Fungsi kepekatan peluang distribusi Kumaraswamy adalah sebagai berikut:
\[f(x|5,3)=15x^{4}(1-x^{(5)})^{2}\]
\[E(X)=15x^{5}(1-x^{(5)})^{2}\]
Panjang sub interval adalah sebagai berikut:
\[h=\frac {b-a}{n}=\frac{1-0}{4}=\frac{1}{4}\]
Karena n=4 maka bentuk dari Metode Trapezoidal adalah sebagai berikut:
\[T_4=\frac{h}{2}[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+f(x_4)]\]
Menghitung fungsi f(x) dari setiap \(x_i\)
\[f(x_0)=f(0)=0\]
\[f(x_1)=f(0.25)=0.015\]
\[f(x_2)=f(0.5)=0.44\]
\[f(x_3)=f(0.75)=2.071\]
\[f(x_4)=f(1)=0\]
Karena \(h=\frac{1}{4}\) diperoleh
\[T_4=\frac{1}{8}[0+(2*0.015)+(2*0.44)+(2*2.071)+0]=0.631\]
Sehingga
\[\int\limits_{0}^{1}{15x^{5}(1-x^{(5)})^{2}}\,dx ≈ T_4=0.631\]
Jika dibandingkan dengan hasil yang didapatkan secara exact sebagai berikut:
\[\int\limits_{0}^{1}{15x^{5}(1-x^{(5)})^{2}}\,dx=0.71\]
Panjang sub interval adalah sebagai berikut:
\[h=\frac {b-a}{n}=\frac{1-0}{4}=\frac{1}{4}\]
Karena n=4 maka bentuk dari Metode Trapezoidal adalah sebagai berikut:
\[T_4=\frac{h}{2}[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+f(x_4)]\]
Menghitung fungsi f(x) dari setiap \(x_i\)
\[f(x_0)=f(0)=0\]
\[f(x_1)=f(0.25)=0.015\]
\[f(x_2)=f(0.5)=0.44\]
\[f(x_3)=f(0.75)=2.071\]
\[f(x_4)=f(1)=0\]
Karena \(h=\frac{1}{4}\) diperoleh
\[T_4=\frac{1}{12}[0+(4*0.015)+(2*0.44)+(4*2.071)+0]=0.768\]
Sehingga
\[\int\limits_{0}^{1}{15x^{5}(1-x^{(5)})^{2}}\,dx ≈ T_4=0.768\]
Jika dibandingkan dengan hasil yang didapatkan secara exact sebagai berikut:
\[\int\limits_{0}^{1}{15x^{5}(1-x^{(5)})^{2}}\,dx=0.71\]
\[x=\frac{1}{2}[t(b-a)+a+b]\]
\[x=\frac{1}{2}[t(1-0)+0+1]\]
\[x=\frac{1}{2}[t+1]\]
dan
\[dx=\frac{1}{2}(b-a)dt\]
\[dx=\frac{1}{2}dt\]
Sehingga integralnya menjadi
\[\int\limits_{0}^{1}{15x^{5}(1-x^{(5)})^{2}}\,dx=\int\limits_{0}^{1}{15\frac{1}{2}[t+1]^{5}(1-\frac{1}{2}[t+1]^{(5)})^{2}}\,\frac{1}{2}dt\]
Kemudian, menggunakan nilai-nilai koefisien dan titik gauss saat \(n=4\)
\[\int\limits_{0}^{1}{\frac{15}{4}[t+1]^{5}(1-\frac{1}{2}[t+1]^{(5)})^{2}}\,dt ≈ I=C_1f(t_1)+C_2f(t_2)+C_3f(t_3)+C_4f(t_4)\]
Menghitung fungsi \(f(x)\) pada setiap \(x_i\)
\(f(-0.8611363)=\frac{15(\frac{1}{2}[-0.8611363+1])^{5}(1-(\frac{1}{2}[-0.8611363+1])^{(5)})^{2}}{2}=0.0000121\)
\(f(-0.3399810)=\frac{15(\frac{1}{2}[-0.3399810+1])^{5}(1-(\frac{1}{2}[-0.3399810+1])^{(5)})^{2}}{2}=0.0291\)
\(f(0.3399810)=\frac{15(\frac{1}{2}[0.3399810+1])^{5}(1-(\frac{1}{2}[0.3399810+1])^{(5)})^{2}}{2}=0.7576\)
\(f(0.8611363)=\frac{15(\frac{1}{2}[0.8611363+1])^{5}(1-(\frac{1}{2}[0.8611363+1])^{(5)})^{2}}{2}=0.4775\)
Sehingga
\(I=0.347858*0.0000121+0.6521452*0.0291+0.6521452*0.7576+ 0.347858*0.4775=0.679\)
Jika dibandingkan dengan hasil yang didapatkan secara exact sebagai berikut:
\[\int\limits_{0}^{1}{15x^{5}(1-x^{(5)})^{2}}\,dx=0.71\]
Berikut adalah user-defined function untuk metode Trapezoidal
trapezoid <- function(ftn, a, b, n = 100) {
h <- (b-a)/n
x.vec <- seq(a, b, by = h)
f.vec <- sapply(x.vec, ftn) # ftn(x.vec)
Trap <- h*(f.vec[1]/2 + sum(f.vec[2:n]) + f.vec[n+1]/2)
return(Trap)
}Berikut adalah user-defined function untuk metode Simpson
simpson_n <- function(ftn, a, b, n = 100) {
