complete Computer Based Numerical Techniques Course

operations of normalized floating point point, . Errors and Floating point Representation Pitfall of floating point Representation, Bisection Method, Regula-Falsi Method, Program of Bisection Method , Algorithm of Bisection Method, Secant Method, Newton-Raphson Method, Gauss Elimination Method, Gauss Seidel Iterative method, Calsulus of finite Differences, Relations b/w the operation, Newton Forward & backward Difference formula, and many more..

Sunday, May 20, 2018

Question. Use Bessel’s Interpolation formula find y25? given y20=24, y24=32, y28=35 and y32=40

Question. Use Bessel’s Interpolation formula find y25?

given y20=24, y24=32, y28=35 and y32=40

Sol:

u x y ∆ f(x) ∆²f(x) ∆³f( x)

-1 20 24

0 24 32 -5

3 7

1 28 35 2

2 32 40

x = a + uh

25 = 24 + u(4)

u = 25 – 24

[u = 0.25]

f(u) = { 32 + 35 } + (0.25 – ½ ) 3

2 2

+ 0.25(0.25 – 1) { -5 + 2 }

2 × 1 2

+ 0.25(0.25 – 1)(0.25 – ½ ) (7)

3× 2× 1

= 33.5 – 0.75 + 0.25(-0.75) (-3)

2 2

+ 0.25(-0.75) (-0.25)(7)

= 33.5 – 0.75 + 0.140625 + 0.05468

= 32.9453

Saturday, May 19, 2018

Bessel's Interpolation Formula

Formula:

f(u) = { f(0) + f(1) } + ( u – 1 ) ∆f(0)

2 2

+ u(u – 1) { ∆² f(-1) + ∆² f(0) }

2i 2

+ u(u – 1) (u – ½) ∆³ f(-1)

+ u(u + 1)(u – 1)(u – 2) { ∆⁴f(-2) + ∆⁴f(-1) }

4i 2

+. . . . . . . . . .. .. .. . . . .. . . . . .

Friday, May 18, 2018

Laplace – Everett’s Formula

Formula:

F(u) = { u f( 1) + u(u – 1)(u + 1) ∆² f(0)

1i 3i

+ u(u – 1)(u + 1)(u – 2)(u + 2) ∆⁴f(-1)}

+{ w f(0) + w(w – 1)(w + 1) ∆²f(-1)

1i 3i

+ w(w – 1)(w + 1)(w – 2)(w + 2) ∆⁴f(-2)}

+………………………..

Thursday, May 17, 2018

Using gauss backward difference formula estimate the no. of person. Wages(Ro.) belwo40 40-60 60-80 80-100 100-120 No. of person 250 120 100 70 50

Using gauss backward difference formula estimate the no. of person.

Wages(Ro.) belwo40 40-60 60-80 80-100 100-120

No. of person 250 120 100 70 50

Sol:

u x f(x) ∆f(x) ∆²f(x) ∆³f(x) ∆⁴f(x)

-2 40 250

-130

-1 60 120 110

-20 -120

0 80 100 -10 140

-30 20

1 100 70 10

-20

2 120 50

x = a + uh

70 = 80 + u (20)

u = 70 – 80

[ u = -0.5]

f(u) = 100 + (-5)(-30) + (-0.5)(-0.5 – 1) (-10)

+ (-0.5)(-0.5 – 1)(-0.5 + 1) (20)

+ (-0.5)(-0.5 – 1)(-0.5 + 1)(-0.5 – 2) (140)

= 100 + 15 + (-3.75) + 1.25 + (-5.46875)

= 107.03

Wednesday, May 16, 2018

Gauss Backward Difference formula

Formula:

F(u) = f(0) + u ∆f(0) + u(u – 1) ∆² f(-1)

1i 2i

+ u(u – 1)(u + 1) ∆³ f(-1)

+ u(u – 1)(u + 1)(u – 2) ∆⁴f(-2)

Tuesday, May 15, 2018

Using gauss backward difference formula estimate the no. of person Wagrs(Ro.) below40 40-60 60-80 80-100 100-120 No. of peoples 250 120 100 70 50 (in thousands)

Sol:

u x f(x) ∆f(x) ∆²f(x) ∆³f(x) ∆⁴f(x)

-2 40 250

-130

-1 60 120 110

-20 -120

0 80 100 -10 140

-30 20

1 100 70 10

-20

2 120 50

x = a +uh

70 = 80 + u(20)

u = 70 - 80

[u = - 0.5]

f(u)= 100 + (-0.5)(-30) + (-0.5)(-0.5 - 1) (-10)

+ (-0.5)(-0.5 - 1)(-0.5 + 1) (20)

+ (-0.5)(-0.5 - 1)(-0.5 + 1)(0.5 - 2) (140)

= 100 + 15 + (-3.75) + 1.25 + (-5.46875)

= 107.03

Monday, May 14, 2018

Gauss Backward Difference Formula

Formula:

f(u) = f(0) + u ∆f(0) + u(u – 1) ∆²f(-1)

1i 2i

+ u(u – 1)( u + 1) ∆³f(-1)

+ u(u – 1)(u + 1)(u – 2) ∆⁴f(-2)

