Study On The Applications Of Numerical Analysis Computer Science Essay

Study On The Applications Of Numerical Analysis Computer Science Essay It finds applications in all fields of engineering and the physical sciences, but in the 21st  century, the life sciences and even the arts have adopted elements of scientific computations.  Ordinary differential equations  appear in the  movement of heavenly bodies (planets, stars and galaxies);  optimization  occurs in portfolio management;  numerical linear algebra  is important for data analysis;  stochastic differential equations  and  Markov chains  are essential in simulating living cells for medicine and biology. Before the advent of modern computers numerical methods often depended on hand  interpolation  in large printed tables. Since the mid 20th century, computers calculate the required functions instead. The interpolation  algorithms  nevertheless may be used as part of the software for solving  differential equations. INTRODUCTION TO NUMERICAL ANALYSIS AND METHODS The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following. Advanced numerical methods are essential in making  numerical weather prediction  feasible. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of  ordinary differential equations. Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving  partial differential equations  numerically. Hedge funds  (private investment funds) use tools from all fields of numerical analysis to calculate the value of stocks and derivatives more precisely than other market participants. Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. This field is also called  operations research. Insurance companies use numerical programs for  actuarial  analysis. The rest of this section outlines several important themes of numerical analysis. History of Numerical Analysis The field of numerical analysis predates the invention of modern computers by many centuries.  Linear interpolation  was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like  Newtons method,  Lagrange interpolation polynomial,Gaussian elimination, or  Eulers method. To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions. The canonical work in the field is the  NIST  publication edited by  Abramowitz and Stegun, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy. The  mechanical calculator  was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done. Direct and iterative methods Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in  infinite precision arithmetic. Examples include  Gaussian elimination, the  QR  factorization method for solving  systems of linear equations, and the  simplex method  of  linear programming. In practice,  finite precision  is used and the result is an approximation of the true solution (assuming  stability). In contrast to direct methods,  iterative methods  are not expected to terminate in a number of steps. Starting from an initial guess, iterative methods form successive approximations that  converge  to the exact solution only in the limit. A  convergence test  is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include  Newtons method, the  bisection method, and  Jacobi iteration. In computational matrix algebra, iterative methods are generally needed for large problems. Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g.  GMRES  and the  conjugate gradient method. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method. Discretization Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called  discretization. For example, the solution of a  differential equation  is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum. Different Areas And Methods under Numerical Analysis The field of numerical analysis is divided into different disciplines according to the problem that is to be solved. One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the  Horner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control  round-off errors  arising from the use of  floating point  arithmetic. Interpolation, extrapolation, and regression Interpolation  solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points? Extrapolation  is very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points. Regression  is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function. The  least squares-method is one popular way to achieve this. Solving equations and systems of equations Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation  2x  + 5 = 3  is linear while  22  + 5 = 3  is not. Much effort has been put in the development of methods for solving  systems of linear equations. Standard direct methods, i.e., methods that use some  matrix decomposition  are  Gaussian elimination,  LU decomposition,  Cholesky decomposition  for  symmetric  (or  hermitian) and  positive-definite matrix, and  QR decomposition  for non-square matrices.  Iterative methods  such as the  Jacobi method,  Gauss-Seidel method,  successive over-relaxation  and  conjugate gradient method  are usually preferred for large systems. Root-finding algorithms  are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is  differentiable  and the derivative is known, then  Newtons method  is a popular choice.  Linearization  is another technique for solving nonlinear equations. Solving eigenvalue or singular value problems Several important problems can be phrased in terms of  eigenvalue decompositions  or  singular value decompositions. For instance, thespectral image compression  algorithm  is based on the singular value decomposition. The corresponding tool in statistics is calledprincipal component analysis. Optimization Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy someconstraints. The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance,  linear programming  deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the  simplex method. The method of  Lagrange multipliers  can be used to reduce optimization problems with constraints to unconstrained optimization problems. Evaluating integrals Numerical integration, in some instances also known as numerical  quadrature, asks for the value of a definite  integral. Popular methods use one of the  Newton-Cotes formulas  (like the midpoint rule or  Simpsons rule) or  Gaussian quadrature. These methods rely on a divide and conquer strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use  Monte Carlo  or  quasi-Monte Carlo methods  (see  Monte Carlo integration), or, in modestly large dimensions, the method of  sparse grids. Differential equations Numerical analysis is also concerned with computing (in an approximate way) the solution of  differential equations, both ordinary differential equations and  partial differential equations. Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a  finite element method, a  finite differencemethod, or (particularly in engineering) a  finite volume method. The theoretical justification of these methods often involves theorems from  functional analysis. This reduces the problem to the solution of an algebraic equation. Applications Of Numerical Analysis Methods and Its Real Life Implementations, Advantages Etc. NEWTON RAPHSON METHOD: ORDER OF CONVERGENCE: 2 ADVANTAGES: 1. The advantage of the method is its order of convergence is quadratic. 2. Convergence rate is one of the fastest when it does converges 3. Linear convergence near multiple roots. REGULA FALSI METHOD: ORDER OF CONVERGENCE: 1.618 ADVANTAGES: 1. Better-than-linear convergence near simple root 2. Linear convergence near multiple root 3. No derivative needed DISADVANTAGES 1. Iterates may diverge 2. No practical rigorous error bound GAUSS ELIMINATION METHOD: ADVANTAGES: It is the direct method of solving linear simultaneous equations. 2. It uses back substitution. 3. It is reduced to equivalent upper triangular matrix.: 1. It requires right vectors to be known. GAUSS JORDAN: ADVANTAGES: 1. It is direct method. 2. The roots of the equation are found immediately without using back substitution. . It is reduced to equivalent identity matrix. The additional steps increase round off errors. 2. It requires right vectors to be known. GAUSS JACOBI METHOD: 1. It is iterative method. 2. The system of equations must be diagonally dominant. 3. It suits better for large numbers of unknowns 4. It is self correcting method. GAUSS SEIDEL METHOD: 1. It is iterative method. 2. The system of equations must be diagonally dominant. 3. It suits better for large numbers of unknowns 4. It is self correcting method. 5. The number of iterations is less than Jacobi method. Real life Applications Area of mathematics and computer science. Applications of algebra Geometry Calculus Variables which vary continuously. Problems(application areas) 1. Natural sciences 2. Social sciences 3. Engineering 4. Medicine 5. Business.(in financial industry) Tools of numerical analysis Most powerful tools of numerical analysis à  Computer graphics à  Symbolic mathematical computations à  Graphical user interfaces Numerical analysis is needed to solve engineering problems that lead to equations that cannot be solved analytically with simple formulas. Examples are solutions of large  systems  of algebraic equations, evaluation of integrals, and solution of differential equations. The finite element method is a numerical method that is in widespread use to solve partial differential equations in a variety of engineering fields including stress analysis, fluid dynamics, heat transfer, and electro-magnetic fields. In hydro static pressure processing In high hydrostatic pressure (HHP) processing, food and biotechnological substances are compressed up to 1000 M Pa to achieve various pressure-induced conversions such as microbial and enzyme inactivations, phase transitions of proteins, and solid-liquid state transitions. From the point of view of thermodynamics, Heat transfer leads to space-time-dependent temperature fields that affect many pressure-induced conversions and produce undesired process non uniformities Effects related to HHP processing can be studied appropriately by use of numerical analysis because in situ measurement techniques are barely available, optical accessibility is hardly possible, and technical equipment is expensive. This reports on two examples, where numerical analysis is applied successfully and delivers substantial insights into the phenomenon of high-pressure processing. Calculation E.g TSP problem (traveling salesman problem) to travel no. of cities in such a way that the expenses on traveling are minimized. à   NP-complete problem. à   optimal solution we have to go through all possible routes à   numbers of routes increases exponential with the numbers of cities. Modern Applications and Computer Software Sophisticated numerical analysis software is being embedded in popular software packages e.g. spreadsheet programs. Buisness Applications:- Modern business makes much use of optimization methods in deciding how to allocate resources most efficiently. These include problems such as inventory control,scheduling, how best to locate manufacturing storage facilities, investment strategies,and others. In Financial Industry Quantitative analysts developing financial applications have specialized expertise in their area of analysis. Algorithms used for numerical analysis range from basic numerical functions to calculate interest income to advanced functions that offer specialized optimization and forecasting techniques. Sample Finance Applications Three common examples from the financial services industry that require numerical algorithms are: à ¢Ã¢â€š ¬Ã‚ ¢ Portfolio selection à ¢Ã¢â€š ¬Ã‚ ¢ Option pricing à ¢Ã¢â€š ¬Ã‚ ¢ Risk management   In market Given the broad range of numerical tools available a financial services provider can develop targeted applications that address specific market needs. For example, quantitative analysts developing financial applications have specialized expertise in their area of analysis.