n <- max(c(2*(n %/% 2), 4))
h <- (b-a)/n
x.vec1 <- seq(a+h, b-h, by = 2*h) # ganjil
x.vec2 <- seq(a+2*h, b-2*h, by = 2*h) # genap
f.vec1 <- sapply(x.vec1, ftn) # ganjil
f.vec2 <- sapply(x.vec2, ftn) # genap
S <- h/3*(ftn(a) + ftn(b) + 4*sum(f.vec1) + 2*sum(f.vec2))
return(S)
}\[𝐸(𝑋) = \frac{𝑏Γ(1+ 1/a )Γ(𝑏)}{Γ(1+ 1/a +𝑏)}=15x^{4}(1-x^{(5)})^{2}\]
Mendefinisikan fungsi f(x)
f <- function(x){
15*x^5*(1-x^5)^2
}Menghitung integral menggunakan fungsi trapezoid
trapezoid(f,0,1,n = 4)## [1] 0.6312869Hitunglah menggunakan R fungsi trapzfun dari package pracma
trapzfun(f,0,1,maxit = 4)## $value
## [1] 0.7098069
##
## $iter
## [1] 4
##
## $rel.err
## [1] 0.005901199Hitunglah menggunakan R hingga hasil integral mendekati hasil exact-nya dengan toleransi 0.0001
exact_value=0.7102273
tol <- 0.0001
err <- 1
n = 4
while(err>tol){
res_trap <- trapezoid(f,0,1,n = n)
err <- abs(res_trap-exact_value)
cat("n=",n,", result=",res_trap,", error=",err,"\n",sep = "")
n=n+1
if(n==1000){
break
}
}## n=4, result=0.6312869, error=0.07894042
## n=5, result=0.6738123, error=0.03641497
## n=6, result=0.6914928, error=0.01873454
## n=7, result=0.6997126, error=0.01051473
## n=8, result=0.7039057, error=0.006321603
## n=9, result=0.7062116, error=0.004015656
## n=10, result=0.7075597, error=0.002667593
## n=11, result=0.7083885, error=0.001838822
## n=12, result=0.7089199, error=0.001307439
## n=13, result=0.7092729, error=0.0009544148
## n=14, result=0.7095147, error=0.000712649
## n=15, result=0.7096846, error=0.0005426739
## n=16, result=0.7098069, error=0.0004204042
## n=17, result=0.7098966, error=0.0003306616
## n=18, result=0.7099637, error=0.0002636072
## n=19, result=0.7100146, error=0.0002127014
## n=20, result=0.7100538, error=0.0001734994
## n=21, result=0.7100844, error=0.0001429188
## n=22, result=0.7101085, error=0.0001187833
## n=23, result=0.7101278, error=9.953088e-05Mendefinisikan fungsi f(x)
f2 <- function(x){
15*x^5*(1-x^5)^2
}Menghitung integral menggunakan metode simpson
simpson_n(f2,0,1,n = 4)## [1] 0.7683974Hitunglah menggunakan R hingga hasil integral mendekati hasil exact-nya dengan toleransi 0.0001
exact_value=0.7102273
tol <- 0.0001
err <- 1
n = 4
while(err>tol){
res_simp <- simpson_n(f2,0,1,n = n)
err <- abs(res_simp-exact_value)
cat("n=",n,", result=",res_simp,", error=",err,"\n",sep = "")
n=n+1
if(n==1000){
break
}
}## n=4, result=0.7683974, error=0.05817006
## n=5, result=0.7683974, error=0.05817006
## n=6, result=0.7497082, error=0.03948094
## n=7, result=0.7497082, error=0.03948094
## n=8, result=0.728112, error=0.01788467
## n=9, result=0.728112, error=0.01788467
## n=10, result=0.7188088, error=0.008581532
## n=11, result=0.7188088, error=0.008581532
## n=12, result=0.7147289, error=0.004501593
## n=13, result=0.7147289, error=0.004501593
## n=14, result=0.712782, error=0.002554712
## n=15, result=0.712782, error=0.002554712
## n=16, result=0.711774, error=0.001546662
## n=17, result=0.711774, error=0.001546662
## n=18, result=0.7112144, error=0.0009870756
## n=19, result=0.7112144, error=0.0009870756
## n=20, result=0.7108852, error=0.0006578651
