Descriptive Essay on My Pet

Expressive Essay on My Pet Expressive Essay on My Pet My pet is a pooch named Tipsy. Delirious is a charming earthy colored pooch that has a couple of dark spots spread over his body and tail. Drunk is a kelpie crossed with an outskirt collie, and he has cushy smooth ears. Despite the fact that Tipsy has an extremely solid body outline, he has an exceptionally delicate face and is consistently a neighborly canine to those whom he knows. In the event that a more unusual methodologies out house, in any case, Tipsy can get forceful. He generally barks noisily to stand out for us to the moving toward stranger. Blasted loves numerous things. Among these is to nestle his wet nose in my grasp and in the hands on my folks and kin. He needs consideration more often than not on the grounds that he is terrified of being relinquished or disregarded. I really ran over Tipsy while he was as yet a little dog. It shows up his proprietor had relinquished him out and about. I discovered him pondering in our neighborhood. I educated my folks about the little dog. I needed to keep him. They spoke with the nearby specialists so the specialists could permit us to receive the pet. Loaded loves food, particularly bones. When we have taken care of him his normal food, we generally give him a couple of bones on which to bite. Dazed can really spend entire evenings biting bones since he adores them to such an extent. At whatever point Tipsy is concerned, he takes a gander at us with frantic eyes that seem like he is pitiful. He does as such while swaying his tail from one side to the next. At whatever point we see him showing these signs, we promptly set him up a snappy supper and a few bones for him to eat. Dazed has additionally made it a propensity to play with our feline, Toppy. In some cases, Tipsy plays with and spills out the cat’s water, something that consistently leaves the feline giving him a horrendous glare. In some cases, the feline even howls as though to tell Tipsy that he isn't dazzled with Tipsy for spilling out his water. At whatever point, my pooch plays with the cat’s water, I see him lifting his head as a portion of the cat’s water spills out from his tongue, similar to the manner in which water drops from a cascade. Blasted additionally loves the chipping sound made by the fowls that live on the trees in our compound. At whatever point Tipsy hears these sounds, he raises his ears and focuses them towards the bearing where the trilling sound is beginning. One can generally watch the manner in which his eyes light up with fervor at whatever point the fowls start making their mitigating commotions. My Pet expressive paper composing tips: Since this is a depiction exposition, one should portray the one of a kind attributes of one’s most loved pet which for this situation is a pooch. Since most mutts have a name, it is judicious that one starts this portrayal by giving the name of the canine followed by the species to which the pooch has a place. When this is done, one can start depicting the things that the canine preferences and those that it doesn't appreciate. For example in this exposition, the author has depicted the manner in which the canine loves food and what it does at whatever point it needs to eat some food. Prepared to pay for exposition help on the web? Dont waver to contact composing administration now!

Evaluate the importance of regular exercise Essay Example for Free

Assess the significance of customary exercise Essay Lately, the familiarity with the significance of general wellbeing has expanded altogether and keeping up a solid body and psyche is without a doubt everyone’s day by day wishes. Doing exercise is one of the approaches to help individuals to satisfy their fantasies as it can reinforce our body and improve our brain. The adequacy of the guides from the state to help accomplishing the objective effectively has additionally been put on the public’s conversation table. In this exposition, will initially look at the significance of activity, following by investigating how both Hong Kong and British government assumes their job in guaranteeing its residents practice consistently and proposing some subsequent activities. In the first place, doing activity can assist with improving our wellbeing condition and makes us more grounded. Exercise can be separated into four classifications: continuance, quality, equalization, and adaptability which will give us benefits on the off chance that we can complete them all. (Go4Life n.d.) Researches. (Mayo center 2014) has indicated that working out will on one hand lower pressure and then again in decrease muscle to fat ratio which will makes us look fitter and more beneficial. When muscle to fat ratio is decreased, it can likewise bring down the pulse and keep up a smooth roundabout blood stream. Dangers presented to our body because of hypertension can be stopped for instance strokes. In addition, practice shows a negative relationship with miseries. Endorphin is a hormone discharged when doing exercise which positively affects our state of mind by making us more joyful and decreasing paces of despondency. This shows the more we work out, the higher the degree of joy we can pick up. Endorphins will decrease the view of agony in mind will prompts positive emotions. (WebMD 2014) Social circle can likewise be extended when doing exercise as some of them require collaboration and players should co-work with one another which expands their unions and lift their connections. Accept playing b-ball for instance, cooperative individuals willâ have to convey and confide in one another and pass the ball to each other in order to pick up objectives. Solid social help which came about can lessen the opportunity of despondency as ones consideration has been expanded. On the other hand, with no work out may prompt self destruction in some extraordinary cases. Studies done by Dh aval and Inas shows that there is an immediate connection between overweight status, burdensome scatters, and self-destructive practices. Overweight youths will have a low confidence as they are named as â€Å"fat boy† which makes them having a feeble self-assurance and body disappointment. (Dhaval Inas 2009) As practicing can identify with life and demise issues, it demonstrates that it is of high significance that customary activities are expected to keep up our body wellbeing. Next, the administration ought to likewise assume a significant job in managing social insurance issue of residents. As indicated by ongoing reports, (Ko 2010; Chapman 2014) both Hong Kong and Britain are confronting an issue of expanded passing rates because of heftiness. This is predominantly because of the expanding individual crowded and changing in way of life attributes with diminished in physical action however increment in food consumption which brings about incessant ailments. Regarding the Hong Kong government’s official site, it has made a stage forward to handle this issue by presenting the National Fitness Day on eighth August yearly (GovHK 2010). This offers its neighborhood residents in complimentary wearing involvement with designated sports focus in 18 areas of Hong Kong for instance moving play-in, wellness corner and some wellness and wellbeing talk and workshops. Parent-kid exercises are likewise accessible which can advance their connections. Through this occasion, it can without a doubt stir the publics’ consideration in doing sports and even expands their own enthusiasm for building up particular sort of sports. In any case, British government just did constrained activities in creating citizens’ sportsmanship except for London Olympics 2012 (Olympic.org 2013) which uncovers more Britain are dead because of overweight and hefty continued expanding and the figure is even 50 percent more when contrasting with France as Chapman said. It is accepted the British government should take Hong Kong as a kind of perspective to guarantee its residents include in sports all the more frequently. By presenting reward framework can build their consideration in doing sports and they will be all the more ready to do as such. Additionally, open games places with low participation expenses canâ also be set up in order to help low pay families to appreciate this diversion in minimal effort. Advancements and commercials are required so more individuals will think about it and raise their attention to it. Because of the high authority of the administration, it will ideally assist with advancing activity i n Britain all the more successfully and vouch for the accomplishment of Hong Kong in elevating sports to its habitations. Likewise, residents keeping up great wellbeing will likewise profit the state. With an unforeseen weakness condition, laborer will bring about truant from work and will diminish the efficiency of the firm. (Krol at el., 2012) During the missing time frame, it might be hard for the organization to discover a flawlessly reasonable substitute as the efficiency misfortune will be capricious as laborers are not allocated to the correct occupations. Benefits will be lost and lessens the monetary development of the country (Wei at el., 2011). Likewise, as referenced prior, self destruction occurrences of laborers can likewise influence the organization. Foxconn, the creation industrial facility of Apple items is one of where frequently knew about specialists there ended it all because of poor working conditions and solitariness ( BBC 2010). This will in no uncertainty harm the notoriety of the firm and influence deals. Be that as it may, if there can be sufficient work out for the work, they will feel not so much focused but rather more glad to work. Undesirable mishaps can be maintained a strategic distance from. One of the principle obligations of residents is to add to the general public that we should deliver to the benefit boosting yield to help keeping up the work power of our place to look after productivity. With great wellbeing, laborers can work all the more proficiently as they can perform well and show their own gifts. Profitability will at that point increment because of invigorating monetary development which will benefits the general public. With better monetary conditions, individuals can bear the cost of their own lives and have a better quality of living. So as to abstain from stopping creation procedure and decreasing productivity of firms, the legislature to a high degree ought to be dependable to manage rules to help its residents in general medical problems. Notwithstanding, some may contend that it is our own obligation to remain solid which implies we ought not depend on the state. Wellbeing is a drawn out speculation which can be determinates without anyone else and we are personallyâ responsible day by day. Some may likewise say that (Sherman 2012) great wellbeing will be achieved if exertion is given and we organize it well. Long haul exertion is required to adjust our body, brain and exertion. Exercise is an individual movement which relies upon the individual ability. On the off chance that the administration compel them to work out, they may feel malcontented and clashes between the two gatherings will exist. Costs will be engaged with managing the debates which endeavors interests of them. Besides, individuals may imagine that they will have their own training to keep up wellbeing for instance controlling their eating regimen, for example, following the food pyramid (GovHK 2012). Guardians likewise assume indispensable job in m anaging the wellbeing states of their kids. Solid way of life practices ought to be sustained since little so they will pay more goal to that in any event, when they grow up. Furthermore, bolsters from schools ought to likewise be given to understudies for instance setting up progressively physical training classes and furthermore extra-round exercises identified with sports in order to build their measure of physical exercise. It is undouble that educating from guardians to their kids is the best method to pass on the possibility of wellbeing to them as most kids will in general tune in to their folks more than any other individual. Along these lines some accept that it isn't the state’s obligation and it ought not mediate with people’s every day schedules. To finish up, albeit a few people might be disappointed if the administration interfere with their activity design, it is of foremost significance which the state ought to be capable to manage the medical problems of its residents as this is one of its duties. On the off chance that the state help to safeguard the public’s practicing musicality, it can diminish the ailment related issues and it is a lot simpler for the state to mediate because of its high power. Simultaneously, it will likewise bring positive effects like expanding profitability which benefits the general public. Subsequently, to an enormous degree the state should help in guaranteeing its residents practice normally so as to keep up a decent wellbeing. Reference List 1. Whiten, B, 2010. Foxconn suicides: Workers feel very forlorn. BBC on the web, [online] 28 May 2010. Accessible at: http://www.bbc.co.uk/news/10182824 [16 May 2014]. 2. Chapman, J, 2014. Englands weight passing rate. [online] Available at: [Accessed 04 May 2014] 3. Dhaval, D and Inas, R., 2009. Overweight status, self-discernment, and self-destructive practices among young people. Sociology Medicine, [e-journal]. 68(9) Available through: Lancaster University Library site [Accessed 15 February 2014]. 4. Go4Life n.d., 4 Types of Exercise. [online] Available at: [Accessed 12 May 2014] 5. GovHK, 2010. Game For All Day on August 8 advances advantages of standard exercise. [online] Available at: [Accessed 10 6. GovHK, 2012. The Food Pyramid - A Guide to a Balanced Diet. [online] Available at: http://www.cheu.gov.hk/eng/data/exercise_07.htm [Accessed 10 May 2014] 7. Krol, M ; Brouwer, W ; Severens, J ; Kaper, J ; Evers, S., 2002. Profitability cost counts in wellbeing monetary assessments: Cor

Support MIT Sports!!

