library(dplyr)
library(kableExtra)
library(nortest)
library(knitr)
library(gridExtra)
library(numDeriv)
knitr::opts_chunk$set(echo = TRUE)
options(knitr.table.format = "html")
options(sciepn = 999)

Question1.

  • (a)Write a computer program using the Mid-Square Method using 6 digits to generate 10,000 random numbers ranging over [0, 999999]. Use the Kolmogorov-Smirnov Goodness-of-fit test to see if the random numbers that you create are uniformly distributed. (Note: You must notify the initial seed number used, and you may adapt 0.05 as the α value. Also, you may find warning messages for conducting the Goodness-of-fit test, and comment on the Goodness-of-fit test. )

  • (b)Consider the combination of 3 multiplicative congruential generators, i.e.,\(u_i=\frac{x_i}{30269}+\frac{y_i}{30307}+\frac{z_i}{30323}(mod 1)\) with \(x_i =171x_{i－1}(mod30269), y_i =172 y_{i－1} (mod30307), z_i =170z_{i－1}(mod30323)\). Use both the \(\chi2\) and Kolmogorov-Smirnov Goodness-of-fit tests to check if the data are from \(U(0,1)\) distribution.

  • (c)The calculators use \(U_{n+1} = ({\pi} +U_n)^5 (mod 1)\) to generate random numbers between 0 and 1.
    Compare the result with those in (a) & (b), and discuss your finding based on the comparison.

(a) The mid-Square Method

# N 為模擬次數, x 為起始值(亂數決定)#
x <- floor(runif(1)*10^6)
N <- 10000
u <-  NULL
for( i in 1:N){
  y <-  x^2 %>% as.vector()
  while(nchar(y) != 12){
    y <- paste0(0,y)}
    x <- substr(y,4,9) %>% as.numeric()
    u[i] <- substr(y,4,9)}
u <- matrix(u,ncol = 5) 
kable(u) %>% kable_styling() %>% scroll_box(height = "200px")
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num<-as.numeric(u)
hist(num/1000000, main = " The mid-square method ", breaks = 10000)

P <- ks.test(num/1000000,'punif')
## Warning in ks.test(num/1e+06, "punif"): ties should not be present for the
## Kolmogorov-Smirnov test
P$p.value
## [1] 0
print(ifelse(P$p.value <= 0.05,"Reject H0; This is not an uniform distribution","Do not reject H0; This is an uniform distribution"))
## [1] "Reject H0; This is not an uniform distribution"

(b) The combination of 3 m.c.g.

