MIT2_007s09_lec21

# MIT2_007s09_lec21 - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 2.007 Design and Manufacturing I Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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2.007 –Design and Manufacturing I Optimization and Solution of Systems Dan Frey 28 APR 2009 -2 -1 0 1 2 3 -4 -3 0 1 f g h a b c d e 0 x 1 x 2 x * x
Today’s Agenda • Seeding and impounding procedures • Methods for Solving Systems – Newton-Raphson – Secant – Bisection • Examples related to mechanism design

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Seeding • Run on the table unopposed • Timing and set-up as in the actual contest • Three tries – best of three counts • Your “seeding card” is essential – Get your scores recorded and initialed – Don’t lose your card • “In-lab” competition – Basically a way to get round 1 partly finished – Same as next Weds but not broadcast
Impounding • A way to bring the work to an end • Your machine is checked –Safety – Wiring – Rules issues • Your “seeding card” is essential – Your impound checks are recorded – Your card goes in the WOODEN BOX

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Linear Systems (Back Solving) A=[1 1 1; 0 2 3; 0 0 6]; b=[3; 1; 4]; x(3)=b(3)/A(3,3) x(2)=(b(2)-x(3)*A(2,3))/A(2,2) x(1)=(b(1)-x(2)*A(1,2)-x(3)*A(1,3))/A(1,1); norm(b-A*x') What will happen when I run this code?
Linear Systems (Solving) A=[1 1 1; 1 2 3; 1 3 6]; b=[3; 1; 4]; x=A\b b=[5; 0; -10]; x=A\b What will happen when I run this code?

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Linear Systems (Existence of Soln) A=[1 1 1; 1 2 3; 1 3 6; -1 -1 1]; b=[3; 1; 4; 7]; x=A\b; norm(b-A*x) What will happen when I run this code?
Linear Systems (Existence of Soln) A=[1 1 1; 1 2 3; 1 3 6; -1 -1 1]; b=[3; 1; 4; 6 ]; x=A\b; norm(b-A*x) What will happen when I run this code?

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Linear Systems (Multiple Solutions) A=[1 1 1; 1 2 3; 1 3 6; -1 -1 1]; b1=[3; 1; 4; 7]; x1=A\b1; norm(b1-A*x1) b2=[5; 0; -10; -15]; x2=A\b2; norm(b2-A*x2) What will happen when I run this code? b3=5*b1-2*b2; x3=A\b3; norm(b3-A*x3) norm(x3-(5*x1-2*x2))
Comparisons Linear Systems • Sometimes solved sequentially • # of equations = # of unknowns • # of equations > # of unknowns • When we can find two solutions Nonlinear systems •?

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Newton-Raphson Method • Make a guess at the solution • Make a linear approximation of a function by e.g., finite difference • Solve the linear system • Use that solution as a new guess • Repeat until some criterion is met Initial guess 0 Next estimate
Newton-Raphson Method ) ( ) ( 1 k k k k x f x f x x + = + 0 x 1 x 2 x ( )( ) ) ( 1 k k k k F x F x x x J = + If one equation in one variable Solve this system for x k+ 1 Generalizing to systems of equations * x

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A Fundamental Difficulty x guess 10 := x root root y x guess () x guess , := 5051 0 1 5 40 20 0 20 yx root xx root , • If there are many solutions, which solution you find will depend on the initial guess x guess 3 := x root root y x guess x guess , := 0 1 5 40 20 0 20 root root ,
If you seek to find a root of a function f ( x ) , and you use the Newton-Raphson method.

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MIT2_007s09_lec21 - MIT OpenCourseWare http/ocw.mit.edu...

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