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Comparison of Iterative Methods:

Guaranteed

convergence

112. False

Position

Guaranteed

convergence

1gain of

one bit

 per

iteration

1.

Bisection

Reliability

of

Convergence

Evaluatio

ns of

functions

 per

iteration

order of

converg

ence

Iterative formulaMethod 

11

1

2

 and enclose root

−

+

−

+=   i i

i

i i

 x x x

 x x

i 1 i i i 1i+1

i i 1

x f(x ) x f(x )x

f(x ) f(x )− −

−

−=

−

1, enclose root

−i i x xEasy to program.

 No guarantee of

convergence

115.

Iterative

method 

 No guarantee of

convergence if

not near root.

Economical

method 

11.624. Secant

Sensitive to

starting value,

Convergence fast

if starting point

near the root.

223.

 Newton-

Raphson

Reliability

of

Convergence

Evaluati

ons of

functions

per

iteration

order of

converg

ence

Iterative formulaMethod 

i

i+1 ii

f(x )

x x f (x )= −

′

i 1 i i i 1i+1

i i 1

x f(x ) x f(x )x

f(x ) f(x )

− −

−

−=

−

1, need not

enclose root

−i i x x

1( )−

=i i

 x g x

INTERPOLATION

In numerical Analysis, the method of finite

differences enables us to establish the relation

y = f(x), between two variables, using a given set of

(n + 1) observations as :

x: a a + h a + 2h … … a + nh

y: y0

f(a)

y1

f(a + h)

y2

f(a +3h)… …

yn

f(a + nh)

Where f(x) is a continuous function. The relation

can then be used to find the value of y (the

dependent variable) for values of x (the independent

variable) for a < x < a +nh.

The values of (dy/dx) and (d 2y/dx

2) etc. can also be

calculated.

In numerical analysis, we say that x is argument and 

y = f(x) is entry.

Operators:

(1) Identity Operator : I{f(x)} = f(x)

(2) Shift Operator : If values of x be

given to be at a, a + h, a + 2h …. …. at a

regular interval ‘h’, then E{f(x)} = f(x + h) and

E2 {f(x)} = f(x + 2h) and in general

Er  f(x) = f(x + rh).

Taking it to be true for all possible

negative and fractional values of r

E-1

 f(x) = f(x − h) and E-1/2

 f(x) = f(x − (1/2) h)

  Em

 En f(x) = f{x + (m + n) h}

(3) Forward Difference Operator (delta) ∆

  ∆ f(x) = f(x + h) − f(x)

∆2  f(x) = ∆  {∆  f(x)} = ∆  f(x + h) −  ∆  f(x)

  …

…

∆n f(x) = ∆

n−1 {∆ f(x)} = ∆

n−1  f(x + h) − ∆

n−1f(x)

 Now

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∆ f(x) = E f(x) − I f(x) = (E − I) f(x)

or ∆ ≡ E − I or E ≡ ∆ + I and thus

∆2(f(x)) = (E − I)

2 f(x) = (E

2 − 2 E + I ) f(x)

= f(x + 2h) − 2 f(x + h) + f(x).In general, Newton’s formula for n

th  forward 

difference is as follows:

∆n f(x) = (E − I)

n f(x) =

0

n

r =

∑  n

r C   E

n − r  (−1)

r  f(x)

=0

n

r =

∑  n

r C   (−1)

r  f (x + (n − r) h)

= f(x + nh) −  1

nC   f(x + (n − 1)h) + … + (−1)

n f(x)

(3) Backward Difference Operator (Nebla) ∇ 

∇ f(x) = f(x) − f(x − h)

Relations between ∆, ∇ and E :

(a)  E ≡ ∆ + I or ∆ ≡ E − I

(b)  E ∆  ≡ ∇ E

(c)  E ∇  ≡ ∇ E ≡  ∆ 

(d)  ∇  ≡  ∆ E−1

 

(e)  ∇  ≡ I − E−1

 or E−1

  ≡ I − ∇ 

(1) ∆ − ∇  ≡  ∆ ∇ 

Proof : ∆ f(x) = f(x + h) − f(x) and

∇ f(x) = f(x) − f(x − h)

 Now (∆∇) f(x) = ∆ { ∇ f(x)}

= ∆ { f(x) − f(x − h)} = ∆ f(x) − ∇ f(x)

Or (∆∇) f(x) = (∆ − ∇) f(x) ∴  ∆ − ∇  ≡  ∆ ∇.

(2)  (I + ∆) (I −  ∇) ≡ I

Proof : (I + ∆) (I −  ∇) f(x)

= (I + ∆) { (I −  ∇) f(x)}

= (I + ∆) {f(x) − ∇ f(x)}