# N 為模擬次數, (x, y, z) 為任意給定值 #
N = 10000
x <- 5; y <- 8; z <- 34
u <- NULL
for(i in 1:N){
  x <- (x * 171 ) %% 30269
  y <- (y * 172 ) %% 30307
  z <- (z * 170 ) %% 30323
  temp <- (x/30269 + y/30307 + z/30323) %% 1
  u <- c(u , temp)}
u <- matrix(u , ncol = 5)
kable(u) %>% kable_styling() %>% scroll_box(height = "200px")
0.2642632 0.0711310 0.7140377 0.2520377 0.6910384
0.0437878 0.0006039 0.8295787 0.7819919 0.3591055
0.8924169 0.7349981 0.7490125 0.6607122 0.7366934
0.0218728 0.9673492 0.9635072 0.8553598 0.0160317
0.2485370 0.1582377 0.3026961 0.6884113 0.9252578
0.8729845 0.3005099 0.7603075 0.7612790 0.8984499
0.6573878 0.7279885 0.2506518 0.1723766 0.6488055
0.9872906 0.1663427 0.5972115 0.3035168 0.2507577
0.6322797 0.2335053 0.4782565 0.7062420 0.6420172
0.4218715 0.4258172 0.8251468 0.2326660 0.6183557
0.0696737 0.9186240 0.0462698 0.4906426 0.7631394
0.2928525 0.5184204 0.6882126 0.9790809 0.6583009
0.6765218 0.3065429 0.3645330 0.5827639 0.8929791
0.7115257 0.2991902 0.4702463 0.5554481 0.2054306
0.2023078 0.4250717 0.2106141 0.2455437 0.4925026
0.3386329 0.4029453 0.7658363 0.7335160 0.9352451
0.6576473 0.7683270 0.4766342 0.2161075 0.4588521
0.0154808 0.5958452 0.9738957 0.4830718 0.4966042
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0.8789925 0.1204347 0.2877668 0.5674735 0.2245776
0.0090702 0.3599978 0.0731874 0.2910573 0.5759542
0.5336686 0.1111488 0.6035613 0.9881365 0.6168966
0.8656320 0.7210236 0.1135605 0.8708210 0.5785344
0.8513992 0.3230243 0.7771304 0.3148467 0.2244940
0.3156356 0.3173314 0.4492368 0.3549244 0.8895086
0.1791312 0.4086482 0.1231824 0.1471914 0.4097955
0.7773834 0.1667499 0.4188330 0.0905467 0.5923223
0.5872194 0.8532782 0.9854688 0.8781409 0.3090686
0.6367347 0.5025195 0.8113075 0.5266796 0.9125349
0.4513426 0.9126370 0.4210689 0.9828865 0.3142195
0.3453624 0.1254252 0.0855907 0.2757137 0.1536807
0.2458732 0.5414834 0.8245046 0.8529241 0.7317605
0.1441601 0.5055150 0.2389854 0.3823161 0.5024120
0.4751865 0.1126121 0.4163379 0.6193783 0.0540594
0.3456362 0.5524858 0.4028073 0.6023126 0.7980278
0.3067640 0.9867541 0.9617922 0.4578481 0.3953332
0.0463785 0.2948513 0.4233525 0.0416346 0.2019290
0.7600580 0.4539947 0.6780236 0.1161482 0.1971724
0.6948629 0.9634845 0.7370385 0.0413442 0.2872697
0.0184617 0.2010322 0.1810285 0.9765839 0.0175242
0.9748881 0.1941192 0.4033415 0.6075149 0.1516195
0.9375700 0.8416996 0.6425394 0.3211000 0.7068152
0.9444609 0.5429957 0.2181008 0.8568852 0.9601441
0.6211771 0.7719459 0.7418136 0.7096113 0.0649156
0.3765111 0.1690699 0.0587290 0.1031397 0.5948072
0.4023512 0.2506021 0.6856612 0.5690039 0.3134688
0.3588546 0.6323919 0.6235390 0.3334897 0.8966653
0.6357665 0.2432632 0.8676035 0.4867448 0.6681890
0.7076785 0.5754086 0.5991517 0.5082227 0.2947922
0.7943223 0.5192528 0.5194354 0.4941086 0.6930069
0.2894028 0.1249889 0.1078086 0.3986197 0.4287144
0.9228871 0.3715353 0.9617810 0.3166686 0.7908775
0.6030095 0.9537682 0.9958348 0.6171619 0.4223289
0.4664308 0.9592533 0.9869175 0.4820325 0.4831841
0.5564606 0.9802133 0.2035735 0.3102689 0.4231449
0.7537597 0.1630226 0.0472432 0.2682713 0.3842194
0.5233953 0.1466860 0.5272697 0.2220132 0.6808557
0.4317697 0.4095395 0.9874287 0.6789081 0.1315577
0.1011349 0.1553323 0.3000135 0.0301768 0.5863315
0.8158566 0.9902222 0.2908716 0.9916222 0.1814111
0.7581830 0.5545537 0.5384836 0.0892575 0.6606723
0.7868458 0.0644111 0.1314392 0.6707796 0.9800863
0.1728483 0.2175640 0.4367371 0.8102136 0.2793303
0.4121994 0.9838617 0.4958283 0.3913825 0.9419512
0.1924176 0.4919633 0.3970736 0.2536245 0.4304753
0.9401282 0.7274800 0.8424539 0.6469732 0.8873747
0.5629508 0.0725039 0.2789188 0.6001516 0.6736856
0.5159773 0.4078377 0.3601519 0.9901775 0.1878607
0.1137230 0.1419848 0.4738255 0.1722401 0.3580220
0.0697818 0.0721236 0.2299485 0.8676342 0.0902411
0.8323698 0.5536295 0.0633515 0.9923250 0.7875911
0.1810271 0.9339114 0.4522868 0.8756288 0.3844846
0.3865158 0.3097594 0.2493550 0.0171484 0.2839435
0.2952634 0.1592293 0.6571583 0.4275339 0.7060936
0.2890029 0.2677973 0.2139833 0.8420630 0.5083632
0.4779159 0.0760181 0.7436501 0.7239669 0.6386504
0.0972679 0.1258604 0.0291783 0.6031549 0.0894728
0.1554308 0.4387621 0.4784671 0.2558719 0.2950370
0.9449053 0.8744346 0.3392744 0.7862080 0.2973152
0.6445620 0.4727309 0.7296716 0.4816120 0.6642019
0.9039175 0.3389841 0.9587158 0.1715007 0.3246062
0.6161272 0.8512951 0.3592115 0.6321854 0.0507153
0.3670785 0.5695557 0.9547610 0.2755416 0.7486131
0.8558900 0.8539778 0.3100242 0.0175213 0.1794662
0.7016238 0.0044356 0.1606027 0.3554882 0.7385204
0.8063731 0.8244028 0.5092033 0.2907042 0.0008299
0.1240647 0.0332868 0.2910119 0.3240325 0.2181300
0.9864364 0.9040059 0.0361975 0.2842740 0.6710754
0.6266324 0.9437259 0.1748146 0.8146495 0.8297192
0.5991665 0.0333946 0.2520695 0.3058311 0.1927628
0.2111053 0.2486570 0.3769592 0.4354740 0.0124480
0.0330294 0.5150719 0.1609652 0.5360640 0.0643975
0.7477765 0.0749829 0.7421342 0.0559096 0.7441084
0.0133913 0.1720000 0.7991940 0.5081535 0.9306289
0.9283228 0.6445998 0.6970208 0.6356201 0.5844337
0.8184866 0.7972048 0.4648356 0.3056473 0.1916226
0.7299974 0.1670689 0.2381330 0.7186886 0.1919457
0.5179211 0.9017721 0.7041861 0.6840190 0.8040958
0.6276346 0.8774607 0.1879882 0.9198135 0.2928164
0.0890511 0.8237631 0.5810288 0.1778060 0.2767724
0.4705275 0.3639818 0.0592634 0.6791765 0.7689874
0.8559066 0.0333443 0.8717344 0.2963203 0.9971285
0.6521810 0.1015764 0.7124537 0.3960171 0.7222274
0.8276268 0.8244433 0.7222435 0.8558789 0.9371977
0.0273563 0.3745588 0.5151505 0.8849845 0.9653683
0.0595485 0.1852580 0.7015081 0.8354735 0.3115112
0.8955350 0.3981255 0.0909711 0.9315021 0.5320463
0.4763787 0.7740738 0.1203649 0.3093250 0.5604425
0.2087459 0.9215359 0.1849930 0.4869022 0.7811225
0.8949867 0.8637084 0.4918154 0.9784730 0.5609511
0.9949014 0.2200435 0.8069310 0.2536747 0.0014324
0.3698287 0.9416510 0.0979469 0.1431324 0.7026315
0.1391032 0.7737633 0.3735596 0.6124447 0.5305674
0.2788114 0.8995517 0.9364969 0.7423358 0.9828322
0.8466829 0.3076771 0.4426110 0.9051737 0.2699022
0.9069480 0.0293071 0.6483581 0.2526094 0.7507819
0.7062820 0.3740131 0.0315178 0.7967327 0.2108915
0.5796922 0.6143681 0.4693605 0.9217809 0.0377228
0.3408757 0.4712506 0.3801204 0.9912314 0.0115194
0.3650851 0.7559628 0.9060654 0.4146970 0.9003860
0.1968729 0.8984992 0.2395786 0.0934573 0.0867776
0.0144887 0.1516010 0.0627039 0.4154813 0.5394467
0.6657052 0.9707449 0.1204122 0.4981994 0.2642718
0.8332563 0.3040652 0.0315750 0.7595130 0.7610582
0.3419948 0.1981970 0.1691565 0.9159351 0.8701060
0.3887535 0.9373380 0.6420612 0.9082610 0.2640703
0.8001726 0.5948276 0.7485018 0.7401804 0.4460510
0.1693894 0.8001088 0.2382195 0.6017756 0.8723611
0.3504890 0.4767856 0.9553782 0.7509672 0.3110872
0.7917635 0.0300165 0.8993516 0.9777189 0.1068504
0.3274581 0.9525412 0.7753758 0.0747968 0.0410720
0.0261998 0.4040625 0.0078792 0.0927604 0.5008888
0.2012436 0.1854190 0.5707201 0.6423308 0.4182352
0.6455568 0.1825887 0.8615652 0.4750520 0.1721947
0.7690682 0.8545246 0.6873186 0.0812381 0.8343027
0.0261047 0.8721772 0.4512186 0.8414524 0.2714838
0.1508710 0.7480685 0.0175726 0.1747833 0.0754294
0.4516891 0.7663001 0.8832736 0.7444727 0.3527271
0.5234005 0.7051982 0.4382496 0.1273925 0.1422303
0.2291997 0.8647419 0.8682420 0.0307138 0.0246089
0.4356116 0.6199665 0.7139253 0.1140595 0.7298192
0.2627743 0.1360300 0.7914424 0.8456574 0.0143827
0.5326963 0.0007395 0.9558081 0.5294534 0.8579996
0.8095950 0.6305768 0.3155755 0.7510804 0.8035717
0.0594631 0.0683521 0.9602401 0.6153009 0.7052255
0.1726139 0.3101112 0.3940828 0.6563805 0.1656554
0.5375380 0.3750748 0.1147255 0.2726777 0.7880659
0.5937348 0.0313343 0.8689982 0.9299393 0.3610039
0.1188469 0.2180888 0.4314961 0.1206954 0.9282924
0.8318868 0.4145573 0.9985504 0.3023612 0.4330053
0.1236710 0.2244524 0.3396329 0.2186499 0.5217667
0.7363152 0.8551591 0.9899931 0.1734172 0.7226142
0.4118923 0.2697023 0.4588313 0.4843741 0.1464625
0.0019103 0.1290129 0.1465379 0.0786841 0.3201288
0.5110554 0.9854821 0.6499469 0.9665435 0.1651902
0.4144948 0.4651146 0.6982143 0.6117811 0.5313664
0.3784096 0.0434096 0.7174369 0.5153199 0.4679122
0.3169152 0.5289410 0.9524300 0.0887638 0.2532637
0.5437301 0.8599169 0.8767987 0.3997077 0.7804379
0.7590813 0.0405789 0.0546117 0.6141715 0.2191507
0.8099851 0.2894114 0.6221151 0.3959905 0.6667355
0.4458065 0.7924230 0.1931385 0.4978207 0.2516071
0.9531968 0.8143003 0.6689831 0.2270434 0.0900375
0.9616451 0.1977844 0.2964236 0.0756666 0.0851844
0.3974922 0.1772078 0.5512156 0.4361852 0.9575318
0.7953207 0.0211303 0.9105513 0.4263936 0.9828047
0.2107296 0.5337992 0.4540871 0.7008013 0.0487345
0.8831548 0.2710966 0.5970616 0.8637687 0.8959278
0.5993139 0.3366956 0.6012032 0.4872798 0.4551529
0.7765560 0.7090268 0.4249890 0.1200184 0.5789308
0.5155743 0.1422878 0.9795000 0.5595786 0.1509995
0.0792509 0.0688126 0.6350603 0.4483721 0.0516453
0.4668457 0.6784940 0.0372176 0.6447032 0.7031273
0.3392785 0.4258072 0.3631033 0.9881923 0.0641549
0.1079120 0.4109652 0.3011489 0.6842931 0.7435470
0.5861508 0.3270649 0.0734065 0.4294337 0.5205312
0.7177778 0.0602058 0.4514055 0.5914789 0.5185751
0.4463946 0.4973861 0.8843189 0.6217523 0.9162321
0.5450783 0.3298035 0.9960966 0.0159491 0.2219478
0.6047435 0.8096090 0.4223060 0.1392953 0.4481289
0.7348944 0.6331269 0.8041932 0.5235158 0.4759070
0.0847227 0.1752158 0.8839453 0.1975059 0.3128624
0.9466861 0.2077144 0.7619910 0.5765363 0.4861613
0.8734571 0.0169025 0.5213064 0.0173138 0.4424608
0.6672945 0.6499754 0.6711322 0.2558654 0.3243044
0.2554454 0.8586047 0.5682756 0.8922655 0.6636506
0.0218404 0.7037057 0.2065064 0.2921336 0.7495687
0.2857361 0.4222085 0.5944025 0.9287440 0.9389040
0.1598085 0.2122840 0.4287806 0.1817614 0.4537382
0.3460875 0.4820717 0.2712238 0.7337711 0.7411728
0.7690563 0.7954132 0.6937981 0.9011367 0.0063677
0.9412083 0.0790322 0.8979334 0.7035713 0.0778096
0.1332566 0.9843809 0.2532528 0.1859826 0.0048569
0.9196117 0.0804397 0.7304962 0.4037158 0.4176690
0.5263212 0.5938508 0.9366062 0.9992829 0.2063393
0.3423929 0.1101609 0.9174431 0.3004182 0.8324580
0.8632543 0.3028107 0.1143679 0.4096634 0.1497958
0.7057358 0.3723647 0.1782384 0.2935551 0.4947131
0.7656632 0.0109114 0.8174364 0.9451001 0.5324968
0.3003782 0.8638048 0.8173005 0.1905125 0.8182253
0.8646783 0.2235210 0.0617919 0.0997807 0.2009482
0.5753826 0.5292517 0.0709130 0.4708593 0.1369352
0.8474091 0.1300952 0.8655580 0.5202799 0.0301907
0.2337738 0.4273672 0.4641895 0.7043252 0.1601409
0.5433995 0.8850733 0.3869131 0.5809051 0.2255277 
h = hist(u,main='The combination of 3 multiplicative congruential generators')