Support MIT Sports!! Another blogget for you all. MIT Mens Heavyweight Crew is a finalist for Crew of the Week and you can all help our crew team be recognized by going here and voting on the right hand side. This site is a little smarter than some other ones weve come across, so we could use your support. With all the recent varsity team stuff, it cant hurt to show our teams a little support! For those of you unfamiliar, Crew is sort of a big deal round these parts. For those who are REALLY unfamiliar, Crew is the sport where a lot of people sit in a very small skinny rowboat and row super fast. Crew at MIT is one of the largest sports, and our only Division 1 team. This year especially, the teams have been enjoying a lot of success, beating Princeton for the first time in 33 years. Sports are perhaps a surprisingly large part of many students lives here. And our school supports our athletic endeavors not because we win all the time (though we have had some awesome teams), but just because they recognize the importance of sports. Budget cuts around the country may be forcing a lot of colleges to reevaluate the number of sports they have, but MIT sports will always be a big part of many students lives. Many people are having their first chance to play varsity, after picking up the sport not long if at all before coming here. If youre interested in athletics here, or student lifeor heck you just want to congratulate Crew for beating PrincetonHave at it!

Numerical differential equation analysis package - Free Essay Example

Sample details Pages: 31 Words: 9220 Downloads: 1 Date added: 2017/06/26 Category Statistics Essay Did you like this example? The Numerical Differential Equation Analysis package combines functionality for analyzing differential equations using Butcher trees, Gaussian quadrature, and Newton-Cotes quadrature. Butcher Runge-Kutta methods are useful for numerically solving certain types of ordinary differential equations. Deriving high-order Runge-Kutta methods is no easy task, however. There are several reasons for this. The first difficulty is in finding the so-called order conditions. These are nonlinear equations in the coefficients for the method that must be satisfied to make the error in the method of order O (hn) for some integer n where h is the step size. The second difficulty is in solving these equations. Besides being nonlinear, there is generally no unique solution, and many heuristics and simplifying assumptions are usually made. Finally, there is the problem of combinatorial explosion. For a twelfth-order method there are 7813 order conditions! Don’t waste time! Our writers will create an original "Numerical differential equation analysis package" essay for you Create order This package performs the first task: finding the order conditions that must be satisfied. The result is expressed in terms of unknown coefficients aij, bj, and ci. The s-stage Runge-Kutta method to advance from x to x+h is then where Sums of the elements in the rows of the matrix [aij] occur repeatedly in the conditions imposed on aij and bj. In recognition of this and as a notational convenience it is usual to introduce the coefficients ci and the definition This definition is referred to as the row-sum condition and is the first in a sequence of row-simplifying conditions. If aij=0 for all ij the method is explicit; that is, each of the Yi (x+h) is defined in terms of previously computed values. If the matrix [aij] is not strictly lower triangular, the method is implicit and requires the solution of a (generally nonlinear) system of equations for each timestep. A diagonally implicit method has aij=0 for all ij. There are several ways to express the order conditions. If the number of stages s is specified as a positive integer, the order conditions are expressed in terms of sums of explicit terms. If the number of stages is specified as a symbol, the order conditions will involve symbolic sums. If the number of stages is not specified at all, the order conditions will be expressed in stage-independent tensor notation. In addition to the matrix a and the vectors b and c, this notation involves the vector e, which is composed of all ones. This notation has two distinct advantages: it is independent of the number of stages s and it is independent of the particular Runge-Kutta method. For further details of the theory see the references. ai,j the coefficient of f(Yj(x)) in the formula for Yi(x) of the method bj the coefficient of f(Yj(x)) in the formula for Y(x) of the method ci a notational convenience for aij e a notational convenience for the vector (1, 1, 1, ) Notation used by functions for Butcher. RungeKuttaOrderConditions[p,s] give a list of the order conditions that any s-stage Runge-Kutta method of order p must satisfy ButcherPrincipalError[p,s] give a list of the order p+1 terms appearing in the Taylor series expansion of the error for an order-p, s-stage Runge-Kutta method RungeKuttaOrderConditions[p], ButcherPrincipalError[p] give the result in stage-independent tensor notation Functions associated with the order conditions of Runge-Kutta methods. ButcherRowSum specify whether the row-sum conditions for the ci should be explicitly included in the list of order conditions ButcherSimplify specify whether to apply Butchers row and column simplifying assumptions Some options for RungeKuttaOrderConditions. This gives the number of order conditions for each order up through order 10. Notice the combinatorial explosion. In[2]:= Out[2]= This gives the order conditions that must be satisfied by any first-order, 3-stage Runge-Kutta method, explicitly including the row-sum conditions. In[3]:= Out[3]= These are the order conditions that must be satisfied by any second-order, 3-stage Runge-Kutta method. Here the row-sum conditions are not included. In[4]:= Out[4]= It should be noted that the sums involved on the left-hand sides of the order conditions will be left in symbolic form and not expanded if the number of stages is left as a symbolic argument. This will greatly simplify the results for high-order, many-stage methods. An even more compact form results if you do not specify the number of stages at all and the answer is given in tensor form. These are the order conditions that must be satisfied by any second-order, s-stage method. In[5]:= Out[5]= Replacing s by 3 gives the same result asRungeKuttaOrderConditions. In[6]:= Out[6]= These are the order conditions that must be satisfied by any second-order method. This uses tensor notation. The vector e is a vector of ones whose length is the number of stages. In[7]:= Out[7]= The tensor notation can likewise be expanded to give the conditions in full. In[8]:= Out[8]= These are the principal error coefficients for any third-order method. In[9]:= Out[9]= This is a bound on the local error of any third-order method in the limit as h approaches 0, normalized to eliminate the effects of the ODE. In[10]:= Out[10]= Here are the order conditions that must be satisfied by any fourth-order, 1-stage Runge-Kutta method. Note that there is no possible way for these order conditions to be satisfied; there need to be more stages (the second argument must be larger) for there to be sufficiently many unknowns to satisfy all of the conditions. In[11]:= Out[11]= RungeKuttaMethod specify the type of Runge-Kutta method for which order conditions are being sought Explicit a setting for the option RungeKuttaMethod specifying that the order conditions are to be for an explicit Runge-Kutta method DiagonallyImplicit a setting for the option RungeKuttaMethod specifying that the order conditions are to be for a diagonally implicit Runge-Kutta method Implicit a setting for the option RungeKuttaMethod specifying that the order conditions are to be for an implicit Runge-Kutta method $RungeKuttaMethod a global variable whose value can be set to Explicit, DiagonallyImplicit, or Implicit Controlling the type of Runge-Kutta method in RungeKuttaOrderConditions and related functions. RungeKuttaOrderConditions and certain related functions have the option RungeKuttaMethod with default setting $RungeKuttaMethod. Normally you will want to determine the Runge-Kutta method being considered by setting $RungeKuttaMethod to one of Implicit, DiagonallyImplicit, and Explicit, but you can specify an option setting or even change the default for an individual function. These are the order conditions that must be satisfied by any second-order, 3-stage diagonally implicit Runge-Kutta method. In[12]:= Out[12]= An alternative (but less efficient) way to get a diagonally implicit method is to force a to be lower triangular by replacing upper-triangular elements with 0. In[13]:= Out[13]= These are the order conditions that must be satisfied by any third-order, 2-stage explicit Runge-Kutta method. The contradiction in the order conditions indicates that no such method is possible, a result which holds for any explicit Runge-Kutta method when the number of stages is less than the order. In[14]:= Out[14]= ButcherColumnConditions[p,s] give the column simplifying conditions up to and including order p for s stages ButcherRowConditions[p,s] give the row simplifying conditions up to and including order p for s stages ButcherQuadratureConditions[p,s] give the quadrature conditions up to and including order p for s stages ButcherColumnConditions[p], ButcherRowConditions[p], etc. give the result in stage-independent tensor notation More functions associated with the order conditions of Runge-Kutta methods. Butcher showed that the number