Ralph Waldo Emerson Essay Example for Free

Ralph Waldo Emerson Essay Learning is a natural ability that is wired into many animals DNA; the way that humans should learn has been debated by the various educators because of the endless ways to teach. Teachers and parents take this matter seriously like Ralph Waldo Emerson in â€Å"From Education† and Todd Gitlin in â€Å"The Liberal Arts in an Age of Info-Glut† who created essays on education; and Billy Collins in â€Å"The History Teacher† entail for then and who wrote a poem concerned with the status of education. These people show what the importance of education is entailing what learning should and should not involve; a teacher should respect and have patience for children; a teacher should also let a child have creativity and lessons of the past. Many teens do not respect their teacher which might be because the teacher does not respect them; as the golden rule goes, â€Å"treat others the way you would treat yourself†; Ralph Waldo Emerson in â€Å"From Education† would probably agree with this quote because he strongly believes that, â€Å"the secret of [e]ducation lies in respecting the pupil,† (page 102). Shows that Emerson understood what children are thinking then did something to help them by simply respecting them. Another big issue with education is patience; some people do not have enough patience to guide children to what they are supposed to learn. Emerson also believes that patience is an important part in education, â€Å"to regard the young [children, they require] no doubt, rare patience: a patience that nothing but faith in medial forces of the soul can give,† Emerson was trying to say that someone has to really care in order to teach information to children (page 105). Some people do not see the point in having art classes or history classes; the reason why is because art assists students to express themselves in a way that words cannot along with helping them find out what kind of person they truly are; and history is needed to teach children where they originated from; and to show them mistakes that other people in the past made so they will not make them again. Todd Gitlin in â€Å"The Liberal Arts in an Age of Info-Glut† agrees with this â€Å"[students] need some orientation to philosophy, history, language, literature, music, and arts that have lasted more than 15 minutes,† (page 156) because students need the outlet to express themselves; or if they do not they would not learn morals then they would become exactly  what people were trying to stop; As shown in â€Å"The History Teacher† by Billy Collins who stated, â€Å"the children would leave his classroom for the playground to torment the weak and the smart,† all because â€Å"[the teacher tries] to protect his students’ innocence he told them the Ice Age was really just the Chilly Age, a period of a million years when everyone had to wear sweaters. And the Stone Age became the Gravel Age, named after the long driveways of time,† this demonstrates how not teaching children lessons of the past which are the brood violence’ that others committed in history; by not teaching this history it affects children’s behavior in real life (page 143). Morals are right and wrong many people’s morals will differ; this is why teaching them is tricky because the teacher’s morals might not be the same as the parent’s morals. This is why some of the history in textbooks is sometimes just the summary of what really happened but not enough for the main point to be set across unlike in â€Å"The History Teacher† where, â€Å"the Spanish Inquisition [is] nothing more than an outbreak of questions such as â€Å"How far is it from here to Madrid? †,† and â€Å"the War of the Roses took place in a garden, and the Enola Gay dropped one tiny atom on Japan,† explains when a teacher goes too far in trying to protect student’s mind from bad; when only good is taught and everything else is censored then the students become the opposite; because they do not know the consequences of doing bad, which is why history is taught. As teachers and parents they have to let their students make their own choices as shown in â€Å"From Education† Emerson says, â€Å"it is not for you to choose what he shall know, [or] what he shall do,† shows that sometimes a student needs to find out the consequences the hard way, and let them choose what they want to learn (page 143). Education is needed in modern day society to help students and children grow and help the nation; by giving students proper education the teacher are not really helping the students but they are helping themselves. Teaching creativity helps a student express themselves; respect should be given so the student can give it back; patience is needed to help them understand; morals need to be taught through history so the student will make fewer mistakes; these are essential to a good education and an even better future: as shown by the writings of Billy Collins, Ralph Waldo Emerson, and Todd Gitlin.

