Newtons Method :: Newton-Raphson Method

Finding roots of a function is often a task which faces mathematicians. For simple functions, such as linear ones, the task is simple. When functions become more complex, such as with cubic and quadratic functions, mathematicians call upon more convoluted methods of finding roots. For many functions, there exist formulas which allow us to find roots. The most common such formula is, perhaps, the quadratic formula. When functions reach a degree of five and higher, a convenient, root-finding formula ceases to exist. Newton’s method is a tool used to find the roots of nearly any equation. Unlike the cubic and quadratic equations, Newton’s method – more accurately, the Newton-Raphson Method – can help to find roots of nearly any type of function, including all polynomial functions.   Ã‚  Ã‚  Ã‚  Ã‚  Newton’s method use derivative calculus to find the roots of a function or relation by first taking an approximation and then improving the accuracy of that approximation until the root is found. The idea behind the method is as follows. Given a point, P(Xn,Yn), on a curve, a line tangent to the curve at P crosses the X axis at a point whose X coordinate is closer to the root than Xn. This X coordinate, we will call Xn+1. Repeating this process using Xn+1 in place of Xn will return a new Xn+1 which will be closer to the root. Eventually, our Xn will equal our Xn+1. When this is the case, we have found a root of the equation. This method may be unnecessarily complex when we are solving a quadratic or cubic equation. However, the Newton-Raphson Method compensates for its complexity in its breadth. The following examples show the versatility of the Newton Raphson Method.   Ã‚  Ã‚  Ã‚  Ã‚  Example 1 is a simple quadratic function. The most practical approach to finding the roots of this equation would be to use the quadratic equation or to factor the polynomial. However, the Nowton-Raphson method still works and allows us to find the roots of the equation. The initial number, Xn, 3, is a relatively poor approximation. The choice of 3 illustrates that the initial guess can be any number. However, as the initial approximation worsens, the calculation becomes more laborious.   Ã‚  Ã‚  Ã‚  Ã‚  Example 2 demonstrates one of the advantages to Newton’s method. Function 2 is a Quintic function. Mathematician, Niels Henrik Abels proved that there exists no convenient equation, such as the cubic equation, which can help us find the function’s roots.

Dark Brown Dog by Stephen Crane Essay

_A Dark Brown Dog_ is a short story by the praised realistic author Stephen Crane. Within this piece, Crane takes a different approach to the boy and dog theme by creating an atmosphere of abusiveness. This trait is well incorporated by Crane and can be easily found within the characters that Crane had carefully crafted. Specifically, the abusive trait can be found in varying levels within the father and the young boy. And while the prominence of this sadistic trait remains in the story, the trait helps create a a change in boy and dog’s relationship. In the story, the father’s presence is accompanied by and is synonymous with sadism and abuse. The first incident in which the father appears is when the boy first brings the dark brown dog to his house. Naturally, the family clamors about the dog and scrutinize it to the point where the boy is protesting loudly as to keep the dog. At that moment the father walks into the house in the midst of the yelling and, â€Å"perce ives that it would amaze and anger everybody if such a dog were allowed to remain, he decided that it should be so† (Crane 3). This reflects how the father takes enjoys forcing the family to put up with the presence of the unwanted dog. The incident is also representative of how the father finds pleasure in others suffering, a tenet of sadism. While this incident may not seem to define the father as an abuser, it remains as a hint of the full extent of the father’s sadistic nature. In fact, the entire magnitude of the father’s abusiveness is encapsulated by the turning point of the story in which the boy and the dog come home from one of their adventures only to arrive to the sight of the father throwing cooking utensils in a drunken rage. The sight drove the boy to seek shelter from under a table while the dog trounced around the room excited by the spectacle, unaware of the danger. The dog’s vulnerability and naivety led the father to take advantage of it as he beat the dog with a pan, force the dog into submission, and eventually, â€Å"swung him two or three times hilariously about his head, and then flung him with great accuracy through the window† (Crane 6). The father was clearly enlightened by his ability to prey upon a defenseless dog and to murder him without a second thought. Not only that, but this incident showed how father truly wanted to make the dog suffer as the father unnecessarily twirled the dog around before killing him. To a lesser extent, the boy also exhibits the abusive trait that is found in  his father and instead of inflicting pain on everyone, the boy abuses the dog. For instance, when the boy and the dog first encounter one another, the dog acts playfully with the boy who promptly hit the dog and sent him into a prayer like stance showing the dog’s submission. This submission was foud comical as it was stated that, â€Å"the child was greatly amused and gave him little taps repeatedly, to keep him so† (Crane 1). Superficially, this depicts the child simply amusing himself. However, there exists an underlying layer that reveals the sadistic trait of the child as he finds delight in the dog’s pain and fright. It shows how the child extracts a feeling of superiority over the dog which Crane further establishes in his story by describing the child as a terrible despot and the dog as a subject. In addition to this occurrence, the child had displayed his abusive nature on a separate instance. Crane describes this instance as a general and recurring case as he describes how, â€Å"Sometimes, too, the child would beat the dog, although it is not known that he ever had what truly could be called a just cause† (Crane 4). What is established here is that the child harms the dog for no reason. Through the conclusion derived previously, it is inferred that the child had done this deed because he found pleasure in exerting his might over the dog. This reestablishes how the child exhibits the sadistic traits which his father also bears, only to a lesser degree. Pointedly, Crane crafts the boy to be reflective of his father’s habits. This is done to show the cyclical nature of abuse and how the ones being abused may develop sadistic traits themselves. Regardless, of this fact Crane proceeds to simultaneously illustrate how abuse can shape a relationship. Specifically, the relationship being altered one existing between the boy and the dog. As noted, the boy and the dog first start off with an abusive relationship with power and dominance being held by the boy and a subservient role taken upon by the dog. This is perfectly encapsulated by the fact the dysfunctional family of the boy would often go out of their way to harm the dog and as a result, â€Å"The child became a guardian and friend† (Crane 3), to the dog. This is descriptive of how the abusive nature of the boy’s home had forced him to take upon the role of a protector. What’s more, is that the abusive atmosphere of his home had led the boy to grow a fondness for his new companion and to strive to protect to dog instead of  letting him befall to harm. What solidifies this conclusion is the ending of the story, after the father threw the dog out the window, the family had found the boy, â€Å"seated by the body of his dark brown friend† (Crane 6). When taken a step back, what this shows is how the abuse and sadism surrounding the boy and the dog had altered their relationship from an abusive one to one based upon loyalty and friendship. In conclusion, within the 6 short pages of a _Dark Brown Dog_, Stephen Crane creates characters with a shared and prominent trait of abusiveness. These characters were primarily the father, and to a lesser, but still significant extent, the boy. And while this trait had shaped the characters, it also played a role in defining the course of the boy and dog’s relationship. Case in point, the abusive trait in the story had been integral to _Dark Brown Dog._