## n=21, result=0.7108852, error=0.0006578651
## n=22, result=0.7106819, error=0.0004545628
## n=23, result=0.7106819, error=0.0004545628
## n=24, result=0.7105511, error=0.0003237819
## n=25, result=0.7105511, error=0.0003237819
## n=26, result=0.710464, error=0.0002366803
## n=27, result=0.710464, error=0.0002366803
## n=28, result=0.7104042, error=0.0001769138
## n=29, result=0.7104042, error=0.0001769138
## n=30, result=0.7103621, error=0.0001348299
## n=31, result=0.7103621, error=0.0001348299
## n=32, result=0.7103318, error=0.0001045198
## n=33, result=0.7103318, error=0.0001045198
## n=34, result=0.7103096, error=8.225045e-05Hitunglah menggunakan R dengan fungsi gaussLegendre dengan menggunakan domain [−1,1]
Mendefinisikan fungsi hasil transformasi f(t)
ft <- function(t){
(15*(((t+1)/2)^5)*(((1-(((t+1)/2)^5))^2)))/2
}Legendre order 4
gL <- gaussLegendre(n = 4,a = -1,1)Mendefinisikan koefisien dan gauss point
Ci <- gL$w # koefisien
Ci## [1] 0.3478548 0.6521452 0.6521452 0.3478548ti <- gL$x # gauss point
ti## [1] -0.8611363 -0.3399810 0.3399810 0.8611363ft(ti)## [1] 1.210189e-05 2.912641e-02 7.575892e-01 4.779089e-01Menghitung integral
I <- sum(Ci * ft(ti))
I## [1] 0.6792999Hitunglah menggunakan R dengan fungsi gaussLegendre dengan menggunakan domain asal
Mendefinisikan fungsi f(x)
f3 <- function(x){
15*x^5*((1-(x^5))^2)
}Legendre order 4 dengan domain [0,1]
gL <- gaussLegendre(n = 4,a = 0,1)Mendefinisikan koefisien dan gauss point
Ci <- gL$w # koefisien
Ci## [1] 0.1739274 0.3260726 0.3260726 0.1739274xi <- gL$x # gauss point
xi## [1] 0.06943184 0.33000948 0.66999052 0.93056816f3(xi)## [1] 2.420379e-05 5.825281e-02 1.515178e+00 9.558177e-01Menghitung integral
I <- sum(Ci * f3(xi))
I## [1] 0.6792999Hitunglah menggunakan R hingga hasil integral mendekati hasil exact-nya dengan toleransi 0.00001
exact_value=0.7102273
tol <- 0.0001
err <- 1
n = 4
while(err>tol){
gL <- gaussLegendre(n = n,a = 0,1)
Ci <- gL$w # koefisien
xi <- gL$x # gauss point
res_gl <- sum(Ci * f3(xi))
err <- abs(res_gl-exact_value)
cat("n=",n,", result=",res_gl,", error=",err,"\n",sep = "")
n=n+1
if(n==1000){
break
}
}## n=4, result=0.6792999, error=0.03092738
## n=5, result=0.707375, error=0.002852279
## n=6, result=0.7101422, error=8.507726e-05exact <- 0.7102273
t <- trapezoid(f,0,1,4)
s <- simpson_n(f2,0,1,4)
g <- I
nilai <- rbind(t,s,g)
exact <- 0.7102273
error_t <- abs(trapezoid(f,0,1,4)-exact)
error_s <- abs(simpson_n(f2,0,1,4)-exact)
error_g <- abs(I-exact)
error <- rbind(error_t,error_s,error_g)
perbandingan<- c("Metode Trapezoidal","Metode Simpson","Metode Gauss Quadrature")
hasil <- data.frame(perbandingan,nilai,error)
hasil## perbandingan nilai error
## t Metode Trapezoidal 0.6312869 0.07894042
## s Metode Simpson 0.7683974 0.05817006
## g Metode Gauss Quadrature 0.6792999 0.03092738Berdasarkan ketiga metode tersebut yaitu, metode Trapezoidal, metode Simpson, dan metode Gauss Quadrature. Nilai yang mendekati dari nilai fungsi E(x) exact atau sebenarnya sebesar 0.71 adalah metode Gauss Quadrature dengan nilai sebesar 0.679 dengan nilai error 0.0309.