# Kolmogorov-Smirnov #
p1 <- ks.test(u,'punif')
p1$p.value
## [1] 0.4986677
print(ifelse(p1$p.value <= 0.05,"Reject H0; This is not an uniform distribution","Do not reject H0; This is an uniform distribution"))
## [1] "Do not reject H0; This is an uniform distribution"
# Chisq test #
p2 <- chisq.test(h$counts,p = rep(1/length(h$counts),length(h$counts)))
p2$p.value
## [1] 0.6036862
print(ifelse(p2$p.value <= 0.05,"Reject H0; This is not an uniform distribution","Do not reject H0; This is an uniform distribution"))
## [1] "Do not reject H0; This is an uniform distribution"

(c) \(U_{n+1} = ({\pi} +U_n)^5 (mod 1)\)

# N 為模擬次數 #
u0 <- runif(1,0,1)
N <- 10000
U <- NULL
for(i in 1:N){
  if(i == 1){U[i] <- ((pi+u0)^5) %% 1}
  else{
  U[i] <- ((pi+U[i-1])^5) %% 1}
  U <-c(U,U[i])
}
U <- matrix(U[-1] , ncol = 5)
kable(U) %>% kable_styling() %>% scroll_box(height = "200px")
0.8210746 0.6823540 0.5146645 0.8065452 0.1252945
0.0978449 0.6343959 0.4067907 0.3157353 0.1094023
0.7369365 0.6324625 0.5390046 0.9726329 0.1461647
0.6781305 0.6692625 0.4473344 0.8009743 0.1481815
0.1289659 0.7337697 0.4196634 0.5667067 0.3278663
0.2050381 0.1009464 0.8171062 0.2517132 0.6998747
0.7966811 0.4479506 0.2150029 0.8990180 0.5381388
0.3914945 0.9309777 0.0838345 0.0479532 0.6532821
0.5183124 0.3222401 0.0890251 0.0990468 0.0226329
0.6729194 0.6370967 0.9069526 0.3991789 0.2029200
0.5974949 0.3816725 0.6645689 0.5312226 0.4698938
0.8498539 0.9085367 0.7964270 0.3368365 0.3699130
0.0983739 0.7941924 0.0860045 0.2321206 0.9085277
0.0282953 0.4018776 0.2648974 0.0597659 0.7819833
0.0512466 0.6553067 0.7072941 0.2571762 0.8445438
0.8068169 0.1243139 0.6478232 0.5322134 0.3772689
0.6458006 0.5513105 0.3783082 0.2388177 0.5241871
0.2951767 0.8147418 0.3213524 0.4150115 0.9601001
0.4595223 0.3151456 0.9984556 0.0856947 0.9555803
0.5986825 0.5514856 0.2616790 0.0968172 0.5732326
0.0112463 0.9775488 0.5444933 0.1714393 0.4437587
0.5365033 0.8603495 0.4987113 0.1426327 0.4594292
0.1552792 0.4883641 0.2757207 0.0891610 0.5203791
0.5025608 0.2418864 0.0408688 0.9809733 0.5290587
0.6629393 0.4222472 0.4493174 0.7978696 0.3704103
0.0878164 0.8981279 0.0663695 0.8219884 0.2867308
0.2491368 0.8621491 0.7395970 0.2249673 0.5969729
0.1936491 0.7984001 0.6925466 0.4459113 0.3398620
0.7020333 0.4610223 0.5895095 0.2400980 0.4506461
0.8911219 0.8610395 0.0789265 0.2515401 0.1718426
0.5652229 0.3735596 0.4411292 0.7842389 0.3856356
0.8499060 0.6865369 0.2900541 0.5203617 0.9688215
0.1644216 0.1160958 0.8969143 0.5134538 0.3508586
0.9331453 0.9000332 0.2470653 0.3257523 0.5792382
0.3068543 0.4016330 0.8262202 0.1702323 0.1807984
0.6607044 0.4625765 0.4582097 0.4160962 0.8123469
0.7494149 0.1711906 0.4958041 0.9539991 0.3849330
0.8880996 0.9928328 0.7271619 0.3470061 0.4252285
0.5745294 0.0247410 0.6742458 0.7198045 0.3068287
0.6794177 0.2609705 0.0025246 0.4645283 0.6426017
0.4999622 0.0694311 0.2512562 0.8197376 0.0097075
0.3748847 0.3638872 0.5961084 0.4505681 0.7769659
0.6988977 0.3432438 0.4958597 0.1069234 0.9143522
0.4748261 0.9395199 0.7758563 0.7637711 0.6407771
0.5766751 0.1210498 0.6069746 0.4634292 0.1407031
0.7278182 0.6982621 0.1615941 0.8909697 0.9680524
0.4095889 0.7837866 0.2471959 0.3639160 0.2535268
0.7605832 0.9832860 0.9123693 0.3649816 0.1025923
0.7617123 0.1417253 0.9601782 0.1700321 0.3585883
0.0714192 0.5616443 0.0661588 0.2956584 0.3545294
0.4219755 0.4781979 0.6280350 0.7956010 0.3154820
0.6790543 0.4657753 0.1886018 0.0930755 0.7917108
0.1127016 0.8748310 0.5886998 0.1185143 0.4283681
0.9926033 0.1956123 0.2947014 0.2639637 0.8520324
0.6894289 0.9180744 0.1280909 0.0789720 0.8660502
0.2261384 0.6866178 0.7047352 0.4655788 0.8202613
0.1985594 0.2029552 0.8437385 0.7084401 0.0954630
0.7489581 0.4919220 0.3610229 0.9060345 0.4273637
0.3647171 0.3365643 0.1841687 0.4319259 0.0368219
0.9701133 0.0329579 0.8697624 0.7472374 0.3789718
0.1959015 0.4120401 0.6171964 0.3952693 0.8309293
0.0974914 0.7123763 0.3083122 0.4654608 0.3080682
0.5423057 0.2390550 0.6923856 0.6085850 0.5196160
0.4818109 0.5699920 0.4155413 0.7528381 0.8431786
0.5734448 0.3635033 0.5096335 0.8183104 0.6550170
0.6458685 0.0534625 0.9237135 0.6945752 0.8232685
0.3650795 0.9598173 0.3663225 0.7837411 0.8056179
0.2440375 0.5554946 0.1844739 0.9292027 0.1896002
0.8335713 0.7143719 0.0564749 0.8829981 0.2030320
0.6021925 0.4426672 0.5323603 0.8656234 0.5399794
0.4524025 0.5581314 0.3726202 0.2698830 0.3422210
0.6356069 0.1810443 0.9698134 0.0738168 0.1857648
0.8642451 0.9621471 0.7673779 0.7014774 0.8469939
0.4941974 0.8554949 0.6663055 0.2845983 0.4739219
0.3221738 0.2775117 0.6203401 0.1258992 0.8037775
0.5893395 0.2634434 0.4512287 0.4539284 0.9579625
0.9142087 0.7291640 0.6570389 0.9096227 0.9335764
0.4465666 0.9190520 0.9260318 0.2560420 0.9011867
0.7830355 0.0149324 0.5359024 0.7759849 0.9413889
0.0920021 0.3619213 0.6056284 0.7583935 0.7158349
0.5313935 0.8606482 0.8335237 0.2259722 0.0610525
0.4923805 0.8714770 0.5428093 0.0916582 0.9334218
0.7363060 0.8388615 0.9456414 0.3434014 0.6880661
0.9650505 0.2246458 0.6372540 0.0557293 0.7594120
0.9784035 0.2394411 0.5420113 0.1425814 0.4046902
0.0910868 0.8222121 0.2107320 0.0593691 0.8760430
0.0312880 0.5010153 0.3800040 0.0488696 0.7736172
0.5649589 0.3013360 0.6242002 0.5735025 0.9733054
0.6006593 0.7713515 0.3247348 0.7008738 0.7647126
0.9476687 0.3144275 0.4353084 0.6265761 0.5590799
0.4689222 0.0390601 0.5104370 0.7168202 0.0700320
0.5439198 0.5227006 0.6381093 0.1523358 0.6834431
0.9694447 0.6191271 0.4144598 0.7669772 0.7993711
0.2407071 0.2372429 0.6444664 0.1985952 0.6315485
0.6499799 0.3878228 0.9235868 0.7712416 0.7425374
0.6043809 0.6636363 0.1932766 0.1856145 0.0337397
