STAT Simulation HW3
Kajal Chokshi
9/26/2019
Excercise 9.4
The minimum value of n must be 48 in order to have an error at most of 10^-16 for log1.5.
The minimum value of n must be 38 in order to have an error at most of 10^-16 for log2.
A better way to calculate is to initialize x as log2 = 0.414214.
#log1.5
eps <- 1e-16
x <- 0.5
n <- 0
log1x <- 0
while (n == 0 || abs(last.term) > eps) {
n <- n + 1
last.term <- (x^n)/n
log1x <- log1x + last.term
}
n## [1] 48#log2 (as log2 = 2*log sqrt(2), and sqrt(2) = 1.414214, we initiate at 0.414214)
eps <- 10^-16
x <- 0.414214
n <- 0
log1x <- 0
while (n == 0 || abs(last.term) > eps) {
n <- n + 1
last.term <- (x^n)/n
}
n## [1] 38Excercise 9.5
Using the system time function, we see that the second method takes less time. Also, just by looking at the two formulas, we see that the second has less costly calculations! With the first, we must do a subtraction before taking the power of it.
We demonstrate catastrophic cancellation below as well! We see that the second method is more efficient but there is a difference between the two calculations.
x<- c(1:1000000)
xbar <- mean(x)
y <- 0
n <- length(x)
system.time(for (i in 1:n){
y <- (x[i] -xbar)^2 + y
}
)## user system elapsed
## 0.115 0.001 0.117y <-0
system.time(
for(i in 1:n){
y <- (x[i]^2) + y
}
)## user system elapsed
## 0.096 0.000 0.097options(digits=22)
x <- c(100.5371293010230012, 104.23849132410213)
x_bar <- mean(x)
n <- length(x)
ay <- sum((x - x_bar)^2)/(n-1)
be <- (sum(x^2) - n*(x_bar^2))
ay-be## [1] 1.697308960046939319e-12Excercise 9.9
Vector operations is much quicker than for loops.
library(stats)
x <- c(1:50)
y <- c(2:51)
n <- length(x)
indexSum <- c()
k <- 1
#Convultion using for loops
convol_FL <- function(x, y){
n <- length(x)
indexSum <- c()
k <- 1
for (k in 1:((2*n)-1)){
indexSum[k] <- 0
for (i in max(c(1, k-n+1)):min(c(k:n))) {
indexSum[k] <- x[i]*y[k-i+1] + indexSum[k]
}
}
return(indexSum)
}
system.time(convol_FL(x,y))## user system
## 0.017000000000000126121336 0.001000000000000000888178
## elapsed
## 0.017000000000000126121336#Convultion using vector operations programming
convol_VB <- function(k, x, y){
n <- length(x)
i <- max(c(1, k-n+1))
j <- min(c(k, n))
sum(x[i:j]*y[(k-i+1):(k-j+1)])
}
n <- length(x)
system.time(unlist(lapply(as.list(1:(2*n-1)),convol_VB, x = x, y= y)))## user system
## 0.01000000000000000888178 0.00000000000000000000000
## elapsed
## 0.01099999999999989874766Excercise 9.10
n <- 10000
x <- rnorm(0, n)
j <- system.time(for (i in 1:n) x[i] <- i^2)
hist(j)
n <- 10000
x <- rnorm(0, n)
j <- system.time(for (i in 1:n) x[i] <- i^2)
hist(j)
n <- 10000
x <- rep(0, n)
k <- system.time(for (i in 1:n) x[i] <- i+i)
hist(k)
n <- 10000
x <- rep(0, n)
l <- system.time(for (i in 1:n) x[i] <- i^20)
hist(l)
Excercise 10.3
Newton Raphson is the faster method. It converges with the point in the root finding methodologies. It has a continous derivative for the function that is defined.
