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    Analisis Numerik sederhana

    Metoda iterasi utk mencari akar suatuFungsi

    Fungsi Polynome pangkat 2 mudah dicari

    akar2nya

    di Engineering: Polynome pangkat 3, 4,

    Metoda iterasi sederhana:

    ! "isection Method

    2 Ne#ton$s Method

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    Analisis Numerik sederhanaMetode Ne#ton

    Metode iterasi:Akar suatu Fungsi didekati dgn persamaan:

    %imana : &n adalah harga a#al yg dipilih

    'harga yg dianggap dekat dengan akar ygdicari(

    )('

    )(

    n

    n

    n xf

    xf

    xx =

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    Analisis Numerik sederhana

    Metoda iterasi utk mencari akar suatuFungsi

    )ontoh :

    4186124)(6.

    271832)(5.

    6132)(4.

    534)(3.

    72)(2.

    65)(1.

    234

    23

    23

    3

    3

    2

    ++=

    +=

    ++=

    +==

    +=

    xxxxxfNo

    xxxxfNo

    xxxxfNo

    xxxfNo

    xxxfNo

    xxxfNo

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    Fungsi kuadrat

    21

    2

    ,:

    0)(

    xxakar

    xdgncbxaxxf ++=

    omplex04

    04

    04

    24,

    212

    212

    212

    2

    21

    cxxacb

    realxxacb

    realxxacb

    aacbbxx

    =

    65)(

    1.

    2+= xxxf

    No

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    *urunan Fungsi

    dx

    dumuu

    dx

    dno

    mxx

    dx

    dno

    xdxdno

    cdx

    dkcbilano

    mm

    mm

    1

    1

    )(f(x)ubila4.

    3.

    12.

    0

    onst1.

    ==

    =

    =

    ==

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    27.

    .6.

    )(5.

    f(x):w,,4.

    v

    dx

    dvu

    dx

    duv

    v

    u

    dx

    dno

    dx

    dvu

    dx

    duvvu

    dx

    dno

    dx

    dw

    dx

    dv

    dx

    duwvu

    dx

    dno

    udx

    dccu

    dx

    duno

    =

    +=

    ++=++

    =*urunan

    Fungsi

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    +oal atihan*urunan

    Fungsi

    52)(5.

    )44()(4.

    1

    22)(3.

    72)(2.

    65)(1.

    32

    3

    2

    +=

    ++=

    +=

    =

    +=

    xxfNo

    xxxfNo

    x

    xxfNo

    xxxfNo

    xxxfNo

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    Pers grs +inggung, grs Normal

    -ur.a suatu /ungsi: /'&(

    *itik P '&o,yo( 0erada pada kura /'&(

    Pers grs +inggung: y 1 yo m '& &o(

    %engan: m tg /$ '&o(

    Pers grs Normal: y 1 yo !5m '& &o(

    %engan: m tg /$ '&o(

    xxxf

    soal

    4)(

    :

    2=

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    *itik ma&imum1minimum suatu Fungsi

    -ur.a suatu /ungsi: /'&(

    *itik ma&imum1minimumnya:

    /$ '&( 6 , &! , &2

    /$$ '&!(

    /$$ '&2(

    "ila /$$ '&!( 76 maka &! titikminimum

    "ila /$$ '&!( 86 maka &! titik

    ma&imum

    1)(

    :

    23++=

    xxxxf

    soal

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    *urunan Parsiil

    )f

    !)(

    x

    f(

    x

    !

    ..................!

    f

    ..................

    x

    f

    :"a#siil$u#unan)(

    32

    =

    =

    =

    ++= yxyxxf

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    )f

    %)(

    !

    f(

    !

    %

    )f

    %

    )(x

    f

    (x

    %

    )f

    !)(

    x

    f(

    x

    !

    ......%

    f......

    !

    f......

    x

    f

    :"a#siil$u#unan

    0),,(:3&"e#muaan"e#s

    =

    =

    =

    =

    =

    =

    =zyxf

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    9ntegral suatu Fungsi

    ika /$'&( adalah turunan dari se0uah

    /ungsi,

    Maka /ungsi asalnya , F'&( adalah :

    = dxxfxF )(')(

    inte#alonstanta:sini*i

    )()(' CxFdxxf +=

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    )ontoh aplikasi

    +)1(+)65(

    +4

    1

    4

    3

    3

    2

    2

    5

    43

    ==++

    =+=

    dxxxdxxx

    x

    dxCxdxx

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    ;umus2

    cxn

    dxxNo

    dxkkdxNo

    cxdxNo

    nn +=

    =

    +=

    +

    113.

    2.

    1.

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    +=

    +=

    =

    =

    +=

    cxdxxNo

    cxdxxNo

    dxxgdxxfdxxgdxxfNo

    dxxfkxfdkNo

    cxfxfdNo

    sin,os8.

    ,ossin7.

    )()()()(6.

    )()((5.

    )()((4.

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    +istem Persamaan inier

    3

    -41o.

    =+

    =+

    yx

    yx

    2

    -32

    4

    4o.

    =

    =++

    =++

    zyx

    zyx

    zyx

    432

    01

    43o.

    =+

    =+

    =+

    xyz

    zx

    yx

    7674

    423

    1432

    2o.

    ==

    =+

    zyx

    zyx

    zyx

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    Matri&

    +olom

    +

    87-3

    14321

    =

    =

    =

    kjumlah

    barisjumlah

    A

    +6,0+

    ++

    4-

    50

    -321

    ==

    ==+

    =

    =

    AAB

    BABA

    B

    A

    +

    11

    12

    03

    430201

    =

    ==

    AB

    BA

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    Matri&

    *et

    )()(*et

    fe

    cbg

    ih

    cbd

    ih

    feaA

    ihg

    fed

    cba

    A

    bcadA

    d

    baA

    ++=

    =

    =

    =

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    Matri&

    )()(*et

    *et

    idbhfageccdhbfgaeiA

    ihg

    fed

    cba

    ihg

    fed

    cba

    A

    atau

    ihg

    fed

    cba

    A

    ++++=

    =

    =

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    Matri&

    /

    *et

    1

    )....(b

    '

    )()(*et

    a

    bd

    AAAInvers

    kolombarisd

    caAATransose

    bcadA

    dbaA

    ==

    ==

    =

    =

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    +istem Persamaan inier

    7674

    423

    1432

    =

    ==+

    zyx

    zyx

    zyx

    7

    4

    1

    6/7/4

    2/1/3

    4/32

    =

    z

    y

    x

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    +istem Persamaan inier

    B!A

    z

    y

    x

    7

    4

    1

    6/7/4

    2/1/3

    4/32

    =

    =

    A

    Az

    A

    Ay

    A

    Ax

    *et

    )3*et(

    *et

    )2*et(

    *et

    )1*et(

    =

    =

    =

    6/7/7/

    2/1/4

    4/31

    *et)1*et( =A

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    7/

    4

    1

    6/7/4

    2/1/3

    4/32

    =

    z

    y

    x

    .....6/7/7/2/1/4

    4/31

    *et)1*et( ==A

    ......

    6/7/4

    2/43

    4/12

    *et)2*et( ==A

    7/7/4

    41/3

    132

    *et)3*et( ==A