0.6044897 0.8182386 0.4715930 0.7548886 0.8092056
0.7116049 0.6063731 0.8165986 0.1790924 0.5521771
0.3885311 0.5679363 0.5918195 0.7741006 0.6209140
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0.2371419 0.9046789 0.6427368 0.1465060 0.7580564
0.3220046 0.6137946 0.1482506 0.3476430 0.8360590
0.4676197 0.9191989 0.3683161 0.1916385 0.7119379
0.4380322 0.2146354 0.6955998 0.4595573 0.7555463
0.7431847 0.8506455 0.8937879 0.6281574 0.9373861
0.7705126 0.1033159 0.0953486 0.3121383 0.1644620
0.3316318 0.7595532 0.3645349 0.4082989 0.9573083
0.4338001 0.5681833 0.8324227 0.7355410 0.0100909
0.2769560 0.6489417 0.1689636 0.1003317 0.9660754
0.8837927 0.5321507 0.6535548 0.1083098 0.4366230
0.9082855 0.1817114 0.3054585 0.5364009 0.5871559
0.4561794 0.3688606 0.6745493 0.0615706 0.8011492
0.7930154 0.1089668 0.3243986 0.2060614 0.7779832
0.9906033 0.9030093 0.1926401 0.4388861 0.1142741
0.7710764 0.3779125 0.0781095 0.4445363 0.8753296
0.9920975 0.0177025 0.0019285 0.1021813 0.8445305
0.9508609 0.7393278 0.9601192 0.1310070 0.3604748
0.9389756 0.3872126 0.9827368 0.3741566 0.7717954
0.3663368 0.1904109 0.3469516 0.1424126 0.8349299
0.1953689 0.7024504 0.6794239 0.9611919 0.2995223
0.7671817 0.3463345 0.5065318 0.5015865 0.4984516
0.4372011 0.2226148 0.1721362 0.8042850 0.0477709
0.0611339 0.9370345 0.5626064 0.5729009 0.0047263
0.9762572 0.6779710 0.3833975 0.1280527 0.3285539
0.0022289 0.9592379 0.2388549 0.6829128 0.1982386
0.1067859 0.7360261 0.4392998 0.2319627 0.5493767
0.6872476 0.6486190 0.7845291 0.9575145 0.0183052
0.8794330 0.1990660 0.8650264 0.3010383 0.0396697
0.1972721 0.0643727 0.5004253 0.5621928 0.8347701
0.9484616 0.6835713 0.7821998 0.9940667 0.0998605
0.5778086 0.9366086 0.1010213 0.8283950 0.8481462
0.8118150 0.0886946 0.4893347 0.1563778 0.9329733
0.7349886 0.7269943 0.0849237 0.1519970 0.0697656
0.4763813 0.4865205 0.6788429 0.5675590 0.5417341
0.9079551 0.6430259 0.8874747 0.0579389 0.9555898
0.0118487 0.4447315 0.7509130 0.2987777 0.5865690
0.8342475 0.2635801 0.6064212 0.9766661 0.2340466
0.4467615 0.8210189 0.6153502 0.5902219 0.3087296
0.9445370 0.0291278 0.4674534 0.7694676 0.9881369
0.0970172 0.4717546 0.2969654 0.1084362 0.1798900
0.2814201 0.9543016 0.7084910 0.6069108 0.2592123
0.9401871 0.7726825 0.9619845 0.0985915 0.8923422
0.0467185 0.8756914 0.6248740 0.1482053 0.1799247
0.4606286 0.3156698 0.0024970 0.3418072 0.2803558
0.5294967 0.9258366 0.2377877 0.8810607 0.2100333
0.7680107 0.2688315 0.7429727 0.3266044 0.9389632
0.4052440 0.3621595 0.5291745 0.7863445 0.3491290
0.3141041 0.0401081 0.4755040 0.0237390 0.2938967
0.8084444 0.0593237 0.1566934 0.7577163 0.5672920
0.6251696 0.0250290 0.3387108 0.4429830 0.8052683
0.2999507 0.4057574 0.6055582 0.8187783 0.7654236
0.7988122 0.7204204 0.7642727 0.2700152 0.3870886
0.9576228 0.1494198 0.0469584 0.1633565 0.0942283
0.4540102 0.0535941 0.5845669 0.2973695 0.7499862
0.9780338 0.0283654 0.3022317 0.9910681 0.5430627
0.5586197 0.0866371 0.4009600 0.4487786 0.1791921
0.6385762 0.6082886 0.9323880 0.6186011 0.8346907
0.8910294 0.4597818 0.2634632 0.7113226 0.0005415
0.4428511 0.8169323 0.7424419 0.0774529 0.2835143
0.7099102 0.0014730 0.9250170 0.6492493 0.3794681
0.5222217 0.7377812 0.1476795 0.8496822 0.2121859
0.1874627 0.6343823 0.0339382 0.8804035 0.2988921
0.8886832 0.6186939 0.9101237 0.4664339 0.0567823
0.3441427 0.8041243 0.9309435 0.4326260 0.6932071
0.6027282 0.3781792 0.2753051 0.3182909 0.3034471
0.9787605 0.2223788 0.7575972 0.8010361 0.2562892
0.6055159 0.7859412 0.3052427 0.6413908 0.9407312
0.7225631 0.5439029 0.5222313 0.7689339 0.8021222
0.5353808 0.9538680 0.1961951 0.4841574 0.9542398
0.1287733 0.1626470 0.2796295 0.5984206 0.6857572
0.0948945 0.8743312 0.7123525 0.7550157 0.2791833
0.1153346 0.5454213 0.2128387 0.3255688 0.4067311
0.4715718 0.3557506 0.7120224 0.0376142 0.4917639
0.7985266 0.2283696 0.8487425 0.7834592 0.1988339
0.6134564 0.6354895 0.6886662 0.5946636 0.9198456
0.5830010 0.7447174 0.4050021 0.0870240 0.0940749
0.7942209 0.5173091 0.1226723 0.8184482 0.6659470
0.4360954 0.7733068 0.6184416 0.8640889 0.2435137
0.1548091 0.6086660 0.5519719 0.2930607 0.4895636
0.2249666 0.8329407 0.4300249 0.9853369 0.2839017
0.4454477 0.8150937 0.1988243 0.1134260 0.6461104
0.8562323 0.7463156 0.9138532 0.3990348 0.6140343
0.2189584 0.3416951 0.9656222 0.4180053 0.1576388
0.6003317 0.7984893 0.7916970 0.4848982 0.8984407
0.6265614 0.5684112 0.4117782 0.2387978 0.2787621
0.7019176 0.8648055 0.5036292 0.4020657 0.1184235
0.7648284 0.2158787 0.6056576 0.8036727 0.2127244
0.6938615 0.6400283 0.8622585 0.8309890 0.6396370
0.0112097 0.3746922 0.9390173 0.3823199 0.9747133
0.5184177 0.5517460 0.4241311 0.4074575 0.7844094
0.7673387 0.2197427 0.4192034 0.0678042 0.7228359
0.6204749 0.1006791 0.4472945 0.4999958 0.8395644
0.5862046 0.3001987 0.3865285 0.4044564 0.1072322
0.8821835 0.9728581 0.6602151 0.6911738 0.9357659
0.7974045 0.1236964 0.2381611 0.1071609 0.9237718
0.2619677 0.2001660 0.9865085 0.8960318 0.4459244
0.7381749 0.7497839 0.8135717 0.0738081 0.2509472
0.0802625 0.3110696 0.8826153 0.6968673 0.3914106
0.1603075 0.6485010 0.3635689 0.2687524 0.4529557
0.4819301 0.0773074 0.1029804 0.3086576 0.0972958
0.6761874 0.5711123 0.5736010 0.9370850 0.4346649
0.0628955 0.4271749 0.7946156 0.7477464 0.9839468
0.9040239 0.8837031 0.9097215 0.9774891 0.0984950
0.7362733 0.7906370 0.3890838 0.7744076 0.0950120
0.9280610 0.1440055 0.6427273 0.9022952 0.1798010
0.3163080 0.8884451 0.1385020 0.4227827 0.2050571
0.3818939 0.0300721 0.6923840 0.3302175 0.8085989
0.0790700 0.9492486 0.4138378 0.4055264 0.8133030
0.5183229 0.6794315 0.1472908 0.5375627 0.8133030 
h <- hist(U)