Fungsi kepekatan peluang distribusi eksponensial adalah sebagai berikut:
\[𝑓(𝑥|𝜆) = 𝜆e^{−𝜆𝑥}=2e^{-2x}\]
Fungsi sebaran kumulatif distribusi eksponensial adalah sebagai berikut:
\[F(𝑥|𝜆) = 1-e^{-2x}\]
Nilai exact dari FSK distribusi eksponensial
Ft <- function(x){
2*exp(-2*x)
}
F <- integrate(Ft,0,4)
F## 0.9996645 with absolute error < 1.1e-14Mendefinisikan fungsi f(x)
f <- function(x){
2*exp(-2*x)
}Menghitung integral menggunakan fungsi trapezoid
trapezoid(f,0,4,n = 50)## [1] 1.001796Hitunglah menggunakan R fungsi trapzfun dari package pracma
trapzfun(f,0,4,maxit = 5)## $value
## [1] 1.004866
##
## $iter
## [1] 5
##
## $rel.err
## [1] 0.01553891Hitunglah menggunakan R hingga hasil integral mendekati hasil exact-nya dengan toleransi 0.0001
exact_value=0.9996645
tol <- 0.0001
err <- 1
n = 50
while(err>tol){
res_trap <- trapezoid(f,0,4,n = n)
err <- abs(res_trap-exact_value)
cat("n=",n,", result=",res_trap,", error=",err,"\n",sep = "")
n=n+1
if(n==1000){
break
}
}## n=50, result=1.001796, error=0.002131746
## n=51, result=1.001714, error=0.002049003
## n=52, result=1.001635, error=0.001970985
## n=53, result=1.001562, error=0.001897339
## n=54, result=1.001492, error=0.001827745
## n=55, result=1.001426, error=0.00176191
## n=56, result=1.001364, error=0.001699569
## n=57, result=1.001305, error=0.001640479
## n=58, result=1.001249, error=0.001584417
## n=59, result=1.001196, error=0.001531181
## n=60, result=1.001145, error=0.001480583
## n=61, result=1.001097, error=0.001432452
## n=62, result=1.001051, error=0.001386631
## n=63, result=1.001007, error=0.001342973
## n=64, result=1.000966, error=0.001301345
## n=65, result=1.000926, error=0.001261623
## n=66, result=1.000888, error=0.001223692
## n=67, result=1.000852, error=0.001187446
## n=68, result=1.000817, error=0.001152787
## n=69, result=1.000784, error=0.001119624
## n=70, result=1.000752, error=0.001087871
## n=71, result=1.000722, error=0.00105745
## n=72, result=1.000693, error=0.001028287
## n=73, result=1.000665, error=0.001000315
## n=74, result=1.000638, error=0.000973468
## n=75, result=1.000612, error=0.0009476878
## n=76, result=1.000587, error=0.0009229182
## n=77, result=1.000564, error=0.0008991072
## n=78, result=1.000541, error=0.000876206
## n=79, result=1.000519, error=0.0008541686
## n=80, result=1.000497, error=0.0008329523
## n=81, result=1.000477, error=0.0008125168
## n=82, result=1.000457, error=0.0007928242
## n=83, result=1.000438, error=0.0007738389
## n=84, result=1.00042, error=0.0007555275
## n=85, result=1.000402, error=0.0007378584
## n=86, result=1.000385, error=0.000720802
## n=87, result=1.000369, error=0.0007043303
## n=88, result=1.000353, error=0.0006884168
## n=89, result=1.000338, error=0.0006730365
## n=90, result=1.000323, error=0.000658166
## n=91, result=1.000308, error=0.000643783
## n=92, result=1.000294, error=0.0006298663
## n=93, result=1.000281, error=0.0006163961
## n=94, result=1.000268, error=0.0006033534
## n=95, result=1.000255, error=0.0005907204
## n=96, result=1.000243, error=0.00057848
## n=97, result=1.000231, error=0.0005666162
## n=98, result=1.00022, error=0.0005551136
## n=99, result=1.000208, error=0.0005439578
## n=100, result=1.000198, error=0.0005331349
## n=101, result=1.000187, error=0.0005226319
