |f(3.2969)| In this case, the solution we found was not as good as the solution we found using the bisection method (f(3.2963) = 0.000034799) however, we only used six instead of eleven iterations. Source code for Bisection method #include #include #define epsilon 1e-6 main() { double g1,g2,g,v,v1,v2,dx; int found,converged,i; found=0; printf( enter the first guessn); scanf(%lf,g1); v1=g1*g1*g1-15; printf(value 1 is %lfn,v1); while (found==0) { printf(enter the second guessn); scanf(%lf,g2); v2=g2*g2*g2-15; printf( value 2 is %lfn,v2); if (v1*v2>0) {found=0;} else found=1; } printf(right guessn); i=1; while (converged==0) { printf(n iteration=%dn,i); g=(g1+g2)/2; printf(new guess is %lfn,g); v=g*g*g-15; printf(new value is%lfn,v); if (v*v1>0) { g1=g; printf(the next guess is %lfn,g); dx=(g1-g2)/g1; } else { g2=g; printf(the next guess is %lfn,g); dx=(g1-g2)/g1; } if (fabs(dx)less than epsilon {converged=1;} i=i+1; } printf(nth calculated value is %lfn,v); } Example 1 Consider finding the root of f(x) = x2 3. Let ÃŽÂ µstep = 0.01, ÃŽÂ µabs = 0.01 and start with the interval [1, 2]. Table 1. Bisection method applied to f(x)  =  x2 3. a b f(a) f(b) c  =  (a  +  b)/2 f(c) Update new b à ¢Ã‹â€ Ã¢â‚¬â„¢ a 1.0 2.0 -2.0 1.0 1.5 -0.75 a = c 0.5 1.5 2.0 -0.75 1.0 1.75 0.062 b = c 0.25 1.5 1.75 -0.75 0.0625 1.625 -0.359 a = c 0.125 1.625 1.75 -0.3594 0.0625 1.6875 -0.1523 a = c 0.0625 1.6875 1.75 -0.1523 0.0625 1.7188 -0.0457 a = c 0.0313 1.7188 1.75 -0.0457 0.0625 1.7344 0.0081 b = c 0.0156 1.71988/td> 1.7344 -0.0457 0.0081 1.7266 -0.0189 a = c 0.0078 Thus, with the seventh iteration, we note that the final interval, [1.7266, 1.7344], has a width less than 0.01 and |f(1.7344)| Example 2 Consider finding the root of f(x) = e-x(3.2 sin(x) 0.5 cos(x)) on the interval [3, 4], this time with ÃŽÂ µstep = 0.001, ÃŽÂ µabs = 0.001. Table 1. Bisection method applied to f(x)  = e-x(3.2 sin(x) 0.5 cos(x)). a b f(a) f(b) c  =  (a  +  b)/2 f(c) Update new b à ¢Ã‹â€ Ã¢â‚¬â„¢ a 3.0 4.0 0.047127 -0.038372 3.5 -0.019757 b = c 0.5 3.0 3.5 0.047127 -0.019757 3.25 0.0058479 a = c 0.25 3.25 3.5 0.0058479 -0.019757 3.375 -0.0086808 b = c 0.125 3.25 3.375 0.0058479 -0.0086808 3.3125 -0.0018773 b = c 0.0625 3.25 3.3125 0.0058479 -0.0018773 3.2812 0.0018739 a = c 0.0313 3.2812 3.3125 0.0018739 -0.0018773 3.2968 -0.000024791 b = c 0.0156 3.2812 3.2968 0.0018739 -0.000024791 3.289 0.00091736 a = c 0.0078 3.289 3.2968 0.00091736 -0.000024791 3.2929 0.00044352 a = c 0.0039 3.2929 3.2968 0.00044352 -0.000024791 3.2948 0.00021466 a = c 0.002 3.2948 3.2968 0.00021466 -0.000024791 3.2958 0.000094077 a = c 0.001 3.2958 3.2968 0.000094077 -0.000024791 3.2963 0.000034799 a = c 0.0005 Thus, after the 11th iteration, we note that the final interval, [3.2958, 3.2968] has a width less than 0.001 and |f(3.2968)| Convergence Rate Why dont we always use false position method? There are times it may converge very, very slowly. Example: What other methods can we use? Comparison of rate of convergence for bisection and false-position method

