Direct and iterative method

Direct and iterative method INTRODUCTION TO DIRECT AND ITERATIVE METHOD Many important practical problems give rise to systems of linear equations written as the matrix equation Ax = c, where A is a given n Ãâ€" nnonsingular matrix and c is an n-dimensional vector; the problem is to find an n-dimensional vector x satisfying equation . Such systems of linear equations arise mainly from discrete approximations of partial differential equations. To solve them, two types of methods are normally used: direct methods and iterative methods. Directapproximate the solution after a finite number of floating point operations. Since computer floating point operations can only be obtained to a given precision, the computed solution is usually different from the exact solution. When a square matrix A is large and sparse, solving Ax = c by direct methods can be impractical, and iterative methods become a viable alternative. Iterative methods, based on splitting A into A = M−N, compute successive approximations x(t) to obtain more accurate solutions to a linear system at each iteration step t. This process can be written in the form of the matrix equation x(t) = Gx(t−1) + g, where an n Ãâ€" n matrix G = M−1N is the iteration matrix. The iteration process is stopped when some predefined criterion is satisfied; the obtained vector x(t) is an approximation to the solution. Iterative methods of this form are called linear stationary iterative methods of the first degree. The method is of the first degree because x(t) depends explicitly only on x(t−1) and not on x(t−2), . . . , x(0). The method is linear because neither G nor g depends on x(t−1), and it is stationary because neither G nor g depends on t. In this book, we also consider linear stationary iterative methods of the second degree, represented by the matrix equation x(t) = Mx(t−1) − Nx(t−2) + h. HISTORY OF DIRECT AND ITERATIVE METHOD Ø Direct methods to solve linear systems Direct methods for solving the linear systems with the Gauss elimination method is given byCarl Friedrich Gauss (1777-1855). Thereafter the Choleski gives method for symmetric positive definite matrices. Ø Iterative methods for non-linear equations The Newton_Raphson method is an iterative method to solve nonlinear equations. The method is defined byIsaac Newton (1643-1727)andJoseph Raphson (1648-1715). Ø Iterative methods for linear equations The standard iterative methods, which are used are the Gauss-Jacobi and the Gauss-Seidel method.Carl Friedrich Gauss (1777-1855)is a very famous mathematician working on abstract and applied mathematics.Carl Gustav Jacob Jacobi (1804-1851)is well known for instance for the Jacobian the determinant of the matrix of partial derivatives. He has also done work on iterative methods leading to the Gauss-Jacobi method. Another iterative method is the Chebyshev method. This method is based on orthogonal polynomials bearing the name ofPafnuty Lvovich Chebyshev (1821-1894). The Gauss-Jacobi and Gauss-Seidel method use a very simple polynomial to approximate the solution. In the Chebyshev method an optimal polynomial is used. DIRECT AND ITERATIVE METHOD 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 ininfinite precision arithmetic. Examples includeGaussian elimination, theQRfactorization method for solvingsystems of linear equations, and thesimplex methodoflinear programming. In contrast to direct methods,iterative methodsare not expected to terminate in a number of steps. Starting from an initial guess, iterative methods form successive approximations thatconvergeto the exact solution only in the limit. Aconvergence criterionis 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 includeNewtons method, thebisection method, andJacobi 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.GMRESand theconjugate 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. In the case of asystem of linear equations, the two main classes of iterative methods are thestationary iterative methods, and the more generalKrylov subspacemethods. Ø Stationary iterative methods Stationary iterative methods solve a linear system with anoperatorapproximating the original one; and based on a measurement of the error (the residual), form acorrection equationfor which this process is repeated. While these methods are simple to derive, implement, and analyse, convergence is only guaranteed for a limited class of matrices. Examples of stationary iterative methods are theJacobi method,Gauss–Seidel methodand theSuccessive over-relaxation method. Ø Krylov subspace methods Krylov subspacemethods form anorthogonal basisof the sequence of successive matrix powers times the initial residual (theKrylov sequence). The approximations to the solution are then formed by minimizing the residual over the subspace formed. The prototypical method is theconjugate gradient method(CG). Other methods are the generalized minimal residual method and the biconjugate gradient method EXAMPLE OF DIRECT METHOD GAUSS ELIMINATION METHOD :- Inlinear algebra,Gaussian eliminationmethod is analgorithmfor solvingsystems of linear equations, finding therankof amatrix, and calculating the inverse of aninvertible square matrix. Gaussian elimination is named after German