Role of War in Society Essay Example for Free

Role of War in Society Essay Both The Prince by Niccolo Machiavelli and Utopia by Thomas Moore examine the role and the importance of war to maintain a strong and successful society. More (via Hythloday) believes that war should be the last option and is not needed in a safe and happy government without the help of auxiliaries and mercenaries while on the other hand, Machiavelli believes that war plays a crucial war in a prosperous and thriving society using other troops and people to avoid war. Machiavelle uses his belief that war is the way to have a strong successful government to shape the society of Italy while More uses the belief of peace and forbearance of war to shape his utopian society. More and Machiavelli have opposing opinions on the values and how the way it can maintain a stability and prosperity in a society. Machiavelli believes that power is attained in a government through the conquering of war and that good law and government follows naturally from good military as we can see in Chapter 12. â€Å"The presence of sound military forces indicates the presence of sound law† (Machiavelli 37). This shows that he believes that a strong developing state directly involves the conquering of war. He believed that successful war is the very foundation upon which all states are built. Machiavelli praises Alexandar the Great, Cyrus, Scipio and Caesar as leaders who gained power through their conquering wars. â€Å"Anyone who reads Xenophon’s life of Cyrus must realize how close Scipio modeled himself on Cyrus, how much that imitation contributed to his glory, and how closely he conformed, in temperance, affability, humanity, and liberty to the thing that Xenophon wrote about Cyrus† (Machiavelli 42). In contrast, More shows his views on war through the Utopians. â€Å"Nothing more inglorious than that glory that is gained by war (More 64). The Utopians viewed war as the last resort and would avoid it at all costs. They hated war and found no glory in the practice of killing others. They would only engage in war to protect themselves and their people, to defend allies, and to relieve oppression. â€Å"No man is to be esteemed our enemy that has never injured us; and that the partnership of the human nature is instead of a league. And that kindness and good-nature unite men more effectually and with greater strength than any agreements whatsoever; since thereby the engagements of mens hearts become stronger than the bond and obligation of words† (More 64). This shows that agreement and unity was very important to the Utopians and they would do anything just to maintain it. They would also try their best to preserve good terms with the people or societies around them. In Machiavelle, all of the issues in the government are in a military point of view because the successfulness in war is crucial to the preservation of the government. â€Å"A prince must have no other objective, no other thought, nor take up any profession but that of war, its methods and its discipline, for that is the only art expected of a ruler. And it is of such great value that it not only keeps hereditary princes in power, but often raises men of lowly condition to that rank† (More 40). Military defense and the strategies required are primary to conquer in war and therefore sustain a happy society. Machiavelle uses the Prince to communicate the importance of war to the audience. â€Å"As for physical training, apart from keeping his troops well disciplined and exercised, he should do a great deal of hunting, and thus harden his body to strenuous exercise, meanwhile learning to read terrain.† This shows that he believed that having brute force would help his army in war to conquer the enemy. War is also what helps a prince to attain power and fortune therefore military and war should be his main priority while ruling. â€Å"..in times of peace he should think about it even more than in wartime. He can do this in two ways, by training the body and training the mind. As for physical training, apart from keeping his troops well and disciplined and exercised, he should do a great deal of hunting, and this harden his body to strenuous exercise..† (Machievelle 41) Princes should always and only think about war especially in times of peace. They need to always be prepared and be in a good state to go to war at any time. â€Å"As for exercising the mind, a prince should read history and reflect on the actions of great men.† Princes should examine the tactics that the past rulers used and put them into action. He should learn from them and learn from both their successes and loses and put them as an example for himself. More in his utopian society would rather use cunning to win rather than brute force because he considered strength to be trait belonging to animals. Unlike Machiavelle, when war is made, the utopians would try to use any technique just to stop war. First one is propaganda in which they would secretly post posters in the enemy territory offering big rewards to people that would assassinate the enemy leaders. The second technique they would use is to promise the throne to the enemy’s rulers brother if he helps the utopians. They would never let things debase to the point where a massacre would occur and would prevent it by not destroying too much of the enemy’s land. nothing more inglorious than that glory that is gained by war â€Å" (More 64). This shows how much the utopians are against war and how much they want to avoid it because they believe that it can ruin their utopian society. Machievelle also believed that one using one own army is best as seen through the example of Cesare Borgia. â€Å"I am never reluctant to cite Cesare Borgia and his deeds. The duke entered the Romagna with auxiliary troops, consisting entirely of Frenchmen; and with them he took Imola and Forli. But then when he found they were not to be trusted, he adopted mercenaries as less dangerous, and hired the Orsini and Vitelli. When he found they too were undependable, treacherous, and dangerous to his service, he got rid of them, and turned to the troops of his own.† (Machievelle 38) The danger with the auxiliaries was because they will always be loyal to their ruler in the end and will probably wait to attack you. Mercenaries on the other hand are better than the auxiliaries but still possess characteristics that are tricky and untrustworthy. Therefore, in the end, Machievelle states that using one’s own troops is the best as Cesare Borgia did in the end. In contrast, the Utopians use spies and bribery to try to stop war. They believe that mercenaries and auxiliaries are tricky but at the same time useful. â€Å"Their second preference is to make use of troops belonging to the nation they are trying to defend; and after them, they enlist as auxiliaries squadrons drawn from their other allies. Lastly they enroll their own citizens. But they always appoint one of their own men who has demonstrated his military capacity to be commander in chief of the whole army.† (More 140) The utopians would do anything to avoid war and they usually would use troops of other nations to fight in war and their troops could be the last resort. They would also avoid fighting on their own land and give the best care to their citizens. A counterargument would be that if ever an intruder invades the Utopian society, war would be initiated. â€Å"For you’ll never find yourself at wat except by your own choice, and peace, not war, ought always to be your first priority† (More 66) Yes if peace cannot be made, war would have to be initiated in the Utopian society but even so, peace is always their foremost choice before they initiate anytime of violence or way on another society. In conclusion, Machievelle’s and More’s contrasting views on war and the strategies and tactics used in war serve one purpose which is to create a stable and prosperous society. Machievelle believes that war using one’s own army is the most crucial factor in maintaining a happy society while More argues that avoiding war using bribery and mercenaries is what the government should be doing no matter what kind of tactics are needed to be used.