Bluefield health plan Essay

Arc Electric employees were opting for their health insurance plan. Arc Electric had expanded their workforce due to which more employees were enrolling for Bluefield’s health insurance plan for the benefits. But when Bluefield released that the utilization of their physician services had tremendously increased in the last 6 months because of which their profits were being affected, they had to find out the cause for this. Soon they realized that the main cause for the increase in the utilization of their physician services was the increase in the number of new employees who were opting for the health insurance plan. Exhibit 1 clearly shows that the number of Arc Electric employees using Bluefield’s Health insurance plan has increased from 3912 in July, 2006 to 4137 in August, 2006. Thus, in only one month the increase has been of 225 people, which is comparatively quite high. Also, in 2006, the total cost incurred by Arc Electric for inpatient and outpatient hospital services were 203425 and 182440 in July and 212250 and 180700 in August, and for surgical services were 101250 and 103400 in July and August. Thus, the total cost incurred for hospital services and surgical services were 487115 and 496350 in July and August. While on the other hand the total cost incurred by Arc Electric for visits to physician’s office was only 337900 and 391450 in July and August. Therefore we can see that the difference is almost of 147215 and 104900 in July and August. As Bluefield’s contract with Arc Electric was about to expire the next month, they had to renegotiate the terms in their contract with Arc Electric and request for an increase in their premium rate in order to maintain their profit. They had realized that the main reason for their erosion of profits was the increasing number of Arc Electric employees who had opted for their health insurance plan. But, Bluefield were also aware of the fact that during renegotiations if they tried to increase the fixed premium which they charged every employee of Arc Electric per month, then they may refuse to do anymore business with them and sign a contract with some other health insurance company. This, Bluefield was not ready to risk. Thus, Bluefield wanted the staff members and directors to devise a renegotiating strategy which they could present before Arc Electric and maintain their contract with them while at the same time see to it that their profitability remains at par. After much consideration and results from various studies, including Exhibit 1, the employees of Bluefield realized that simply by increasing their copayment charges they will not be able to bring about a decrease in the number of physician visits since people do not actually like to visit the physicians but rather do it in order to remain healthy and fit. The only way they can reduce their costs is by paying less to their health care providers, like the physicians. Thus, they first needed to negotiate with the physicians and ask them to decrease the costs of services supplied by them. If they simply asked the physicians to lower their cost of service by around 10% or 25% they might do it with the fear that they may loose all of their patients and also be left out of Bluefield’s health insurance plan. But this may have certain negative effects as in return of a lower fee per visit the physicians may also lessen the quality of care that they give to their clients. This is the reason why Bluefield required a further analysis of physicians visit. Out of the $250 fixed premium that Bluefield charged each employee of Arc Electric every month, the total premium revenue was portioned out as 55% for the hospital and surgical services and 30% for the physician visits. Thus while $137. 5 went for the hospital and surgical services only $75 went in for the physician visits. Thus for every premium collected, the profitability of hospital and surgical services was almost $62. 5 more than the profitability of the physician services. Thus, when compared to physician’s services, hospital and surgical services have a profitability of almost 45% more than the former.

Find Out Whos Concerned About Psychology Compare and Contrast Essay Topics and Why You Should Pay Attention

Find Out Who's Concerned About Psychology Compare and Contrast Essay Topics and Why You Should Pay Attention Merely mentioning the similarities and differences isn't enough if one may not analyze the key ideas. A Conclusion is the previous portion of the essay that summarizes the evidence and each one of the key points. In the conclusion, it's important to recap all the essential points. If you are making your own essay topic, then there should be an obvious foundation for comparison and the comparison should make logical sense. The Foolproof Psychology Compare and Contrast Essay Topics Strategy The most significant thing you need so as to create a winning paper is to get a good topic. The very first thing you ought to do is identify the form of compare and contrast essay which you're handling. You will need to read up the many journals and case files to have a very good point. Use critical thinking and have a look at the familiar thing at a wholly new angle. You must find a good deal of resources to use to reveal the total picture of the issue, and the structure of an essay needs to be well planned. You also know how to come across sources and the very best kind to utilize in your paper to allow it to be relevant and interesting. The model for this kind of essay may also be utilized to analyze candidates for a true life political race or any competition. Structurally, the post would then list five unique varieties of dog food. Psychology Compare and Contrast Essay Topics: No Longer a Mystery Another possibility is that there's some type of genetic factor involved with the growth of both PTSD and depression. It is not unusual for somebody to have both ADHD and OCD. For those who have PTSD, it's important to get treatment when possible. If you presently have PTSD and depression, additionally, it is important to get treatment after possible. The majority of the time our teachers would give us a particular undertaking and instructi ons to follow, so it wasn't really an extremely hard issue to do. Students are encouraged that when selecting a topic, you think beyond the box as this will probably earn you better grades. In organizing compare-contrast paragraphs, using either of the above methods, they may find it helpful to create a compare-contrast-prewriting chart. Various students have various reasons for picking a specific subject and one statement would not be sufficient to say why all students who pick psychology choose it. Writing a paper in psychology is among the most intriguing tasks for students who are interested in the matter. Evaluation is among the principal aims of these kinds of essays as you will need to figure out the things which are best and desirable, since the writer should recognize the advantages and pitfalls. To start with, develop a thesis with a purpose. Psychology Compare and Contrast Essay Topics - Dead or Alive? The above mentioned compare and contrast essay topics are just a couple of the numerous topics you are able to decide to go over in your essay. It is possible to also order a fully written compare and contrast essay and alleviate the quantity of work you must do. You must research the selected topic and discover facts to contradict your first thesis. You can begin with the form of topic you select for your compare and contrast essay. If you're writing an essay, then think of how you'll make a bigger point about both subjects being compared. If you're looking for an essay topic to receive your assignment off the ground, the above mentioned choices can prove to be useful. If you wish to be certain that you have a special topic for your essay, you are able to look through the overall directions of our topics and hunt to find out more online. There are lots of others persuasive compare and contrast essay topics regarding education if you are prepared to write about doing it. In the start, you will have lots of suggestions to compose a topic. The first thing which you have to do before you get started writing is to pick an acceptable topic to write about. You're not restricted to anything, and you may pick any compare and contrast writing topics you're passionate about. For example an individual may select a topic like, life in the shoes of a werewolf. With the web, it's easy to look for lists of transitions and the relationships they show. You don't need to discuss major issues and changes in education in case you don't want. Secondly, communication might be a topic of prime significance in any marriage. An excellent effective communication plays a crucial part in keeping the couple's relationship with time.

Financial Crisis And Basel Capital Adequacy Accords

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