and complexity of the order conditions can be reduced considerably at high orders by the adoption of so-called simplifying assumptions. For example, this reduction can be accomplished by adopting sufficient row and column simplifying assumptions and quadrature-type order conditions. The option ButcherSimplify in RungeKuttaOrderConditions can be used to determine these automatically. These are the column simplifying conditions up to order 4. In[15]:= Out[15]= These are the row simplifying conditions up to order 4. In[16]:= Out[16]= These are the quadrature conditions up to order 4. In[17]:= Out[17]= Trees are fundamental objects in Butchers formalism. They yield both the derivative in a power series expansion of a Runge-Kutta method and the related order constraint on the coefficients. This package provides a number of functions related to Butcher trees. f the elementary symbol used in the representation of Butcher trees ButcherTrees[p] give a list, partitioned by order, of the trees for any Runge-Kutta method of order p ButcherTreeSimplify[p,,] give the set of trees through order p that are not reduced by Butchers simplifying assumptions, assuming that the quadrature conditions through order p, the row simplifying conditions through order , and the column simplifying conditions through order all hold. The result is grouped by order, starting with the first nonvanishing trees ButcherTreeCount[p] give a list of the number of trees through order p ButcherTreeQ[tree] give True if the tree or list of trees tree is valid functional syntax, and False otherwise Constructing and enumerating Butcher trees. This gives the trees that are needed for any third-order method. The trees are represented in a functional form in terms of the elementary symbol f. In[18]:= Out[18]= This tests the validity of the syntax of two trees. Butcher trees must be constructed using multiplication, exponentiation or application of the function f. In[19]:= Out[19]= This evaluates the number of trees at each order through order 10. The result is equivalent to Out[2] but the calculation is much more efficient since it does not actually involve constructing order conditions or trees. In[20]:= Out[20]= The previous result can be used to calculate the total number of trees required at each order through order10. In[21]:= Out[21]= The number of constraints for a method using row and column simplifying assumptions depends upon the number of stages. ButcherTreeSimplify gives the Butcher trees that are not reduced assuming that these assumptions hold. This gives the additional trees that are necessary for a fourth-order method assuming that the quadrature conditions through order 4 and the row and column simplifying assumptions of order 1 hold. The result is a single tree of order 4 (which corresponds to a single fourth-order condition). In[22]:= Out[22]= It is often useful to be able to visualize a tree or forest of trees graphically. For example, depicting trees yields insight, which can in turn be used to aid in the construction of Runge-Kutta methods. ButcherPlot[tree] give a plot of the tree tree ButcherPlot[{tree1,tree2,}] give an array of plots of the trees in the forest {tree1, tree2,} Drawing Butcher trees. ButcherPlotColumns specify the number of columns in the GraphicsGrid plot of a list of trees ButcherPlotLabel specify a list of plot labels to be used to label the nodes of the plot ButcherPlotNodeSize specify a scaling factor for the nodes of the trees in the plot ButcherPlotRootSize specify a scaling factor for the highlighting of the root of each tree in the plot; a zero value does not highlight roots Options to ButcherPlot. This plots and labels the trees through order 4. In[23]:= Out[23]= In addition to generating and drawing Butcher trees, many functions are provided for measuring and manipulating them. For a complete description of the importance of these functions, see Butcher. ButcherHeight[tree] give the height of the tree tree ButcherWidth[tree] give the width of the tree tree ButcherOrder[tree] give the order, or number of vertices, of the tree tree ButcherAlpha[tree] give the number of ways of labeling the vertices of the tree tree with a totally ordered set of labels such that if (m, n) is an edge, then mn ButcherBeta[tree] give the number of ways of labeling the tree tree with ButcherOrder[tree]-1 distinct labels such that the root is not labeled, but every other vertex is labeled ButcherBeta[n,tree] give the number of ways of labeling n of the vertices of the tree with n distinct labels such that every leaf is labeled and the root is not labeled ButcherBetaBar[tree] give the number of ways of labeling the tree tree with ButcherOrder[tree] distinct labels such that every node, including the root, is labeled ButcherBetaBar[n,tree] give the number of ways of labeling n of the vertices of the tree with n distinct labels such that every leaf is labeled ButcherGamma[tree] give the density of the tree tree; the reciprocal of the density is the right-hand side of the order condition imposed by tree ButcherPhi[tree,s] give the weight of the tree tree; the weight (tree) is the left-hand side of the order condition imposed by tree ButcherPhi[tree] give (tree) using tensor notation ButcherSigma[tree] give the order of the symmetry group of isomorphisms of the tree tree with itself Other functions associated with Butcher trees. This gives the order of the tree f[f[f[f] f^2]]. In[24]:= Out[24]= This gives the density of the tree f[f[f[f] f^2]]. In[25]:= Out[25]= This gives the elementary weight function imposed by f[f[f[f] f^2]] for an s-stage method. In[26]:= Out[26]= The subscript notation is a formatting device and the subscripts are really just the indexed variable NumericalDifferentialEquationAnalysis`Private`$i. In[27]:= Out[27]//FullForm= It is also possible to obtain solutions to the order conditions using Solve and related functions. Many issues related to the construction Runge-Kutta methods using this package can be found in Sofroniou. The article also contains details concerning algorithms used in Butcher.m and discusses applications. Gaussian Quadrature As one of its methods, the Mathematica function NIntegrate uses a fairly sophisticated Gauss-Kronrod-based algorithm. The Gaussian quadrature functionality provided in Numerical Differential Equation Analysis allows you to easily study some of the theory behind ordinary Gaussian quadrature which is a little less sophisticated. The basic idea behind Gaussian quadrature is to approximate the value if an integral as a linear combination of values of the integrand evaluated at specific points: Since there are 2n free parameters to be chosen (both the abscissas xi and the weights wi) and since both integration and the sum are linear operations, you can expect to be able to make the formula correct for all polynomials of degree less than about 2n. In addition to knowing what the optimal abscissas and weights are, it is often desirable to know how large the error in the approximation will be. This package allows you to answer both of these questions. GaussianQuadratureWeights[n,a,b] give a list of the pairs (xi, wi) to machine precision for quadrature on the interval a to b GaussianQuadratureError[n,f,a,b] give the error to machine precision GaussianQuadratureWeights[n,a,b,prec] give a list of the pairs (xi, wi) to precision prec GaussianQuadratureError[n,f,a,b,prec] give the error to precision prec Finding formulas for Gaussian quadrature. This gives the abscissas and weights for the five-point Gaussian quadrature formula on the interval (-3, 7). In[2]:= Out[2]= Here is the error in that formula. Unfortunately it involves the tenth derivative of f at an unknown point so you dont really know what the error itself is. In[3]:= Out[3]= You can see that the error decreases rapidly with the length of the interval. In[4]:= Out[4]= Newton-Cotes As one of its methods, the Mathematica function NIntegrate uses a fairly sophisticated Gauss-Kronrod based algorithm. Other types of quadrature formulas exist, each with their own advantages. For example, Gaussian quadrature uses values of the integrand at oddly spaced abscissas. If you want to integrate a function presented in tabular form at equally spaced abscissas, it wont work very well. An alternative is to use Newton-Cotes quadrature. The basic idea behind Newton-Cotes quadrature is to approximate the value of an integral as a linear combination of values of the integrand evaluated at equally spaced points: In addition, there is the question of whether or not to include the end points in the sum. If they are included, the quadrature formula is referred to as a closed formula. If not, it is an open formula. If the formula is open there is some ambiguity as to where the first abscissa is to be placed. The open formulas given in this package have the first abscissa one half step from the lower end point. Since there are n free parameters to be chosen (the weights) and since both integration and the sum are linear operations, you can expect to be able to make the formula correct for all polynomials of degree less than about n. In addition to knowing what the weights are, it is often desirable to know how large the error in the approximation will be. This package allows you to answer both of these questions. NewtonCotesWeights[n,a,b] give a list of the n pairs (xi, wi) for quadrature on the interval a to b NewtonCotesError[n,f,a,b] give the error in the formula Finding formulas for Newton-Cotes quadrature. option name default value QuadratureType Closed