Inverse Matrix Condition Number

Inverse Matrix Condition Number Inverse Matrix and Condition No. Saswati Rakshit Contents (Jump to) Aim Scope/Applications Introduction/Basics Objective System Flow Mathematics Figure/Descriptions Future Works References Aim: Consider 2 random matrices B and C of size 8Ãâ€"8 and write a cpgm / matlab to find A to satisfy the bellow condition: If AÃâ€"B = C Prove A = CÃâ€"B-1 And repeat the pgm for matrix of size 32Ãâ€"32 and 128Ãâ€"128. Scope/Application: In many applications we require inversion of matrix. In Linear Algebra, if AÃâ€"B=C, and from B and C we can compute A where A=CÃâ€"B-1. Stimulus-Response Computations In this framework, a system is provided with an input, called a  stimulus, and the resulting response of the system is measured. Some typical examples of stimuli are visual scenes i.e. if we increase incident light’s intensity then scene’s brightness will increase. The general goal is to find a  function  that accurately describes the relation between stimulus and response. Many systems can be modeled as a linear combination of equations, and thus written as a matrix equation: [Interactions]{response}= {stimuli} The system response can thus be found using the matrix inverse. Sometimes in image processing application if we have noisy image matrix and if we know what the noise matrix was added we can find the clear image by multiplying noisy image matrix with inverted noise matrix. Intro/Basics: We have considered two 8Ãâ€"8 matrices B and C. We suppose AÃâ€"B = C. Now by performing matrix multiplication on A and B we get C. Now we have to compute A from B and C. So AÃâ€"B = C and we have to proof A = CÃâ€"B-1. It is conceptually easy to compute AÃâ€"B = C and to find A = CB-1 for 2 dimensional matrices. But for large dimensional matrices it is not possible to easily compute because there is some round off errors in A which is the result of B-1 related to B’s condition number. Thecondition numberof a function with respect to an argument measures how much the output value of the function can change for a small change in the input argument. The condition number of a regular (square) matrix is the product of the norm of the matrix and the norm of its inverse and hence depends on the kind of matrix-norm. Condition number of a square nonsingular (invertible) matrix A is defined by: cond () = |||| ·|||| where the || ·|| above could be any of the norms defined for matrices. The numerical value of the condition number of an nÃâ€"n matrix depends on the particular norm used .The norm of a square matrix A is a non-negative real number denoted by ||A||. These matrix norms have the following properties: 1. ||A|| à ¯Ã¢â€š ¬Ã‚ ¾Ãƒ ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã¢â€š ¬Ã‚ ° if A ≠  0 2. ||à ¯Ã‚ Ã‚ §A|| à ¯Ã¢â€š ¬Ã‚ ½Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ §Ãƒ ¯Ã‚ Ã‚ ¼Ã‚ ·Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ ¼A|| for any scalar value à ¯Ã‚ Ã‚ §Ãƒ ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã¢â€š ¬Ã‚   à ¯Ã¢â€š ¬Ã‚ ³Ãƒ ¯Ã¢â€š ¬Ã‚ ®Ãƒ ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ ¼A|| à ¯Ã¢â€š ¬Ã‚ «Ãƒ ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ ¼B|| ≠¤ à ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ ¼A|| à ¯Ã¢â€š ¬Ã‚ «Ãƒ ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ ¼B||à ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã¢â€š ¬Ã‚   à ¯Ã¢â€š ¬Ã‚ ´Ãƒ ¯Ã¢â€š ¬Ã‚ ®Ãƒ ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ ¼AB|| ≠¤ à ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ ¼A|| ·Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ ¼B||à ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã¢â€š ¬Ã‚   à ¯Ã¢â€š ¬Ã‚ µÃƒ ¯Ã¢â€š ¬Ã‚ ®Ãƒ ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ ¼Ax|| ≠¤ à ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ ¼A|| ·Ãƒ ¯Ã‚ Ã‚ ¼Ãƒ ¯Ã‚ Ã‚ ¼||à ¯Ã¢â€š ¬Ã‚  Ãƒ ¯Ã¢â€š ¬Ã‚  for any vector The norm of a matrix is a measure of how large its elements are. It is a way of determining the â€Å"size† of a matrix that is not necessarily related to how many rows or columns the matrix has. Three commonly used norms are: 1. The 1-norm: = This is the maximum absolute column sum where simply we sum the absolute values down each column and then take the biggest answer. 2. The inifinity-norm: = This is the maximum absolute row sum where simply we sum the absolute values along each row and then take the biggest answer. 