Generalizes to higher dimension need the derivative can diverge you can only

# Generalizes to higher dimension need the derivative

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(+) Generalizes to higher dimension (-) need the derivative (-) can diverge (you can only control the initial guess) In the case where we cannot or don’t want to calculate f’(x), we can use forward difference, backward difference, or central difference to calculate f’(x) Secant Method use backward difference (because we only know two previous guesses) to approximate the derivative , where *lose quadratic convergence, but we do not have to calculate the derivative *need two initial guesses recall that the smaller the step size, the more accurate the approximate *When the Secant method is close to converging, it will behave like the Newton-Raphson method Better than linear convergence, but not quadratic Advantages and Disadvantages (+) Only need function evaluation (no derivatives needed) (+) Generally have better than linear convergence (but no better than Newton-Raphson) (+/-) Can be generalized to higher dimensions...but it is intense (-) Can diverge The best method, out of the so far talked about open/bracketed methods, is the Newton- Raphson method, but the only downside is you have to calculate the derivative Matlab Functions MATLAB has the function fzeros( ) which used both open and bracketed methods when calculating a root of a function , * Note can be an initial guess or a bracket Nonlinear Systems of Equations
let , then We could use Fixed-Point Iteration, such that but as before, we can get better performance by using Newton-Raphson Consider a system of two equations, Apply Taylor’s Theorem, In general, compare with linear system J is known as the Jacobian. We want to find the root, so solve for , therefore is the Newton Raphson for multi dimensions *Convergence is still quadratic, but J has elements To avoid calculating J, we could apply a multi-dimensional Secant method
Optimization many engineering problems involve maximizing or minimizing functions Examples minimizing fuel to transfer between two orbits minimizing drag on an aircraft (altitude and velocity) maximizing profit To solve an optimization problem, we need a function (performance index) and a set of independent variables Example Consider a scalar function f(x) When a function has maximum and minimum points, or optimization points, f’(x)=0 MAX MIN f(x) is concave down f(x) is concave up f’(x)=0 f’(x)=0 f’’(x)<0 f’’(x)>0 When f’’(x)=0, we do not know if there is a min or max *Note that functions can have many local min/max, but only one global min/max 1) Find an optimal point by solving f’(x)=0

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