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Design and Optimization of Thermal Systems Episode 2 Part 1

Tham khảo tài liệu 'design and optimization of thermal systems episode 2 part 1', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 222 Design and Optimization of Thermal Systems containing the root. The iterative process is continued reducing the interval at each step until the change in the approximation to the root from one iteration to the next is less than a chosen convergence criterion as given by x 1 1 - x l E or x l 1 - x l x l e 4.11 where x l 1 and x l represent approximations to the root after the l 1 th and l th iterations respectively and e is the chosen convergence parameter. Probably the most important and widely used method for root solving is the Newton-Raphson method in which the iterative approximation to the root Xị is used to calculate the next iterative approximation to the root Xị 1 as x x. - f Xi 4.12 i 1 i f x . where f x is the derivative off x at x Xị. This equation gives an iterative process for finding the root starting with an initial guess x1. The process is terminated when the convergence criterion given by Equation 4.11 is satisfied. The Newton-Raphson method can be used for real as well as complex roots employing complex algebra for the functions their derivatives and for x. It can also be used for multiple roots where the graph of f x versus x is tangent to the x-axis with no sign change in f x . When the scheme converges it converges very rapidly to the root. It can be shown that it has a second-order convergence implying that the error in each iteration varies as the square of the error in the previous iteration and thus reduces very rapidly. However the iteration process may diverge depending on the initial guess and nature of the equation. Figure 4.5 shows graphically the iterative process in a convergent case. The tangent to the curve at a given approximation is used to obtain the next approximation to the root. Figure 4.6 shows a few cases in which the method diverges. If the scheme diverges a new starting point is chosen and the process repeated. A method similar to the Newton-Raphson method is the secant method which uses interpolation and .