Newton_Rapson method.pdf
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The document describes Newton's method for finding the roots of a function. It works by using the tangent line at a given point as a linear approximation of the function. The x-intercept of this line is then used as the next estimate in the sequence. The process is repeated, using the new point's tangent line each time, with the estimates getting closer to a root as long as the linear approximations remain accurate. While faster than bisection, Newton's method can fail if the derivative approaches zero at a turning point, causing the linear model to become flat and send the estimates far off course.
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numerical calculation. hence we know the fee of the underlying characteristic f(x) at the set
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The following document presents some novel numerical methods valid for one and several variables, which
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real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear
lassomodel, sparsity, multivariate modeling, NIR spectroscopy, biodiesel from...
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible that the method has at most an order of convergence (at least) linear. Keywords: Iteration Function, Order of Convergence, Fractional Derivative.
Book chapter-5
The document discusses Markov chains and their relationship to random walks on graphs and electrical networks. Some key points:
- A Markov chain is a process that transitions between a finite set of states based on transition probabilities that depend only on the current state.
- For a strongly connected Markov chain, there exists a unique stationary distribution that the long-term probabilities of the chain converge to, regardless of the starting state.
- Random walks on undirected graphs can be modeled as Markov chains, where the transition probabilities are proportional to edge conductances in an analogous electrical network. The stationary distribution of such a random walk is proportional to vertex degrees or conductances.
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- 1. 1 Example
- 2. 2 Newton - Rapson Method Supposewe know the function f and its derivative f’ at any point xc. The tangent line drawn at xc is defined by: can be thoughtof as a linear model ' c c c c L x f x x x f x Newton’s Method cont. Thezero of the linear model Lc is given by: If Lc is a good approximationto f over a wide interval then x+ should be a good approximation to a root of f ' c c c f x x x f x Newton’s Method cont. Repeat the formulato create an algorithm: If at each step the linear model is a good approximationto f then xn should get closer to a root of f as n increases. 1 ' n n n n f x x x f x
- 3. 3 (x1,f(x1)) (x2=x1-f(x1)/f’(x1),0) First stage: (x2,f(x2)) (x3=x2-f(x2)/f’(x2),0) Notice: we are getting closer Zoom in for second stage Convergence of Newton’s Method Wecan show that the rate of convergenceis much faster than the bisection method. However– as always, there is a catch. The method uses a local linear approximation,which clearly breaks down neara turning point. Small f’(xn) makes the linear model very flat and will send the search far away …
- 4. 4