3. The Euclidean norm: = This is the square root of the sum of all the squares. However, regardless of the norm, this condition number is always greater or equal to 1. If it is close to one, the matrix is well conditioned which means its inverse can be computed with good accuracy. If the condition number is large, then the matrix is said to be ill-conditioned. Practically, such a matrix is almost singular (not invertible), and the computation of its inverse or solution of a linear system of equations is prone to large numerical errors. A matrix that is not invertible has the condition number equal to infinity. Mathematically, if the condition number is less than ∞, the matrix is invertible. Numerically, there are roundoff errors which occur. A high condition number means that the matrix is almost non-invertible. The higher the condition number, the greater is the error in the calculation. This condition number helps to estimate how difficult a matrix will be to numerically invert. This condition number has certain properties: 1. For any matrix A, cond (A) ≠¥Ãƒ ¯Ã¢â€š ¬Ã‚  1 2. For identity matrix, cond (I) = 1 3. For any matrix A and scalar à ¯Ã‚ Ã‚ §, cond à ¯Ã¢â€š ¬Ã‚ ¨Ãƒ ¯Ã‚ Ã‚ §Ãƒ ¯Ã¢â€š ¬Ã‚  A) = cond (A) 4. For any diagonal matrix D = Diag(di), cond (D) = (max |di|)/(min|di|) A matrix A is ill-conditioned if relatively small changes in the input (in the matrix A) can cause large change in the output (the solution of Ax = b), i.e. the solution is not very accurate if input is rounded. Otherwise it is well-conditioned. If a matrix is ill-conditioned, then a small roundoff error can have a drastic effect on the output. However, if the matrix is well-conditioned, then the computerized solution is quite accurate. Thus the accuracy of the solution depends on the conditioning number of the matrix. Objective: To know how to determine the matrix inverse in an efficient manner. If AÃâ€"B=C and we have to prove A=CÃâ€"B-1 where A, B and C are nÃâ€"n matrices (n = 8, 32, 128) and find out the condition number of matrix using norms and finding accuracy. System flow: Steps performed: 1. Taking two matrices B and C of order 8Ãâ€"8. 2. Performing Matrix multiplication and result is stored in matrix A (performed using C Code) 3. Now calculate B-1 (performed using C Code) 4. Now again multiplying C and B-1. We get result matrix which is not accurate. 5. We need to calculate norms and condition number of a matrix (B) so we need to find norms of B and B-1. We can calculate norms in different way. Here we have used most popularly used 3 types of norms to calculate condition number of that matrix (B) which we need to get in inverse form. The norms are: 1-norm = Infinity-norm = iii) Euclidean norm = 6. Now we use norms to find condition number of matrix B by using formula cond (B) = |||| ·|||| Flow Diagram yes no Math For 22 Matrix First we consider a 22 matrix such that A= B= So by multiplying A and B we ge a 22 matrix C = Now We need to prove A=CB-1 So we need to find B-1 B-1 = 0.800 -0.200 -0.600 0.400 So now by doing CxB-1 = =A (proved) Before finding B-1 we can calculate condition number of B for the correctness of above proof, As we know cond (B) = |||| ·|||| Condition number using the 1-norm and inifinity-norm: Formula used Row Sum taking absolute values B = 2 13 3 47 Column sum 5 5 (taking absolute values) (max) Row sum B-1 = 0.800 -0.200 1.000 -0.600 0.400 1.000 Col Sum 1.4 .6 Applying 1-Norm = = maximum absolute column sum = 5, 1 = 1.4, So, cond1 (B) =  ·1 = 5Ãâ€"1.4= 7 Applying infinity-norm = = max absolute row sum = 7, ∞ = 1 So, cond∞ (B) =  ·Ã¢Ë†Å¾ = 7 Like this way we have also found condition number using the Euclidean norm which is = =5.47 = 1.095 CondE (B) =  ·E = 5.82 Here cond(B) is low in all cases.so we successfully get A =C. Because of low condition number of B,the inverse of B is acceptable. For 88 Matrix A = 1 2 3 4 1 2 2 1 2 3 1 4 3 4 2 1 4 1 3 2 3 3 1 2 2 2 1 4 2 2 2 1 3 2 1 4 3 1 2 1 1 1 2 3 1 2 2 1 1 2 1 2 1 2 1 2 2 2 3 3 2 1 2 2 B= 4 1 3 2 3 3 1 2 2 3 1 4 3 4 2 1 2 2 1 4 2 2 2 1 1 1 2 3 1 2 2 1 2 2 3 3 2 1 2 2 1 2 3 4 1 2 2 1 1 2 1 2 1 2 1 2 3 3 1 3 2 3 1 1 C=AÃâ€"B=27 30 28 52 27 37 28 20 35 38 42 64 35 46 35 27 42 35 41 59 37 43 31 27 29 29 32 49 28 37 27 22 34 30 35 50 32 39 28 25 22 24 24 41 21 29 22 17 23 25 22 39 22 30 20 15 34 33 30 53 32 40 28 23 B-1= -0.016 -0.429 0.063 0.524 0.063 -0.397 -0.222 0.587 -0.365 0.143 -0.540 0.048 0.460 -0.127 -0.111 0.508 0.095 0.071 -0.381 -0.143 0.119 0.381 -0.167 -0.024 0.270 -0.214 0.921 -0.905 -0.579 0.746 0.278 -0.484 0.206 0.571 0.175 -0.810 0.175 0.159 -0.111 -0.635 0.079 0.143 -0.317 0.381 -0.317 -0.016 0.111 0.063 -0.571 0.071 -0.714 1.857 0.786 -1.286 -0.500 0.643 0.159 -0.214 0.365 -0.238 -0.135 -0.032 0.722 -0.373 A=CÃâ€"B-1 =0.995 1.983 3.029 3.987 1.029 1.984 2.006 0.979 1.992 2.975 1.035 3.983 3.035 3.980 2.005 0.972 3.989 0.971 3.029 1.984 3.029 2.981 1.006 1.970 1.993 1.980 1.027 3.987 2.027 1.984 2.004 0.977 2.991 1.976 1.027 3.986 3.027 0.983 2.004 0.974 0.996 0.986 2.022 2.990 1.022 1.987 2.004 0.983 0.994 1.986 1.021 1.991 1.021 1.988 1.005 1.982 1.992 1.979 3.028 2.987 2.028 0.983 2.007 1.975 Relative Error for A11=(1-.995)=.005,A12= 0.017 and so on When we perform CÃâ€" B-1, we do not get original value of A because of B-1. If B-1 is not accurate we will not get accurate A. To get accuracy of A-1 we need to find condition number of