mathematician and scientistCarl Friedrich Gauss. Elementary row operationsare used to reduce a matrix torow echelon form.Gauss–Jordan elimination, an extension of this algorithm, reduces the matrix further toreduced row echelon form. Gaussian elimination alone is sufficient for many applications. EXAMPLE Suppose that our goal is to find and describe the solution(s), if any, of the followingsystem of linear equations: The algorithm is as follows :eliminate x from all equations below L1 and then eliminate y from all equations below L2 .This will form a triangular form.Using the back substitution Each unknown can be solved . In the example, x is eliminated from l2 by adding 3/2L1to L2. X is then eliminatedmfrom L3 by adding L1 to L3 The result is: Nowyis eliminated fromL3by adding− 4L2toL3: The result is: This result is a system of linear equations in triangular form, and so the first part of the algorithm is complete.The second part, back-substitution, consists of solving for the unknowns in reverse order. It can be seen that Then,zcan be substituted intoL2, which can then be solved to obtain Next,zandycan be substituted intoL1, which can be solved to obtain The system is solved. Some systems cannot be reduced to triangular form, yet still have at least one valid solution: for example, ifyhad not occurred inL2andL3after the first step above, the algorithm would be unable to reduce the system to triangular form. However, it would still have reduced the system toechelon form. In this case, the system does not have a unique solution, as it contains at least onefree variable. The solution set can then be expressed parametrically . In practice, one does not usually deal with the systems in terms of equations but instead makes use of theaugmented matrix(which is also suitable for computer manipulations). The Gaussian Elimination algorithm applied to theaugmented matrixof the system above, beginning with:which, at the end of the first part of the algorithm That is, it is inrow echelon form. At the end of the algorithm, if theGauss–Jordan eliminationis applied: That is, it is inreduced row echelon form, or row canonical form. EXAMPLE OF ITERATIVE METHOD OF SOLUTION A. JACOB METHOD :- The Jacobi method is a method of solving amatrix equationon a matrix that has no zeros along its main diagonal (Bronshtein and Semendyayev 1997, p.892). Each diagonal element is solved for, and an approximate value taken in. The process is then iterated until it converges. This algorithm is a stripped-down version of theJacobi transformationmethod ofmatrix diagonalization. The Jacobi method is easily derived by examining each of the equations in thelinear system of equationsAx=b in isolation. If, in theith equation solve for the value ofwhile assuming the other entries ofremain fixed. This gives which is the Jacobi method. In this method, the order in which the equations are examined is irrelevant, since the Jacobi method treats them independently. The definition of the Jacobi method can be expressed withmatricesas B. Stationary Iterative Methods Iterative methods that can be expressed in the simple form Where neighter B nor c depend upon the iterative count k) are called stationary iterative method. The four main stationary iterative method : the Jacobi method, the Gauss Seidel method ,Successive Overrelaxation method and the symmetric Successive Overrelaxation method C. The Gauss-Seidel Method We are considering an iterative solution to the linear system where is ansparse matrix,xandbare vectors of lengthn, and we are solving forx. Iterative solvers are an alternative to direct methods that attempt to calculate an exact solution to the system of equations. Iterative methods attempt to find a solution to the system of linear equations by repeatedly solving the linear system using approximations to the vector. Iterations continue until the solution is within a predetermined acceptable bound on the error. Iterative methods for general matrices include the Gauss-Jacobi and Gauss-Seidel, while conjugate gradient methods exist for positive definite matrices. Use of iterative methods is the convergence of the technique. Gauss-Jacobi uses all values from the previous iteration, while Gauss-Seidel requires that the most recent values be used in calculations. The Gauss-Seidel method has better convergence than the Gauss-Jacobi method, although for dense matrices, the Gauss-Seidel method is sequential. The convergence of the iterative method must be examined for the application along with algorithm performance to ensure that a useful solution to can be found. The Gauss-Seidel method can be written as: where: ¯ is theunknown in during theiteration,and, is the initial guess for theunknown in, is the coefficient ofin therow andcolumn, is thevalue in. or where: ¯ K(k)is theiterative solution to is the initial guess atx Dis the diagonal ofA Lis the of strictly lower triangular portion ofA Uis the of strictly upper triangular portion ofA bis right-hand-side vector. EXAMPLE. 