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...
Yasser F. O. MohammadAssiut University
Egypt
Previously in NMBracketing Methods
BisectionFalse Position
Fixed Point Iteration
Local Convergence MethodsNewton RaphsonSecant
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
Graphical Initial ApproximationDraw the functionLocate approximate roots
PROBLEMWhat you see may be misleading
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?)
Initial Approximation Tablexi f f'
+ -- +
≈0
+ -
≈0
- +
1( )if x ( )if x 1( ) ( )i if x f x 1( ) ( )i if x f x
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?
Matlab CodeSee page 68