B. As we know cond (B) = |||| ·|||| Condition number using the 1-norm and inifinity-norm: Formula used Row Sum taking absolute values B = 4 1 3 2 3 3 1 2 19 2 3 1 4 3 4 2 1 20 (max) 2 2 1 4 2 2 2 1 16 1 1 2 3 1 2 2 1 13 2 2 3 3 2 1 2 2 18 1 2 3 4 1 2 2 1 16 1 2 1 2 1 2 1 2 12 3 3 1 3 2 3 1 1 17 Column sum 16 16 15 25 16 19 13 11 (taking absolute values) (max) B-1 = For B-1, Row sum (max) taking absolute values = 6.428 (7th row) and column sum(max) taking absolute values = 4.906 (4th column) Applying 1-Norm = = maximum absolute column sum = 25, 1 = 4.906, So, cond1 (B) =  ·1 = 25Ãâ€"4.906 = 122.65 Applying infinity-norm = = max absolute row sum = 20, ∞ = 6.428 So, cond∞ (B) =  ·Ã¢Ë†Å¾ = 20Ãâ€"6.428 = 128.56. Like this way we have also found condition number using the Euclidean norm which is = 17.83. So here we can say that as the condition number of matrix B is high for all three cases, therefore the inverse of this matrix is showing numerical roundoff errors. Concept of Relative Error and Condition Number assume A is nonsingular and Ax = b if we change b to b + à ¢Ã‹â€ Ã¢â‚¬  b, the new solution is x + à ¢Ã‹â€ Ã¢â‚¬  x with A(x + à ¢Ã‹â€ Ã¢â‚¬  x) = b + à ¢Ã‹â€ Ã¢â‚¬  b the change in x is à ¢Ã‹â€ Ã¢â‚¬  x = A-1à ¢Ã‹â€ Ã¢â‚¬  b ‘condition’ of the solution †¢ the equations are well-conditioned if small à ¢Ã‹â€ Ã¢â‚¬  b results in small à ¢Ã‹â€ Ã¢â‚¬  x †¢ the equations are ill-conditioned if small à ¢Ã‹â€ Ã¢â‚¬  b can result in large à ¢Ã‹â€ Ã¢â‚¬  x [Singular matrix:A square matrix is called singular matrix if it’s determinant is zero.i.e. a singular matrix is not invertible] Example: Consider the linear system Ax = b with So = So here we easily find x= Now ,we change a small in b.let change in b is à ¢Ã‹â€ Ã¢â‚¬  b= So changed value= and solving the system A = we get =A= where x= changed to = due to small change in b. Now to calculate least condition number of the system we need to find Relative Error in the output and relative error in the input. Here we have relative error in the input/relative residual. = 0.01 Relative Error in the output =1 As we know, If condition number is closed to 1 then relative error and relative residual will be close. The condition number is defined by: Relative error in the output =Condition number Ãâ€" Relative error in the input. So,condition number= 1/.01=100 A matrix has high condition number is related to the fact that A is close to the singular matrix B= The following result shows that 1/cond(A) indicates how close A is to a singular matrix.Here cond(A) is 100 so, 1/cond(A)=.01 which is close enough. Description: The condition number associated with the  linear equation  Ax=bgives a bound on how inaccurate the solutionxwill be after approximation. This is before the effects of  round-off error  are taken into account; conditioning is a property of the matrix. Weshould think of the condition number as being the rate at which the solution,x, will change with respect to a change inb. Thus, if the condition number is large, even a small error inbmay cause a large error inx. On the other hand, if the condition number is small then the error inxwill not be much bigger than the error inb. The condition number may also be infinite, but this implies that the problem does not possess a unique, well-defined solution for each choice of data that is, the matrix is not invertible, and no algorithm can be expected to reliably find a solution. For large dimensional matrix such as for 3232 and 128128, the condition number is high and so inverse of that large dimensional matrix will give much error in output. Codes and Output Matrix multiplication int main() { int m, n, p, q, c, d, k, sum = 0; int A[10][10], B[10][10], C[10][10]; printf(Enter rows and columns of An); scanf(%d%d, m, n); printf(Enter the elements of An); for (c = 0; c for (d = 0; d scanf(%d, A[c][d]); printf(Enter rows and columns of Bn); scanf(%d%d, p, q); printf(Enter the elements of Bn); for (c = 0; c for (d = 0; d scanf(%d, B[c][d]); for (c = 0; c for (d = 0; d for (k = 0; k sum = sum + A[c][k]*B[k][d]; } C[c][d] = sum; sum = 0; } } for (c = 0; c for (d = 0; d printf(%dt, C[c][d]); printf(n); } getch(); } Matrix inverse #include #include int main() { float a[10][10],b[10][10],tem=0,temp=0,temp1=0,temp2=0,temp4=0,temp5=0; int n=0,m=0,i=0,j=0,p=0,q=0; printf(Enter size of 2d array(Square matrix) : ); scanf(%d,n); for(i=0;i { for(j=0;j { printf(Enter element no. %d %d :,i,j); scanf(%f,a[i][j]); if(i==j) b[i][j]=1; else b[i][j]=0; } } for(i=0;i { temp=a[i][i]; if(temp temp=temp*(-1); p=i; for(j=i+1;j { if(a[j][i] tem=a[j][i]*(-1); else tem=a[j][i]; if(temp temp=temp*(-1); if(tem>temp) { p=j; temp=a[j][i]; } } //row exchange in both the matrix for(j=0;j { temp1=a[i][j]; a[i][j]=a[p][j]; a[p][j]=temp1; temp2=b[i][j]; b[i][j]=b[p][j]; b[p][j]=temp2; } //dividing the row by a[i][i] temp4=a[i][i]; for(j=0;j { a[i][j]=(float)a[i][j]/temp4; b[i][j]=(float)b[i][j]/temp4; } //making