101−x2+ 23= 6, −x1+ 112−x3+ 34= 25, 21−x2+ 103−x4= − 11, 32−x3+ 84= 15. Solving forx1,x2,x3andx4gives: x1=x2/ 10 −x3/ 5 + 3 / 5, x2=x1/ 11 +x3/ 11 − 34/ 11 + 25 / 11, x3= −x1/ 5 +x2/ 10 +x4/ 10 − 11 / 10, x4= − 32/ 8 +x3/ 8 + 15 / 8 Suppose we choose(0,0,0,0)as the initial approximation, then the first approximate solution is given by x1= 3 / 5 = 0.6, x2= (3 / 5) / 11 + 25 / 11 = 3 / 55 + 25 / 11 = 2.3272, x3= − (3 / 5) / 5 + (2.3272) / 10 − 11 / 10 = − 3 / 25 + 0.23272 − 1.1 = − 0.9873, x4= − 3(2.3272) / 8 + ( − 0.9873) / 8 + 15 / 8 = 0.8789. x1 x2 x3 x4 0.6 2.32727 − 0.987273 0.878864 1.03018 2.03694 − 1.01446 0.984341 1.00659 2.00356 − 1.00253 0.998351 1.00086 2.0003 − 1.00031 0.99985 The exact solution of the system is (1,2,-1,1) APPLICATION OF DIRECT AND ITERATIVE METHOD OF SOLUTION FRACTIONAL SPLITING METHOD OF FIRST ORDER FOR LINEAR EQUATION First we describe the simplest operator-splitting, which is calledsequential operator-splitting, for the following linear system of ordinary differential equations:(3.1)where the initial condition is. The operators and are linear and bounded operators in a Banach space The sequential operator-splitting method is introduced as a method that solves two subproblems sequentially, where the different subproblems are connected via the initial conditions. This means that we replace the original problem with the subproblemswhere the splitting time-step is defined as. The approximated solution is. The replacement of the original problem with the subproblems usually results in an error, calledsplitting error. The splitting error of the sequential operator-splitting method can be derived as whereis the commutator ofAandB The splitting error iswhen the operatorsA andB do not commute, otherwise the method is exact. Hence the sequential operator-splitting is called thefirst-order splitting method. THE ITERATIVE SPLITING The following algorithm is based on the iteration with fixed splitting discretization step-size. On the time intervalwe solve the following subproblems consecutively for:(4.1)where is the known split approximation at the time level. We can generalize the iterative splitting method to a multi-iterative splitting method by introducing new splitting operators, for example, spatial operators. Then we obtain multi-indices to control the splitting process; each iterative splitting method can be solved independently, while connecting with further steps to the multi-splitting method

The Ethical Issues behind Cosmetic Surgery

The article I chose is entitled â€Å"Ethnic Differences Emerge in Plastic Surgery† and was written by Sam Dolnick for the New York Times on February 19, 2011. It explains that cosmetic plastic surgery is no longer an opportunity limited to wealthy suburbanites. In New York City, there has been a surge of immigrants having plastic surgeries that are correlated with their ethnic beauty standards.Furthermore, doctors practicing in various ethnic enclaves throughout the city have seen; Italians having their knees reshaped, Egyptians getting face-lifts, Iranians getting nose jobs, Dominican women getting their buttocks’ lifted, Asians having â€Å"double eye-lid surgery† or receiving a crease in their eyelids to make their eyes appear rounder, and Russians getting breast implants.The article goes on to contrast the cultural beauty standards or tastes between one group of wealthy Long Island suburbanites with Washington Heights’ ancestrally Dominican citizens, where, in Long Island suburbs they want fat removed from their behinds, and in Washington Heights they want their rear ends enlarged and rounded. There are three ethical issues present in this article, the first being the usage of surgery to improve patients’ psychological self-esteem issues versus the usage of surgery for traditional health reasons.The second issue is the fact that doctors are willing to offer layaway plans to help patients afford operations, but even more precarious, unlicensed practitioners are performing illegal surgery throughout most ethnic enclaves. The third is the belief that American pop culture and media have a role in affecting people’s personal awareness and ideals of beauty, and this is making them much more willing to have unnecessary cosmetic plastic surgery performed. John Stuart Mill’s Utilitarian doctrine is the first ethical theory I chose to apply to these ethical problems.Mill argued that morality is based on the consequenc es you take, and that consequences must be a means to increase utility or pleasure (Utilitarianism: Of What Sort of Proof the, para 3,9). To the first ethical issue, Mill would most likely say that people who undergo cosmetic plastic surgery are moral because they are acting accordance with increasing happiness. However, they are acting upon a lower pleasure, which is vanity, and although this may bring some joy, it is something that can become an endless pursuit.Towards the second issue, Mill would most likely say that the legality of the operations, and how the surgery is paid for, is superfluous to morality. If there is a tendency for patients to come out feeling and looking better, than the surgeon has done his or her job and there is nothing immoral going on. However, if there is a tendency for people to leave these places, where layaway is acceptable and the surgeons do not have licenses to practice, in pain and vexation, there is something immoral about this particular situat ion.One could further argue, that performing surgery illegally and allowing layaway for