Essay --

Should Marijuana be legalized ? In all honesty, should it be? Ask yourself again. How is alcohol legal. Marijuana is much safer and has medicinal use capability . According to Cbsnews.com, in a recent poll in the New England Journal of Medicine, 76% of doctors approve Marijuana for medical use. 1446 doctors asked in 72 total countries. Why is marijuana so widely supported? Because it’s medical capabilities are legit. (I’ll address medical capabilities below) Alcohol totally impairs you and and causes your frontal lobes to malfunction. Its stupid, alcohol, which is actually dangerous is legal and people are still killing themselves under the influence, but Marijuana is good for you yet illegal? It seems as if the government is trying to kill us with such a thing being legal (I don’t drink or smoke so this isn’t biased) . Besides, who really cares? --> If â€Å"potheads† want to smoke, why not just let them smoke? Prisons are for criminals and not for people who smoke a plant that makes them happy. A significant aspect of this controversy is that freedom is being seized. In 2007, arrests for Marijuana alone totalled up to 775,138, dramatically surpassing arrests for all violent crime combined. That’s just ridiculous. People should not be getting incarcerated for smoking a plant that is beneficial to health. Most people just smoke it to reduce stress. Weed smokers are not criminals; rapists, killers, and thieves are criminals. Seriously people, prisons are being flooded. Sticking someone in prison isn’t something that’s cheap. According to Ministry of Justice, the cost of keeping someone in prison is roughly $25,000 a year. These people can’t get jobs, in many states, possession of Marijuana can be ruled a misdemeanor or a felony. A ... ...mount of our population suffers from at least one of these conditions. Moreover, non-cannabis supporters’ strongest argument is the abuse of the plant. The opposition’s argument states that Marijuana is addictive, but recent studies by a professor at the University of Cambridge in England named Leslie L. Iverson, who also wrote a book called â€Å"The Science Of Marijuana†, show that only 9% of Marijuana users develop serious dependency of the plant. Those are overwhelming statistics for anyone trying to keep the plant illegal. In conclusion, Marijuana should definitely be decriminalized. It benefits human health, it will boost the economy if taxed, it will massively reduce the profit of international crime gangs, it will bring down the ever so high crime rate, and will result in prisons not being packed with people who don’t belong there. Support Marijuana legalization!