the type of quadrature, Open or Closed Option for NewtonCotesWeights and NewtonCotesError. Here are the abscissas and weights for the five-point closed Newton-Cotes quadrature formula on the interval (-3, 7). In[2]:= Out[2]= Here is the error in that formula. Unfortunately it involves the sixth derivative of f at an unknown point so you dont really know what the error itself is. In[3]:= Out[3]= You can see that the error decreases rapidly with the length of the interval. In[4]:= Out[4]= This gives the abscissas and weights for the five-point open Newton-Cotes quadrature formula on the interval (-3, 7). In[5]:= Out[5]= Here is the error in that formula. In[6]:= Out[6]= Runge-Kutta Methods From Wikipedia, The Free Encyclopedia Jump to: navigation, search In numerical analysis, the Runge-Kutta methods (German pronunciation:[kta]) are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians C. Runge and M.W. Kutta. See the article on numerical ordinary differential equations for more background and other methods. See also List of Runge-Kutta methods. Contents 1 The common fourth-order Runge-Kutta method 2 Explicit Runge-Kutta methods o 2.1 Examples 3 Usage 4 Adaptive Runge-Kutta methods 5 Implicit Runge-Kutta methods 6 References 7 External links The Common Fourth-Order Runge-Kutta Method One member of the family of Runge-Kutta methods is so commonly used that it is often referred to as RK4, classical Runge-Kutta method or simply as the Runge-Kutta method. Let an initial value problem be specified as follows. Then, the RK4 method for this problem is given by the following equations: where yn + 1 is the RK4 approximation of y(tn + 1), and Thus, the next value (yn + 1) is determined by the present value (yn) plus the product of the size of the interval (h) and an estimated slope. The slope is a weighted average of slopes: k1 is the slope at the beginning of the interval; k2 is the slope at the midpoint of the interval, using slope k1 to determine the value of y at the point tn + h / 2 using Eulers method; k3 is again the slope at the midpoint, but now using the slope k2 to determine the y-value; k4 is the slope at the end of the interval, with its y-value determined using k3. In averaging the four slopes, greater weight is given to the slopes at the midpoint: The RK4 method is a fourth-order method[needs reference], meaning that the error per step is on the order of h5, while the total accumulated error has order h4. Note that the above formulae are valid for both scalar- and vector-valued functions (i.e., y can be a vector and f an operator). For example one can integrate Schrdingers equation using the Hamiltonian operator as function f. Explicit Runge-Kutta Methods The family of explicit Runge-Kutta methods is a generalization of the RK4 method mentioned above. It is given by where (Note: the above equations have different but equivalent definitions in different texts). To specify a particular method, one needs to provide the integer s (the number of stages), and the coefficients aij (for 1 j i s), bi (for i = 1, 2, , s) and ci (for i = 2, 3, , s). These data are usually arranged in a mnemonic device, known as a Butcher tableau (after John C. Butcher): 0 c2 a21 c3 a31 a32 cs as1 as2 as,s 1 b1 b2 bs 1 bs The Runge-Kutta method is consistent if There are also accompanying requirements if we require the method to have a certain order p, meaning that the truncation error is O(hp+1). These can be derived from the definition of the truncation error itself. For example, a 2-stage method has order 2 if b1 + b2 = 1, b2c2 = 1/2, and b2a21 = 1/2. Examples The RK4 method falls in this framework. Its tableau is: 0 1/2 1/2 1/2 0 1/2 1 0 0 1 1/6 1/3 1/3 1/6 However, the simplest Runge-Kutta method is the (forward) Euler method, given by the formula yn + 1 = yn + hf(tn,yn). This is the only consistent explicit Runge-Kutta method with one stage. The corresponding tableau is: 0 1 An example of a second-order method with two stages is provided by the midpoint method The corresponding tableau is: 0 1/2 1/2 0 1 Note that this midpoint method is not the optimal RK2 method. An alternative is provided by Heuns method, where the 1/2s in the tableau above are replaced by 1s and the bs row is [1/2, 1/2]. If one wants to minimize the truncation error, the method below should be used (Atkinson p.423). Other important methods are Fehlberg, Cash-Karp and Dormand-Prince. Also, read the article on Adaptive Stepsize. Usage The following is an example usage of a two-stage explicit Runge-Kutta method: 0 2/3 2/3 1/4 3/4 to solve the initial-value problem with step size h=0.025. The tableau above yields the equivalent corresponding equations below defining the method: k1 = yn t0 = 1 y0 = 1 t1 = 1.025 k1 = y0 = 1 f(t0,k1) = 2.557407725 k2 = y0 + 2 / 3hf(t0,k1) = 1.042623462 y1 = y0 + h(1 / 4 f(t0,k1) + 3 / 4 f(t0 + 2 / 3h,k2)) = 1.066869388 t2 = 1.05 k1 = y1 = 1.066869388 f(t1,k1) = 2.813524695 k2 = y1 + 2 / 3hf(t1,k1) = 1.113761467 y2 = y1 + h(1 / 4 f(t1,k1) + 3 / 4 f(t1 + 2 / 3h,k2)) = 1.141332181 t3 = 1.075 k1 = y2 = 1.141332181 f(t2,k1) = 3.183536647 k2 = y2 + 2 / 3hf(t2,k1) = 1.194391125 y3 = y2 + h(1 / 4 f(t2,k1) + 3 / 4 f(t2 + 2 / 3h,k2)) = 1.227417567 t4 = 1.1 k1 = y3 = 1.227417567 f(t3,k1) = 3.796866512 k2 = y3 + 2 / 3hf(t3,k1) = 1.290698676 y4 = y3 + h(1 / 4 f(t3,k1) + 3 / 4 f(t3 + 2 / 3h,k2)) = 1.335079087 The numerical solutions correspond to the underlined values. Note that f(ti,k1) has been calculated to avoid recalculation in the yis. Adaptive Runge-Kutta Methods The adaptive methods are designed to produce an estimate of the local truncation error of a single Runge-Kutta step. This is done by having two methods in the tableau, one with order p and one with order p 1. The lower-order step is given by where the ki are the same as for the higher order method. Then the error is which is O(hp). The Butcher Tableau for this kind of method is extended to give the values of : 0 c2 a21 c3 a31 a32 cs as1 as2 as,s 1 b1 b2 bs 1 bs The Runge-Kutta-Fehlberg method has two methods of orders 5 and 4. Its extended Butcher Tableau is: 0 1/4 1/4 3/8 3/32 9/32 12/13 1932/2197 7200/2197 7296/2197 1 439/216 8 3680/513 -845/4104 1/2 8/27 2 3544/2565 1859/4104 11/40 16/135 0 6656/12825 28561/56430 9/50 2/55 25/216 0 1408/2565 2197/4104 1/5 0 However, the simplest adaptive Runge-Kutta method involves combining the Heun method, which is order 2, with the Euler method, which is order 1. Its extended Butcher Tableau is: 0 1 1 1/2 1/2 1 0 The error estimate is used to control the stepsize. Other adaptive Runge-Kutta methods are the Bogacki-Shampine method (orders 3 and 2), the Cash-Karp method and the Dormand-Prince method (both with orders 5 and 4). Implicit Runge-Kutta Methods The implicit methods are more general than the explicit ones. The distinction shows up in the Butcher Tableau: for an implicit method, the coefficient matrix aij is not necessarily lower triangular: The approximate solution to the initial value problem reflects the greater number of coefficients: Due to the fullness of the matrix aij, the evaluation of each ki is now considerably involved and dependent on the specific function f(t,y). Despite the difficulties, implicit methods are of great importance due to their high (possibly unconditional) stability, which is especially important in the solution of partial differential equations. The simplest example of an implicit Runge-Kutta method is the backward Euler method: The Butcher Tableau for this is simply: It can be difficult to make sense of even this simple implicit method, as seen from the expression for k1: In this case, the awkward expression above can be simplified by noting that so that from which follows. Though simpler then the raw representation before manipulation, this is an implicit relation so that the actual solution is problem dependent. Multistep implicit methods have been used with success by some researchers. The combination of stability, higher order accuracy with fewer steps, and stepping that depends only on the previous value makes them attractive; however the complicated problem-specific implementation and the fact that ki must often be approximated iteratively means that they are not common. References J. C. Butcher, Numerical methods for ordinary differential equations, ISBN 0471967580 George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler. Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977. (See Chapter 6.) Ernst Hairer, Syvert Paul Nrsett, and Gerhard Wanner. Solving ordinary differential equations I: Nonstiff problems, second edition. Berlin: Springer Verlag, 1993. ISBN 3-540-56670-8. William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Sections 16.1 and 16.2.) Kaw, Autar; Kalu, Egwu (2008), Numerical Methods with Applications (1st ed.), www.autarkaw.com. Kendall E. Atkinson. An Introduction to Numerical Analysis. John Wiley Sons 1989 F. Cellier, E. Kofman. Continuous System Simulation. Springer Verlag, 2006. ISBN 0-387-26102-8. External links Runge-Kutta Runge-Kutta 4th Order Method Runge Kutta Method for O.D.E.s Numerical integration First order methods Euler method Backward Euler Semi-implicit Euler Exponential Euler Second order methods Verlet integration Velocity Verlet Crank-Nicolson method Beemans algorithm Midpoint method Heuns method Newmark-beta method Leapfrog integration Higher order methods Runge-Kutta methods List of Runge-Kutta methods Linear multistep method Retrieved from https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods Categories: Numerical differential equations | Runge-Kutta methods This page was last modified on 28 November 2009 at 11:21. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. See Terms of Use for details. Wikipedia is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Contact us Privacy policy About Wikipedia Disclaimers Higher Order Taylor Methods Marcelo Julio Alvisio Lisa Marie Danz May 16, 2007 Introduction Differential equations are one of the building blocks in science or engineering. Scientists aim to obtain numerical solutions to differential equations whenever explicit solutions do not exist or when they are too hard to find. These numerical solutions are approximated though a variety of methods, some of which we set out to explore in this project. We require two conditions when computing differential equations numerically. First, we require that the solution is continuous with initial value. Otherwise, numerical error introduced in the representation of the number in computer systems would produce results very far from the actual solution. Second, we require that the solution changes continuously with respect to the differential equation itself. Otherwise, we cannot expect the method that approximates the differential equation to give accurate results. The most common methods for computing differential equations numerically include Eulers method, Higher Order Taylor method and Runge-Kutta methods. In this project, we concentrate on the Higher Order Taylor Method. This method employs the Taylor polynomial of the solution to the equation. It approximates the zeroth order term by using the previous steps value (which is the initial condition for the first step), and the subsequent terms of the Taylor expansion by using the differential equation. We call it Higher Order Taylor Method, the lower order method being Eulers Method. Under certain conditions, the Higher Order Taylor Method limits the error to O(hn), where n is the order used. We will present several examples to test this idea. We will look into two main parameters as a measure of the effectiveness of the method, namely accuracy and efficiency. Theory of the Higher Order Taylor Method Definition 2.1 Consider the differential equation given by y0(t)= f(t,y), y(a)= c. Then for ba, the nth order Taylor approximation to y(b) with K steps is given by yK, where {yi} is defined recursively as: t0 = a y0 = y(a)= c ti+1 = ti + h h2 f hn n1f yi+1 = yi + hf(ti,yi)+ (ti,yi)+ +(ti,yi) 2 t n! tn1 with h =(b a)/K. It makes sense to formulate such a definition in view of the Taylor series expansion that is used when y(t) is known explicitly. All we have done is use f(t,y) for y0(t), ft(t,y) for y00(t), and so forth. The next task is to estimate the error that this approximation introduces. We know by Taylors Theorem that, for any solution that admits a Taylor expansion at the point ti, we have h2 hn h(n+1) y(ti+1)= y(ti)+ hy0(ti)+ y00(ti)+ + y(n)(ti)+ y(n+1)() 2 n!(n + 1)! where is between ti and ti+1 Using y0 = f(t,y), this translates to h2 f hn (n1)fh(n+1) (n)f y(ti+1)= y(ti)+hf(ti,yi)+ (ti,yi)++(ti,yi)+ (,y()) 2 t n! t(n1) (n + 1)! t(n) Therefore, the local error, that is to say, the error introduced at each step if the values calculated previously were exact, is given by: 1 (n)f Ei =(hn+1)(,y()) (n + 1)! tn which means that 1 (n)f max (hn+1)(,y()) Ei [a,b] (n + 1)! tn 23 We can say Ei = O(hn+1). Now, since the number of steps from a to b is proportional to 1/h, we multiply the error per step by the number of steps to find a total error E = O(hn). In Practice: Examples We will consider differential equations that we can solve explicitly to obtain an equation for y(t) such that y0(t)= f(t,y). This way, we can calculate the actual error by subtracting the exact value for y(b) from the value that the Higher Order Taylor method predicts for it. To approximate values in the following examples, the derivatives of f(t,y) were computed by hand. MATLAB then performed the iteration and arrived at the approximation. Notice that the definitions given in the previous section could also have been adapted for varying step size h. However, for ease of computation we have kept the step size constant. In our computations, we have chosen step size of (b a)/2k, which resulted in K =2k evenly spaced points in the interval. Example 3.1 We consider the differential equation 1+ t y0(t)= f(t,y)= 1+ y with initial condition y(1) = 2. It is clear that y(t)= t2 +2t +6 1 solves this equation. Thus we calculate the error for y(2) by subtracting the approximation of y(2) from y(2), which is the exact value. Recall that we are using h =2k because (b a)=1. The following table displays the errors calculated. k = 1 k = 2 k = 3 k = 4 order = 1 .0333 .0158 .0077 .0038 order = 2 .0038 .0009 .0002 .0001 order = 3 .0003269 .0000383 .0000046 .0000006 Runge-Kutta Methods The Taylor methods in the preceding section have the desirable feature that the F.G.E. is of order O(hN ), and N can be chosen large so that this error is small. However, the shortcomings of the Taylor methods are the a priori determination of N and the computation of the higher derivatives, which can be very complicated. Each Runge-Kutta method is derived from an appropriate Taylor method in such a way that the F.G.E. is of order O(hN ). A trade-off is made to perform several function evaluations at each step and eliminate the necessity to compute the higher derivatives. These methods can be constructed for any order N. The Runge-Kutta method of order N = 4 is most popular. It is a good choice for common purposes because it is quite accurate, stable, and easy to program. Most authorities proclaim that it is not necessary to go to a higher-order method because the increased accuracy is offset by additional computational effort. If more accuracy is required, then either a smaller step size or an adaptive method should be used. The fourth-order Runge-Kutta method (RK4) simulates the accuracy of the Taylor series method of order N = 4. The method is based on computing yk+1 as follows: (1) yk+1 = yk + w1k1 + w2k2 + w3k3 + w4k4, where k1, k2, k3, and k4 have the form (2) k1 = h f (tk , yk ), k2 = h f (tk + a1h, yk + b1k1), k3 = h f (tk + a2h, yk + b2k1 + b3k2), k4 = h f (tk + a3h, yk + b4k1 + b5k2 + b6k3). By matching coefficients with those of the Taylor series method of order N = 4 so that the local truncation error is of order O(h5), Runge and Kutta were able to obtain the 490 CHAP. 9 SOLUTION OF DIFFERENTIAL EQUATIONS following system of equations: (3) b1 = a1, b2 + b3 = a2, b4 + b5 + b6 = a3, w1 + w2 + w3 + w4 = 1, w2a1 + w3a2 + w4a3 = 1 2, w2a2 1 + w3a2 2 + w4a2 3 = 1 3 , w2a3 1 + w3a3 2 + w4a3 3 = 1 4 , w3a1b3 + w4(a1b5 + a2b6) = 1 6 , w3a1a2b3 + w4a3(a1b5 + a2b6) = 1 8 , w3a2 1b3 + w4(a2 1b5 + a2 2b6) = 1 12 , w4a1b3b6 = 1 24 The system involves 11 equations in 13 unknowns. Two additional conditions must be supplied to solve the system. The most useful choice is (4) a1 = 1 2 and b2 = 0. Then the solution for the remaining variables is (5) a2 = 1 2 , a3 = 1, b1 = 1 2 , b3 = 1 2 , b4 = 0, b5 = 0, b6 = 1, w1 = 1 6 , w2 = 1 3 , w3 = 1 3 , w4 = 1 6 The values in (4) and (5) are substituted into (2) and (1) to obtain the formula for the standard Runge-Kutta method of order N = 4, which is stated as follows. Start with the initial point (t0, y0) and generate the sequence of approximations using (6) yk+1 = yk + h( f1 + 2 f2 + 2 f3 + f4) 6 , SEC. 9.5 RUNGE-KUTTA METHODS 491 where (7) f1 = f (tk , yk ), f2 = f tk + h 2 , yk + h 2 f1 , f3 = f tk + h 2 , yk + h 2 f2 , f4 = f (tk + h, yk + h f3). Discussion about the Method The complete development of the equations in (7) is beyond the scope of this book and can be found in advanced texts, but we can get some insights. Consider the graph of the solution curve y = y(t) over the first subinterval [t0, t1]. The function values in (7) are approximations for slopes to this curve. Here f1 is the slope at the left, f2 and f3 are two estimates for the slope in the middle, and f4 is the slope at the right (a)). The next point (t1, y1) is obtained by integrating the slope function (8) y(t1) y(t0) = _ t1 t0 f (t, y(t)) dt. If Simpsons rule is applied with step size h/2, the approximation to the integral in (8) is (9) _ t1 t0 f (t, y(t)) dt h 6 ( f (t0, y(t0)) + 4 f (t1/2, y(t1/2)) + f (t1, y(t1))), where t1/2 is the midpoint of the interval. Three function values are needed; hence we make the obvious choice f (t0, y (t0)) = f1 and f (t1, y(t1)) f4. For the value in the middle we chose the average of f2 and f3: f (t1/2, y(t1/2)) f2 + f3 2 . These values are substituted into (9), which is used in equation (8) to get y1: (10) y1 = y0 + h 6 f1 + 4( f2 + f3) 2 + f4 . When this formula is simplified, it is seen to be equation (6) with k = 0. The graph for the integral in (9) is shown in Figure 9.9(b). 