surgery is based in greed. To this I believe Mill would still say that it is the patients responses that matter, for they are the ones who choose to have the surgery illegally, and if they have gained more pleasure through the experience, the actions taken by the illegal surgeons or greedy doctors was moral (Utilitarianism: Of What Sort of Proof, para 6).To the last part, is media and pop culture responsible for unnecessary surgery, and people unhealthy personal beauty obsessions, he would say, generally no. Although he might say that certain parts of media are potent at changing the way people view their bodies and other people’s bodies, and may make people obsessively drawn into the lower pleasures of cosmetic bodily improvement. Immanuel Kant’s Deontological ethics is the second theory I chose to administer to the article. Kant argues that for an action to have genuine moral wort h, it must be done out of duty, or from pure intentions.People or semi-rational beings must not allow subjective experiences and past events, or particular circumstances, determine what actions are morally right, but must act to achieve the categorical imperative, thus making every intention and every action together an end in itself (Groundwork: Moving from popular, para 2-4, 6-7). To the first, again being the usage of surgery for traditional reasons, versus the usage of surgery for patients who have psychological self-esteem issues; they believe that having cosmetic surgery will result in looking better and therefore bring happiness.Kant would strongly see this as immoral, as it is first off not being used to preserve one’s life, and it is seeking out happiness through picking means which Kant believes will only lead to pain (Groundwork: Moving from popular moral, para 15-16). Also cosmetic plastic surgery is not universafiable because it means that our self-love is based off of subjective contingent circumstances, and not based off of self-love because we are semi-rational beings.To the second ethical theory being layaway plans and illegal surgery, Kant would probably see the layaway plans as bad in that if everyone had layaway, there would not be any true ownership, and everyone would be in debt to someone else. Illegal surgery is also not universifiable in that, if it were universifiable, there would be no legality, and the state laws for surgical practices would not exist. As to whether Kant would see people receiving cosmetic plastic surgery as victims of our hyper-idealized beauty popular culture and media, he would probably say yes, there is a problem.Pop culture, the media, and social networks certainly do have an insidiously furtive part in how we behave, and what we believe is beautiful. To go out of your way to make yourself more beautiful because someone or something tells you to do so, especially if it is instilled upon you in a covert w ay, is not acting in accordance with freedom. It is dually leading you to use surgery as a mere means to happiness, and using you as a mere means to waste your money on advertised beauty/ diet products, more surgery, and more repetitive mindless television shows.In terms of which ethical theory best resolves the ethical issues spurred by this article, I would have to say Kant’s theory of good intentions is most critical and affirmative. All of these ethical issues I have raised have questionable intentions behind them. If I were to go with Mill, I certainly would be a lot more lenient with these issues. That is not necessarily a bad thing, since these peoples actions are not necessarily questionable. Their actions are somewhat good, because whether you are the patient, the doctor, or the pop culture media (entertainment) you are aiming at or supposedly affecting happiness (in a Mill sense).But their intentions are somewhat convoluted, and in my opinion, impure. I believe most of the doctors have greed as their intention, especially those who are willing to perform surgery illegally. Sure, the illegal surgeons may be acting in a proper way, that is, they may perform the surgery properly. But behind it all is an intention of surreptitiously avoiding the rules of the state to save money, meanwhile being completely liable to malpractice and putting in jeopardy their lives and the lives of their patients.The patients may know that these doctors are quacks, but if they don’t, then in fact they are being lied to and are being placed under the knives of surgeons who are dishonest and hence treating the patients as a mere means. Furthermore, I believe anyone who is getting cosmetic plastic surgery based off of established beauty standards, unless in some horrific accident or bearing some horrific deformity is impure in their reasoning.They are giving into standards that are not their own, but have been environmentally, and artificially created, beefed up and branded. Thus, in a Kantian sense, people are acting not under a categorical imperative, but under empirical knowledge that has been conjured through culturally relative or subjective circumstances. In sum, Kant would say be happy with what you look like, and don’t attempt to change it based off of your cultural preferences or media created beauty bombardments, because you are not acting freely and with duty.

