materi kuliah mo
TRANSCRIPT
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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
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=
=
A
Az
A
Ay
A
Ax
*et
)3*et(
*et
)2*et(
*et
)1*et(
=
=
=
6/7/7/
2/1/4
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*et)1*et( =A
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7/
4
1
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=
z
y
x
.....6/7/7/2/1/4
4/31
*et)1*et( ==A
......
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*et)2*et( ==A
7/7/4
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132
*et)3*et( ==A