## n=102, result=1.000177, error=0.0005124362
## n=103, result=1.000167, error=0.000502536
## n=104, result=1.000157, error=0.00049292
## n=105, result=1.000148, error=0.0004835774
## n=106, result=1.000139, error=0.0004744979
## n=107, result=1.00013, error=0.0004656717
## n=108, result=1.000122, error=0.0004570896
## n=109, result=1.000113, error=0.0004487425
## n=110, result=1.000105, error=0.000440622
## n=111, result=1.000097, error=0.00043272
## n=112, result=1.00009, error=0.0004250287
## n=113, result=1.000082, error=0.0004175406
## n=114, result=1.000075, error=0.0004102487
## n=115, result=1.000068, error=0.0004031462
## n=116, result=1.000061, error=0.0003962266
## n=117, result=1.000054, error=0.0003894836
## n=118, result=1.000047, error=0.0003829112
## n=119, result=1.000041, error=0.0003765039
## n=120, result=1.000035, error=0.0003702561
## n=121, result=1.000029, error=0.0003641625
## n=122, result=1.000023, error=0.0003582181
## n=123, result=1.000017, error=0.0003524181
## n=124, result=1.000011, error=0.0003467579
## n=125, result=1.000006, error=0.0003412329
## n=126, result=1, error=0.000335839
## n=127, result=0.9999951, error=0.0003305719
## n=128, result=0.9999899, error=0.0003254278
## n=129, result=0.9999849, error=0.0003204029
## n=130, result=0.99998, error=0.0003154935
## n=131, result=0.9999752, error=0.000310696
## n=132, result=0.9999705, error=0.0003060072
## n=133, result=0.9999659, error=0.0003014237
## n=134, result=0.9999614, error=0.0002969424
## n=135, result=0.9999571, error=0.0002925604
## n=136, result=0.9999528, error=0.0002882747
## n=137, result=0.9999486, error=0.0002840824
## n=138, result=0.9999445, error=0.000279981
## n=139, result=0.9999405, error=0.0002759677
## n=140, result=0.9999365, error=0.0002720401
## n=141, result=0.9999327, error=0.0002681958
## n=142, result=0.9999289, error=0.0002644324
## n=143, result=0.9999252, error=0.0002607477
## n=144, result=0.9999216, error=0.0002571395
## n=145, result=0.9999181, error=0.0002536057
## n=146, result=0.9999146, error=0.0002501442
## n=147, result=0.9999113, error=0.0002467531
## n=148, result=0.9999079, error=0.0002434306
## n=149, result=0.9999047, error=0.0002401747
## n=150, result=0.9999015, error=0.0002369837
## n=151, result=0.9998984, error=0.0002338558
## n=152, result=0.9998953, error=0.0002307895
## n=153, result=0.9998923, error=0.0002277832
## n=154, result=0.9998893, error=0.0002248352
## n=155, result=0.9998864, error=0.000221944
## n=156, result=0.9998836, error=0.0002191083
## n=157, result=0.9998808, error=0.0002163266
## n=158, result=0.9998781, error=0.0002135976
## n=159, result=0.9998754, error=0.0002109198
## n=160, result=0.9998728, error=0.0002082921
## n=161, result=0.9998702, error=0.0002057133
## n=162, result=0.9998677, error=0.000203182
## n=163, result=0.9998652, error=0.0002006972
## n=164, result=0.9998628, error=0.0001982577
## n=165, result=0.9998604, error=0.0001958624
## n=166, result=0.999858, error=0.0001935102
## n=167, result=0.9998557, error=0.0001912002
## n=168, result=0.9998534, error=0.0001889313
## n=169, result=0.9998512, error=0.0001867026
## n=170, result=0.999849, error=0.0001845131
## n=171, result=0.9998469, error=0.0001823618
## n=172, result=0.9998447, error=0.000180248
