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