492 CHAP. 9 SOLUTION OF DIFFERENTIAL EQUATIONS y t m1 = f1 m2 = f3 m3 = f4 m4 = f4 (t0, y0) y = y(t) (t1, y(t1)) t0 t1/2 t1 (a) Predicted slopes mj to the solution curve y = y(t) z t (t0, f1) (t1/2, f2) (t1/2, f3) (t1, f4) t0 t1/2 t1 (b) Integral approximation: h 6 y(t1) y0 = ( f1 + 2f2 + 2f3 + f4) Figure 9.9 The graphs y = y(t) and z = f (t, y(t)) in the discussion of the Runge-Kutta method of order N = 4. Step Size versus Error The error term for Simpsons rule with step size h/2 is (11) y(4)(c1) h5 2880 . If the only error at each step is that given in (11), after M steps the accumulated error for the RK4 method would be (12) _M k=1 y(4)(ck) h5 2880 b a 5760 y(4)(c)h4 O(h4). The next theorem states the relationship between F.G.E. and step size. It is used to give us an idea of how much computing effort must be done when using the RK4 method. Theorem 9.7 (Precision of the Runge-Kutta Method). Assume that y(t) is the solution to the I.V.P. If y(t) C5[t0, b] and {(tk , yk)}M k=0 is the sequence of approximations generated by the Runge-Kutta method of order 4, then (13) |ek| = |y(tk ) yk| = O(h4), |_k+1| = |y(tk+1) yk hTN (tk , yk)| = O(h5). SEC. 9.5 RUNGE-KUTTA METHODS 493 In particular, the F.G.E. at the end of the interval will satisfy (14) E(y(b), h) = |y(b) yM| = O(h4). Examples 9.10 and 9.11 illustrate Theorem 9.7. If approximations are computed using the step sizes h and h/2, we should have (15) E(y(b), h) Ch4 for the larger step size, and (16) E y(b), h 2 C h4 16 = 1 16 Ch4 1 16 E(y(b), h). Hence the idea in Theorem 9.7 is that if the step size in the RK4 method is reduced by a factor of 12 we can expect that the overall F.G.E. will be reduced by a factor of 1. Example 9.10. Use the RK4 method to solve the I.V.P. y_ = (t y)/2 on [0, 3] with y(0) = 1. Compare solutions for h = 1, 12 , 14 , and 18 . Table 9.8 gives the solution values at selected abscissas. For the step size h = 0.25, a sample calculation is f1 = 0.0 1.0 2 = 0.5, f2 = 0.125 (1 + 0.25(0.5)(0.5)) 2 = 0.40625, f3 = 0.125 (1 + 0.25(0.5)(0.40625)) 2 = 0.4121094, f4 = 0.25 (1 + 0.25(0.4121094)) 2 = 0.3234863, y1 = 1.0 + 0.25 0.5 + 2(0.40625) + 2(0.4121094) 0.3234863 6 = 0.8974915. _ Example 9.11. Compare the F.G.E. when the RK4 method is used to solve y_ = (ty)/2 over [0, 3] with y(0) = 1 using step sizes 1, 12 , 14 , and 18 Table 9.9 gives the F.G.E. for the various step sizes and shows that the error in the approximation to y(3) decreases by about 1 16 when the step size is reduced by a factor of 1/2. E(y(3), h) = y(3) yM = O(h4) Ch4 where C = 0.000614. _ A comparison of Examples 9.10 and 9.11 and Examples 9.8 and 9.9 shows what is meant by the statement The RK4 method simulates the Taylor series method of order N = 4. For these examples, the two methods generate identical solution sets {(tk , yk)} 494 CHAP. 9 SOLUTION OF DIFFERENTIAL EQUATIONS Table 9.8 Comparison of the RK4 Solutions with Different Step Sizes for y_ = (t y)/2 over [0, 3] with y(0) = 1 yk tk h = 1 h = 12 h = 14 h = 18 y(tk ) Exact 0 1.0 1.0 1.0 1.0 1.0 0.125 0.9432392 0.9432392 0.25 0.8974915 0.8974908 0.8974917 0.375 0.8620874 0.8620874 0.50 0.8364258 0.8364037 0.8364024 0.8364023 0.75 0.8118696 0.8118679 0.8118678 1.00 0.8203125 0.8196285 0.8195940 0.8195921 0.8195920 1.50 0.9171423 0.9171021 0.9170998 0.9170997 2.00 1.1045125 1.1036826 1.1036408 1.1036385 1.1036383 2.50 1.3595575 1.3595168 1.3595145 1.3595144 3.00 1.6701860 1.6694308 1.6693928 1.6693906 1.6693905 Table 9.9 Relation between Step Size and F.G.E. for the RK4 Solutions to y_ = (t y)/2 over [0, 3] with y(0) = 1 Step size, h Number of steps, M Approximation to y(3), yM F.G.E. Error at t = 3, y(3) yM O(h4) Ch4 where C = 0.000614 1 3 1.6701860 0.0007955 0.0006140 12 6 1.6694308 0.0000403 0.0000384 14 12 1.6693928 0.0000023 0.0000024 18 24 1.6693906 0.0000001 0.0000001 over the given interval. The advantage of the RK4 method is obvious; no formulas for the higher derivatives need to be computed nor do they have to be in the program. It is not easy to determine the accuracy to which a Runge-Kutta solution has been computed. We could estimate the size of y(4)(c) and use formula (12). Another way is to repeat the algorithm using a smaller step size and compare results. A third way is to adaptively determine the step size, which is done in Program 9.5. In Section 9.6 we will see how to change the step size for a multistep method. SEC. 9.5 RUNGE-KUTTA METHODS 495 Runge-Kutta Methods of Order N = 2 The second-order Runge-Kutta method (denoted RK2) simulates the accuracy of the Taylor series method of order 2. Although this method is not as good to use as the RK4 method, its proof is easier to understand and illustrates the principles involved. To start, we write down the Taylor series formula for y(t + h): (17) y(t + h) = y(t) + hy_ (t) + 1 2 h2 y__ (t) + CT h3 + , where CT is a constant involving the third derivative of y(t) and the other terms in the series involve powers of h j for j 3. The derivatives y_ (t) and y__ (t) in equation (17) must be expressed in terms of f (t, y) and its partial derivatives. Recall that (18) y_ (t) = f (t, y). The chain rule for differentiating a function of two variables can be used to differentiate (18) with respect to t, and the result is y__ (t) = ft (t, y) + fy(t, y)y_ (t). Using (18), this can be written (19) y__ (t) = ft (t, y) + fy(t, y) f (t, y). The derivatives (18) and (19) are substituted in (17) to give the Taylor expression for y(t + h): y(t + h) = y(t) + h f (t, y) + 1 2 h2 ft (t, y) + 1 2 h2 fy(t, y) f (t, y) + CT h3 + . (20) Now consider the Runge-Kutta method of order N = 2, which uses a linear combination of two function values to express y(t + h): (21) y(t + h) = y(t) + Ah f0 + Bhf1, where (22) f0 = f (t, y), f1 = f (t + Ph, y + Qhf0). Next the Taylor polynomial approximation for a function of two independent variables is used to expand f (t, y) (see the Exercises). This gives the following representation for f1: (23) f1 = f (t, y) + Phft (t, y) + Qhfy(t, y) f (t, y) + CPh2 + , 496 CHAP. 9 SOLUTION OF DIFFERENTIAL EQUATIONS where CP involves the second-order partial derivatives of f (t, y). Then (23) is used in (21) to get the RK2 expression for y(t + h): y(t + h) = y(t) + (A + B)h f (t, y) + BPh2 ft (t, y) + BQh2 fy(t, y) f (t, y) + BCPh3 + . (24) A comparison of similar terms in equations (20) and (24) will produce the following conclusions: h f (t, y) = (A + B)h f (t, y) implies that 1 = A + B, 1 2 h2 ft (t, y) = BPh2 ft (t, y) implies that 1 2 = BP, 1 2 h2 fy(t, y) f (t, y) = BQh2 fy(t, y) f (t, y) implies that 1 2 = BQ. Hence, if we require that A, B, P, and Q satisfy the relations (25) A + B = 1 BP = 1 2 BQ = 1 2 , then the RK2 method in (24) will have the same order of accuracy as the Taylors method in (20). Since there are only three equations in four unknowns, the system of equations (25) is underdetermined, and we are permitted to choose one of the coefficients. There are several special choices that have been studied in the literature; we mention two of them. Case (i): Choose A = 12 . This choice leads to B = 12 , P = 1, and Q = 1. If equation (21) is written with these parameters, the formula is (26) y(t + h) = y(t) + h 2 ( f (t, y) + f (t + h, y + h f (t, y))). When this scheme is used to generate {(tk , yk)}, the result is Heuns method. Case (ii): Choose A = 0. This choice leads to B = 1, P = 12 , and Q = 12 . If equation (21) is written with these parameters, the formula is (27) y(t + h) = y(t) + h f t + h 2 , y + h 2 f (t, y) . When this scheme is used to generate {(tk , yk)}, it is called the modified Euler-Cauchy method. Numerical Methods Using Matlab, 4th Edition, 2004 John H. Mathews and Kurtis K. Fink ISBN: 0-13-065248-2 Prentice-Hall Inc. Upper Saddle River, New Jersey, USA https://vig.prenhall.com/ Deriving the Runge-Kutta Method Deriving the midpoint method The Taylor method is the gold standard for generating better numerical solutions to first order differential equations. A serious weakness in the Taylor method, however, is the need to compute a large number of partial derivatives and do other symbolic manipulation tasks. For example, the second order Taylor method for the equation y( t) = f(t,y(t)) is yi+1 = yi + h f(ti ,yi ) + h2 2 f t ( ti ,yi ) + f ( ti ,yi ) f y ( ti ,yi ) Higher order formulas get even uglier. The Midpoint method arises from an attempt to replace the second order Taylor method with a simpler Euler-like formula yi+1 = yi + h f(ti + ,yi + ) We can solve for the best values for and by applying a first order Taylor expansion to the term f(ti + ,yi + ): yi+1 = yi + h f ( ti ,yi ) + f t ( ti ,yi ) + f y ( ti ,yi ) + 2f t y ( ti ,yi ) The choices of and that make this look as close as possible to the second order Taylor formula above are = h2 = h2 f(ti ,yi ) leading to the so-called midpoint rule: 1 yi+1 = yi + h f(ti + h2 ,yi + h2 f(ti ,yi)) This formula has a simple interpretation. Essentially what we are doing here is driving an Euler estimate half way across the interval [ti , ti+1] and computing the slope f(ti + h2 ,yi + h2 f(ti ,yi)) at that midpoint. We then rewind back to the point ( ti ,yi ) and drive an Euler estimate all the way across the interval to ti+1 using this new midpoint slope in place of the old Euler slope. The Runge-Kutta Method The textbook points out that it is possible to derive similar methods by starting with more complex Euler-like formulas with more free parameters and then trying to match those Euler-like methods to higher order Taylor formulas. The Runge-Kutta method is essentially an attempt to match a more complex Euler-like formula to a fourth order Taylor method. The problem with this is that the Euler-like formula needed to match all the complexity of the fourth order Taylor method formula is quite complex. The textbook states in exercise 31 at the end of section 5.4 that the formula required is yi+1 = yi + h6 f(ti ,yi ) + h3 f(ti + 1 h,yi + 1 h f(ti ,yi)) + h3 f (ti + 2 h,yi + 2 h f (ti + 2 h, yi + 3 h f ( ti ,yi ))) + h6 f (ti + 3 h, yi + 3 h f (ti + 4 h, yi + 5 h f ( ti + 6 h,yi+ 7 h f ( ti ,yi)))) It is very messy to do so, but this form can expanded out and matched against the Taylor formula of order four. This allows us to solve for all the unknown coefficients. A somewhat cleaner alternative derivation is based on the following argument. Another way to solve for yi+1 is to compute this integral !t i+1 t i y( t) dt = y(ti+1) y(ti ) = yi+1 yi We can imagine beginning to compute the integral by noting that y( t) = f(t,y(t)) !t i+1 t i y( t) dt = !t i+1 t i f ( t,y( t)) dt 2 Unfortunately, we can not do the integral on the right hand side exactly, because we dont know what y(t) is. That is, after all, the unknown we are trying to solve for. Even though we cant compute the integral on the right exactly, we can estimate it. For example, applying Simpsons rule to the integral produces the estimate !t i+1 t i f ( t,y( t)) dt h3 f ( ti ,y( ti))+4 f ti+ti+1 2 ,y ti+ti+1 2 + f ( ti+1,y( ti+1)) The Runge-Kutta method takes this estimate as a starting point. The thing we need to do to make this estimate work is to find a way to estimate the unknown terms y((ti + ti+1) /2) and y(ti+1) . The first step is to rewrite the estimate as h3 f ( ti ,y( ti))+2 f ti+ti+1 2 ,y ti+ti+1 2 +2 f ti+ti+1 2 ,y ti+ti+1 2 + f ( ti+1,y( ti+1)) We write the middle term twice because we are going to develop two different estimates for y((ti + ti+1) /2). The thinking is that the mistakes we make in developing those two interior estimates may partly cancel each other out. Here is how we will develop our estimates. 