7 Things Confident Entrepreneurs Never Do - 1190 Words

Sujan Patel, a self-proclaimed entrepreneur and marketer and VP of Marketing at When I Work, wrote an article for Entrepreneur.com titled â€Å"7 Things Confident Entrepreneurs Never Do.† While this article is not an exhaustive list of mistakes entrepreneurs can make, it gets the main point across—entrepreneurs need to be confident because self-confidence is key in the decision-making process (Schermerhorn). Patel says they need to be tenacious and determined and have faith in order to be successful, but he points out seven things to avoid at all costs. The first thing he says entrepreneurs should avoid is â€Å"second guessing themselves and their employees.† As a manager in any situation it is extremely important to trust yourself and your team.†¦show more content†¦It is important to dedicate time to the business and its employees, but everyone has important priorities like friends, family, and things done for enjoyment, like exercise or reading. Hobbies often make people happier than money because they help reduce stress and improve confidence Entrepreneurs must make time in their busy schedule to address these priorities—when quality of life outside of work decreases, the quality of the business may decrease, as well. The third thing Patel says entrepreneurs should avoid is â€Å"refusing to learn new skills.† In an ever-changing business market, companies and the people that run them need to learn how to adapt. In today’s society, technology is rapidly advancing and companies are changing in parallel. An entrepreneur should be open to learning something new because it will diversify their background and open up their mind to new ideas. It can also provide several opportunities that would not have presented themselves if they did not learn new skills. For example, a confident entrepreneur can take on the challenge of learning graphic design or coding if they do not have the resources to hire a programmer. Patel mentions several websites that offer to teach entrepreneurs a variety of skills at a low price point. Taking advantage of these resources can allow an entrepreneur to become more confident in running their business. The fourth thing he

Noras Discovery of Self in Ibsens A Dolls House Essay

Noras Discovery of Self in Henrik Ibsens A Dolls House Ibsens play, A Doll House, involves a woman who begins the play as a common housewife and through a series of joyous occurrences and catastrophes becomes a self-liberating woman. Nora Helmer is transformed and decides to abandon her family and home in search of her true self. She arrives at this point because of several factors. Her refusal to submit to her husband and her self-realization is brought on by the way she has been taught to act by her husband and her father, and the contradicting demands the situations that she has had to deal with gave her. Her true devotion to herself is discovered because of the false devotion she felt towards her husband and her†¦show more content†¦The culmination of this theme occurs when Torvald finally finds out about the borrowed money, and when it threatens his career. Torvald immediately renounces his love for her, despite the fact that the money was borrowed for him, because of her devotion to him. Upon finding out that the mone y is no longer an issue, Torvald reneges on his decisions, and is ready to accept Nora back into his home. But Nora will have none of it. She realizes that what is best for herself in not there, in that mans presence. She refuses to again submit to his decisions, and admits to Torvald that there marriage as it has been is over. (Ibsen 967-1023) So Noras inability to play the character that Torvald has written for her in the situations she must endure is what sets her upon the path of refusal to obey her husbands wishes. Torvalds dissatisfaction with Noras portrayal of her designated role causes him to renounce his love for her, which in turn allows her to disregard all further stipulations from him. In effect, it is not Noras decision to ignore Torvald, but really it is Torvalds own fault. His unreasonable expectation of her to bend any way he pleases causes her to begin to find a new path for herself. Besides a refusal to submit, Nora also experiences a self-realization, an awakening to her real place in humanity, and the idea that she too, is a rational, capable person who is deserving of the respect that entitles. This is what Ibsen really wrote theShow MoreRelated Essay on Lies and Self-realization in A Dolls House1162 Words   |  5 PagesLies and Self-realization in A Dolls House    In Ibsens play,   A Dolls House,   the characters willingly exist in a situation of untruth or inadequate truth that conceals conflict.   Noras independent nature is in contradiction to the tyrannical authority of Torvald.   This conflict is concealed by the way they both hide their true selves from society, each other, and ultimately themselves.   Just like Nora and Torvald, every character in this play is trapped in a situation of untruth. 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