p1 <- ks.test(U,'punif')
## Warning in ks.test(U, "punif"): ties should not be present for the
## Kolmogorov-Smirnov test
p1$p.value
## [1] 0.06796908
print(ifelse(p1$p.value <= 0.05,"Reject H0; This is not an uniform distribution","Do not reject H0; This is an uniform distribution"))
## [1] "Do not reject H0; This is an uniform distribution"
p2 <- chisq.test(h$counts,p = rep(1/length(h$counts),length(h$counts)))
p2$p.value
## [1] 0.4032896
print(ifelse(round(p2$p.value, digits = 3)<= 0.05,"Reject H0; This is not an uniform distribution","Do not reject H0; This is an uniform distribution"))
## [1] "Do not reject H0; This is an uniform distribution"

Result and Discuss

Do you ever shine?

Question2.

emsp;There are several ways for checking the goodness-of-fit for empirical data. In specific, there are a lot of normality tests available in R. Generate a random sample of size 10, 50, and 100 from N(0,1) and t-distribution (with degrees 10 and 20) in R. You may treat testing random numbers from t-distribution as the power. For a level of significance \(α = 0.05\)test, choose at least four normality tests in R (“nortest” module) to check if this sample is from \(N(0,1)\). Tests used can include the Kolmogorov-Smirnov test and the Cramer-von Mises test. Note that you need to compare the differences among the tests you choose.

Question3.

Question4.

Question5.

Use the bisection, false positions, and secant methods to find the roots, and check your answers with the functions in R.

You need to specify the starting points, convergence criterion, and number of iterations.

 

 the question of (a) is \(f(x)=e-\frac{1}{3.5+x}\)

 

 the question of (b) is \(f(x)=\frac{e^x}{\sqrt{1+x^2}}-0.5\)

 the question of (c) is the eigenvalue of matrix A \(\begin{bmatrix}2&3&4\\3&4&5\\4&5&6\end{bmatrix}\)

二分法“Bisection Method”

#bisection method
bisection.method<-function(f,a,b){
fa <- f(a) ; fb <- f(b)
c <- (a+b)/2 ; fc <- f(c)
na=c();nb=c();nc=c();nfa=c();nfb=c();nfc=c()
if((fa*fb)>0){
  print("Wrong Input")
}
i <- 0
W1 <- c(a,b,fa,fb,c,fc)
while (abs(b-a) > 0.000001){
  i<-i+1
  fa <- f(a) ; fb <- f(b) ; fc <- f(c)
  if((fa*fc)<0){ 
    a = a ; fa = fa
    b = c ; fb = fc
    na = c(na,a) ; nb = c(nb,b)
    nfa = c(nfa,fa) ; nfb = c(nfb,fb)
  }
  else{ 
    a = c ; fa = fc
    b = b ; fb = fb
    na = c(na,a) ; nb = c(nb,b) ; nfa = c(nfa,fa) ; nfb = c(nfb,fb)
  }
  c = (a+b)/2 ; fc <- f(c)
  nc = c(nc,c) ; nfc = c(nfc,fc)
}
W2 <- c(na,nb,nfa,nfb,nc,nfc)
W2 <- matrix(data = W2,ncol = 6,byrow = F)
W2 <- rbind(W1,W2)
W <- data.frame("a"=W2[,1],"b"=W2[,2],"func.a"=W2[,3],"func.b"=W2[,4],"c"=W2[,5],"func.c"=W2[,6])
b=round(b,digits = 6)
print('Bisection Method')
print('covegence criterion : |b-a|<1e-6')
print(paste('number of iterations : ',i))
print(paste('approximation root :', b))
print(knitr::kable(W,"rst",digits = 7))
kable(W,"html") %>% kable_styling(bootstrap_options = "striped", full_width = F) %>% add_header_above(c(" Bisection Method" = 6))
}

