Newton's Method is a mathematical tool often used in numerical analysis, which serves to approximate the zeroes or roots of a function (that is, all x:f(x)=0).
The method is constructed as follows: given a function f(x) defined over the domain of real numbers x, and the derivative of said function (f'(x)), one begins with an estimate or "guess" as to where the function's root might lie. For example, suppose one is presented with the function f(x)=x2+x−2.5. This is similar to another function g(x)=x2+x−2, whose roots are x=1 and x=−2. Thus, thanks to this similarity, one might use x=1 or x=−2 as guesses to start Newton's Method with f(x).
(Alternately, if a graphical representation is available but the exact root is not listed, an acceptable approximation might be the nearest whole number to the root).
Whatever method used, we declare this initial guess to be x0. We arrive at a better approximation, x1, by employing the Method: x1=x0−f(x0)f'(x0). Essentially, by utilizing the derivative, one is able to increment closer to the actual value. In the above example, f(x)=x2+x−2.5, if we assume x0=1, then x1=1−f(1)f'(1)=1−−.53=76or≈1.16667.
Often, one may be able to find the root another way (by using a graphing calculator, for example), and an exam item or textbook problem may demand a certain degree of accuracy (such as within 1% of the actual value). In such a case, if x1 is not an accurate enough approximation, one performs the iteration again, as often as needed for the desired degree of accuracy. The formula to find the general xn, then, is xn=xn−1−f(xn−1)f'(xn−1)
pls answer e. 5. Newton's Method a. Discuss the use of Newton's method to approximate solutions...
d. Use the solution found in part e as an initial guess with Newton's method and a = 0.1, to obtain a good estimate for the solution to the nonlinear system -200u1 + 100u2 = sin(0.1) + au ? 1001 - 20042 + 100u3 = sin(0.2) + auz? 100uz – 200uz + 100u4 = sin(0.3) + auz 100uz – 20014 + 100u5 = sin(0.4) + au42 10004 - 200us + 1000g = sin(0.5) + aus? 100ug - 2004+ 100u, =...
in matlab -Consider the equation f(x) = x-2-sin x = 0 on the interval x E [0.1,4 π] Use a plot to approximately locate the roots of f. To which roots do the fol- owing initial guesses converge when using Function 4.3.1? Is the root obtained the one that is closest to that guess? )xo = 1.5, (b) x0 = 2, (c) x.-3.2, (d) xo = 4, (e) xo = 5, (f) xo = 27. Function 4.3.1 (newton) Newton's method...
(b) . Write the k-th step of the trapezoidal method as a root-finding problem Ğ = is Y+1 where the unknown (e)Find the Jacobian matrix of the vector function from the previous part. (dWrite a function in its own file with definition [Y] dampedPendulum(L, T) function alpha, beta, d, h, that approximates the solution to the equivalent system you derived in part (a) with L: the length of the pendulum string alpha: the initial displacement beta: the initial velocity d:...
please answer all prelab questions, 1-4. This is the prelab manual, just in case you need background information to answer the questions. The prelab questions are in the 3rd photo. this where we put in the answers, just to give you an idea. Lab Manual Lab 9: Simple Harmonic Oscillation Before the lab, read the theory in Sections 1-3 and answer questions on Pre-lab Submit your Pre-lab at the beginning of the lab. During the lab, read Section 4 and...
I need Summary of this Paper i dont need long summary i need What methodology they used , what is the purpose of this paper and some conclusions and contributes of this paper. I need this for my Finishing Project so i need this ASAP please ( IN 1-2-3 HOURS PLEASE !!!) Budgetary Policy and Economic Growth Errol D'Souza The share of capital expenditures in government expenditures has been slipping and the tax reforms have not yet improved the income...