other elements 0 in order to make the matrix a[][] an indentity matrix and obtaining a inverse b[][] matrix for(q=0;q { if(q==i) continue; temp5=a[q][i]; for(j=0;j { a[q][j]=a[q][j]-(temp5*a[i][j]); b[q][j]=b[q][j]-(temp5*b[i][j]); } } } printf(nnn); printf(Inverse of the matrix using Guass jordan elimination method:nn); for(i=0;i { for(j=0;j { printf(%.3f,b[i][j]); } printf(n); } getch(); } Matrix Condition Number #include #include int main() { int i,j,n,p,x=0,m=0,q,z=0,i1,j1; float Cond_A,poo,a[5][5],b[5],c[5],A[50][50],B[50][50],k[50],l[50]; printf(n n); printf(Program to find condition number of a matrix using infinity-norm); printf(n nn); printf(Enter rows and columns of An); scanf(%d%d, m, n); printf(Enter the elements of An); for (i = 0; i for (j = 0; j scanf(%f, A[i][j]); for(i=0;i { b[x]=0;c[x]=0; for(j=0;j { b[x]=b[x]+A[i][j]; } ++x; } for(i=0;i //FINDING LARGEST { if(b[i]>m) m=b[i]; } printf(largest row sum is %d,m); printf(nnEnter rows and columns of inv[A]n); scanf(%d%d, p, q); printf(Enter the elements of [A]n); for (i1 = 0; i1 for (j1 = 0; j1 scanf(%f, B[i1][j1]); for(i1=0;i1 { k[z]=0;l[z]=0; for(j1=0;j1 { k[z]=k[z]+B[i1][j1]; } ++z; } poo = k[0]; for(i1=1;i1 //FINDING LARGEST { if(k[i1]>poo) poo=k[i1]; } printf(largest row sum is %f,poo); Cond_A=m*poo; printf(nnCondition number of A is %f,Cond_A); //return 0; getch(); } Future works: If we work with a foggy image matrix(C) and we know the fog matrix(B) added to that image and the relation AÃâ€"B = C exist we will know whether it is possible to get the clear image matrix(A) by doing CÃâ€"B-1 calculating condition number of matrix B. If the condition number of matrix B is high then it is not possible to get accurate A from CÃâ€"B-1 as roundoff errors will increase. References: Matrix Inverse and Condition, Berlin Chen, Department of Computer Science Information Engineering, National Taiwan Normal University. Inversion error, condition number, and approximate inverses of uncertain matrices,  Laurent El Ghaoui, Department of Electrical Engineering and Computer Science,  University of California at Berkeley, Berkeley, CA 94720, USA. faculty.nps.edu/rgera/MA3042/2009/ch7.4.pdf www.rejonesconsulting.com/CS210_lect07.pdf http://teal.gmu.edu/ececourses/ece699/notes/note4.html Weisstein, Eric W. Matrix Norm. From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/MatrixNorm.html

Essay on Pointing the Finger in John Milton’s Paradise Lost

Pointing the Finger in Paradise Lost  Ã‚   After the fall in Milton’s Paradise Lost, Adam and Eve bicker and blame one another for their decent. First, Adam accuses Eve for her physical act of accepting the apple from Satan and eating it, thus defying God’s decree not to eat from the Tree of Knowledge. In retaliation, Eve responds and attempts to not only justify her act, but also to place the blame on Adam. Eve’s reaction is typical of someone who does not like to admit he is wrong. Eve begins by challenging Adam with an argument that he would have done the same thing had he been in her situation. "[Had’st] thou been there,/ Or here th’ attempt, thou couldst not have discern’d/ Fraud in the Serpent, speaking as he spake" (IX 1148-1150). She is trying to justify her action in Adam’s mind by making him realize he would have acted the same way, and in effect she also hopes to gain his sympathy. This tactic is often effective because we do not tend to choose to recognize faults in others when we realize we are susceptible to the same mistakes. Aristotle recognizes the relationship between eliciting sympathy and making the audience relate to the situation in his Poetics when he describes the ideal character as one who is "true to life" (81). An audience must be able to relate to a falling character, or else they will not pity his plight. In other words, if a speaker wants sympathy from his audience, he must make them "feel his pain." Eve proceeds in her rebuttal with justification for her action based on the circumstances of the scenario. She argues, "No ground of enmity between us known,/ Why hee should me ill of seek to harm" (II 1151-1152). She seems to imply that a less trusting person would not have listened to the se... ...ccusation that he did not try hard enough to keep her at his side when he asks, "What could I more?/ I warn’d thee, I admonish’d thee, foretold/ The danger, and the lurking Enemy/ That lay in wait" (IX 1170-1172). No matter how developed any one piece of Aristotle’s triangle seems, it is useless without the other two parts. If you take a step back and observe this scene of Paradise Lost with your own experiences in mind, you realize how petty the "blame game" can be. Eve tries very hard to use persuasion as a "finger pointing" tool so she can alleviate any guilt of her own. However, she fails. We all must know what it feels like to fail in an argument, yet we still test situations like these sometimes when we do not want to accept full responsibility for something that has gone wrong. Works Cited   Milton, John. Paradise Lost. New York: W.W. Norton, 1993.   