library(spuRs)## Loading required package: MASS## Warning: package 'MASS' was built under R version 3.5.2## Loading required package: lattice#Fixed Point
fixpoint <- function(x) {return(cos(x))}
system.time(fixedpoint(fixpoint, 0))## At iteration 1 value of x is: 1
## At iteration 2 value of x is: 0.5403023058681397650105
## At iteration 3 value of x is: 0.8575532158463934351289
## At iteration 4 value of x is: 0.6542897904977791334602
## At iteration 5 value of x is: 0.7934803587425656168364
## At iteration 6 value of x is: 0.7013687736227565139302
## At iteration 7 value of x is: 0.7639596829006541911511
## At iteration 8 value of x is: 0.722102425026707850364
## At iteration 9 value of x is: 0.7504177617637604047474
## At iteration 10 value of x is: 0.7314040424225098924182
## At iteration 11 value of x is: 0.7442373549005568689196
## At iteration 12 value of x is: 0.7356047404363472708155
## At iteration 13 value of x is: 0.7414250866101093073723
## At iteration 14 value of x is: 0.7375068905132426966276
## At iteration 15 value of x is: 0.7401473355678758192511
## At iteration 16 value of x is: 0.7383692041223230972946
## At iteration 17 value of x is: 0.7395672022122561362067
## At iteration 18 value of x is: 0.7387603198742112020625
## At iteration 19 value of x is: 0.7393038923969059705854
## At iteration 20 value of x is: 0.7389377567153443404635
## At iteration 21 value of x is: 0.7391843997714937497534
## At iteration 22 value of x is: 0.739018262427412198079
## At iteration 23 value of x is: 0.739130176529671056862
## At iteration 24 value of x is: 0.7390547907469173782857
## At iteration 25 value of x is: 0.7391055719265362533577
## At iteration 26 value of x is: 0.7390713652989449045094
## At iteration 27 value of x is: 0.739094407379091267174
## At iteration 28 value of x is: 0.7390788859949920519554
## At iteration 29 value of x is: 0.73908934140339277441
## At iteration 30 value of x is: 0.739082298522402258989
## At iteration 31 value of x is: 0.7390870426953322080976
## At iteration 32 value of x is: 0.7390838469650001796296
## At iteration 33 value of x is: 0.7390859996481299321047
## At iteration 34 value of x is: 0.7390845495752126348421
## At iteration 35 value of x is: 0.7390855263619244830409
## At iteration 36 value of x is: 0.7390848683867141666326
## At iteration 37 value of x is: 0.7390853116067619410501
## At iteration 38 value of x is: 0.7390850130484203361902
## At iteration 39 value of x is: 0.7390852141609169922987
## At iteration 40 value of x is: 0.7390850786891229873277
## At iteration 41 value of x is: 0.739085169944554420951
## At iteration 42 value of x is: 0.7390851084737986020912
## At iteration 43 value of x is: 0.7390851498812395226068
## At iteration 44 value of x is: 0.7390851219886893641586
## At iteration 45 value of x is: 0.7390851407774466919065
## At iteration 46 value of x is: 0.7390851281211138212868
## At iteration 47 value of x is: 0.7390851366465718275123
## At iteration 48 value of x is: 0.7390851309037207439445
## At iteration 49 value of x is: 0.7390851347721743014318
## At iteration 50 value of x is: 0.7390851321663374218218
## At iteration 51 value of x is: 0.739085133921660530909
## At iteration 52 value of x is: 0.7390851327392538028249
## At iteration 53 value of x is: 0.7390851335357372375867
## Algorithm converged## user system
## 0.002000000000000223820962 0.000000000000000000000000
## elapsed
## 0.002999999999999669597628#Newton Raphson
nwtnraph <- function(x){
fx <- cos(x)-x
dfx <- -sin(x)-1
return(c(fx,dfx))
}
system.time(newtonraphson(nwtnraph, 0, max.iter = 1000))## At iteration 1 value of x is: 1
## At iteration 2 value of x is: 0.7503638678402438921822
## At iteration 3 value of x is: 0.7391128909113616751725
## At iteration 4 value of x is: 0.7390851333852839211147
## Algorithm converged## user system
## 0.006000000000000227373675 0.000000000000000000000000
## elapsed
## 0.005999999999999783284466Excercise 10.10
Part A.
