Iterasi

Nama : Mutiara Arsyillah

NIM : 230605110132

Jurusan : Teknik Informatika

library(mosaicCalc)

## Loading required package: mosaic

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library(r2symbols)

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Metode Iterasi

Metode Iterasi adlah metode yang memisahkan χ dengan sebagian χ lainnya sehinhga diperoleh χ = g(χ). Dikenal juga dengan metode χ = g(χ). Bentuk iterasi dapat juga dituliskan dalam bentuk

*χi + 1 = g(χi)*
Dimana i = 0,1,2,3,...

Mengulangi suatu tindakan berarti melakukan tindakan itu berulang kali. (“Iterate” berasal dari kata Latin iterum , yang berarti “lagi.”) Seekor burung mengulangi seruannya, menyanyikannya berulang-ulang. Dalam matematika, “iterasi” memiliki keunikan.

Berikut adalah langkah-langkah dalam memecahkan soal dengan metode iterasi.

Contoh Soal:

Hitung akar persamaan berikut dengan Metode Iterasi.

f(x) = χ³ + χ² - 3χ - 3 = 0

1. Bentuk kemungkinan persamaan

1. 3χ = χ³ + χ² - 3 → χ = (χ³ + χ² - 3)/3

2. χ³ = -χ² + 3χ + 3 → χ = (-χ² + 3χ + 3 )⅓

3. χ² = -χ³ + 3χ + 3 → χ = (-χ³ + 3χ + 3)½

2. Cek Konvergensi

library(EBImage)

Image <- readImage("C://Users/WINDOWS 11/Pictures/Konvergensi.png")
print(Image)

## Image 
##   colorMode    : Color 
##   storage.mode : double 
##   dim          : 1824 726 4 
##   frames.total : 4 
##   frames.render: 1 
## 
## imageData(object)[1:5,1:6,1]
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    1    1    1    1    1
## [2,]    1    1    1    1    1    1
## [3,]    1    1    1    1    1    1
## [4,]    1    1    1    1    1    1
## [5,]    1    1    1    1    1    1

Image1 <- Image + 0

par(mfrow= c(1,1))
plot(Image1)

3. Tentukan Perkiraan χ awal

ditentukan perkiraan awal χ = 2

χi + 1(2) = (-χ² + 3χ + 3)⅓
χi + 1(2) = (-2² + 3.2 + 3)⅓ = 1,70998

4. Tentukan Error

Image <- readImage("C://Users/WINDOWS 11/Pictures/rms_error.png")
print(Image)

## Image 
##   colorMode    : Color 
##   storage.mode : double 
##   dim          : 412 154 4 
##   frames.total : 4 
##   frames.render: 1 
## 
## imageData(object)[1:5,1:6,1]
##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]
## [1,] 0.8862745 0.8862745 0.8862745 0.8823529 0.8823529 0.8823529
## [2,] 0.8862745 0.8862745 0.8862745 0.8823529 0.8823529 0.8823529
## [3,] 0.8862745 0.8862745 0.8862745 0.8823529 0.8823529 0.8823529
## [4,] 0.8823529 0.8823529 0.8823529 0.8784314 0.8784314 0.8784314
## [5,] 0.8823529 0.8823529 0.8823529 0.8784314 0.8784314 0.8784314

Image1 <- Image + 0

par(mfrow= c(1,1))
plot(Image1)

Setelahnya lakukan iterasi berikutnya. Iterasi berikutnya semakin mendekati akar persamaan (konvergen).