mase201sp09_Mar31_PRINTABLE

mase201sp09_Mar31_PRINTABLE - Finding zeros of nonlinear...

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Finding zeros of nonlinear functions ackground nd pproaches • Background and approaches • Bisection method (bracketed interval) • Newton’s method (open interval with analytic derivative) ecant method (open interval with • Secant method (open interval with approximate derivative) ATLAB built functions zero ec 10 1 • MATLAB built-in functions fzero (sec 10.1 in your text) 3/31/2009 1 MASE201 Spring 2009 Review our example For a given applied F , coefficient of iction and mass find t friction , and mass m , find at which the block will start to move Solution method: g m F The sliding block – Guess –C a l c u l a t e F using equation ompare to applied F Compare to applied – Adjust to reduce error between calculated and applied F equires repeated calculation of g m Requires repeated calculation of cos and sin functions sin cos F 3/31/2009 2 MASE201 Spring 2009 Problem Solution Method? sin cos g m F What’s a good starting guess for ? Divide F by mg Consider limiting cases: = 0 or 90 deg sin cos g m F 1 0 n s 0 90 sin cos 3/31/2009 3 MASE201 Spring 2009 Problem Approach (cont’d) • We can plot solution at g m specific values of • The shape of the F - curve hanges for each value of sin cos F changes for each value of • There may be 0, 1, or 2 solutions for depending upon the ratio F/(mg) and 3/31/2009 4 MASE201 Spring 2009
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Finding f(x) = 0 ifferent possibilities for number of solutions • Different possibilities for number of solutions (roots): x) 0 f( x 0 0 0 No solution One solution One solution Several solutions x x x x 3/31/2009 5 MASE201 Spring 2009 Approach Make an initial estimate for the numeric solution Bracketing Methods pen Methods (approximate solution) Use the approximate solution to find a better Open Methods numeric solution Stop when the solution is precise enough 3/31/2009 6 MASE201 Spring 2009 Figure 3-5 from Gilat, “Numerical Methods for Engineers and Scientists” Bracketing and Open Methods • Bracketing Methods – Will converge to a solution • Open Methods – May not find a solution – May be less efficient • We will discuss: isection method – May require fewer iterations e will discuss: – Bisection method • Other methods: Regula falsi method We will discuss: – Newton’s method – Secant method
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mase201sp09_Mar31_PRINTABLE - Finding zeros of nonlinear...

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