Lecture 16 - Hand Calculation Example FalsePosition iter 1...

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( ) 1 2.0 3.2 0.2461 2 2.9375 3.2 2.9968 0.01207 3 2.99698 3.2 2.999856 0.000576 4 2.999856 3. 2.93 2 2.99999315 0 7 . 00 5 0 0274 l u r r iter x x x f x - - - - Hand Calculation Example [ ] [ ] 2 3 0 2 x , x estimeates initial 0 3 x 2 x x f Example u l 2 . , . ) ( : = = - - = False- Position 3.2 2 3.2 0.84* 0.84 ( 3) r x - = - - - f(2) = - 3, f(3.2) = 0.84
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2.55 2.6 2.65 2.7 2.75 2.8 2.85 2.9 2.95 3 3.05 3.1 0 2 4 6 8 xr---F.P. bisection n x r [ ] [ ] 2 3 0 2 x , x estimeates initial 0 3 x 2 x x f Example u l 2 . , . ) ( : = = - - = Hand Calculation Example
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Linear interpolation False- position (Regula- Falsi) Linear Interpolation
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Why don ' t we always use false- position method? There are times it may converge very, very slowly. Example: What other methods can we use? 0 4 x 3 x ) x ( f 4 = - + = Convergence Rate
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Convergence slower than bisection method Bisection root [ ] [ ] 3 0 x x 0 4 x 3 x x f u l 4 , , ) ( = = - + = 1 2 3 1 2 3
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0 4 x 3 x ) x ( f 4 = - + = Bisection Method False-Position Method » xl = 0; xu = 3; es = 0.00001; maxit = 100; » [xr,fr]=bisect2(inline( ‘x^4+3*x-4’ )) Bisection method has converged step xl xu xr f(x) 1.0000 0 3.0000 1.5000 5.5625 2.0000 0 1.5000 0.7500 -1.4336 3.0000 0.7500 1.5000 1.1250 0.9768 4.0000 0.7500 1.1250 0.9375 -0.4150 5.0000 0.9375 1.1250 1.0312 0.2247 6.0000 0.9375 1.0312 0.9844 -0.1079 7.0000 0.9844 1.0312 1.0078 0.0551 8.0000 0.9844 1.0078 0.9961 -0.0273 9.0000 0.9961 1.0078 1.0020 0.0137 10.0000 0.9961 1.0020 0.9990 -0.0068 11.0000 0.9990 1.0020 1.0005 0.0034 12.0000 0.9990 1.0005 0.9998 -0.0017 13.0000 0.9998 1.0005 1.0001 0.0009 14.0000 0.9998 1.0001 0.9999 -0.0004 15.0000 0.9999 1.0001 1.0000 0.0002 16.0000 0.9999 1.0000 1.0000 -0.0001 17.0000 1.0000 1.0000 1.0000 0.0001 18.0000 1.0000 1.0000 1.0000 0.0000 19.0000 1.0000 1.0000 1.0000 0.0000 » xl = 0; xu = 3; es = 0.00001; maxit = 100; » [xr,fr]=false_position(inline( ‘x^4+3*x-4’ )) False position method has converged step xl xu xr f(xr) 1.0000 0 3.0000 0.1333 -3.5997 2.0000 0.1333 3.0000 0.2485 -3.2507 3.0000 0.2485 3.0000 0.3487 -2.9391 4.0000 0.3487 3.0000 0.4363 -2.6548 5.0000 0.4363 3.0000 0.5131 -2.3914 6.0000 0.5131 3.0000 0.5804 -2.1454 7.0000 0.5804 3.0000 0.6393 -1.9152
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Lecture 16 - Hand Calculation Example FalsePosition iter 1...

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