Autar Kaw Benjamin Rigsby. Transforming Numerical Methods Education for STEM Undergraduates

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1 Autr Kw Bejmi Rigsby Trsformig Numericl Methods Eductio for STEM Udergrdutes

2

3 . solve set of simulteous lier equtios usig Nïve Guss elimitio,. ler the pitflls of the Nïve Guss elimitio method,. uderstd the effect of roud-off error whe solvig set of lier equtios with the Nïve Guss elimitio method,

4 4. ler how to modify the Nïve Guss elimitio method to the Gussi elimitio with prtil pivotig method to void pitflls of the former method,. fid the determit of squre mtri usig Gussi elimitio, d 6. uderstd the reltioship betwee the determit of coefficiet mtri d the solutio of simulteous lier equtios.

5 A method to solve simulteous lier equtios of the form [A][X][C] Two steps. Forwrd Elimitio. Bck Substitutio

6 The gol of forwrd elimitio is to trsform the coefficiet mtri ito upper trigulr mtri

7 A set of equtios d ukows b b b (-) steps of forwrd elimitio

8 Step For Equtio, divide Equtio by by. d multiply ( b ) b

9 Subtrct the result from Equtio b b... b b + + or b

10 Repet this procedure for the remiig equtios to reduce the set of equtios s b b b Ed of Step b

11 Step Repet the sme procedure for the rd term of Equtio b b + " " "... + b b " " "... Ed of Step

12 At the ed of (-) Forwrd Elimitio steps, the system of equtios will look like b b " " " b ( ) ( ) b Ed of Step (-)

13 ) ( " ) ( " " b b b b

14 Solve ech equtio strtig from the lst equtio Emple of system of equtios

15 b b " " b " ( ) ( ) b

16 Strt with the lst equtio becuse it hs oly oe ukow b ( ) ( )

17 ( ) ( ) ( ) ( ) ( ),..., for...,,, i b i ii i i i i i i i i i i i i i ( ) ( ) ( ),..., for + i b i ii i j j i ij i i i ) ( ) ( b

18 THE END

19 For ll resources o this topic such s digitl udiovisul lectures, primers, tetbook chpters, multiple-choice tests, worksheets i MATLAB, MATHEMATICA, MthCd d MAPLE, blogs, relted physicl problems, plese visit html

20

21 The upwrd velocity of rocket is give t three differet times Tble Velocity vs. time dt. Time, t ( s) Velocity, v ( m/s) The velocity dt is pproimted by polyomil s: v ( ) t t + t +, t. Fid the velocity t t6 secods.

22 ( ). t, t t t v + + Assume v v v t t t t t t Results i mtri templte of the form: Usig dt from Tble, the mtri becomes:

23 Forwrd Elimitio. Bck Substitutio

24 Forwrd Elimitio. Bck Substitutio

25 Forwrd Elimitio

26 Number of steps of forwrd elimitio is (-)(-)

27 Divide Equtio by d 64 multiply it by 64,.6. [ 6.8].6 [ ] Subtrct the result from Equtio [ ] [ ] [ ] Substitute ew equtio for Equtio

28 [ 6.8].76 [ ] Subtrct the result from Equtio Substitute ew equtio for Equtio [ ] [ ] [ ] Divide Equtio by d 44 multiply it by 44,

29 Divide Equtio by 4.8 d multiply it by 6.8, [ ]. [ ] [ ] Subtrct the result from Equtio [ ] [.7.76] Substitute ew equtio for Equtio

30 Bck Substitutio

31 Solvig for

32 Solvig for

33 Solvig for

34

35 Solutio The solutio vector is The polyomil tht psses through the three dt poits is the: v ( t) t + t +.947t t +.87, t v ( ) ( ) ( 6) m/s. +.87

36 THE END

37

38 Pitfll#. Divisio by zero

39

40 Divisio by zero is possibility t y step of forwrd elimitio

41 Pitfll#. Lrge Roud-off Errors Ect Solutio

42 Pitfll#. Lrge Roud-off Errors Solve it o computer usig 6 sigifict digits with choppig

43 Pitfll#. Lrge Roud-off Errors Solve it o computer usig sigifict digits with choppig Is there wy to reduce the roud off error?

