Numerical Differential Equation Analysis Package

Numerical Differential Equation Analysis Package 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! 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 i≠¤j 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 i 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 m 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 desi

Arce Dairy Marketing Plan Essay

A. Product Performance The company still has an edge compared to the other ice cream industries, because they have maintained their quality since then, together with variety of flavours. It’s not too late if they will expand more their business and will make more effort to increase their market share. Background Arce Dairy is very proud of their high quality ice cream product. They strictly follow their tradition of using ONLY fresh CARABAO MILK and other fresh ingredients in making their ice cream. Its ice cream was well-known for its creaminess, authentic flavours, and unique gold can packaging—qualities that remained throughout the years. Arce Dairy Ice Cream is made from fresh carabao’s milk that has a richer and flavourful since they are exposed to the tropical condition which helps them produce better milk. It is an advantage for the Arce’s since they own a 21 hectare dairy farm in Novaliches which is also the only large and the only private commercial breeder of milking carabaos in the Philippines. Aside from their milk quality, they also have their fresh ingredients such as fruits, nuts, chocolates and cheese that are being stored on large chillers. Long-term objectives Arce Dairy tend to branch out their product not just on large scale stores like, malls, supermarkets and etc., but also on small retail stores. Because it is more accessible and convenient that you can just easily buy the goods that you want in just a walking distant store. Arce Dairy Ice Cream aims to provide the best all-natural ice cream products to consumers both here and abroad, provide good and natural quality products for the health conscious ones thereby improving the quality of life. Showing patriotism to its fellow Filipinos in patronizing Filipino-made products by manufacturing and marketing consistently high quality ice cream, thus, giving consumers added value for their money. Lastly, to ensure in surpassing the global quality standard of becoming the source of pride for the Philippine Dairy Industry. Problems/Opportunities Problems How will Arce Dairy Ice Cream compete in the ice cream industry and regain success using its new brand name? Low market share, wherein, Arce Dairy Ice Cream belongs to the remaining 20% of the ice cream market. Poor promotional and distribution strategies. They should compete with the other ice cream industry, and will not stick on their captured market, they should expand their market. They should try branching out especially on small retail stores. They should put much effort on their promotional and advertising activities so people will notice the product. Opportunities People’s acceptance to its new brand name. New Ice cream flavours offered. They could introduce new product line using their fresh milk. Health and wellness campaigns encouraging people to go for the healthier choice. Social media network serving as effective vehicle in product promotion. Marketing Objectives Increase sales Brand Awareness/Exposure. Brand Image establishment. Marketing Programs Arce Dairy must conduct surveys and increase it sales through the help of retail partners. Its sales must be compared to the sales projections to monitor the progress of their sales. The company must attain its projected sales or beyond it. Positioning Statement Target Market â€Å"Yuppies† or young urban professionals. – Health-conscious, self-indulgent young urban professionals or â€Å"yuppies† who have the money to lavishly spend on anything they want, and also who are fickle in choosing their brand choices, depending on what is trending or getting good reviews. 20-26 years old, class A-B, male and female, living  and working in/around NCR. Product strategy Arce Dairy started from three generations. They produce different variety of flavours like Mango, Chocolate, Blueberry Cheesecake, Buko Lechias, Avocado, Quezo Real, Vanilla, Strawberry, Dark Chocolate, Cheese, Ube, Almonds and Chips and many more. Arce Dairy actually has their own farm where they get most of their fresh ingredients especially the â€Å"CARABAO MILK† which they use as base for their ice cream. Pricing strategy Arce Dairy Ice Cream Container SizePrice range 425 mlPhp 70.00 (classic) Php 80.00 (supreme) Php 108.00 (sugar-free)750 mlPhp 120.00 (classic) Php 205.00 (supreme)1.5 LPhp 228.00 (classic) Php 408.00 (supreme, sugar-free) Advertising strategy Launch print ads that will capture the target market’s attention. Execute an effective guerrilla ad that stays faithful to Arce Dairy’s new image. Release a TVC to be aired during primetime. Document the active guerrilla campaign and post it as a viral video. Use imagery that implies. National Trigger the curiosity of the target market and make them try the product. Promote Arce Dairy’s new â€Å"sexy† image.  Cover up Arce Dairy’s â€Å"local/low class† image to people. Expose people to the brand. Cooperative Trade Promotion strategy Since Arce Dairy has no promotional straregies except on relying through word  of mouth strategy, the Unfortunately, Arce Dairy has almost NO promotion. They’ve produced no TV ad, very few and unmemorable print ads. No promotional efforts are done, and they rely on word of mouth to get cutomers. Field sales strategy Distribution strategy Product support Training Requirements Marketing research 4/5 of the people who are familiar with the brand â€Å"Arce Dairy† have claimed that it’s their favourite ice cream in the Philippines. Even though they have flavour favourites, they try the different flavours every now and then. Though they love Arce Dairy, they consume other brands more often because Arce Dairy is not available on many local stores. But whenever there is Arce Dairy available, it’s their first choice. What they love about Arce Dairy is the consistency of the ice cream, the smooth creamy texture, and the mild sweetness of the flavours. Most of them found out about Arce Dairy first from personal references like friends, and relatives. Once they tasted Arce, it was immediately their â€Å"new† favourite ice cream. According to them, the price doesn’t bother them, since the taste is worth the price they pay. They also mentioned that they wish Arce was available to more stores. 5/5 were NOT aware that Arce Dairy uses CARABAO MILK in their ice cream. 2/5 of the people who aren’t familiar with Arce Dairy said that their favourite brand of ice cream is Selecta (particularly Magnum) To the yuppies, quality of ice cream and product feedback is vital. If it hits home, they might actually stick to it. All they need is a good push to try the product. Financial summary Strengths More expensive than other brands.  Maintained a clean reputation for making quality ice cream, proven by several awards given to Arce Dairy. Arce Dairy ice cream is scientifically proven much healthier compared to leading ice cream brands. Has variety of unique ice cream flavours not available in other brands. Weaknesses Has very limited advertising/promotion materials, leading to unpopularity amongst the masses. Product is poorly distributed, making it available to very limited stores and locations. Not top of the mind when asked about an ice cream product. Has a â€Å"local† image, therefore viewed as â€Å"cheap†. Performance History Arcefoods Corporation has been honored the Hall of Fame Award as â€Å"Outstanding Ice Cream Manufacturer† by the Consumers Union of the Philippines while ArceDairy Ice Cream has been awarded the 1999 Dangal Ng Bayan â€Å"Outstanding Ice Cream Product† and the New Millennium Excellence Award â€Å"Most Outstanding Ice Cream Brand† by the Parangal ng Bayan Awards Foundation. The ArceDairy type of ice cream, the â€Å"all-natural† ice cream, has captured the hearts of Manilanians through the decades of its uniqueness and its old fashioned manufacturing process. Tenacious requests from prominent clients, captive loyal followers and food connoisseurs, demand for its return.