Journal for Advanced Lifespan Development

As far as disciplines are concerned in such a vast scientific world that people circulate in, when psychology was conceived and eventually became a separate field of concentration, it has become perhaps one of the most popular courses and career any individual would want to venture in. The main concerns however are left for aficionados to speculate that the study of human behavior, its scientific explorations on the human mind and its mental processes is also a very complex and quite difficult arena (Papalia et al, 2002).There are theories and theorists that abound to posit on the description, explanation or understanding and prediction (and if possible, modification or control) of behavior among human organisms. Despite the modern breakthroughs, there lies a tension between what has been known and what are still unknown on the creature called man (Bernstein, 1991). This paper is anchored on the understanding that psychology actually pertains to the human psyche which is the equivale nt of the soul and/or spirit where insights, knowledge and understanding reside.Where this is the central issue, spirituality similarly is attracted to the human mind where the attention that much of the Christian doctrine is also supposed to attend to or has been prepared to change. Baumrind’s theory of parenting styles Introduction Diana Baumrind developed a very widely known theory of parenting which created a great impact on the idea of parenting styles that most parents adhere to and follow today. The concept of main concepts include the authoritative, permissive and authoritarian models (Atkinson et al, 1993).The main idea concerns these styles and their impact on the resulting developing person. It is considered influential because it is perhaps the easiest to remember and where most parents or adults readily relate to in most cases (Morgan, 1977). The theory indeed has inspired a lot of consequent studies especially the implied and emphasized acceptance of the superio rity of the authoritative model over and against the rest of the other styles.In a sense, Baumrind and her theory or studies convey the clear message that there is such a correct way of rearing children and this she mentioned as the authoritative way (Thomson, 2005). Critiques of course abound as to the validity and applicability of her theory at all times or in all set-ups or situations. The studies many argue had only been correlation which means that there is no such direct â€Å"cause and effect† as one would have wanted to appear (Papalia et al, 2002, p. 301). Implications to CounselingCounseling in this perspective would imply that the counselor adhere to Baumrind’s understanding that there is the right way of bringing up children (Baumrind, 1959). However, as studies also reveal, parents never behave as consistently as they should as they relate with their children in all aspects of their lives (Baumrind, 1971). Here is the complication here. Counseling then, wo uld serve as guidance to parents and would-be parents as to what would emerge as more successful parenting by introducing and comparing the different parenting styles.Despite the fact that what was employed was correlational type of research methodology does not mean that the results have become altogether invalid or insignificant. It means that for most of those who advocate the authoritative style, children have emerged as more adjusted and more capable of getting into different stresses that life in reality has to offer (Bootzin, 1991). Implications to Culture Culture is of course influenced to the degree that people inhabiting certain localities or communities believe on the Baumrind model.It means that parents and adults who are primary caregivers adjust to the authoritative standpoint for a society to make a better chance of surviving and minimizing the chaos that people bring on themselves by virtue of living in a dysfunctional world (Baumrind, 1975; Bernstein, 1991). Culture then must give in to the idea that even if parents come from different ethnic groups or race, that does not deter them to adopt the â€Å"right† way of bringing up or rearing children. Implications to the Biblical worldview The Judeo-Christian worldview is definitely perhaps the primary reason that these styles have emerged.The backdrop of the styles mimic with the typical home where the bible and its influence may be peculiarly absent or obviously the hub from which the family must revolve around. The authoritarian is by and large a translation of the Judeo-Christian worldview of bringing up children and way of raising a family where cooperation is not taken for granted. It goes to show also that the Bible or Scriptures must take the front seat in all endeavors and decision-making processes. Conclusion Many get their inspiration for adopting their way of raising their family based on the Baumrind model.Indeed, it has been effective to a degree because it can easily be remem bered and understood. Usually, anyone coming from a background of any of the styles can immediately relate. Eph 6:1 admonishes children: â€Å"Children, obey your parents in the Lord: for this is right, and reinforced in Col 3:20, â€Å"Children, obey your parents in all things: for this is well pleasing unto the Lord† (KJV). Another is found in Proverbs 4:1, â€Å"Hear, my children, the instruction of a father, and give attention to know understanding† (NKJV).The parents then must see to it that the guidance must be contained in the context of the Christian home. Again, though, these must be taken in the context of the Scriptural admonition that wisdom comes from God through His Word. The day to day efforts and energies expended on raising children in the context of the Christian household entail more than an accumulation of experiences and knowledge. It takes the wisdom that God through His Spirit infuses as the parent or guardian waits upon in meditation and in sub mission to God and His word. References 1. Atkinson, R. L. , R. C. Atkinson, E.E. Smith, D. J. Bem, and S. Nolen-Hoeksema, 1993. Introduction to Psychology, 13th ed. New York: Harcourt College Publishers. 2. Baumrind, Diana. 1959. Conceptual issues involved in evaluating improvement due to psychotherapy. Psychiatry, 22, 341-348. 3. Baumrind, Diana. 1975. The contributions of the family to the development of competence in children. Schizophrenia Bulletin, 14, 12-37. 4. Baumrind, Diana, 1971. Harmonious parents and their preschool children. Developmental Psychology, 4(1), 99-102. 5. Bernstein, D. A. , E. J. Roy, T. K. Srull, and C. D. Wickens, 1991. Psychology.New Jersey: Houghton Mifflin Company. 6. Bootzin, R. R. 1991. Psychology. New York: Gilford Press. 7. Morgan, Clifford T. 1977. A Brief Introduction to Psychology. 2nd ed. New York: McGraw-Hill Book Company. 8. Papalia, Diane E. , S. W. Olds. , RD Feldman. 2002. Human Development. 8th Ed. , International Edition. McGraw-Hill. 9. Thompson, Ross A. â€Å"Child Development. † Microsoft ® Encarta ® 2006 [DVD]. Redmond, WA: Microsoft Corporation, 2005. 10. The Holy Bible. 2001. New King James Version. Power BibleCD Online Publishing, Inc. 127 N. Matteson Street PO Box 21, Bronson, MI 49028

Cyberbullying Is A Recent Problem Among Youths And Their...

Cyberbullying has become a recent problem among youths and their families. Children go online, and through social media are harassed, degraded, and ostracized, usually by their classmates. This is an epidemic that needs to be stopped, but demanding the schools to do the bidding of other, more qualified individuals is not the way to fix the situation. Cyberbullying is not the school’s issue to solely handle because it happens out of their jurisdiction, it causes further issues for both parties, and there are better solutions to this problem other than a school suspension. Schools should focus on what happens in the school and on school grounds, not about what happens when the student goes home. The only reason schools should do something about cyberbully is if it happens on one of their computers or over their Wi-Fi. Moreover, students mainly do the cyberbullying, while on their devices at home, on their own Wi-Fi. This means that schools no longer have any jurisdiction or say so in what goes on in the student’s life after they step off of campus. This is because the responsibility lies with local authority and the child’s parents/guardians, not the school’s authority. The school’s main priority should be to provide quality education and a safe, productive environment for everyone on campus. Parents are responsible for their children while they are under their care and they are the people who must take action in lieu of their children, either going to law enforcement orShow MoreRelatedSocial Media Bullying And Cyberbullying1383 Words   |  6 Pagesincreasing problem in our society for some time, social media has made it possible to have access to individuals with no respite. There seems to be no escape from constant ridicule and body shaming, we are seeing an increase in many psychological issues not only in our young people but adults as well. 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