How to Conjugate the Spanish Verb Oír

Oà ­r is a common Spanish verb that typically means to hear. It is highly irregular, in part because of its use of an accent in its ending. The only other verbs that are conjugated in the same way as oà ­r are those few verbs based on oà ­r, namely desoà ­r (to ignore), entreoà ­r (to half-hear or hear indistinctly) and trasoà ­r (to mishear). Irregular forms are shown below in boldface. Translations are given as a guide and in real life may vary with context. Infinitive of Oà ­r oà ­r (to hear) Gerund of Oà ­r oyendo (hearing) Participle of Oà ­r oà ­do (heard) Present Indicative of Oà ­r yo oigo, tà º oyes, usted/à ©l/ella oye, nosotros/as oà ­mos, vosotros/as oà ­s, ustedes/ellos/ellas oyen (I hear, you hear, he hears, etc.) Preterite of Oà ­r yo oà ­, tà º oà ­ste, usted/à ©l/ella oyà ³, nosotros/as oà ­mos, vosotros/as oà ­steis, ustedes/ellos/ellas oyeron (I heard, you hears, she hears, etc.) Imperfect Indicative of Oà ­r yo oà ­a, tà º oà ­as, usted/à ©l/ella oà ­a, nosotros/as oà ­amos, vosotros/as oà ­ais, ustedes/ellos/ellas oà ­an (I used to hear, you used to hear, he used to hear, etc.) Future Indicative of Oà ­r yo oirà ©, tà º oirà ¡s, usted/à ©l/ella oirà ¡, nosotros/as oiremos, vosotros/as oirà ©is, ustedes/ellos/ellas oirà ¡n (I will hear, you will hear, he will hear, etc.) Conditional of Oà ­r yo oirà ­a, tà º oirà ­as, usted/à ©l/ella oirà ­a, nosotros/as oirà ­amos, vosotros/as oirà ­ais, ustedes/ellos/ellas oirà ­an (I would hear, you would hear, she would hear, etc.) Present Subjunctive of Oà ­r que yo oiga, que tà º oigas, que usted/à ©l/ella oiga, que nosotros/as oigamos, que vosotros/as oigà ¡is, que ustedes/ellos/ellas oigan (that I hear, that you hear, that she hear, etc.) Imperfect Subjunctive of Oà ­r que yo oyera (oyese), que tà º oyeras (oyeses), que usted/à ©l/ella oyera (oyese), que nosotros/as oyà ©ramos (oyà ©semos), que vosotros/as oyerais (oyeseis), que ustedes/ellos/ellas oyeran (oyesen) (that I hears, that you hears, that he hears, etc.) Imperative of Oà ­r oye tà º, no oigas tà º, oiga usted, oigamos nosotros/as, oà ­d vosotros/as, no oigà ¡is vosotros/as, oigan ustedes (hear, dont hear, hear, lets hear, etc.) Compound Tenses of Oà ­r The perfect tenses are made by using the appropriate form of haber and the past participle, oà ­do. The progressive tenses use estar with the gerund, oyendo. Sample Sentences Showing Conjugation of Oà ­r and Related Verbs Tenemos dos objetivos: erradicar las barreras de la discriminacià ³n que enfrentan los sordos y ofrecer empleo a los que no pueden oà ­r. (We have two goals: to eradicate the discriminatory barriers facing the deaf and to offer work to those who cannot hear. Infinitive.) Todos hemos oà ­do que  «lo que cuenta es lo que està ¡ dentro ». (Weve all heard that what counts is whats inside. Present perfect.) Desoyes todo lo que no te interesa. (Youre ignoring everything that doesnt interest you. Present indicative.) Entreoyà ³ una conversacià ³n al otro lado de la puerta. (She half-heard a conversation on the other side of the door. Preterite.) Aquella noche yo oà ­a la lluvia desde la cama y pensaba en ti. (That night I heard the rain from the bed and thought about you. Imperfect.) Es cierto que lo oirà © cada vez que pase por aquà ­. (It is certain that I will hear it every time it passes by here. Future.) Los dispositivos permiten restaurar la audicià ³n en personas que no oirà ­an de otro modo. (The devices provide for the restoration of hearing in people who wouldnt hear any other way. Conditional.)  ¡Desgraciados de los que desoigan mis palabras! (How wretched are those who mishear my words! Present subjunctive.) Yo no querà ­a que oyeras esto. (I didnt want you to hear this. Imperfect subjunctive.)  ¡Oye, oye! (Hear ye, hear ye! Imperative.)