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# lecture3 - Lecture 3 Newtons Method and Loops Solving...

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Lecture 3 Newton’s Method and Loops Solving equations numerically For the next few lectures we will focus on the problem of solving an equation: f ( x ) = 0 . (3.1) As you learned in calculus, the final step in many optimization problems is to solve an equation of this form where f is the derivative of a function, F , that you want to maximize or minimize. In real engineering problems the function you wish to optimize can come from a large variety of sources, including formulas, solutions of differential equations, experiments, or simulations. Newton iterations We will denote an actual solution of equation (3.1) by x . There are three methods which you may have discussed in Calculus: the bisection method, the secant method and Newton’s method. All three depend on beginning close (in some sense) to an actual solution x . Recall Newton’s method. You should know that the basis for Newton’s method is approximation of a function by it linearization at a point, i.e. f ( x ) f ( x 0 ) + f ( x 0 )( x - x 0 ) . (3.2) Since we wish to find x so that f ( x ) = 0, set the left hand side ( f ( x )) of this approximation equal to 0 and solve for x to obtain: x x 0 - f ( x 0 ) f ( x 0 ) . (3.3) We begin the method with the initial guess x 0 , which we hope is fairly close to x . Then we define a sequence of points { x 0 ,x 1 ,x 2 ,x 3 ,... } from the formula: x i +1 = x i - f ( x i ) f ( x i ) , (3.4) which comes from (3.3). If f ( x ) is reasonably well-behaved near x and x 0

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