We get VERY close!
library(spuRs)
kajal <- function(x){
fx <- sin(x)
dfx <- cos(x)
return(c(fx,dfx))
}
newtonraphson(kajal,3,max.iter=1000)## At iteration 1 value of x is: 3.142546543074277831664
## At iteration 2 value of x is: 3.141592653300476989386
## Algorithm converged## [1] 3.141592653300476989386pi## [1] 3.141592653589793115998Part B.
The graph definitely looks like the sin(x) curve.
kajalsin <- function(x,n){
sum_start <- 0
for (k in 0:n){
sum_start <- (((-1)^k)*(x^(2*k+1)))/(factorial(2*k+1)) + sum_start
}
return(sum_start)
}
xvals <- seq(0,7,0.1)
y <- c()
for (i in 1:length(xvals)){
y[i] <- kajalsin(xvals[i], 10)
}
plot(xvals,y, main="Sin Curve Estimation")
Part C.
We use the secant method and get fairly close.
kajalsin <- function(x,n){
sum_start <- 0
for (k in 0:n){
sum_start <- (((-1)^k)*(x^(2*k+1)))/(factorial(2*k+1)) + sum_start
}
return(sum_start)
}
secantmeth <- function(x0,x1,f,n=10){
x <- c()
x[1] <- x0
x[2] <- x1
for (i in (2:n)) {
x[i+1] <- x[i] - kajalsin(x[i],n)*(x[i]-x[i-1])/(kajalsin(x[i], n)-kajalsin(x[i-1], n))
print(x[i+1])
}
}
secantmeth(3,3.1,funcSample)## [1] 3.141773092017887236693
## [1] 3.141592601790574423859
## [1] 3.141592653600141282766
## [1] 3.141592653600141726855
## [1] 3.141592653600141282766
## [1] 3.141592653600141282766
## [1] NaN
## [1] NaN
## [1] NaNpi## [1] 3.141592653589793115998Excercise 7 Matthews
Below, I have plotted the function in Wolfram Alpha to see the approximate roots. We see that around 300, 750, and 900 the curve crosses the x axis. These are all roots.
Underline Test
When using the Newton Raphson method though, we find that the algorithm does not converge for the second root. I hypothesize it has something to do with the derivative at that point exactly as it bends back up quickly.
The three roots are 312, 773, and 872.
library(spuRs)
#Closer Graph to Get Better Estimates
ogfunc <- function(x){x^3 - 1957*x^2 + 1187296*x - 210305472}
curve(ogfunc, ylim = c(-1,10), xlim =c(300,900))
#Need the function and deriviatve for NR
kfunc <- function(x){
fx <- x^3 - 1957*x^2 + 1187296*x - 210305472
dfx <- 3*x^2 - 3914 + 1187296
return(c(fx,dfx))
}
#Converges to 312
x0 <- 300
newtonraphson(kfunc, x0, max.iter=100)## At iteration 1 value of x is: 302.2338738198216105957
## At iteration 2 value of x is: 304.0312544233837002139
## At iteration 3 value of x is: 305.4843732570101337842
## At iteration 4 value of x is: 306.663655501942457704
## At iteration 5 value of x is: 307.6236335545912652378
## At iteration 6 value of x is: 308.4070165692093041798
## At iteration 7 value of x is: 309.0475657194317022913
## At iteration 8 value of x is: 309.5721722714416728195
## At iteration 9 value of x is: 310.0023892331710726467
## At iteration 10 value of x is: 310.3555795593104562613
## At iteration 11 value of x is: 310.6457895939144577824
## At iteration 12 value of x is: 310.8844219228392944387
## At iteration 13 value of x is: 311.0807593336186300803
## At iteration 14 value of x is: 311.2423766152629127646
## At iteration 15 value of x is: 311.3754667605620056747
## At iteration 16 value of x is: 311.4851010905982207078
## At iteration 17 value of x is: 311.5754378573031999622
## At iteration 18 value of x is: 311.6498903233285204806
## At iteration 19 value of x is: 311.7112627307860179826
## At iteration 20 value of x is: 311.7618606604520437031
## At iteration 21 value of x is: 311.8035808544436235934
## At iteration 22 value of x is: 311.8379844936840186165
## At iteration 23 value of x is: 311.8663570932353650278
## At iteration 24 value of x is: 311.8897575379587010502
## At iteration 25 value of x is: 311.9090582809555485255
## At iteration 26 value of x is: 311.9249783338329962135
## At iteration 27 value of x is: 311.9381103660977032632
## At iteration 28 value of x is: 311.9489429824366766297
## At iteration 29 value of x is: 311.9578790474173501934
## At iteration 30 value of x is: 311.9652507667108807254
## At iteration 31 value of x is: 311.9713321042522125026
## At iteration 32 value of x is: 311.9763490095569409277
## At iteration 33 value of x is: 311.9804878438505966187
## At iteration 