假位法“False Position Method”

#false position method
false.position.method <- function(f,a,b){
fa <- f(a) ; fb <- f(b)
c <- a-(fa*(b-a)/(fb-fa)) ; fc <- f(c)
e=abs(c-b)
na=c();nb=c();nc=c();ni=c();nfa=c();nfb=c();nfc=c()
if((fa*fb)>0){
  print("Wrong Input")
}
i=0
W1<-c(a,b,c,fa,fb,fc)
while(e>0.000001){
  i <- i+1
  fa <- f(a) ; fb <- f(b)
  c <- b-(fb*(b-a)/(fb-fa)) ;  fc <- f(c)
  if(sign(fc) == sign(fa)){
    e <- abs(c-a)
    a <- c ; fa <- fc
    b <- b ; fb <- fb
    na = c(na,a) ; nb = c(nb,b)
    nfa=c(nfa,fa) ; nfb = c(nfb,fb)
  }
  else{
    e <- abs(c-b)
    a <- a ; fa <- fa
    b <- c ; fb <- fc
    na = c(na,a) ; nb = c(nb,b)
    nfa = c(nfa,fa) ; nfb = c(nfb,fb)
  }
  nfc = c(nfc,fc) ; nc = c(nc,c)
}
W2<-c(na,nb,nc,nfa,nfb,nfc)
W2<-matrix(data=c(na,nb,nc,nfa,nfb,nfc),ncol=6,byrow = F)
W<-rbind(W1,W2)
W<-data.frame("a"=W2[,1],"b"=W2[,2],"c"=W2[,3],"func.a"=W2[,4],"func.b"=W2[,5],"func.c"=W2[,6])
b=round(b,digits = 6)
print('False Position Method')
print('covegence criterion : |Cn-Cn-1|<1e-6')
print(paste('number of iterations : ',i))
print(paste('approximation root :', c))
print(knitr::kable(W,"rst",digits = 7))
kable(W,"html") %>% kable_styling(bootstrap_options = "striped", full_width = F) %>% add_header_above(c(" False Position Method" = 6))
}

割線法“Secant Method”

#割線法"secnat.method function"
secant.method<-function(f,a,b){
  fa <- f(a) ; fb <- f(b)
  c <- b-((b-a)/(fb-fa))*fb;fc <- f(c)
  d = abs(b-a)
  W1 <- c(a,b,fa,fb,c,fc)
  na=c();nb=c();nc=c();nfa=c();nfb=c();nfc=c()
  i=0
  while(d>0.000001){
    i <- i+1
    fa <- f(a)
    fb <- f(b)
    na = c(na,a) ;  nb = c(nb,b) ; nfa = c(nfa,fa); nfb = c(nfb,fb)
    c <- b-((b-a)/(fb-fa))*fb
    a = b ; b = c
    c2 <- b-((b-a)/(fb-fa))*fb
    nc = c(nc,c2) ; fc <- f(c2) ; nfc = c(nfc,fc)
    d = abs(b-a)
  }
  W2 <- c(na,nb,nc,nfa,nfb,nfc)
  W2 <- matrix(data=c(na,nb,nc,nfa,nfb,nfc),ncol=6,byrow = F)
  W2 <- rbind(W1,W2)
  W <- data.frame("a"=W2[,1],"b"=W2[,2],"c"=W2[,3],"func.a"=W2[,4],"func.b"=W2[,5],"func.c"=W2[,6])
  b = round(b,digits = 6)
  print('Secant Method')
  print('covegence criterion : |Cn-Cn-1|<1e-6')
  print(paste('number of iterations : ',i))
  print(paste('approximation root :', b))
  print(knitr::kable(W,"rst",digits = 7))
  kable(W,"html") %>% kable_styling(bootstrap_options = "striped", full_width = F) %>% add_header_above(c(" Secant Method" = 6))
}

(a)小題

the question of (a)
\(f(x)=e-\frac{1}{3.5+x}\) 

f <- function(x){
  return(exp(1)-(1/(3.5+x)))
}
curve(f,xlim = c(-5,0))
abline(h=0)

當\(x=-3.5\)，其函數值不存在


當\(x<-3.5\)，其函數值皆為正


我們選取\(a=-3.2 ,b=-3\)作為起始點



“R_uniroot”

# We choose start point a=-3.2 b=-3.0
uniroot(f,c(-3.2,-3))
## $root
## [1] -3.132115
## 
## $f.root
## [1] 4.459005e-05
## 
## $iter
## [1] 4
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 6.103516e-05


“Bisection Method”

# We choose start point a=-3.2 b=-3.0
bisection.method(f,-3.2,-3)
## [1] "Bisection Method"
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  18"
## [1] "approximation root : -3.132121"
## 
## 
## =========  =========  ==========  =========  =========  ==========
##         a          b      func.a     func.b          c      func.c
## =========  =========  ==========  =========  =========  ==========
## -3.200000  -3.000000  -0.6150515  0.7182818  -3.100000   0.2182818
## -3.200000  -3.100000  -0.6150515  0.2182818  -3.150000  -0.1388610
## -3.150000  -3.100000  -0.1388610  0.2182818  -3.125000   0.0516152
## -3.150000  -3.125000  -0.1388610  0.0516152  -3.137500  -0.0403389
## -3.137500  -3.125000  -0.0403389  0.0516152  -3.131250   0.0064174
## -3.137500  -3.131250  -0.0403389  0.0064174  -3.134375  -0.0167609
## -3.134375  -3.131250  -0.0167609  0.0064174  -3.132812  -0.0051224
## -3.132812  -3.131250  -0.0051224  0.0064174  -3.132031   0.0006597
## -3.132812  -3.132031  -0.0051224  0.0006597  -3.132422  -0.0022283
## -3.132422  -3.132031  -0.0022283  0.0006597  -3.132227  -0.0007835
## -3.132227  -3.132031  -0.0007835  0.0006597  -3.132129  -0.0000617
## -3.132129  -3.132031  -0.0000617  0.0006597  -3.132080   0.0002991
## -3.132129  -3.132080  -0.0000617  0.0002991  -3.132105   0.0001187
## -3.132129  -3.132105  -0.0000617  0.0001187  -3.132117   0.0000285
## -3.132129  -3.132117  -0.0000617  0.0000285  -3.132123  -0.0000166
## -3.132123  -3.132117  -0.0000166  0.0000285  -3.132120   0.0000060
## -3.132123  -3.132120  -0.0000166  0.0000060  -3.132121  -0.0000053
## -3.132121  -3.132120  -0.0000053  0.0000060  -3.132121   0.0000003
## -3.132121  -3.132121  -0.0000053  0.0000003  -3.132121  -0.0000025
## =========  =========  ==========  =========  =========  ==========
Bisection Method
a b func.a func.b c func.c
-3.200000 -3.000000 -0.6150515 0.7182818 -3.100000 0.2182818
-3.200000 -3.100000 -0.6150515 0.2182818 -3.150000 -0.1388610
-3.150000 -3.100000 -0.1388610 0.2182818 -3.125000 0.0516152
-3.150000 -3.125000 -0.1388610 0.0516152 -3.137500 -0.0403389
-3.137500 -3.125000 -0.0403389 0.0516152 -3.131250 0.0064174
-3.137500 -3.131250 -0.0403389 0.0064174 -3.134375 -0.0167609
-3.134375 -3.131250 -0.0167609 0.0064174 -3.132812 -0.0051224
-3.132812 -3.131250 -0.0051224 0.0064174 -3.132031 0.0006597
-3.132812 -3.132031 -0.0051224 0.0006597 -3.132422 -0.0022283
-3.132422 -3.132031 -0.0022283 0.0006597 -3.132227 -0.0007835
-3.132227 -3.132031 -0.0007835 0.0006597 -3.132129 -0.0000617
-3.132129 -3.132031 -0.0000617 0.0006597 -3.132080 0.0002991
-3.132129 -3.132080 -0.0000617 0.0002991 -3.132105 0.0001187
-3.132129 -3.132105 -0.0000617 0.0001187 -3.132117 0.0000285
-3.132129 -3.132117 -0.0000617 0.0000285 -3.132123 -0.0000166
-3.132123 -3.132117 -0.0000166 0.0000285 -3.132120 0.0000060
-3.132123 -3.132120 -0.0000166 0.0000060 -3.132121 -0.0000053
-3.132121 -3.132120 -0.0000053 0.0000060 -3.132121 0.0000003
-3.132121 -3.132121 -0.0000053 0.0000003 -3.132121 -0.0000025


