Numerical

NUMERICAL FORMULAE Iteration Newton Raphson method for refining an approximate root x0 of f (x) = 0 xn+1 = xn − Particular case to find Secant Method xn+1 = xn − f (xn )/

f (xn ) f 0 (xn )

√ N use xn+1 = 

1 2



f (xn ) − f (xn−1 ) xn − xn−1

xn +



Interpolation

∆fn = fn+1 − fn

,

∇fn = fn − fn−1

,

Gregory Newton Formula

δfn = fn+ 12 − fn− 21  1 µfn = fn+ 12 + fn− 12 2

  p(p − 1) 2 p fp = f0 + p∆f0 + ∆ f0 + ... + ∆r f0 2! r where p =

x − x0 h

Lagrange’s Formula for n points y=

n X

yi `i (x)

i=1

where `i (x) =

Πnj=1,j6=i (x − xj ) Πnj=1,j6=i (xi − xj )

1

N xn



.

Numerical differentiation Derivatives at a tabular point 1 1 µ δf0 − µ δ 3 f0 + µ δ 5 f0 − ... 6 30 1 4 1 6 2 2 00 h f0 = δ f0 − δ f0 + δ f0 − ... 12 90 1 2 1 3 1 1 0 hf0 = ∆f0 − ∆ f0 + ∆ f0 − ∆4 f0 + ∆5 f0 − ... 2 3 4 5 11 4 5 5 2 00 2 3 h f0 = ∆ f0 − ∆ f0 + ∆ f0 − ∆ f0 + ... 12 6 Numerical Integration hf00

=

Z

Trapezium Rule

where

fi = f (x0 + ih), E = −

x0 +h

f (x)dx '

x0

h (f0 + f1 ) + E 2

h3 00 f (a), x0 < a < x0 + h 12

Composite Trapezium Rule Z

x0 +nh

x0

f (x)dx '

h h2 h4 000 {f0 + 2f1 + 2f2 + ...2fn−1 + fn } − (fn0 − f00 ) + (f − f0000 )... 2 12 720 n where f00 = f 0 (x0 ), fn0 = f 0 (x0 + nh), etc Z

Simpson’s Rule

where

E=−

x0 +2h

f (x)dx '

x0

h5 (4) f (a) 90

h (f0 + 4f1 + f2 ) + E 3

x0 < a < x0 + 2h.

Composite Simpson’s Rule (n even) Z

x0 +nh

x0

where

f (x)dx '

h (f0 + 4f1 + 2f2 + 4f3 + 2f4 + ... + 2fn−2 + 4fn−1 + fn ) + E 3

E=−

2

nh5 (4) f (a). 180

x0 < a < x0 + nh

Gauss order 1. (Midpoint) Z

1

f (x)dx = 2f (0) + E

−1

where

E=

2 00 f (a). 3

−1
Gauss order 2. Z

    1 1 f (x)dx = f − √ +f √ +E 3 3 −1 1

where

E=

1 0v f (a). 135

−1
Differential Equations To solve y 0 = f (x, y) given initial condition y0 at x0 , xn = x0 + nh. Euler’s forward method yn+1 = yn + hf (xn , yn )

n = 0, 1, 2, ...

Euler’s backward method yn+1 = yn + hf (xn+1 , yn+1 )

n = 0, 1, 2, ...

Heun’s method (Runge Kutta order 2) yn+1 = yn +

h (f (xn , yn ) + f (xn + h, yn + hf (xn , yn ))). 2

Runge Kutta order 4. yn+1 = yn +

h (K1 + 2K2 + 2K3 + K4 ) 6

where K1

=

K2

=

K3

=

K4

=

f (xn , yn )   hK1 h f xn + , yn + 2 2   hK2 h f xn + , yn + 2 2 f (xn + h, yn + hK3 )

3

Chebyshev Polynomials   Tn (x) = cos n(cos−1 x) To (x) = 1

T1 (x) = x

  sin n(cos−1 x) Tn0 (x) √ Un−1 (x) = = n 1 − x2 Tm (Tn (x)) = Tmn (x). Tn+1 (x) = 2xTn (x) − Tn−1 (x) Z

Un+1 (x) = Tn (x)dx

2xUn (x) − Un−1 (x)   1 Tn+1 (x) Tn−1 (x) = − + constant, 2 n+1 n−1

f (x) =

where

n≥2

1 a0 T0 (x) + a1 T1 (x)...aj Tj (x) + ... 2

aj =

2 π

Z

π

f (cos θ) cos jθdθ 0

R and f (x)dx = constant +A1 T1 (x) + A2 T2 (x) + ...Aj Tj (x) + ... where Aj = (aj−1 − aj+1 )/2j j≥1

4

j≥0