34 value of x is: 311.9839023239046014169
## At iteration 35 value of x is: 311.9867192454786959388
## At iteration 36 value of x is: 311.9890432016346721866
## At iteration 37 value of x is: 311.9909604729663215039
## At iteration 38 value of x is: 311.9925422354377246847
## At iteration 39 value of x is: 311.993847205774500253
## At iteration 40 value of x is: 311.9949238231915842334
## At iteration 41 value of x is: 311.9958120488381041469
## At iteration 42 value of x is: 311.9965448500208822225
## At iteration 43 value of x is: 311.997149424477811408
## At iteration 44 value of x is: 311.9976482102659929296
## At iteration 45 value of x is: 311.9980597188300635025
## At iteration 46 value of x is: 311.998399222226225902
## At iteration 47 value of x is: 311.9986793200467332099
## At iteration 48 value of x is: 311.9989104071103724891
## At iteration 49 value of x is: 311.9991010592943325719
## At iteration 50 value of x is: 311.9992583518388755692
## At iteration 51 value of x is: 311.9993881219447757758
## At iteration 52 value of x is: 311.9994951854155260662
## At iteration 53 value of x is: 311.999583515387200805
## At iteration 54 value of x is: 311.999656389781478083
## At iteration 55 value of x is: 311.9997165129558993613
## At iteration 56 value of x is: 311.9997661160663824376
## At iteration 57 value of x is: 311.9998070398681875304
## At iteration 58 value of x is: 311.9998408030274390512
## At iteration 59 value of x is: 311.9998686584799543198
## At iteration 60 value of x is: 311.9998916399279664802
## At iteration 61 value of x is: 311.9999106002008488758
## At iteration 62 value of x is: 311.9999262429038253686
## At iteration 63 value of x is: 311.9999391485284832015
## At iteration 64 value of x is: 311.9999497959947802883
## At iteration 65 value of x is: 311.999958580423708554
## At iteration 66 value of x is: 311.9999658277997696132
## At iteration 67 value of x is: 311.9999718070684480153
## At iteration 68 value of x is: 311.9999767401166081982
## At iteration 69 value of x is: 311.9999808100062637095
## At iteration 70 value of x is: 311.9999841677684457864
## At iteration 71 value of x is: 311.9999869380073391767
## At iteration 72 value of x is: 311.9999892235246647942
## At iteration 73 value of x is: 311.9999911091344415581
## At iteration 74 value of x is: 311.9999926648102928084
## At iteration 75 value of x is: 311.9999939482823378967
## At iteration 76 value of x is: 311.9999950071793932693
## At iteration 77 value of x is: 311.9999958807962912033
## At iteration 78 value of x is: 311.9999966015524250906
## At iteration 79 value of x is: 311.9999971961944993382
## At iteration 80 value of x is: 311.9999976867892996779
## At iteration 81 value of x is: 311.9999980915424657724
## At iteration 82 value of x is: 311.999998425474132091
## At iteration 83 value of x is: 311.999998700976220789
## At iteration 84 value of x is: 311.9999989282724186523
## At iteration 85 value of x is: 311.9999991157975500755
## At iteration 86 value of x is: 311.9999992705105569257
## At iteration 87 value of x is: 311.9999993981526813513
## At iteration 88 value of x is: 311.999999503460685446
## At iteration 89 value of x is: 311.9999995903424974131
## At iteration 90 value of x is: 311.9999996620221622834
## At iteration 91 value of x is: 311.999999721159724686
## At iteration 92 value of x is: 311.9999997699496816495
## At iteration 93 value of x is: 311.9999998102026097513
## At iteration 94 value of x is: 311.9999998434122971958
## At iteration 95 value of x is: 311.9999998708111661472
## At iteration 96 value of x is: 311.9999998934159748387
## At iteration 97 value of x is: 311.9999999120654479157
## At iteration 98 value of x is: 311.9999999274517676895
## At iteration 99 value of x is: 311.9999999401458694592
## At iteration 100 value of x is: 311.999999950618814637
## Algorithm failed to converge## NULL#The original guess was 770 and I kept playing around with it but the function would not work. However when I used a root calculator, I found 773 is one of the roots and when I plugged in that value the algorithm converged.