Lord Of The Flies Story :: essays research papers

The war was over, but not without casualties. Numerous ships, planes and factories had been bombed with regular bombs, and London was totally destroyed - by a single atomic bomb. Only derelicts lived there now, not knowing of the harmful radiation that surrounded what once was the great capital.Slowly though, things were getting back to normal, shops began opening, and the war started to shift to the back of people's minds. Life was becoming more normal, but the memory of the war was still there, and always would be. Unfortunately, my memories of our time on the island would never shift to the back of my mind - I would never forget what the war meant to me. I might not have been killed, though I wish I had been, I might not have lost my home or seen London go up a huge mushroom cloud of bright red light, but I had lived with people that hardly deserve that title, people that became animals, people that were worse than animals - murderers. And I myself, was one of them.I was slowly getting better, but I never got through a night without hearing the chant and Simon's furtive screams as we jabbed, attacked and mauled him, without seeing Piggy's fat body flying into the air and then crashing down on the rocks. I had never recovered from those experiences and at no time had a job. I live in a small, damp apartment above a coin laundry. The greatest pleasure I had in life was feeding the pigeons in the park. I would get what stale bread I could from the bakery to feed the birds. They would rush at me when they saw me and fight for the small scraps of bread before it had even hit the ground. They were animals, but not like Jack and the others, they were kind animals, they never hurt anyone - they went about their lives regardless of us. Many times I wished that I too could be a pigeon. I wanted to lead the simple life that they did, I wanted to forget what had happened, I wanted to escape the plaguing memories.It was Tuesday morning and I was feeding pigeons in the park. They were fighting for the pathetic scraps of stale bread I threw on the ground as usual, looking at me begging me with their eyes for more.

Themes in A Portrait of the Artist as a Young Man by James Joyce :: A Portrait of the Artist as a Young Man

     Ã‚   James Joyce's A Portrait of the Artist as a Young Man is a novel of complex themes developed through frequent allusions to classical mythology.   The myth of Daedalus and Icarus serves as a structuring element in the novel, uniting the central themes of individual rebellion and discovery, producing a work of literature that illuminates the motivations of an artist, and the development of his individual philosophy.   Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   James Joyce chose the name Stephen Dedalus to link his hero with the mythical Greek hero, Daedalus.   In Greek myth, Daedalus was an architect, inventor, and artisan.   By request of King Minos, Daedalus built a labyrinth on Crete to contain a monster called the Minotaur, half bull and half man.   Later, for displeasing the king, Daedalus and his son Icarus were both confined in this labyrinth, which was so complex that even its creator could not find his way out.   Instead, Daedalus fashioned wings of wax and feathers so that he and his son could escape.   When Icarus flew too high -- too near the sun -- in spite of his father's warnings, his wings melted, and he fell into the sea and drowned.   His more cautious father flew to safety (World Book 3).   By using this myth in A Portrait of the Artist as a Young Man (Portrait of the Artist), Joyce succeeds in giving definitive treatment to an archetype that was well established long before the twentieth century (Beebe 163).   Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   The Daedalus myth gives a basic structure to Portrait of the Artist.   From the beginning, Stephen, like most young people, is caught in a maze, just as his namesake Daedalus was.   The schools are a maze of corridors; Dublin is a maze of streets.   Stephen's mind itself is a convoluted maze filled with dead ends and circular reasoning (Hackett 203): Met her today point blank in Grafton Street.   The crowd brought us together.   We both stopped.   She asked me why I never came, said she had heard all sorts of stories about me.   This was only to gain time.   Asked me, was I writing poems?   About whom?   I asked her.   This confused her more and I felt sorry and mean.   Turned off that valve at once and opened the spiritual-heroic refrigerating apparatus, invented and patented in all countries by Dante Alighieri. (Joyce 246)    Life poses riddles at every turn.   Stephen roams the labyrinth searching his mind for answers (Gorman 204).