“False Position Method”

# We choose start point a=-3.2 b=-3.0
false.position.method(f,-3.2,-3)
## [1] "False Position Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  8"
## [1] "approximation root : -3.13212038131467"
## 
## 
## ====  =========  =========  ==========  =========  =========
##    a          b          c      func.a     func.b     func.c
## ====  =========  =========  ==========  =========  =========
## -3.2  -3.107742  -3.107742  -0.6150515  0.1689375  0.1689375
## -3.2  -3.127622  -3.127622  -0.6150515  0.0328357  0.0328357
## -3.2  -3.131291  -3.131291  -0.6150515  0.0061190  0.0061190
## -3.2  -3.131967  -3.131967  -0.6150515  0.0011311  0.0011311
## -3.2  -3.132092  -3.132092  -0.6150515  0.0002088  0.0002088
## -3.2  -3.132115  -3.132115  -0.6150515  0.0000385  0.0000385
## -3.2  -3.132120  -3.132120  -0.6150515  0.0000071  0.0000071
## -3.2  -3.132120  -3.132120  -0.6150515  0.0000013  0.0000013
## ====  =========  =========  ==========  =========  =========
False Position Method
a b c func.a func.b func.c
-3.2 -3.107742 -3.107742 -0.6150515 0.1689375 0.1689375
-3.2 -3.127622 -3.127622 -0.6150515 0.0328357 0.0328357
-3.2 -3.131291 -3.131291 -0.6150515 0.0061190 0.0061190
-3.2 -3.131967 -3.131967 -0.6150515 0.0011311 0.0011311
-3.2 -3.132092 -3.132092 -0.6150515 0.0002088 0.0002088
-3.2 -3.132115 -3.132115 -0.6150515 0.0000385 0.0000385
-3.2 -3.132120 -3.132120 -0.6150515 0.0000071 0.0000071
-3.2 -3.132120 -3.132120 -0.6150515 0.0000013 0.0000013


“Secant Method”

# We choose start point a=-3.2 b=-3.0
secant.method(f,-3.2,-3)
## [1] "Secant Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  6"
## [1] "approximation root : -3.132121"
## 
## 
## =========  =========  ==========  ==========  ==========  ==========
##         a          b           c      func.a      func.b      func.c
## =========  =========  ==========  ==========  ==========  ==========
## -3.200000  -3.000000  -0.6150515   0.7182818  -3.1077423   0.1689375
## -3.200000  -3.000000  -3.0497003  -0.6150515   0.7182818   0.4975387
## -3.000000  -3.107742  -3.1510652   0.7182818   0.1689375  -0.1475832
## -3.107742  -3.140876  -3.1341706   0.1689375  -0.0662701  -0.0152331
## -3.140876  -3.131540  -3.1320724  -0.0662701   0.0042803   0.0003559
## -3.131540  -3.132107  -3.1321209   0.0042803   0.0001020  -0.0000027
## -3.132107  -3.132121  -3.1321206   0.0001020  -0.0000002   0.0000000
## =========  =========  ==========  ==========  ==========  ==========
Secant Method
a b c func.a func.b func.c
-3.200000 -3.000000 -0.6150515 0.7182818 -3.1077423 0.1689375
-3.200000 -3.000000 -3.0497003 -0.6150515 0.7182818 0.4975387
-3.000000 -3.107742 -3.1510652 0.7182818 0.1689375 -0.1475832
-3.107742 -3.140876 -3.1341706 0.1689375 -0.0662701 -0.0152331
-3.140876 -3.131540 -3.1320724 -0.0662701 0.0042803 0.0003559
-3.131540 -3.132107 -3.1321209 0.0042803 0.0001020 -0.0000027
-3.132107 -3.132121 -3.1321206 0.0001020 -0.0000002 0.0000000

結論與比較

(b)小題

the question of (b)
\({f(x)=\frac{e^x}{\sqrt{1+x^2}}-0.5}\) 

f<-function(x){
  return((exp(-x))/sqrt(1+x^2)-0.5)
}
curve(f,xlim = c(-0,1))
abline(h=0)

由圖形觀察出其根介於(0,1)之間


我們選取\(a=0 ,b=1\)作為起始點



“R_uniroot”

# We choose start point a=0 b=1
uniroot(f,c(0,1))
## $root
## [1] 0.5577421
## 
## $f.root
## [1] -9.543937e-06
## 
## $iter
## [1] 5
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 6.103516e-05


“Bisection Method”