x1 <- 773
newtonraphson(kfunc, x1, max.iter=1000)## Algorithm converged## [1] 773#Converges to 872
x2 <- 870
newtonraphson(kfunc, x2, max.iter=1000)## At iteration 1 value of x is: 870.0313403098131175284
## At iteration 2 value of x is: 870.0621997509597349563
## At iteration 3 value of x is: 870.0925853831881795486
## At iteration 4 value of x is: 870.1225041725120945557
## At iteration 5 value of x is: 870.1519629921427849695
## At iteration 6 value of x is: 870.1809686234211085321
## At iteration 7 value of x is: 870.2095277567518678552
## At iteration 8 value of x is: 870.2376469925371793579
## At iteration 9 value of x is: 870.2653328421104106383
## At iteration 10 value of x is: 870.2925917286698904718
## At iteration 11 value of x is: 870.3194299882109135069
## At iteration 12 value of x is: 870.3458538704576312739
## At iteration 13 value of x is: 870.3718695397924420831
## At iteration 14 value of x is: 870.3974830761842440552
## At iteration 15 value of x is: 870.4227004761140733535
## At iteration 16 value of x is: 870.4475276534979002463
## At iteration 17 value of x is: 870.4719704406074924918
## At iteration 18 value of x is: 870.4960345889871859981
## At iteration 19 value of x is: 870.5197257703682680585
## At iteration 20 value of x is: 870.5430495775786994272
## At iteration 21 value of x is: 870.5660115254498805371
## At iteration 22 value of x is: 870.5886170517187565565
## At iteration 23 value of x is: 870.6108715179256023475
## At iteration 24 value of x is: 870.632780210307487323
## At iteration 25 value of x is: 870.6543483406870791441
## At iteration 26 value of x is: 870.6755810473561041363
## At iteration 27 value of x is: 870.6964833959540328578
## At iteration 28 value of x is: 870.7170603803410813271
## At iteration 29 value of x is: 870.7373169234666647753
## At iteration 30 value of x is: 870.7572578782309165035
## At iteration 31 value of x is: 870.776888028341431891
## At iteration 32 value of x is: 870.7962120891639870024
## At iteration 33 value of x is: 870.8152347085667770443
## At iteration 34 value of x is: 870.8339604677591978543
## At iteration 35 value of x is: 870.852393882123919866
## At iteration 36 value of x is: 870.8705394020423682377
## At iteration 37 value of x is: 870.8884014137146323264
## At iteration 38 value of x is: 870.9059842399717581429
## At iteration 39 value of x is: 870.9232921410824701525
## At iteration 40 value of x is: 870.9403293155522760571
## At iteration 41 value of x is: 870.9570999009167735494
## At iteration 42 value of x is: 870.9736079745265442398
## At iteration 43 value of x is: 870.9898575543272727373
## At iteration 44 value of x is: 871.0058525996308844697
## At iteration 45 value of x is: 871.0215970118812265355
## At iteration 46 value of x is: 871.0370946354116767907
## At iteration 47 value of x is: 871.0523492581961590986
## At iteration 48 value of x is: 871.0673646125932236828
## At iteration 49 value of x is: 871.0821443760821694013
## At iteration 50 value of x is: 871.096692171992913245
## At iteration 51 value of x is: 871.1110115702282428174
## At iteration 52 value of x is: 871.1251060879786791702
## At iteration 53 value of x is: 871.1389791904305184289
## At iteration 54 value of x is: 871.1526342914665974604
## At iteration 55 value of x is: 871.1660747543593288356
## At iteration 56 value of x is: 871.1793038924571419557
## At iteration 57 value of x is: 871.1923249698634208471
## At iteration 58 value of x is: 871.2051412021080523118
## At iteration 59 value of x is: 871.2177557568120391807
## At iteration 60 value of x is: 871.2301717543446102354
## At iteration 61 value of x is: 871.2423922684736226074
## At iteration 62 value of x is: 871.2544203270080060975
## At iteration 63 value of x is: 871.2662589124341820934
## At iteration 64 value of x is: 871.2779109625437286013
## At iteration 65 value of x is: 871.2893793710558156818
## At iteration 