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MHT-CET : Mathematics Entrance Exam

MHT - CET : Mathematics - Numerical Methods Formulae Page 1

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1. Trapezoidal Rules for :

 

=

h

[ ( y0 + yn ) + 2( y1+ y2 + y3 +.+ yn - 1 )]

 

2

 

where h =

b - a

, n = Number of sub-division.

 

n

 

 2. Simpson's Rules for :

 

= [ ( y0 + yn ) + 4( y1+ y3 ++ yn - 1 ) + 2( y2 + y4 + y6 ++ yn - 2 )]

 

where h =

b - a

, n = Even number of sub-division.

 

n

 

 3. Newton's Forward Interpolation Formula:

 

a)

To find the value of f(x) for given x

 

f(x) = f(x0) + u.D f(x0)+

u (u - 1)

D2f(x0)

+

u (u - 1) (u - 2)

D3f(x0) + ....... +

 

 

2!

3!

 

u (u - 1) (u - 2)(u - 3)......(u - n + 1)

Dnf(x0)

 

n!

 

where u =

x - x0

(h > 0) h = xr - xr-1, r = 1, 2, 3

 

h

 

 

b)

To find the polynomial f(x)

 

f(x) = f(x0) +

x - x0

Df(x0)

+

(x - x0)(x - x1)

D2f(x0) +

 

 

h

h22!

 

(x - x0)(x - x1)(x - x2)

D3f(x0) + ........

where h = xr - xr-1, r = 1, 2, 3

 

h32!

 

 4. Newton's Backward Interpolation Formula:

 

To find the value of f(x) for given x

 

f(x) = f(xn) + u. f(xn)+

u (u + 1)

2f(xn)

+

u (u + 1) (u + 2)

3f(xn) + .......

 

 

2!

3!

 

where u =

x - xn

(h > 0) h = xr - xr-1, r = 1, 2, 3

 

h

 

5. Bisection Method:

 

Formula for taking approximate root C

a + b

 

2

 

 

 

6. False position Method:

 

Formula for taking approximate root

 

C  =

af(b) - bf(a)

 

f(b) - f(a)

 

7. Newton-Raphson Method:

 

Formula for taking approximate root

 

xi =  xi - 1 -  

f(xi - 1)

; i = 1, 2,...

 

f '(xi - 1)

 

 

 

8. Relation between D, and E:

a)

Erf(x) = f(x + rh) r is any positive or negative integer.

b)

D = E - 1 or E = D + 1.

c)

= 1 - E-1, or E-1= 1 -

d)

D = E

 

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