Can Doctors Go on Strike?

CAN DOCTORS GO ON STRIKE? The answer is simply yes since strike is a legitimate action to frown on unfavorable conditions. Strike action is legal and any institution that has legal rights can embark on strike when it deems fit by the abiding conditions. If doctors qualify under this democratic legitimacy, then under which conditions should doctors be on strike? * WHY DOCTORS GO ON STRIKE? From our history as Ghanaians, we have seen doctors going on strike for salary increment.These urgencies may be as result of the following: From global perspective, the medical profession is indeed one of the various professions that carry dignity and every doctor has this rooted deep in his or her mind. The cost that involves becoming a medical doctor is less talked about. It really cost. For that matter many doctors don't see it reasonable to leave on a meager salary. Another point might be the gravity of their services to patients. Many doctors do heart transplanting, fix bones, treat cancer, kid ney, liver etc.These works are very delicate and any careless attempt will end up paralyzing or resulting in the death of the patient. They are called deep in the night to respond to emergencies and many other emergency engagements. Doctors have also argued that the risky nature of their profession demands a higher salary. Sometimes they are prone to contagious diseases like flu, HIV/AID's, TB, and over 200 more contagious deadly diseases. For that matter a high salary must be given to encourage or motivate them. All these points are reasonable though debatable. But often when we talk of risky professions, medical sector is the least talked about.We talk about areas like the military, veterinary, police, fire service; prisons etc. are all coupled with high risk and are sectors that need great attention. * POSSIBLE REASONS AGAINST DOCTORS STRIKE ACTION Even though some international researches have proven that, doctors' strike increases mortality rate, nevertheless, we can not overlo ok to project consequences that comes as a result of doctors' strike. 1. Untimely death of patients 2. Prolongation of suffering of patients in severe cases 3. Breach of international code of ethics that doctors themselves have sworn into 4. Put pressure on government . Sensitizes other equally rated professions to follow suite 6. The public might see them as greedy other than selfless civil servants 7. Not ethical in its nature since it involves humans' life 8. Their profession is no greater than others, etc. * THE QUESTION OF ETHICS In most cases, doctors' strike has attracted many protests because the public think they are risking the lives of millions and however not ethical. Can ethics be applied to doctors' strike? First, every doctor is already bonded by code of ethics. Each doctor before taking up the post swears to abide by a code of ethics.One of the lines in the international code of ethics for doctors reads like this, â€Å"I will maintain the utmost respect for human l ife from its beginning even under threat and I will not use my medical knowledge contrary to the laws of humanity;† The international code of ethics further states, â€Å"A physician shall not permit motives of profit to influence the free and independent exercise of professional judgment on behalf of patients. † To the doctor, the life of the patient comes first to all other things. Doctors are also legally required to attend to patients and offer to them an undivided medical attention.This legality exists between the doctor and the government. Though the patient has intent of receiving an excellent care and treatment after paying his NHIS, in cases of strike, the patient has no legal right to sue the doctor in question, unless a private or personal doctor. He can only sue the NHIS or Ministry of Health. The issues of ethics comes into play when we talk of issues of good and evil, live and death, justice and crime, etc. Since the doctors’ primary job is to save lives and that a lost life cannot be revived, it therefore becomes delicate for the doctor when taking any action that will obliterate these ethics.Should a patient loss his or her life as a result of doctors' strike, who would be blamed? The doctor or the government? * THE ISSUE OF EGOTISM When we are talking of labor strike, we must consider the consequential results also. This is what we call utilitarianism. From Wikipedia, â€Å"Utilitarianism is an ethical theory holding that the proper course of action is the one that maximizes the overall â€Å"good† of the greatest number of individuals. It is thus a form of consequentialism, meaning that the moral worth of an action is determined by its resulting outcome. When doctors think an action is right, then it must have a national impact as well. I am yet to offer a round of applause for doctors embarking on a strike action to improve quality health care products, laboratory equipment, enough patient wards, digital and IT in fusion into the health practice, to stop nurses' and medical personnel's migration and others that bring a unanimous good for the majority of the people in the country. Strike actions under such circumstances are justifiable and must be given an immediate support and attention.