66 value of x is: 871.3006669882310006869
## At iteration 67 value of x is: 871.3117766214788844081
## At iteration 68 value of x is: 871.3227110359581502053
## At iteration 69 value of x is: 871.3334729551701229866
## At iteration 70 value of x is: 871.3440650615450522309
## At iteration 71 value of x is: 871.3544899970214601126
## At iteration 72 value of x is: 871.364750363618782103
## At iteration 73 value of x is: 871.3748487240029589884
## At iteration 74 value of x is: 871.3847876020455487378
## At iteration 75 value of x is: 871.3945694833754487263
## At iteration 76 value of x is: 871.4041968159243651826
## At iteration 77 value of x is: 871.4136720104658024866
## At iteration 78 value of x is: 871.4229974411468901963
## At iteration 79 value of x is: 871.4321754460136162379
## At iteration 80 value of x is: 871.4412083275301483809
## At iteration 81 value of x is: 871.4500983530912208153
## At iteration 82 value of x is: 871.4588477555278132058
## At iteration 83 value of x is: 871.4674587336068043442
## At iteration 84 value of x is: 871.4759334525244867109
## At iteration 85 value of x is: 871.4842740443932598282
## At iteration 86 value of x is: 871.4924826087222982096
## At iteration 87 value of x is: 871.5005612128923075943
## At iteration 88 value of x is: 871.5085118926234599712
## At iteration 89 value of x is: 871.5163366524380990086
## At iteration 90 value of x is: 871.5240374661169653336
## At iteration 91 value of x is: 871.5316162771491690364
## At iteration 92 value of x is: 871.539074999176818892
## At iteration 93 value of x is: 871.5464155164333988068
## At iteration 94 value of x is: 871.5536396841766872967
## At iteration 95 value of x is: 871.5607493291155378756
## At iteration 96 value of x is: 871.5677462498310887895
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## At iteration 833 value of x is: 871.9999970186687505702
## At iteration 834 value of x is: 871.9999970663765225254
## At iteration 835 value of x is: 871.9999971133208873653
## At iteration 836 value of x is: 871.9999971595138958946
## At iteration 837 value of x is: 871.999997204967826292
## At iteration 838 value of x is: 871.9999972496943883016
## At iteration 839 value of x is: 871.9999972937051779809
## At iteration 840 value of x is: 871.9999973370116777005
## At iteration 841 value of x is: 871.9999973796252561442
## At iteration 842 value of x is: 871.9999974215569409353
## At iteration 843 value of x is: 871.9999974628176460101
## At iteration 844 value of x is: 871.9999975034180579314
## At iteration 845 value of x is: 871.9999975433687495752
## At iteration 846 value of x is: 871.9999975826801801304
## At iteration 847 value of x is: 871.9999976213624677257
## At iteration 848 value of x is: 871.9999976594258441764
## At iteration 849 value of x is: 871.9999976968800865507
## At iteration 850 value of x is: 871.9999977337348582296
## At iteration 851 value of x is: 871.9999977700000499681
## At iteration 852 value of x is: 871.9999978056848704
## At iteration 853 value of x is: 871.999997840798641846
## At iteration 854 value of x is: 871.9999978753504592532
## At iteration 855 value of x is: 871.9999979093494175686
## At iteration 856 value of x is: 871.9999979428042706786
## At iteration 857 value of x is: 871.9999979757237724698
## At iteration 858 value of x is: 871.9999980081165631418
## At iteration 859 value of x is: 871.9999980399909418338
## At iteration 860 value of x is: 871.9999980713553213718
## At iteration 861 value of x is: 871.9999981022177735213
## At iteration 862 value of x is: 871.9999981325863700476
## At iteration 863 value of x is: 871.9999981624689553428
## At iteration 864 value of x is: 871.9999981918734874853
## At iteration 865 value of x is: 871.9999982208073561196
## At iteration 866 value of x is: 871.9999982492782919508
## At iteration 867 value of x is: 871.9999982772936846231
## At iteration 868 value of