## n=173, result=0.9998427, error=0.0001781708
## n=174, result=0.9998406, error=0.0001761292
## n=175, result=0.9998386, error=0.0001741225
## n=176, result=0.9998366, error=0.00017215
## n=177, result=0.9998347, error=0.0001702108
## n=178, result=0.9998328, error=0.0001683042
## n=179, result=0.9998309, error=0.0001664294
## n=180, result=0.9998291, error=0.0001645858
## n=181, result=0.9998273, error=0.0001627727
## n=182, result=0.9998255, error=0.0001609893
## n=183, result=0.9998237, error=0.0001592352
## n=184, result=0.999822, error=0.0001575095
## n=185, result=0.9998203, error=0.0001558117
## n=186, result=0.9998186, error=0.0001541413
## n=187, result=0.999817, error=0.0001524976
## n=188, result=0.9998154, error=0.00015088
## n=189, result=0.9998138, error=0.0001492881
## n=190, result=0.9998122, error=0.0001477212
## n=191, result=0.9998107, error=0.0001461789
## n=192, result=0.9998092, error=0.0001446606
## n=193, result=0.9998077, error=0.0001431658
## n=194, result=0.9998062, error=0.0001416941
## n=195, result=0.9998047, error=0.000140245
## n=196, result=0.9998033, error=0.000138818
## n=197, result=0.9998019, error=0.0001374127
## n=198, result=0.9998005, error=0.0001360286
## n=199, result=0.9997992, error=0.0001346653
## n=200, result=0.9997978, error=0.0001333224
## n=201, result=0.9997965, error=0.0001319995
## n=202, result=0.9997952, error=0.0001306962
## n=203, result=0.9997939, error=0.0001294122
## n=204, result=0.9997926, error=0.0001281469
## n=205, result=0.9997914, error=0.0001269002
## n=206, result=0.9997902, error=0.0001256715
## n=207, result=0.999789, error=0.0001244606
## n=208, result=0.9997878, error=0.0001232671
## n=209, result=0.9997866, error=0.0001220908
## n=210, result=0.9997854, error=0.0001209311
## n=211, result=0.9997843, error=0.000119788
## n=212, result=0.9997832, error=0.0001186609
## n=213, result=0.999782, error=0.0001175497
## n=214, result=0.999781, error=0.0001164541
## n=215, result=0.9997799, error=0.0001153737
## n=216, result=0.9997788, error=0.0001143083
## n=217, result=0.9997778, error=0.0001132575
## n=218, result=0.9997767, error=0.0001122212
## n=219, result=0.9997757, error=0.0001111991
## n=220, result=0.9997747, error=0.0001101908
## n=221, result=0.9997737, error=0.0001091962
## n=222, result=0.9997727, error=0.0001082151
## n=223, result=0.9997717, error=0.000107247
## n=224, result=0.9997708, error=0.000106292
## n=225, result=0.9997698, error=0.0001053496
## n=226, result=0.9997689, error=0.0001044197
## n=227, result=0.999768, error=0.0001035021
## n=228, result=0.9997671, error=0.0001025965
## n=229, result=0.9997662, error=0.0001017028
## n=230, result=0.9997653, error=0.0001008207
## n=231, result=0.9997644, error=9.995e-05Mendefinisikan fungsi f(x)
f2 <- function(x){
2*exp(-2*x)
}Menghitung integral menggunakan metode simpson
simpson_n(f2,0,4,n = 50)## [1] 0.9996682Hitunglah menggunakan R hingga hasil integral mendekati hasil exact-nya dengan toleransi 0.0001
exact_value=0.9996645
tol <- 0.0001
err <- 1
n = 50
while(err>tol){
res_simp <- simpson_n(f2,0,4,n = n)
err <- abs(res_simp-exact_value)
cat("n=",n,", result=",res_simp,", error=",err,"\n",sep = "")
n=n+1
if(n==1000){
break
}
