Yasser F. O. Mohammad Assiut University Egypt. Previously in NM Bracketing Methods Bisection False...
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Transcript of Yasser F. O. Mohammad Assiut University Egypt. Previously in NM Bracketing Methods Bisection False...
![Page 1: Yasser F. O. Mohammad Assiut University Egypt. Previously in NM Bracketing Methods Bisection False Position Fixed Point Iteration Local Convergence Methods.](https://reader036.fdocuments.us/reader036/viewer/2022082723/5a4d1b497f8b9ab0599a4cf4/html5/thumbnails/1.jpg)
Yasser F. O. MohammadAssiut University
Egypt
![Page 2: Yasser F. O. Mohammad Assiut University Egypt. Previously in NM Bracketing Methods Bisection False Position Fixed Point Iteration Local Convergence Methods.](https://reader036.fdocuments.us/reader036/viewer/2022082723/5a4d1b497f8b9ab0599a4cf4/html5/thumbnails/2.jpg)
Previously in NMBracketing Methods
BisectionFalse Position
Fixed Point Iteration
Local Convergence MethodsNewton RaphsonSecant
![Page 3: Yasser F. O. Mohammad Assiut University Egypt. Previously in NM Bracketing Methods Bisection False Position Fixed Point Iteration Local Convergence Methods.](https://reader036.fdocuments.us/reader036/viewer/2022082723/5a4d1b497f8b9ab0599a4cf4/html5/thumbnails/3.jpg)
Types of ConvergenceLocal convergence
Converges only if the initial approximation is near the root
Newton-Raphson/Secant
Global ApproximationConverges as long as some condition holds for
the initial approximation (even if it is not near)Bracketing methods
A different initial interval is needed for every root
![Page 4: Yasser F. O. Mohammad Assiut University Egypt. Previously in NM Bracketing Methods Bisection False Position Fixed Point Iteration Local Convergence Methods.](https://reader036.fdocuments.us/reader036/viewer/2022082723/5a4d1b497f8b9ab0599a4cf4/html5/thumbnails/4.jpg)
Graphical Initial ApproximationDraw the functionLocate approximate roots
PROBLEMWhat you see may be misleading
![Page 5: Yasser F. O. Mohammad Assiut University Egypt. Previously in NM Bracketing Methods Bisection False Position Fixed Point Iteration Local Convergence Methods.](https://reader036.fdocuments.us/reader036/viewer/2022082723/5a4d1b497f8b9ab0599a4cf4/html5/thumbnails/5.jpg)
HeuristicsA heuristic is a rule that works ‘most of times
in practice’Usually better than nothing
Heuristics for initial approximationA root exists between points at which the
function changes sign (is this a heuristic really?)
A root may exist near points at which f(x) is small and the derivative changes sign (why?)
![Page 6: Yasser F. O. Mohammad Assiut University Egypt. Previously in NM Bracketing Methods Bisection False Position Fixed Point Iteration Local Convergence Methods.](https://reader036.fdocuments.us/reader036/viewer/2022082723/5a4d1b497f8b9ab0599a4cf4/html5/thumbnails/6.jpg)
Initial Approximation Tablexi f f'
+ -- +
≈0
+ -
≈0
- +
1( )if x ( )if x 1( ) ( )i if x f x 1( ) ( )i if x f x
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Convergence ChecksVertical
Absolute
Relative
HorizontalAbsolute
Relative
if p
1i ip p
0if p f p
1
1
i i
i i
p pp p
What happens if:
Vertical check AND Horizontal
check?
Vertical check OR Horizontal check?
Absolute AND Relative check?
Absolute OR Relative check?
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Matlab CodeSee page 68