MA6452 STATISTICS AND NUMERICAL METHODS UNIT IV NUMERICAL DIFFERENTIATION AND INTEGRATION

Transcription

1 MA6452 STATISTICS AND NUMERICAL METHODS UNIT IV NUMERICAL DIFFERENTIATION AND INTEGRATION By Ms. K. Vijayalakshmi Assistant Professor Department of Applied Mathematics SVCE

2 NUMERICAL DIFFERENCE:

3 1.NEWTON S FORWARD DIFFERENCE FORMULA TO COMPUTE THE DERIVATIONS: Newton s forward difference formula for equal intervals is where

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6 Special Case:-

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8 2. NEWTON S BACKWARD DIFFERENCES FORMULA TO COMPUTE THE DERIVATIVES

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10 Special Case:-

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12 3.MAXIMA AND MINIMA OF THE INTERPOLATING POLYNOMIAL

13 Newton s forward interpolation formula is where u=

14 For y to be a maximum or minimum

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16 4. The following data gives the velocity of a particle for 2 seconds at an interval of 5 seconds. Find the initial acceleration using the entire date. time(sec) Volocity(m/sec)

17 Solution: Let v denote velocity and a denote acceleration Then a= Initial acceleration Here x=t, y=v

18 x y y

19 Now h=5

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21 5.From the following table, find the value of x for which y is minimum and find this value ofy. X Y Solution: Since we have equal intervals, we use Newton s forward difference formula.

22 x y Δy

23 For minimum value of y

24 u=1.4

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26 6. Using the following date, find, and the maximum value of X f(x) Solution: Since the x values are not equally spaced, we use Newton s divided difference formula.

27 x

28 By Newton s divided difference formula,

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30 i.e., is maximum if But the roots of this equation are imaginary. Hence there is no extremum value.

31 7.NUMERICAL INTEGRATION The process of computing a definite integral from a set of tabulated values of the integrand is called numerical integration. This process when applied to a function of a single variable is known as quadrature.

32 8.Quadrature formula for equidistant ordinates (or Newton s Cote s formula)

33 9.TRAPEZOIDAL RULE

34 10.ERROR IN TRAPEZOIDAL RULE Hence the error in the Trapezoidalrule is of order

35 11.SIMSON S S ONE THIRD RULE This is known as Simpson s one-third rule.

36 12.ERROR IN SIMPSON S S ONE-THIRD RULE The total error is given by where is the largest of the error in Simpson s rule of order

37 Note: 1.While applying Simpson s rule, the number of sub-intervals should be even. 2.While applying Simpson s rule, the number of sub-intervals should be a multiple of 3

38 . 15.Evaluate using Trapezoidal rule with h=0.2. Hence obtain an approximate value of Solution: Let Here h=0.2 in the interval (0,1)

39 Now, the values of y are given below: x Y By Trapezoidal rule

40 = (1)

41 By actual integration

42 To find the value of from (1) and (2) we get

43 16.Evaluate by dividing the range into six equal parts using Solution: We form the table with

44 Note: X 0 1/6 2/6 3/6 4/6 5/6 1 (Sinx)/x

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49 17.Romerg s Method To evaluate I systematically, we take

50 18.Use Romberg s method to compute correct to 4 decimal places. Solution: We take = 0.5, 0.25 and successively and evaluate the given integral using Trapezoidal rule. i) When h= 0.5, the values of

51 x : Y :

52 (ii) When h=0.25, the values of x : y :

53 (iii) When h=0.125, the values of x: y:

54 Thus we have Now using

55 and

56 The table of these values is Hence the value of the integral=0.7855

57 19. Find the approximate value of using composite trapezoidal rule with and then Romberg s method. Solution: We take h=0.5, 0.25 and successively and evaluate the given integral using Trapezoidal rule.

58 (i) When h=0.5, the value of x 0 1/2 1 y 1 2/3 1/2 By trapezoidal rule

59 (ii) When h=0.25, the value of x 0 1/4 1/2 3/4 1 y 1 4/5 2/3 4/7 1/2 =

60 (iii) When h=0.125, the value of x 0 1/8 2/8 3/8 4/8 5/8 6/8 7/8 1 y 1 8/9 4/5 8/11 2/3 8/13 4/7 8/15 1/2

61 Thus we have Now using

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64 The table of these values is Hence the value of the integral =

65 20.Two-point Gaussian quadrature formula and this is exact for polynomials of degree upto3

66 21.Three- point Gaussian formula which is exact for polynomials of degree upto5

67 Note: by the linear transformation

68 by two-point and three-point Gaussian formula and compare with the exact value. Solution: By two-point Gaussian formula

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70 By Three-point Gaussian formula

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72 The exact value is

73 23.Using 2-point Gaussian quadrature, evaluate Solution: Here a=0, b=1

74 We get equivalent integral

75 By Gaussian two-point formula,

76 23.Double integration Consider the table x y

77 By trapezoidal rule,

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79 by using Trapezoidal rule with h = k = 0.25 Solution: We first form the table for

80 y x

81 By Trapezoidal rule,

82 Applying this rule to each row, we get

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87 Now applying trapezoidal rule to =

88 25.Apply Simpson s s rule to evaluate the integral Take h = 0.2, k = 0.3 Solution: We form the table for

89 y x I

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91 to each row of the table, with h=0.2

92 Now we apply Simpon srule to

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