1. y(ti ) is just yi . We estimate the first y((ti + ti+1) /2) by driving the original Euler slope k1 = f(ti ,yi ) half-way across the interval: 2. k1 = f(ti ,y(ti)) y ti + ti+1 2 yi + h/2 k1 As in the midpoint rule, we compute a second slope at that midpoint we just estimated. We then rewind to the start and drive that slope half-way across the interval again. 3. k2 = f(ti + h/2,yi + h/2 k1 ) y ti + ti+1 2 yi + h/2 k2 We use the second estimated midpoint to compute another slope and then drive that slope all the way across the interval. 4. 3 We use the second estimated midpoint to compute another slope and then drive that slope all the way across the interval. 4. k3 = f(ti + h/2,yi + h/2 k2 ) y(ti+1) = yi + h k3 k4 = f(ti + h,yi + h k3 ) Substituting all of these estimates into the Simpsons rule formula above gives yi+1 yi = !t i+1 t i f ( t,y( t)) dt h3 f ( ti ,y( ti))+ 2 f ti+ti+1 2 ,y ti+ti+1 2 +2 f ti+ti+1 2 ,y ti+ti+1 2 + f ( ti+1,y( ti+1)) or yi+1 = yi + h3 (k1 + 2 k2 + 2 k3 + k4 ) Summary Of The Method k1 = f(ti ,yi ) k2 = f(ti + h/2,yi + h/2 k1 ) k3 = f(ti + h/2,yi + h/2 k2 ) k4 = f(ti + h,yi + h k3 ) yi+1 = yi + h3(k1 + 2 k2 + 2 k3 + k4 ) 4 Taylor Series Methods: To derive these methods we start with a Taylor Expansion: y(t+_t) _ y(t) + _ty0(t) + 1 2 _t2y00(t) + + 1 r! y(r)(t)_tr. Lets say we want to truncate this at the second derivative and base a method on that. The scheme is, then: yn+1 = yn + fn_t + f0 tn 2 _t2. The Taylor series method can be written as yn+1 = yn +_tF (tn, yn,_t) where F = f + 1 2_tf0. If we take the LTE for this scheme, we get (as expected) LTE(t) = y(tn +_t) y(tn) _t f(tn, y(tn)) 1 2 _tf0(tn, y(tn)) = O(_t2). Of course, we designed this method to give us this order, so it shouldnt be a surprise! So the LTE is reasonable, but what about the global error? Just as in the Euler Forward case, we can show that the global error is of the same order as the LTE. How do we do this? We have two facts, y(tn+1) = y(tn) + _tF (tn, y(tn),_t), and yn+1 = yn +_tF (tn, yn,_t) where F = f + 1 2_tf0. Now we subtract these two |y(tn+1) yn+1| = |y(tn) yn +_t (F(tn, y(tn)) F(tn, yn)) + _tLTE| _ |y(tn) yn|+_t |F(tn, y(tn)) F(tn, yn)|+_t|LTE| . Now, if F is Lipschitz continuous, we can say en+1 _ (1 + _tL)en+_t|LTE|. Of course, this is the same proof as for Eulers method, except that now we are looking at F, not f, and the LTE is of higher order. We can do this no matter which Taylor series method we use, how many terms we go forward before we truncate. Advantages And Disadvantages Of The Taylor Series Method: advantages a) One step, explicit b) can be high order c) easy to show that global error is the same order as LTE disadvantages Needs the explicit form of derivatives of f. 4 Runge-Kutta Methods To avoid the disadvantage of the Taylor series method, we can use Runge-Kutta methods. These are still one step methods, but they depend on estimates of the solution at different points. They are written out so that they dont look messy: Second Order Runge-Kutta Methods: k1 = _tf(ti, yi) k2 = _tf(ti + __t, yi + _k1) yi+1 = yi + ak1 + bk2 lets see how we can chose the parameters a,b, _, _ so that this method has the highest order LTE possible. Take the Taylor expansions to express the LTE: k1(t) = _tf(t, y(t)) k2(t) = _tf(t + __t, y + _k1(t) = _t _ f(t, y(t) + ft(t, y(t))__t+ fy(t, y(t))_k1(t) + O(_t2) _ LTE(t) = y(t+_t) y(t) _t a _t f(t, y(t))_t b _t (ft(t, y(t))__t+ fy(t, y(t)_k1(t) + f(t, y(t))_t + O(_t2) = y(t+_t) y(t) _t af(t, y(t)) bf(t, y(t)) bft(t, y(t))_ bfy(t, y(t)_f(t, y(t))+ O(_t2) = y0(t) + 1 2 _ty00(t) (a + b)f(t, y(t)) _t(b_ft(t, y(t))+ b_f(t, y(t))fy(t, y(t)) + O(_t2) = (1 a b)f + ( 1 2 b_)_tft + ( 1 2 b_)_tfyf + O(_t2) So we want a = 1 b, _ = _ = 1 2b . Fourth Order Runge-Kutta Methods: k1 = _tf(ti, yi) (1.3) k2 = _tf(ti + 1 2 _t, yi + 1 2 k1) (1.4) k3 = _tf(ti + 1 2 _t, yi + 1 2 k2) (1.5) k4 = _tf(ti+_t, yi + k3) (1.6) yi+1 = yi + 1 6 (k1 + k2 + k3 + k4) (1.7) The second order method requires 2 evaluations of f at every timestep, the fourth order method requires 4 evaluations of f at every timestep. In general: For an rth order Runge- Kutta method we need S(r) evaluations of f for each timestep, where S(r) = 8 : r for r _ 4 r + 1 for r = 5 and r = 6 _ r + 2 for r _ 7 5 Practically speaking, people stop at r = 5. Advantages of Runge-Kutta Methods 1. One step method global error is of the same order as local error. 2. Dont need to know derivatives of f. 3. Easy for Automatic Error Control. Automatic Error Control Uniform grid spacing in this case, time steps are good for some cases but not always. Sometimes we deal with problems where varying the gridsize makes sense. How do you know when to change the stepsize? If we have an rth order scheme and and r + 1th order scheme, we can take the difference between these two to be the error in the scheme, and make the stepsize smaller if we prefer a smaller error, or larger if we can tolerate a larger error. For Automatic error control yo are computing a useless (r+1)th order shceme . . . what a waste! But with Runge Kutta we can take a fifth order method and a fourth order method, using the same ks. only a little extra work at each step.

Heres What I Know About Essay Topics for Upsc Mains

Here's What I Know About Essay Topics for Upsc Mains The Essay Topics for Upsc Mains Stories It's to be mentioned that the CSAT exam is a qualifying paper and the marks won't be added for the last ranking. The CSAT examination is easily cracked when you have prepared for banking exams. Prelims are a gateway to the most important exam. The candidates who score over the cut-off in the very first stage (Prelims) of the IAS exam is only going to be qualified for the Mains. It is worth it to read great magazines. It is a rather excellent book and gives number of topics that are important for mains 2018. A typical dictionary for reference needs to be kept handy while preparation. Look all what content you'll be able to recall. Even then you're unable to discover the informative and accurate details. It is imperative to return to the answers you may have prepared. Click the ORDER NOW button and complete the form. Start your UPSC exam preparation by going through the various sets of earlier years questions. This can only be reached through a dependable and quality support. It will get clear about what things to study and what not. You also have to take mock tests on a normal basis. If you do this, you are not going to get the wanted marks. This title will get someone who wishes to learn. Make sure you have plenty of time for revision. Give sufficient time for revision too. The majority of the essay topics are picked form the present affairs. It is all about putting into beautiful words, our thoughts and ideas. Essay writing strategies for upsc mains essay ought to be organized in well-structured paragraphs coherent with the stream of the essay. Spontaneous Essay Spontaneous essay doesn't involve brainstorming in the start. Great introduction and excellent conclusion are must. Joy is the easiest type of Gratitude 5. Empowerment alone can't help our women. Poverty anywhere is a danger to prosperity everywhere 4. Issues regarding poverty and hunger. Candidate must compose an essay on a particular topic. Candidates will have to compose an essay on a particular topic. They may be required to write essays on multiple topics. Remember it's a General Essay Paper and you shouldn't be highly technical. Justice must get to the poor 2. On-line price is a little high. Our site provides custom writing help and editing aid. By keeping our writing at the greatest possible level we've achieved a high rate of consumer retention. If you've got to use the very same word at various places, replace the typical word with a new word meaning the exact same. Fortunately our team is made up of professional writers which possess the capability to create remarkable content for you. Analyze your role and duties, together with an ideal plan of action. Consequently, value your time and take pleasure in the training. Fundamentally, an essay is intended to receive your academic opinion on a specific issue. But unfortunately assignments aren't confined to limited ideas you have to do lots of research that is a hectic undertaking. This is definitely the most important factor in regards to essays.

