Rumus MNumerik.xlsx

8/17/2019 Rumus MNumerik.xlsx

http://slidepdf.com/reader/full/rumus-mnumerikxlsx 1/29

0.00001

no a b T f(a) fT faT b-a akar

1 0.00000 1.00000 0.50000 1.00000 3.64872 3.64872 1.00000 lanjut

2 0.50000 1.00000 0.75000 3.64872 5.11700 18.67051 0.50000 lanjut

3 0.75000 1.00000 0.87500 5.11700 5.89888 30.18454 0.25000 lanjut

4 0.87500 1.00000 0.93750 5.89888 6.30359 37.18409 0.12500 lanjut

5 0.93750 1.00000 0.96875 6.30359 6.50965 41.03416 0.06250 lanjut

6 0.96875 1.00000 0.98438 6.50965 6.61364 43.05247 0.03125 lanjut7 0.98438 1.00000 0.99219 6.61364 6.66588 44.08571 0.01563 lanjut

8 0.99219 1.00000 0.99609 6.66588 6.69206 44.60845 0.00781 lanjut

9 0.99609 1.00000 0.99805 6.69206 6.70517 44.87136 0.00391 lanjut

10 0.99805 1.00000 0.99902 6.70517 6.71172 45.00321 0.00195 lanjut

11 0.99902 1.00000 0.99951 6.71172 6.71500 45.06923 0.00098 lanjut

12 0.99951 1.00000 0.99976 6.71500 6.71664 45.10226 0.00049 lanjut

13 0.99976 1.00000 0.99988 6.71664 6.71746 45.11878 0.00024 lanjut

14 0.99988 1.00000 0.99994 6.71746 6.71787 45.12705 0.00012 lanjut

15 0.99994 1.00000 0.99997 6.71787 6.71808 45.13118 0.00006 lanjut

16 0.99997 1.00000 0.99998 6.71808 6.71818 45.13324 0.00003 lanjut

17 0.99998 1.00000 0.99999 6.71818 6.71823 45.13428 0.00002 lanjut

18 0.99999 1.00000 1.00000 6.71823 6.71826 45.13479 0.00001 stop

19 1.00000 1.00000 1.00000 6.71826 6.71827 45.13505 0.00000 stop20 1.00000 1.00000 1.00000 6.71827 6.71828 45.13518 0.00000 stop

f(x) xe^+x = 0

a -1

b 0

epslon 0.00001

no a b T f(a) fT faT b-a akar

1 -1.00000 0.00000 -0.50000 -1.71828 0.17564 -0.30180 1.00000 lanjut



2 -1.00000 -0.50000 -0.75000 -1.71828 -0.58775 1.00992 0.50000 lanjut

3 -0.75000 -0.50000 -0.62500 -0.58775 -0.16765 0.09854 0.25000 lanjut

4 -0.62500 -0.50000 -0.56250 -0.16765 0.01278 -0.00214 0.12500 lanjut

5 -0.62500 -0.56250 -0.59375 -0.16765 -0.07514 0.01260 0.06250 lanjut

6 -0.59375 -0.56250 -0.57813 -0.07514 -0.03062 0.00230 0.03125 lanjut

7 -0.57813 -0.56250 -0.57031 -0.03062 -0.00878 0.00027 0.01563 lanjut

8 -0.57031 -0.56250 -0.56641 -0.00878 0.00204 -0.00002 0.00781 lanjut

9 -0.57031 -0.56641 -0.56836 -0.00878 -0.00336 0.00003 0.00391 lanjut

10 -0.56836 -0.56641 -0.56738 -0.00336 -0.00066 0.00000 0.00195 lanjut

11 -0.56738 -0.56641 -0.56689 -0.00066 0.00069 0.00000 0.00098 lanjut12 -0.56738 -0.56689 -0.56714 -0.00066 0.00001 0.00000 0.00049 lanjut

13 -0.56738 -0.56714 -0.56726 -0.00066 -0.00032 0.00000 0.00024 lanjut

14 -0.56726 -0.56714 -0.56720 -0.00032 -0.00016 0.00000 0.00012 lanjut

15 -0.56720 -0.56714 -0.56717 -0.00016 -0.00007 0.00000 0.00006 lanjut

16 -0.56717 -0.56714 -0.56715 -0.00007 -0.00003 0.00000 0.00003 lanjut

17 -0.56715 -0.56714 -0.56715 -0.00003 -0.00001 0.00000 0.00002 lanjut

18 -0.56715 -0.56714 -0.56714 -0.00001 0.00000 0.00000 0.00001 -0.56714249







Metode Posisi Palsu

!nput

"utput

Langkah-langkah :