x is: 871.9999983048606964076
## At iteration 869 value of x is: 871.9999983319864895748
## At iteration 870 value of x is: 871.9999983586783400824
## At iteration 871 value of x is: 871.9999983849429554539
## At iteration 872 value of x is: 871.9999984107872705863
## At iteration 873 value of x is: 871.9999984362181066899
## At iteration 874 value of x is: 871.9999984612419439145
## At iteration 875 value of x is: 871.9999984858653760966
## At iteration 876 value of x is: 871.999998510094883386
## At iteration 877 value of x is: 871.999998533936604872
## At iteration 878 value of x is: 871.9999985573967933306
## At iteration 879 value of x is: 871.999998580481587851
## At iteration 880 value of x is: 871.9999986031969001488
## At iteration 881 value of x is: 871.9999986255487556264
## At iteration 882 value of x is: 871.9999986475429523125
## At iteration 883 value of x is: 871.9999986691851745491
## At iteration 884 value of x is: 871.9999986904811066779
## At iteration 885 value of x is: 871.9999987114362056673
## At iteration 886 value of x is: 871.9999987320559284854
## At iteration 887 value of x is: 871.9999987523458457872
## At iteration 888 value of x is: 871.9999987723109597937
## At iteration 889 value of x is: 871.999998791956727473
## At iteration 890 value of x is: 871.9999988112880373592
## At iteration 891 value of x is: 871.9999988303098916731
## At iteration 892 value of x is: 871.9999988490275200093
## At iteration 893 value of x is: 871.999998867445583528
## At iteration 894 value of x is: 871.9999988855688570766
## At iteration 895 value of x is: 871.9999989034022291889
## At iteration 896 value of x is: 871.9999989209501336518
## At iteration 897 value of x is: 871.9999989382172316255
## At iteration 898 value of x is: 871.9999989552080705835
## At iteration 899 value of x is: 871.9999989719269706256
## At iteration 900 value of x is: 871.9999989883783655387
## At iteration 901 value of x is: 871.9999990045664617355
## At iteration 902 value of x is: 871.999999020495579316
## At iteration 903 value of x is: 871.9999990361698110064
## At iteration 904 value of x is: 871.9999990515931358459
## At iteration 905 value of x is: 871.9999990667697602476
## At iteration 906 value of x is: 871.999999081703435877
## At iteration 907 value of x is: 871.9999990963981417735
## At iteration 908 value of x is: 871.9999991108577432897
## At iteration 909 value of x is: 871.9999991250858784042
## At iteration 910 value of x is: 871.9999991390862987828
## At iteration 911 value of x is: 871.9999991528627560911
## At iteration 912 value of x is: 871.999999166418774621
## At iteration 913 value of x is: 871.9999991797578786645
## At iteration 914 value of x is: 871.9999991928834788268
## At iteration 915 value of x is: 871.999999205798985713
## At iteration 916 value of x is: 871.9999992185079236151
## At iteration 917 value of x is: 871.9999992310134757645
## At iteration 918 value of x is: 871.9999992433189390795
## At iteration 919 value of x is: 871.9999992554274967915
## At iteration 920 value of x is: 871.9999992673423321321
## At iteration 921 value of x is: 871.999999279066400959
## At iteration 922 value of x is: 871.9999992906028865036
## At iteration 923 value of x is: 871.9999993019547446238
## At iteration 924 value of x is: 871.9999993131250448641
## At iteration 925 value of x is: 871.9999993241165157087
## At iteration 926 value of x is: 871.9999993349321130154
## At iteration 927 value of x is: 871.9999993455745652682
## At iteration 928 value of x is: 871.9999993560467146381
## At iteration 929 value of x is: 871.9999993663512896092
## At iteration 930 value of x is: 871.9999993764910186655
## At iteration 931 value of x is: 871.9999993864685166045
## At iteration 932 value of x is: 871.9999993962862845365
## At iteration 933 value of x is: 871.9999994059470509455
## At iteration 934 value of x is: 871.9999994154530895685
## At