Sunday, May 13, 2018

Apply gauss’s forward formula to find the value of u9 if u0=14, u4=24, u8=32, u12= 35, u16= 40

Sol:- x = a + uh

9 = 8 + u(4)

u = 9 – 8 = 1 = 0.25

4 4

[u = 0.25]

u x f(x) ∆f(x) ∆²f(x) ∆³f(x) ∆⁴f(x)

-2 0 14

-1 4 24 -2

8 -3

0 8 32 -5 10

3 7

1 12 35 2

2 16 40

F(u) = 3.2 + 0.25 + 0.25(0.25 – 1)

1 2

+ 0.25(0.25 – 1)(0.25 + 1) (7)

3 × 2

+ 0.25(0.25 – 1)(0.25 + 1) (0.25 – 2) (10)

4 × 3 × 2

f(u) = 32 + 0.75 + 0.46875 + (-0.2734375) + 0.17089

= 33.1162025

Saturday, May 12, 2018

Gauss forward Difference formula

Formula:

F(u) = f(0) + u + u(u – 1) ∆²f(-1)

1i 2i

+ u(u – 1)(u + 1) ∆³f(-1)

+ u(u – 1)(u + 1)(u – 2) ∆⁴ f(-2)

+ u(u – 1)(u + 1)(u – 2)(u + 2) ∆⁵ f(-2)

+ ………..

Friday, May 11, 2018

The population of town is as follows: x 1921 1931 1941 1951 1961 1971 Population(y) 20 24 29 36 46 51 Estimate the increase in population during the period 1955-1961

Sol:- x = (a + nh) + uh

1955 = 1971 + u(10)

u = 1955 – 1971

[u = -1.6]

x f(x) Ñ f(x) Ñ² f(x) Ñ³f(x) Ñ⁴ f(x) Ñ⁵f(x)

1921 20

1931 24 1

5 1

1941 29 2 0

7 1 -9

1951 36 3 -9

10 -8

1961 46 -5

1971 51

y(a+nh+uh) = 51 + (-1.6)(5) + (-1.6)(-1.6 + 1) (-5)

+(-1.6)(-1.6 + 1)(-1.6 +2) (-8)

3×2

+ (-1.6)(-1.6+1)(-1.6+2)(-1.6+3) ( -9)

4×3×2

+ (-1.6)(-1.6+1)(-1.6+2)(-1.6+3)(-1.6+4) (-9)

4×3×2×1

= 51 - 8 + (-2.4) + (-0.512) + (-0.5375) + (-0.096768)

= 39.453732

Thursday, May 10, 2018

The population of a town as given :- Estimate the population for the year 1925. year(x) : 1891 1901 1911 1921 1931 population: 46 66 81 93 101

Sol:-

x = (a + nh)+uh

1925= 1931 + u(10)

u = 1925 - 1931

u = - 6

[u = - 0.6]

x f(x) Ñf(x) Ñ²f(x) Ñ³f(x) Ñ⁴f(x)

1891 46

1901 66 -5

15 2

1911 81 -3 |-3|

12 |-1|

1921 93 |-4|

|8|

1931 |101|

f(a+nh+uh) = 101 + (-0.6)(8)

+ (-0.6)(- 0.6+1) (-4)

+ (-0.6)(- 0.6+1)(- 0.6+2)(-1)

+ (-0.6)(- 0.6+1)(- 0.6+2)( -0.6+3)(-3)

ð 101 – 4.8 + (-0.3)(0.4)(-4)

+ (-0.6)(0.4)(1.4)(-1)

+ (-0.6)(0.4)(1.4)(2.4)(-3)

ð 101 – 4.8 + 0.48 + 0.056 + 0.1008

= 98.8368

Wednesday, May 9, 2018

Newton's Backward Difference formula

Formula:

f(a+nh+u)= f(a + n) +u Ñf(a + nh)

+ u(u +1) Ñ²f(a + nh)

+ u(u+1)(u+2) Ñ³f(a + nh)

+u(u+1)(u+2)(u+3) Ñ⁴f(a + nh)

[x = (a + nh) + uh]

Tuesday, May 8, 2018

one more question for understanding the Newton Forward D.Method

Ques:- From the following table of year premium for policies maturing at different ages.

estimate the premium for policies maturing at the age of 46.

age: 45 50 55 60 65

premium: 114.84 96.16 83.32 74.48 68.48

Sol:- x = a + uh

46 = 45 + u(5)

u = 46 - 45

|u =0.2|

x f(x) ∆f(x) ∆²f(x) ∆³f(x) ∆⁴f(x)

45 114.84

-18.68

50 96.16 5.84

-12.84 -1.84

55 83.32 4 0.68

-8.84 -1.16

60 74.48 2.84

-6

65 68.48

f(a+uh)=114.84+0.2(-18.68) +0.2(0.2-1) (5.84)

2×1

+ 0.2(0.2-1)(0.2-2) (-1.84)

3×2×1

+ 0.2(0.2-1)(0.2-2)(0.2-3) (0.68)

4×3×2×1

= 114.84 - 3.736 - 0.4672 - 0.08832 -0.022848

= 110.888032