2. #tun$ C=b-f(b) b-a f(x)f((b) - f(a) a

3. %&aC-Clama b

C epslon

'o. a b Clama f(a) f(b) fb(b-a) fb-fa

1 0.00000 1.00000 2.00000 1.00000 -1.28172 -1.28172 -2.28172

2 0.00000 0.43827 0.43827 1.00000 -0.20305 -0.08899 -1.20305

3 *518 0.36430 0.36430 *518 -0.01769 *518 *5184 *518 *518 *518 *518 *518 *518 *518

5 *518 *518 *518 *518 *518 *518 *518

6 *518 *518 *518 *518 *518 *518 *518

f(x) xe^+x = 0

a -1.00000

b 0.00000

epslon 0.00005

'o. a b Clama f(a) f(b) fb(b-a) fb-fa1 -1.00000 0.00000 1.00000 -1.71828 1.00000 1.00000 2.71828

2 -1.00000 -0.36788 -0.36788 -1.71828 0.46854 0.29617 2.18682

3 -1.00000 -0.50331 -0.50331 -1.71828 0.16742 0.08316 1.88570

4 -1.00000 -0.54741 -0.54741 -1.71828 0.05365 0.02428 1.77193

5 -1.00000 -0.56112 -0.56112 -1.71828 0.01658 0.00727 1.73486

6 -1.00000 -0.56531 -0.56531 -1.71828 0.00506 0.00220 1.72334

7 -1.00000 -0.56659 -0.56659 -1.71828 0.00154 0.00067 1.71982

8 -1.00000 -0.56697 -0.56697 -1.71828 0.00047 0.00020 1.71875

9 -1.00000 -0.56709 -0.56709 -1.71828 0.00014 0.00006 1.71842

10 -1.00000 -0.56713 -0.56713 -1.71828 0.00004 0.00002 1.71833

* f(x),a,b,epsilon

* akar

1. entu&an Clama = b-a

,= epslon maka akar = . elesa

4. %&a f(a).f(/) , 0 a&a b=! j&a ta& a=!

5. Clama = C, &ebal &e lan$&a 2



e"x-#x0

1

0.00001

C $ar%!ek akar f! fa&f!

0.43827 3.56344 anjut -0.20305 -0.20305

0.36430 0.20305 anjut -0.01769 -0.01769

*518 *518 *518 *518 *518*518 *518 *518 *518 *518

*518 *518 *518 *518 *518

*518 *518 *518 *518 *518

C $ar%!ek akar f! fa&f!-0.36788 3.71828 anjut 0.46854 -0.80508

-0.50331 0.26909 anjut 0.16742 -0.28767

-0.54741 0.08056 anjut 0.05365 -0.09218

-0.56112 0.02442 anjut 0.01658 -0.02848

-0.56531 0.00742 anjut 0.00506 -0.00870

-0.56659 0.00225 anjut 0.00154 -0.00265

-0.56697 0.00068 anjut 0.00047 -0.00081

-0.56709 0.00021 anjut 0.00014 -0.00024

-0.56713 0.00006 anjut 0.00004 -0.00007

-0.56714 0.00002 -0.56714 0.00001 -0.00002





eton aphson

!nput f(x), f*(x), +o, epsilon, M

"utput akar

Langkah-langkah :

2. #tun$ x=+o - f(+o)f(+o)

3. #tun$ f*(+i),f(+i)

) = 0 a&a $a$al

6. %&a +i -+i

,= epslon a&a a&a = +1 sel+i

7. =+1

as = !teas + 1

&ebal &e lan$&a 2

teas f() f() e&f +1 ae& &a

0 0.00000 1.00000 -3.00000 lanjut 0.33333 1.00000 anjut

1 0.33333 0.06228 -2.60439 lanjut 0.35725 0.06694 anjut

2 0.35725 0.00040 -2.57061 lanjut 0.35740 0.00044 anjut

3 0.35740 0.00000 -2.57039 lanjut 0.35740 0.00000 0.357403

1. !teas =

i = +i-f(+i).f(+i)



fx e-4x

fx e-4

x0 0

epslon 0.00001

"ptonal

anjut

anjut

anjut

top





xi f(xi) /erapa f(x) bila x = 0&1

x0 6 1.79175

x1 7 1.94591 f(x1) - f(x0)

x2 7.5 2.01490 f(x) = f(x0) + ------------------ ( x -

x3 7.7 2.04122 ( x1 - x0 )

x4 8.2 2.10413 3 1 2x5

1.94591 - 1.79175

f(x) = 1.79175 + ------------------ 6.5 -

7 - 6

0.15416

f(x) = 1.79175+ ------------------ 0.