Scarlet Letter Final Exam Expressions of the Transcendentalists Free Essays

Victoria Clark Scarlet Letter Final Exam: Expressions of the Transcendentalists â€Å"Nobody knows this little Rose† by Emily Dickinson expresses how important a rose actually is to its environment and without the rose being of existence will affect the objects that are close to it. Dickinson goes onto say what is affected by the loss of the rose. Also in The Scarlet Letter by Nathaniel Hawthorne he emphasizes how when a situation alters that there is a different way of life that comes with it. We will write a custom essay sample on Scarlet Letter Final Exam: Expressions of the Transcendentalists or any similar topic only for you Order Now Emily Dickinson and Hawthorne use change within an entity to utilize how it can affect the things closest to it. Dickinson uses a rose to express herself,†Nobody knows this little Rose†, to convey how important the rose is be to its environment when it dies. Dickinson says,7 ‘â€Å"Only a bee will miss it†Ã¢â‚¬â„¢ (Dickinson line 5), this means that when the rose should die that the bee will not have somewhere to land to reap the pollen from the rose. Hawthorne uses a black flower to emphasize what is growing upon Chillingworth’s heart. Hawthorne writes,†Ã¢â‚¬â„¢ Let the black flower blossom as it may†Ã¢â‚¬â„¢ (119). The† black flower blossoming† is used to also indicate the evil growing upon Chillingworth’s heart and how it has an affect on the way Chillingworth’s deformity. The authors both use the colors red and black to create an image in the readers mind so that they understand what the colors red and black mean. The image that the red rose puts an image of love in some minds or how miserable the bee might be after the departure of the rose. The word usage that Hawthorne uses to describe the black rose gives the image of death, and the black flower that that was growing over Chillingworth’s heart would be the one that kills him. Dickinson goes on to say â€Å"’ Only a Bird will wonder’†(Dickinson line 9), this line represents how if the bird uses the rose to indicate where food is, it will not be able to get food, and will wonder where the rose is and will have to find a new place to gather food. Hawthorne writes â€Å"’I will keep my secret, as I have this†Ã¢â‚¬â„¢ (53). As Hester and Chillingworth are talking to each other about who her child, Pearl’s father actually is, Chillingworth tells her to keep their connection a secret, along with the secret his true identity. Along with the other secrets, Hester makes a vow to herself that she will never tell anyone who Pearl’s father is. In addition, Hester keeping this secret throughout the book brings a burden upon her heart, as well as Dimmsdale’s. This colossal secret has makes Dimmsdale start to fast and beat himself with a scourge that is hidden in his closet. This change within Hester and Dimmsdale’s minds is affecting Pearl, Dimmsdale, Chillingworth and Hester’s lives. Dickinson and Hawthorne demonstrate how one minute secret or object’s can change from the life of something or someone that is very close to it. Hawthorne says,†Ã¢â‚¬ËœSo speaking she undid the clasp that fastened the scarlet letter, and taking it from her bosom threw to a distance among the withered leaves’†(Hawthorne 138). As Pearl has gone off to play she dress herself up in leaves and makes a scarlet letter of her own and has placed it upon her bosom. While Dimmsdale and Hester discuss their lives, Hester has a sudden outburst of self-assurance and wants to give up the scarlet letter to be free from the bondage it has brought upon her and her relationship with her child, Pearl. Hawthorne goes on to say,†Ã¢â‚¬â„¢ Pearl’, look down at thy feet! There! – Before thee! – on the hither side of the brook! ’†¦ Bring it hither! †¦ Swallow it up for ever! †Ã¢â‚¬â„¢(144). With the scarlet letter being off of Hester’s bosom Pearl does not recognize who Hester is, she sees her as if she is a stranger because Hester has had the scarlet letter on her bosom since Pearl can remember. Pearl also thinks that the scarlet letter is a good thing and that it is beautiful- Pearl wants a scarlet letter of her own. With the Scarlet letter being off of heater’s Bosom Pearl cannot accept the change that her mother has made. Dickinson says,†Ã¢â‚¬ËœOnly a Breeze will sigh†Ã¢â‚¬â„¢(Dickinson line 10) along with the other vital thing that the rose needs to survive the rose is also having an affect on the breeze. The breeze does not have anything to bump against anymore since the rose is not in the spot it was in before when it blew by. Hawthorne and Dickinson use these examples to show that when something or someone is use to seeing or feeling something a certain way; that when it changes they may or may not recognize the difference that has occurred with , in this case, the rose being missing from the breeze’s path and the scarlet letter being gone from Hester’s bosom. Nobody knows this little Rose†, Emily Dickinson concludes her poem by saying, â€Å" Ah Little Rose—how easy/ For such as thee to die! † she understands that the rose meant a lot to the butterfly, bee, breeze ,and bird. She emphasized on how each and every thing was affected by the loss of someth ing that was very dear to them. The Scarlet Letter, Nathaniel Hawthorne uses the Scarlet Letter to call attention to how the Puritan society actually is. He explains how hard it is for a woman who has committed adultery with a secret that she cant reveal until the right time, a reverend who also committed adultery who has to keep the secret of adultery on his heart and sees his adulteress get punished for something that he participated in, and a man who was the seed to the tree that grew within The Scarlet Letter see what it was like to, in actuality, get a taste of his own medicine. Hawthorne began this story with a deep, dark picture of a jail entrance. He gave the reader a image of a gloomy, gray place and he saw it fit to put a rosebush into the story. Later on in the chapter you begin to understand the amazing significance of the rosebush next to the jail. He also dwells on how narrow-minded the puritans were, how they had different religious views. Public Punishment was also put into perspective when it came down to the crookedness of the Puritan society. Lastly, Hawthorne symbolized death and secrets to utilize spiritual breakthrough and mental freedom. Nathaniel Hawthorne has taken me on a emotional and theatrical rollercoaster. Emily Dickinson took a simple rose and highlighted on how natures creatures are affected by an absence of a friend. At the beginning she used the tone of a person who is taking a stroll and picks up a rose and is admiring how beautiful the rose is. As Dickinson goes on she say that she took it from its ways. Which means that after picking the rose she begins to see how the environment around it stop in their tracks and in a sense mourn over the loss of the rose. She also sees how the bee will miss it because of its sweet pollen that it needs to feed its family. Without the bee taking the pollen from the flower the bee cannot help produce for its family and new flowers when it goes to another flower. The Butterfly hastening from its far journey would usually lie down on the rose to rest but now the butterfly will have to lie itself upon another flower that it is not use to. Finally, Dickinson used the breeze to give imagery and a sense of smell; to paint a picture on how the breeze would look brushing up against the rose if it were in its regular spot. Also gives off the sweet smell of the rose. In â€Å"Nobody knows this little Rose†, Emily Dickinson creates a beautiful story in a twelve line poem. From the beginning to end , she creates a full-course dinner with one recipe. Emily Dickinson started off by gathering the ingredients, to slicing and dicing, to mixing all the ingredients together and smelling the beautiful aroma , to finally serving p a stunning creation of a poem. Nathaniel Hawthorne and Emily Dickinson have many similarities in their technique of writing. By reading a piece of their work; I have come to realize that they are two very dynamic authors that bring so many things into prospective about life, death, self-awareness, love and hate, and they put all of their thoughts into one small novel or poem. Works Cited Hawthorne, Nat haniel. The Scarlet Letter. New York: Bantam, 1986. Print. Brooks, Kevin. The Road of the Dead. New York: Push, 2007. Print. How to cite Scarlet Letter Final Exam: Expressions of the Transcendentalists, Papers

Yankee Doodle Went To Town, Riding On A Pony, Essays -

"Yankee Doodle went to town, riding on a pony, Stuck a feather in his hat and called it macaroni." Before beginning my research, I assumed that the song above was a pointless rhyme, with about as much significance as "Mary Had A Little Lamb". However, after much research, I've learned that this poem is a reflection of colonial slang, British fashion, and the classic American tradition of the insult. "Yankee Doodle" was written by British soldiers during the Revolutionary War. While it may be used patriotically today, "Yankee Doodle" was actually a derogatory name given to American colonists by the British. It literally means "Stupid American." In order to understand the rest of this song, you must first have some knowledge of British fashion in the late 1700's. Back in London, many young men were forming social clubs. Each of these clubs had an original name. Members of one such club, the most fashionable in London, called themselves "Macaronis". The Macaronis had toured Italy, and became quite fond of pasta, making it their official dish and trademark. Back in America, the times were changing. Many fads that had been popular for some time were now going out of style. For example, putting a feather in your cap was becoming quite unfashionable. So, when the "Yankee Doodle" in this rhyme put a feather in his cap and called it Macaroni, he was claiming that he was being as fashionable as the British Macaronis, when he was actually quite behind his time. This was laughable to the British, because it was a story confirming the stereotype of a "Yankee Doodle." When I researched this topic, I was quite astounded at how similar today's customs are to customs from colonial times. Cruel insults were expressed in many forms and trendsetters told people what was popular. In this case, history is repeating itself. I would not be surprised if 200 years from now, children were writing reports on how their insults are similar to ours. Unfortunately, the reappearance of insults in history is just an endless repetition of the wrong way of living.