44 Icrese the umber of sigifict digits Decreses roud-off error Does ot void divisio by zero

45 Gussi Elimitio with Prtil Pivotig Avoids divisio by zero Reduces roud off error

46 THE END

47

48 Possible divisio by zero Lrge roud-off errors

49 Icrese the umber of sigifict digits Decreses roud-off error Does ot void divisio by zero

50 Gussi Elimitio with Prtil Pivotig Avoids divisio by zero Reduces roud off error

51 At the begiig of the k th step of forwrd elimitio, fid the mimum of,..., kk, k +, k k If the mimum of the vlues is pk i the p th row, k p, the switch rows p d k.

52 4 b b b b

53 Which two rows would you switch?

54 Switched Rows

55 A method to solve simulteous lier equtios of the form [A][X][C] Two steps. Forwrd Elimitio. Bck Substitutio

56 Sme s ïve Guss elimitio method ecept tht we switch rows before ech of the (-) steps of forwrd elimitio.

57 4 b b b b

58 ) ( " ) ( " " b b b b

59 b b + " " " b ( ) ( ) b......

60 b ( ) ( ) i b ( i) ( i) i j ii ij j i+ ( ) for i,..., i

61 THE END

62

63 Solve the followig set of equtios by Gussi elimitio with prtil pivotig

64 Forwrd Elimitio. Bck Substitutio

65 Forwrd Elimitio

66 Number of steps of forwrd elimitio is (-)(-)

67 Emie bsolute vlues of first colum, first row d below., 64, 44 Lrgest bsolute vlue is 44 d eists i row. Switch row d row

68 [ ].4444 [ ] Subtrct the result from Equtio Divide Equtio by 44 d multiply it by 64, [ ] [ ] [ ] Substitute ew equtio for Equtio

69 [ ].76 [ ] Subtrct the result from Equtio Substitute ew equtio for Equtio Divide Equtio by 44 d multiply it by,.76. [ 6.8] [ ] [ ]

70 Emie bsolute vlues of secod colum, secod row d below..667,.97 Lrgest bsolute vlue is 44 d eists i row. Switch row d row

71 [ ].94 [ ] Subtrct the result from Equtio Substitute ew equtio for Equtio Divide Equtio by.97 d multiply it by.667,.94. [ ] [ ] [..]

72 Bck Substitutio

73 ..... Solvig for

74 Solvig for

75 Solvig for

76

77

78 Cosider the system of equtios I mtri form Solve usig Gussi Elimitio with Prtil Pivotig usig five sigifict digits with choppig

79 Forwrd Elimitio: Step Emiig the vlues of the first colum, -, d or,, d The lrgest bsolute vlue is, which mes, to follow the rules of Prtil Pivotig, we switch row with row Performig Forwrd Elimitio

80 Forwrd Elimitio: Step Emiig the vlues of the first colum -. d. or. d. The lrgest bsolute vlue is., so row is switched with row Performig the row swp

81 Forwrd Elimitio: Step Performig the Forwrd Elimitio results i:

82 Bck Substitutio Solvig the equtios through bck substitutio

83 [ ] X ect [ ] X clculted Compre the clculted d ect solutio. The fct tht they re equl is coicidece, but it does illustrte the dvtge of Prtil Pivotig

84 THE END

85 Determit of Squre Mtri Usig Nïve Guss Elimitio Emple

86 If multiple of oe row of [A] is dded or subtrcted to other row of [A] to result i [B] the det(a)det(b)

87 The determit of upper trigulr mtri [A] det A... ii... give by ( ) i is ii

88 Usig forwrd elimitio to trsform [A] upper trigulr mtri, [U]. to [ A ] [ ] U ( A) det( U ) det

89 Usig ïve Gussi elimitio fid the determit of the followig squre mtri

90 Forwrd Elimitio

91 [ ] [ ] Subtrct the result from Equtio Substitute ew equtio for Equtio. Divide Equtio by d [ 64 8 ] [ ] [ 4.8.6] multiply it by 64,

92 [ ].76 [ ] Subtrct the result from Equtio Substitute ew equtio for Equtio Divide Equtio by d multiply it by 44,.76. [ 44 ] [ ] [ ]

93 ([ 4.8.6] ). [ ] Subtrct the result from Equtio Substitute ew equtio for Equtio Divide Equtio by 4.8 d multiply it by 6.8, [ ] [ ] [.7]

94 After forwrd elimitio ( A) det u u u 84. ( 4.8).7

95 Forwrd Elimitio Bck Substitutio Pitflls Improvemets Prtil Pivotig Determit of Mtri

96 For ll resources o this topic such s digitl udiovisul lectures, primers, tetbook chpters, multiple-choice tests, worksheets i MATLAB, MATHEMATICA, MthCd d MAPLE, blogs, relted physicl problems, plese visit html

97 THE END

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