1

f(x) = 1.79175 + 0.15416 0.

f(x)= &20223

f(x)= &20223


x0 6 1.79175

x1 7 1.94591 f(x1) - f(x0)

x2 7.5 2.01490 f(x) = f(x0) + ------------------ ( x -

x3 7.7 2.04122 ( x1 - x0 )

x4 8.2 2.10413 3 1 2

x5

2.01490 - 1.94591

f(x) = 1.94591 + ------------------ 7.3 -

7.5 - 7

0.06899

f(x) = 1.94591+ ------------------ 0.

0.5

f(x) = 1.94591 + 0.13798 0.



f(x)= &52436

f(x)= &52436



Persamaan 7nterpolasi :

f(x1) - f(x0)

x0 ) f(x) = f(x0) + ------------------ ( x - x0 )

( x1 - x0 )

1.94591 - 1.79175

f(x) = 1.79175+ ------------------ x - 6

6 7 - 6

f(x) = 1.79175 + 0.15416 x - 6

f(x) = 1.79175 + 0.15416 x -0.92496

f(x) = 1.79175 + 0.15416 x + -0.92496

f(x) = 0.86679 + 0.15416 x

f 6.5 = 0.86679 + 0.15416 . 6.5

= 0.86679 + 1.00204

= &20223

Persamaan 7nterpolasi :

f(x1) - f(x0)

x0 ) f(x) = f(x0) + ------------------ ( x - x0 )

( x1 - x0 )

2.01490 - 1.94591

f(x) = 1.94591+ ------------------ x - 7

7 7.5 - 7

f(x) = 1.94591 + 0.13798 x - 7

f(x) = 1.94591 + 0.13798 x -0.96586

f(x) = 1.94591 + 0.13798 x + -0.96586

f(x) = 0.98005 + 0.13798 x



f 7.3 = 0.98005 + 0.13798 . 7.3

= 0.98005 + 1.00725

= &52436




x0 6 1.79175

x1 7 1.94591 x - x1

x2 7.5 2.01490 f(x) = ------------------ f(x0) +

x3 7.7 2.04122 x0 - x1

x4 8.2 2.10413x5 x - 7

f(x) = ------------------ 1.79175 +

6 - 7

1.79175x -7 1.79175

f(x) = ------------------------------------- +

-1

1.79175 x -12.54225

f(x) = ------------------------------------- +-1

f(x) = -1 1.79175 x -12.54225 + 1 1.94591 x

f(x) = -1.79175 x 12.54225 + 1.94591 x

f(x) = 6&1#0 x 6&20045

f 6.5 = 0.15416 6.5 + 0.86679

= 1.00204 + 0.86679= &20223



xi f(xi)

x0 6 1.79175

x- x0 x1 7 1.94591

------------------ f(x1) x2 7.5 2.01490

x1- x0 x3 7.7 2.04122

x4 8.2 2.10413x - 6 x5

------------------ 1.94591

7 - 6

1.94591x -6 1.94591

-------------------------------------

1

1.94591 x -11.67546

-------------------------------------1

-11.67546 f(x) = -2 1.94591 x

-11.67546 f(x) = -3.89182 x

f(x) = 6&3452 x

f 7.3 =

==



/erapa f(x) bila x = 4&3

x - x1 x - x0

f(x) = ------------------ f(x0) + ------------------ f(x1)

x0 - x1 x1 - x0

x - 7.5 x - 7

f(x) = ------------------ 1.94591 + ------------------ 2.01490

7 - 7.5 7.5 - 7

1.94591 x -7.5 1.94591 2.0149 x -7 2.0149

f(x) = ------------------------------------- + -------------------------------------

-0.5 0.5

1.94591 x -14.594325 2.0149 x -14.1043

f(x) = ------------------------------------- + --------------------------------------0.5 0.5

-14.59433 + 2 2.0149 x -14.1043

29.18865 + 4.0298 x -28.2086

6&52661

0.13798 7.3 + 0.98005

1.007254 + 0.98005&52436



1 20 0.93969 18.7938 400 8000 160000 3200000 375.876 1.00590333

1 35 0.81915 28.67025 1225 42875 1500625 52521875 1003.45875 0.79759133

1 50 0.64279 32.1395 2500 125000 6250000 312500000 1606.975 0.58927933

1 65 0.42262 27.4703 4225 274625 17850625 1160290625 1785.5695 0.38096733

1 80 0.17365 13.892 6400 512000 40960000 3276800000 1111.36 0.17265533

1 95 -0.08716 -8.2802 9025 857375 81450625 7737809375 -786.619 -0.03565666

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

8 0 345 2.91074 112.68565 23775 1819875 148171875 12543121875 5096.62025 2.91074

= 6 ( 112.68565 ) - ( 345 ) ( 2.91074 )

