Lecture14 - False-Position(point Method Why bother with...

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False-Position (point) Method Why bother with another method? The bisection method is simple and guaranteed to converge (single root) But the convergence is slow and non- monotonic! The bisection method is a brute force method and makes no use of information about the function
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False-Position Method y(x) x secant line x* x l x u Straight line (linear) approximation to exact curve
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* Equation for the secant: y= f(x l ) + k (x-x l ) o x y x u ,f(x u ) x l ,f(x l ) False-point method x r ,0 Root for the secant line : y= 0 = f(x l ) + k(x r, -x l ) x r = x l - f(x l )/k = x l - f(x l ) (x u -x l )/[f(x u ) - f(x l )] Or: ( )( ) ( ) ( ) u l r u u u l f x x x x x f x f x - = - - Derivation ( ) ( ) ( ) ( ) u l u r u l u r f x f x f x f x slope k x x x x - - = = = - -
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Algorithm for False-Position Method 1. Start with [ x l , x u ] with f ( x l ) . f ( x u ) < 0 (still need to bracket the root) 2. Draw a straight line to approximate the root 3. Check signs of f ( x l ) . f ( x r ) and f ( x r ) . f ( x u ) so that [ x l , x u ] always bracket the root Maybe less efficient than the bisection method for highly nonlinear functions ) ( ) ( ) )( ( ) ( u l u l u u r r x f x f x x x f x x 0 x f - - - = =
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no yes False-point Method Flowchart If f(xm)f(xl) < 0 then xu = xr else xl = xr Solution obtaind xr If f(xr) ? = 0 or abs(xr - pxr) < tolerance Input xl & xu f(xl)*f(xu) < 0 Start end xr =xu - f(xu)(xl -xu)/(f(xl) - f(xu))
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0000274 0 99999315 2 2 3 999856 2 4 000576 0 999856 2 2 3 99698 2 3 01207 0 9968 2 2 3 9375 2 2 2461 0 9375 2 2 3 0 2 1 x f x x x iter r r u l . . . . . . . . . . . . . . . . ) ( - - - - Hand Calculation Example [ ] [ ] 2 3 0 2 x , x estimeates initial 0 3 x 2 x x f Example u l 2 . , .
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