The Negative Effects of Technology on Society Essay

Technology has more negative effects on today’s society than positive. Due to technology in the past few decades Canine Shock Collars have been increasingly popular. Students in school pay more attention to texting than they do their classes. Violent addictive video games have made their way into American homes. Parents encourage their children to not text as much, but them to face the problem of constant communication. The Internet gives the students easier ways to cheat in school, and reinforces laziness. Internet Porn gives every bored male a chance to look at the seediest film in the comfort of his own home. Technology has taken the innocence and mystery away from the American family. Canine shock†¦show more content†¦Through extensive research we are now able to see that multitasking is not an effective learning method, but is a bad thing to do. Video games such as World of Worldcraft and Call of Duty, have become prevalent across America. These games a llow the average American boy to act as if he was a fantasy character or allows him to feel as if he was a military personnel on a killing spree. World of Worldcraft and Call of Duty are extremely violent and addictive video games. It was said that the Columbine killers stated that they got their practice from playing video games. When there are games such as Gran Theft Auto where you get points for raping and pimping women, it numbs the mind to the despicable acts of the video game. When boys that regularly play violent video games are confronted, they have a much higher chance to become increasingly aggressive (Gentile, Lynch, Linder, and Walsh). The children who play video games also have a smaller chance of succeeding academically. Parents are also responsible for this technology epidemic. My parents bought me a cell phone supposedly for my entertainment. I knew that the cell phone was however a technological leash. 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Africas Influence on Western Art Essay - 1858 Words

Africas Influence on Western Art During the mid 19th century up until the Great War of 1914, European countries began to heavily colonize and come into contact with African nations. This was called new imperialism. During this contact, European culture was influenced by Africa. The influence of the African people can be seen in the European society of the time. In the 19th and 20th centuries, modern artists embraced African art for its lack of pretension or formal qualities. In the latter part of the 19th century, the scramble for Africa, consolidated at the Berlin Conference, divided the terrain of the African continent among the numerous European contenders. Fourteen countries were represented by a plethora of†¦show more content†¦Germans claimed Namibia and Tanzania, and Spain was rationed Equatorial Guinea. South of the Sahara Desert, there were three distinct types of societies; nomadic tribes in the desert and steppe regions, sedentary farming cultures located in the savanna and rain-forest fringe areas, and the ancient sophisticated kingdoms of Nigeria and the Guinea coast. All three sectors of the African society had different art traditions. However, all three were similar in certain aspects. These aspects being the similar attention to craftsmanship, a general use of non-permanent materials, use of geometric abstraction, and religious orientation. Religion was at most often marked in masks and sculpture. Masks were used in many ritual ceremonies to embody spiritual forces. Geometric and naturalistic shapes were combined to represent a recognizable human face. As part of the daily ritualistic routine, families would often present offerings to cult figures, full-body images kept in homes as insurance of protection. 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