}## n=50, result=0.9996682, error=3.665977e-06Hitunglah menggunakan R dengan fungsi gaussLegendre dengan menggunakan domain [−1,1]
Mendefinisikan fungsi hasil transformasi f(t)
ft <- function(t){
4*exp(-4*(t+1))
}Legendre order 50
gL <- gaussLegendre(n = 50,a = -1,1)Mendefinisikan koefisien dan gauss point
Ci <- gL$w # koefisien
Ci## [1] 0.002908623 0.006759799 0.010590548 0.014380823 0.018115561 0.021780243
## [7] 0.025360674 0.028842994 0.032213728 0.035459836 0.038568757 0.041528463
## [13] 0.044327504 0.046955051 0.049400938 0.051655703 0.053710622 0.055557745
## [19] 0.057189926 0.058600850 0.059785059 0.060737971 0.061455900 0.061936067
## [25] 0.062176617 0.062176617 0.061936067 0.061455900 0.060737971 0.059785059
## [31] 0.058600850 0.057189926 0.055557745 0.053710622 0.051655703 0.049400938
## [37] 0.046955051 0.044327504 0.041528463 0.038568757 0.035459836 0.032213728
## [43] 0.028842994 0.025360674 0.021780243 0.018115561 0.014380823 0.010590548
## [49] 0.006759799 0.002908623ti <- gL$x # gauss point
ti## [1] -0.99886640 -0.99403197 -0.98535408 -0.97286439 -0.95661096 -0.93665662
## [7] -0.91307856 -0.88596798 -0.85542977 -0.82158207 -0.78455583 -0.74449430
## [13] -0.70155247 -0.65589647 -0.60770293 -0.55715830 -0.50445814 -0.44980633
## [19] -0.39341431 -0.33550025 -0.27628819 -0.21600724 -0.15489059 -0.09317470
## [25] -0.03109834 0.03109834 0.09317470 0.15489059 0.21600724 0.27628819
## [31] 0.33550025 0.39341431 0.44980633 0.50445814 0.55715830 0.60770293
## [37] 0.65589647 0.70155247 0.74449430 0.78455583 0.82158207 0.85542977
## [43] 0.88596798 0.91307856 0.93665662 0.95661096 0.97286439 0.98535408
## [49] 0.99403197 0.99886640ft(ti)## [1] 3.981903530 3.905642252 3.772397335 3.588563207 3.362679687 3.104711626
## [7] 2.825283145 2.534930649 2.243446813 1.959369317 1.689643724 1.439465077
## [13] 1.212281639 1.009929194 0.832856492 0.680401359 0.551081229 0.442869427
## [19] 0.353437578 0.280353308 0.221229896 0.173830135 0.136130230 0.106351372
## [25] 0.082967005 0.064693212 0.050468573 0.039428436 0.030877282 0.024261649
## [31] 0.019145135 0.015186280 0.012119604 0.009739766 0.007888582 0.006444570
## [37] 0.005314632 0.004427521 0.003728748 0.003176647 0.002739352 0.002392480
## [43] 0.002117376 0.001899775 0.001728792 0.001596168 0.001495697 0.001422809
## [49] 0.001374269 0.001347949Menghitung integral
I <- sum(Ci * ft(ti))
I## [1] 0.9996645Hitunglah menggunakan R dengan fungsi gaussLegendre dengan menggunakan domain asal
Mendefinisikan fungsi f(x)
f3 <- function(x){
2*exp(-2*x)
}Legendre order 50 dengan domain [0,4]
gL <- gaussLegendre(n = 50,a = 0,4)Mendefinisikan koefisien dan gauss point
Ci <- gL$w # koefisien
Ci## [1] 0.005817245 0.013519598 0.021181097 0.028761646 0.036231121 0.043560486
## [7] 0.050721347 0.057685987 0.064427456 0.070919671 0.077137513 0.083056926
## [13] 0.088655009 0.093910103 0.098801877 0.103311406 0.107421244 0.111115490
## [19] 0.114379851 0.117201700 0.119570117 0.121475942 0.122911799 0.123872135
## [25] 0.124353233 0.124353233 0.123872135 0.122911799 0.121475942 0.119570117
## [31] 0.117201700 0.114379851 0.111115490 0.107421244 0.103311406 0.098801877
## [37] 0.093910103 0.088655009 0.083056926 0.077137513 0.070919671 0.064427456
## [43] 0.057685987 0.050721347 0.043560486 0.036231121 0.028761646 0.021181097
## [49] 0.013519598 0.005817245xi <- gL$x # gauss point
xi## [1] 0.002267191 0.011936061 0.029291832 0.054271230 0.086778090 0.126686762