iteration 935 value of x is: 871.9999994248070152025
## At iteration 936 value of x is: 871.9999994340113289581
## At iteration 937 value of x is: 871.9999994430684182589
## At iteration 938 value of x is: 871.9999994519805568416
## At iteration 939 value of x is: 871.9999994607500184429
## At iteration 940 value of x is: 871.9999994693791904865
## At iteration 941 value of x is: 871.9999994778702330223
## At iteration 942 value of x is: 871.9999994862254197869
## At iteration 943 value of x is: 871.9999994944469108304
## At iteration 944 value of x is: 871.9999995025368662027
## At iteration 945 value of x is: 871.9999995104973322668
## At iteration 946 value of x is: 871.9999995183304690727
## At iteration 947 value of x is: 871.9999995260382092965
## At iteration 948 value of x is: 871.9999995336225993015
## At iteration 949 value of x is: 871.9999995410855717637
## At iteration 950 value of x is: 871.9999995484292867332
## At iteration 951 value of x is: 871.9999995556553358256
## At iteration 952 value of x is: 871.999999562765765404
## At iteration 953 value of x is: 871.9999995697623944579
## At iteration 954 value of x is: 871.9999995766471556635
## At iteration 955 value of x is: 871.9999995834217543234
## At iteration 956 value of x is: 871.99999959008789574
## At iteration 957 value of x is: 871.9999995966473989029
## At iteration 958 value of x is: 871.9999996031018554277
## At iteration 959 value of x is: 871.9999996094530843038
## At iteration 960 value of x is: 871.9999996157026771471
## At iteration 961 value of x is: 871.9999996218522255731
## At iteration 962 value of x is: 871.9999996279033211977
## At iteration 963 value of x is: 871.9999996338576693233
## At iteration 964 value of x is: 871.999999639716747879
## At iteration 965 value of x is: 871.9999996454821484804
## At iteration 966 value of x is: 871.9999996511552353695
## At iteration 967 value of x is: 871.9999996567374864753
## At iteration 968 value of x is: 871.9999996622304934135
## At iteration 969 value of x is: 871.9999996676355067393
## At iteration 970 value of x is: 871.9999996729540043816
## At iteration 971 value of x is: 871.9999996781874642693
## At iteration 972 value of x is: 871.9999996833371369576
## At iteration 973 value of x is: 871.9999996884043866885
## At iteration 974 value of x is: 871.9999996933905777041
## At iteration 975 value of x is: 871.9999996982969605597
## At iteration 976 value of x is: 871.9999997031247858104
## At iteration 977 value of x is: 871.9999997078754176982
## At iteration 978 value of x is: 871.9999997125501067785
## At iteration 979 value of x is: 871.9999997171498762327
## At iteration 980 value of x is: 871.9999997216760903029
## At iteration 981 value of x is: 871.9999997261298858575
## At iteration 982 value of x is: 871.9999997305123997648
## At iteration 983 value of x is: 871.9999997348247688933
## At iteration 984 value of x is: 871.9999997390682437981
## At iteration 985 value of x is: 871.9999997432437339739
## At iteration 986 value of x is: 871.9999997473523762892
## At iteration 987 value of x is: 871.9999997513953076123
## At iteration 988 value of x is: 871.9999997553734374378
## At iteration 989 value of x is: 871.9999997592879026342
## At iteration 990 value of x is: 871.9999997631398400699
## At iteration 991 value of x is: 871.9999997669301592396
## At iteration 992 value of x is: 871.9999997706598833247
## At iteration 993 value of x is: 871.9999997743298081332
## At iteration 994 value of x is: 871.9999997779410705334
## At iteration 995 value of x is: 871.9999997814944663332
## At iteration 996 value of x is: 871.9999997849910187142
## At iteration 997 value of x is: 871.999999788431637171
## At iteration 998 value of x is: 871.9999997918172311984
## At iteration 999 value of x is: 871.9999997951485966041
## At iteration 1000 value of x is: 871.999999798426642883
## Algorithm failed to converge## NULL