= 676.1139 - 1004.205

6 ( 23775 ) - ( 345 142650 - 119025

= -328.0914

= -6&6323625

= 2.91074 - -0.013887467 ( 345 )

= 2.91074 - -4.791176

= 7.701916

6 6 6

:osentas eo j&a ( 30 ) g(x) = a b(x) =; $ 30 = 1.2836526667 + -0.0138874667 ( 30 < nla e&sa& ( 30 ) = 0.88603

= 0.86702867 - 0.88603

100 = &##110.88603

x> ?> x>.?> x>@ x>A x>B x>C x>@.?> $(x>)

)@

DEDE

t

= ( )G( )( ))H( ∑ ∑ ∑( ^2 )G( )@)( ∑ ∑

=(( . ))∑− ∑ /



n Ix Ix@ I?>

Ix Ix@ IxA = Ix>?>

Ix@ IxA IxB Ix>@?>

J2 x 2 690 47550 3639750

J1 x 115 690 39675 2734125

6 345 23775 2.91074 0 7875 905625

345 23775 1819875 = 112.68565

23775 1819875 148171875 5096.62025 J3 x 2 47550 3639750 296343750

J1 x 7925 47550 2734125 188416875

0 905625 107926875

6 345 23775 2.91074 J3 x 1 0 905625 107926875

0 7875 905625 = -109.3638 J2 x 115 0 905625 104146875

0 905625 107926875 -12874.374 0 0 3780000

6 345 23775 2.91074 3780000 = -297.537

0 7875 905625 = -109.3638 = -4&2439-661

0 0 3780000 -297.537

7875 + 905625 = -109.3638

7875 + 905625 ( -7.87135-005 ) =

7875 + -71.28490625 = -109.3638

= 1.07506191 + -0.0048354 30 + -7.87135-005 900 7875 = -38.07889375

= 1.07506191 + -0.1450625 + -0.0708421429 = -6&66#231#

= 6&21514341

6 + 345 + 23775 = 2.91074

6 + 345 ( -0.004835415 ) + 23775 (

6 + -1.668218202 + -1.8714132738 = 2.91074

6 + -3.539631476 = 2.91074

aL

aE

aM

aL

aE

aM

aL

aE

aM

aL aM

aE a;

aM

aE aM

aE

= a< ax a;x> aEaE

a

aL aE aM

aL

aL

aL



6 = 6.4503714762

= &6416053

aL

a<



pan$an

-0.0662133

0.0215587

0.0535107

0.0416527

0.0009947

-0.0515033

0.0000000

0.0000000

0.0000000

0.00000

&2301004

= 6&20462004



5.3713

4.7351

9.3638

93.2405

67.6145874.374

874.374

576.837

97.537

9.3638

1349-005 ) = 2.91074





1 20 0.93969 18.7938 400 8000 160000

1 35 0.81915 28.67025 1225 42875 1500625

1 50 0.64279 32.1395 2500 125000 6250000

1 65 0.42262 27.4703 4225 274625 17850625

1 80 0.17365 13.892 6400 512000 40960000

1 95 -0.08716 -8.2802 9025 857375 814506250 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

8 0 345 2.91074 112.68565 23775 1819875 148171875

= 6 ( 112.68565 ) - ( 345

6 ( 23775 ) - ( 345

= 2.91074 - -0.0138874667 ( 345 )

=6

:osentas eo j&a ( 30 )g(x) = a b(x) =; $

< nla e&sa& ( 30 ) = 0.88603

= 0.867029 - 0.88603 100 = &##110.88603

n Ix Ix@ IxA I?>

Ix Ix@ IxA IxB = Ix>?>

Ix@ IxA IxB Ix>@?>

IxA IxB IxC

6 345 23775 1819875 2.91074

345 23775 1819875 148171875 = 112.6856523775 1819875 148171875 12543121875 5096.62025