## [7] 0.173842887 0.228064041 0.289140461 0.356835858 0.430888334 0.511011396
## [13] 0.596895063 0.688207069 0.784594146 0.885683391 0.991083710 1.100387330
## [19] 1.213171376 1.328999509 1.447423612 1.567985526 1.690218820 1.813650597
## [25] 1.937803323 2.062196677 2.186349403 2.309781180 2.432014474 2.552576388
## [31] 2.671000491 2.786828624 2.899612670 3.008916290 3.114316609 3.215405854
## [37] 3.311792931 3.403104937 3.488988604 3.569111666 3.643164142 3.710859539
## [43] 3.771935959 3.826157113 3.873313238 3.913221910 3.945728770 3.970708168
## [49] 3.988063939 3.997732809f3(xi)## [1] 1.9909517649 1.9528211259 1.8861986676 1.7942816036 1.6813398436
## [6] 1.5523558128 1.4126415723 1.2674653244 1.1217234063 0.9796846586
## [11] 0.8448218622 0.7197325383 0.6061408194 0.5049645972 0.4164282460
## [16] 0.3402006796 0.2755406146 0.2214347136 0.1767187891 0.1401766540
## [21] 0.1106149481 0.0869150676 0.0680651149 0.0531756862 0.0414835023
## [26] 0.0323466062 0.0252342867 0.0197142180 0.0154386408 0.0121308244
## [31] 0.0095725677 0.0075931400 0.0060598020 0.0048698828 0.0039442911
## [36] 0.0032222851 0.0026573160 0.0022137603 0.0018643738 0.0015883236
## [41] 0.0013696759 0.0011962401 0.0010586881 0.0009498875 0.0008643962
## [46] 0.0007980840 0.0007478483 0.0007114047 0.0006871344 0.0006739744Menghitung integral
I <- sum(Ci * f3(xi))
I## [1] 0.9996645Hitunglah menggunakan R hingga hasil integral mendekati hasil exact-nya dengan toleransi 0.00001
exact_value=0.9996645
tol <- 0.0001
err <- 1
n = 50
while(err>tol){
gL <- gaussLegendre(n = n,a = 0,4)
Ci <- gL$w # koefisien
xi <- gL$x # gauss point
res_gl <- sum(Ci * f3(xi))
err <- abs(res_gl-exact_value)
cat("n=",n,", result=",res_gl,", error=",err,"\n",sep = "")
n=n+1
if(n==1000){
break
}
}## n=50, result=0.9996645, error=3.73721e-08mc_integral <- function(ftn, a, b,m=5000){
#Membangkitkan x berdistribusi U(a,b)
x <- runif(m,a,b)
# Menghitung rata-rata dari output fungsi
Gx <- ftn(x)
Gx_m <- mean(Gx)
theta.hat <- (b-a)*Gx_m
return(theta.hat)
}g <- function(x) {
2*exp(-2*x)
}set.seed(123)
int_g <- mc_integral(g,a=0,b=4,m = 5000)
int_g## [1] 1.018859exact <- 0.9996645
t <- trapezoid(f,0,4,50)
s <- simpson_n(f2,0,4,50)
g <- I
m <- int_g
nilai <- rbind(t,s,g,m)
exact <- 0.9996645
error_t <- abs(trapezoid(f,0,4,50)-exact)
error_s <- abs(simpson_n(f2,0,4,50)-exact)
error_g <- abs(I-exact)
error_m <- abs(int_g-exact)
error <- rbind(error_t,error_s,error_g,error_m)
perbandingan<- c("Metode Trapezoidal","Metode Simpson","Metode Gauss Quadrature","Metode Monte Carlo")
hasil <- data.frame(perbandingan,nilai,error)
hasil## perbandingan nilai error
## t Metode Trapezoidal 1.0017962 2.131746e-03
## s Metode Simpson 0.9996682 3.665977e-06
## g Metode Gauss Quadrature 0.9996645 3.737210e-08
## m Metode Monte Carlo 1.0188585 1.919404e-02Berdasarkan keempat metode tersebut yaitu, metode Trapezoidal, metode Simpson,metode Gauss Quadrature, dan metode Monte Carlo. Nilai yang mendekati dari nilai fungsi exact atau sebenarnya sebesar 0.9996645 adalah metode Gauss Quadrature dengan nilai sebesar 0.9996645 dan error sebesar 3.737210e-08.