1819875 148171875 12543121875 1090182046875 253229.3388

6 345 23775 1819875 2.91074

0 7875 905625 87058125 = -109.3638

x> ?> x>.?> x>@ x>A x>B

DE

DE

Ft

aL

aE

IxC aM

IxN aO Ix>A?>

aL

aEaM

aO

aL

aE

= ( )G( )( ))H( ∑ ∑ ∑( ^2 )G( )@)( ∑ ∑

=(( . ))∑− ∑ /



0 905625 107926875 10663734375 -12874.374

0 87058125 10663734375 1076382421875 -1259268.975

6 345 23775 1819875 2.910740 7875 905625 87058125 = -109.3638

0 0 3780000 652050000 -297.537

0 0 652050000 113954850000 -50252.166

6 345 23775 1819875 2.91074

0 7875 905625 87058125 = -109.3638

0 0 3780000 652050000 -297.537

0 0 0 2952450000 2145.933

K = 0.984372 + 0.001547981 30 + -0.000204092 900 + 7.26831-007 27000

K = 0.984372 + 0.046439418 + -0.1836826984 + 0.0196244444

6&200414

= 4&666666602366966

aM

aO

aLaE

aM

aO

aL

aE

aM

aO

= a< ax a;x>a?x@

a?



x>@.?>

3200000 64000000 375.876 7517.52 1.0059033333

52521875 1838265625 1003.45875 35121.05625 0.7975913333

312500000 15625000000 1606.975 80348.75 0.5892793333

1160290625 75418890625 1785.5695 116062.0175 0.3809673333

3276800000 262144000000 1111.36 88908.8 0.1726553333

7737809375 735091890625 -786.619 -74728.805 -0.0356566670 0 0 0 0

0 0 0 0 0

0 0 0 0 0

12543121875 1090182046875 5096.62025 253229.33875 2.91074

) ( 2.91074 )=

676.1139 - 1004.205

142650 - 119025

= -328.0914

= -6&63224123625

2.91074 - -4.791176=

7.701916= &23014

6 6

30 = 1.2836526667 + -0.013887467 ( 30 ) = 6&204624

J2 x 2 690 47550 3639750 296343750 225.

J1 x 115 690 39675 2734125 209285625 334.

0 7875 905625 87058125 -109.

J3 x 2 47550 3639750 296343750 25086243750 10193

J1 x 7925 47550 2734125 188416875 14422509375 23067

0 905625 107926875 10663734375 -1287

J4 x 2 3639750 296343750 25086243750 2.18036+012 50645

J1 x 606625 3639750 209285625 14422509375 1.10398+012 17657

0 87058125 10663734375 1.07638+012 -12592

J3 x 1 0 905625 107926875 10663734375 -1287

J2 x 115 0 905625 104146875 10011684375 -1257

x>C x>N x>A.?> $(x>)

)@



0 0 3780000 652050000 -297

J4 x 1 0 87058125 10663734375 1.07638+012 -12592

J2 x 11055 0 87058125 10011684375 962427571875 -12090

0 0 652050000 113954850000 -5025

J4 x 2 0 0 1304100000 227909700000 -1005J3 x 345 0 0 1304100000 224957250000 -1026

0 0 0 2952450000 2145

2952450000 = 2145.933

= 4&02341455#59-664

3780000 + 6.5+008 = -297.537

3780000 + 6.5+008 ( 7.26831-007 ) = -297.537

3780000 + 473.9303333 = -297.537

3780000 = -771.4673333

= -6&6666#

7875 + 905625 + 87058125 = -109.3638

7875 + 905625 ( -0.000204092 ) + 87058125 ( 7.26831-007 ) = -10

7875 + -184.83071528 + 63.27656806 = -109.3638

7875 + -121.55414722 = -109.3638

7875 = 12.190347222

= 6&661#452

6 + 345 + 23775 + 1819875 = 2.91074

6 + 345 ( 0.001547981 ) + 23775 ( -0.000204092 ) + 18

6 + 0.5340533069 + -4.852284616 + 1.322742068 = 2.91074

6 + -2.9954892416 = 2.91074

6 = 5.9062292416

= 6&52#341#

aO

a?

aM aO

aM

aM

aM

a;

aE aM aO

aE

aE

aE

aE

a

aL aE aM aO

aL

aL

aL

aL

a<



pan$an

-0.0662133

0.0215587

0.0535107

0.0416527

0.0009947

-0.05150330.0000000

0.0000000

0.0000000

0.0000000

713

351

3638

.2405

.6145

4.374

.6775

7.653

68.975

4.374

6.837



.537

68.975

16.809

2.166

4.3320.265

.933

9.3638

19875 ( 7.26831-007 ) = 2.91074

Date post:	06-Jul-2018
Category:	Documents
Upload:	muhammad-soleh
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Rumus MNumerik.xlsx

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