# We choose start point a=0 b=1
bisection.method(f,0,1)
## [1] "Bisection Method"
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  20"
## [1] "approximation root : 0.557729"
## 
## 
## =========  =========  =========  ==========  =========  ==========
##         a          b     func.a      func.b          c      func.c
## =========  =========  =========  ==========  =========  ==========
## 0.0000000  1.0000000  0.5000000  -0.2398700  0.5000000   0.0424975
## 0.5000000  1.0000000  0.0424975  -0.2398700  0.7500000  -0.1221068
## 0.5000000  0.7500000  0.0424975  -0.1221068  0.6250000  -0.0460992
## 0.5000000  0.6250000  0.0424975  -0.0460992  0.5625000  -0.0033912
## 0.5000000  0.5625000  0.0424975  -0.0033912  0.5312500   0.0191569
## 0.5312500  0.5625000  0.0191569  -0.0033912  0.5468750   0.0077835
## 0.5468750  0.5625000  0.0077835  -0.0033912  0.5546875   0.0021712
## 0.5546875  0.5625000  0.0021712  -0.0033912  0.5585938  -0.0006162
## 0.5546875  0.5585938  0.0021712  -0.0006162  0.5566406   0.0007760
## 0.5566406  0.5585938  0.0007760  -0.0006162  0.5576172   0.0000795
## 0.5576172  0.5585938  0.0000795  -0.0006162  0.5581055  -0.0002685
## 0.5576172  0.5581055  0.0000795  -0.0002685  0.5578613  -0.0000945
## 0.5576172  0.5578613  0.0000795  -0.0000945  0.5577393  -0.0000075
## 0.5576172  0.5577393  0.0000795  -0.0000075  0.5576782   0.0000360
## 0.5576782  0.5577393  0.0000360  -0.0000075  0.5577087   0.0000142
## 0.5577087  0.5577393  0.0000142  -0.0000075  0.5577240   0.0000033
## 0.5577240  0.5577393  0.0000033  -0.0000075  0.5577316  -0.0000021
## 0.5577240  0.5577316  0.0000033  -0.0000021  0.5577278   0.0000006
## 0.5577278  0.5577316  0.0000006  -0.0000021  0.5577297  -0.0000007
## 0.5577278  0.5577297  0.0000006  -0.0000007  0.5577288   0.0000000
## 0.5577278  0.5577288  0.0000006   0.0000000  0.5577283   0.0000003
## =========  =========  =========  ==========  =========  ==========
Bisection Method
a b func.a func.b c func.c
0.0000000 1.0000000 0.5000000 -0.2398700 0.5000000 0.0424975
0.5000000 1.0000000 0.0424975 -0.2398700 0.7500000 -0.1221068
0.5000000 0.7500000 0.0424975 -0.1221068 0.6250000 -0.0460992
0.5000000 0.6250000 0.0424975 -0.0460992 0.5625000 -0.0033912
0.5000000 0.5625000 0.0424975 -0.0033912 0.5312500 0.0191569
0.5312500 0.5625000 0.0191569 -0.0033912 0.5468750 0.0077835
0.5468750 0.5625000 0.0077835 -0.0033912 0.5546875 0.0021712
0.5546875 0.5625000 0.0021712 -0.0033912 0.5585938 -0.0006162
0.5546875 0.5585938 0.0021712 -0.0006162 0.5566406 0.0007760
0.5566406 0.5585938 0.0007760 -0.0006162 0.5576172 0.0000795
0.5576172 0.5585938 0.0000795 -0.0006162 0.5581055 -0.0002685
0.5576172 0.5581055 0.0000795 -0.0002685 0.5578613 -0.0000945
0.5576172 0.5578613 0.0000795 -0.0000945 0.5577393 -0.0000075
0.5576172 0.5577393 0.0000795 -0.0000075 0.5576782 0.0000360
0.5576782 0.5577393 0.0000360 -0.0000075 0.5577087 0.0000142
0.5577087 0.5577393 0.0000142 -0.0000075 0.5577240 0.0000033
0.5577240 0.5577393 0.0000033 -0.0000075 0.5577316 -0.0000021
0.5577240 0.5577316 0.0000033 -0.0000021 0.5577278 0.0000006
0.5577278 0.5577316 0.0000006 -0.0000021 0.5577297 -0.0000007
0.5577278 0.5577297 0.0000006 -0.0000007 0.5577288 0.0000000
0.5577278 0.5577288 0.0000006 0.0000000 0.5577283 0.0000003


“False Position Method”

# We choose start point a=0 b=1
false.position.method(f,0,1)
## [1] "False Position Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  10"
## [1] "approximation root : 0.557728783494413"
## 
## 
## ===  =========  =========  ======  ==========  ==========
##   a          b          c  func.a      func.b      func.c
## ===  =========  =========  ======  ==========  ==========
##   0  0.6757944  0.6757944     0.5  -0.0784769  -0.0784769
##   0  0.5841154  0.5841154     0.5  -0.0185218  -0.0185218
##   0  0.5632506  0.5632506     0.5  -0.0039230  -0.0039230
##   0  0.5588657  0.5588657     0.5  -0.0008098  -0.0008098
##   0  0.5579620  0.5579620     0.5  -0.0001663  -0.0001663
##   0  0.5577765  0.5577765     0.5  -0.0000341  -0.0000341
##   0  0.5577385  0.5577385     0.5  -0.0000070  -0.0000070
##   0  0.5577307  0.5577307     0.5  -0.0000014  -0.0000014
##   0  0.5577291  0.5577291     0.5  -0.0000003  -0.0000003
##   0  0.5577288  0.5577288     0.5  -0.0000001  -0.0000001
## ===  =========  =========  ======  ==========  ==========
False Position Method
a b c func.a func.b func.c
0 0.6757944 0.6757944 0.5 -0.0784769 -0.0784769
0 0.5841154 0.5841154 0.5 -0.0185218 -0.0185218
0 0.5632506 0.5632506 0.5 -0.0039230 -0.0039230
0 0.5588657 0.5588657 0.5 -0.0008098 -0.0008098
0 0.5579620 0.5579620 0.5 -0.0001663 -0.0001663
0 0.5577765 0.5577765 0.5 -0.0000341 -0.0000341
0 0.5577385 0.5577385 0.5 -0.0000070 -0.0000070
0 0.5577307 0.5577307 0.5 -0.0000014 -0.0000014
0 0.5577291 0.5577291 0.5 -0.0000003 -0.0000003
0 0.5577288 0.5577288 0.5 -0.0000001 -0.0000001


“Secant Method”

# We choose start point a=0 b=1
secant.method(f,0,1)
## [1] "Secant Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  6"
## [1] "approximation root : 0.557729"
## 
## 
## =========  =========  =========  ==========  ==========  ==========
##         a          b          c      func.a      func.b      func.c
## =========  =========  =========  ==========  ==========  ==========
## 0.0000000  1.0000000  0.5000000  -0.2398700   0.6757944  -0.0784769
## 0.0000000  1.0000000  0.7809037   0.5000000  -0.2398700  -0.1390305
## 1.0000000  0.6757944  0.4414967  -0.2398700  -0.0784769   0.0882895
## 0.6757944  0.5181506  0.5491327  -0.0784769   0.0288448   0.0061565
## 0.5181506  0.5605205  0.5579664   0.0288448  -0.0019865  -0.0001694
## 0.5605205  0.5577905  0.5577272  -0.0019865  -0.0000440   0.0000011
## 0.5577905  0.5577286  0.5577287  -0.0000440   0.0000001   0.0000000
## =========  =========  =========  ==========  ==========  ==========
Secant Method
a b c func.a func.b func.c
0.0000000 1.0000000 0.5000000 -0.2398700 0.6757944 -0.0784769
0.0000000 1.0000000 0.7809037 0.5000000 -0.2398700 -0.1390305
1.0000000 0.6757944 0.4414967 -0.2398700 -0.0784769 0.0882895
0.6757944 0.5181506 0.5491327 -0.0784769 0.0288448 0.0061565
0.5181506 0.5605205 0.5579664 0.0288448 -0.0019865 -0.0001694
0.5605205 0.5577905 0.5577272 -0.0019865 -0.0000440 0.0000011
0.5577905 0.5577286 0.5577287 -0.0000440 0.0000001 0.0000000

結論與比較

(c)小題

the question of (c)
the eigenvalue of A=\(\begin{bmatrix} 2&3&4\\ 3&4&5\\ 4&5&6 \end{bmatrix}\) 

A<-c(2,3,4,3,4,5,4,5,6)
A<-matrix(A,ncol=3,byrow=T)
eigen(A)
## $values
## [1]  1.248074e+01 -7.216450e-16 -4.807407e-01
## 
## $vectors
##           [,1]       [,2]       [,3]
## [1,] 0.4303622  0.4082483  0.8050601
## [2,] 0.5665422 -0.8164966  0.1111905
## [3,] 0.7027221  0.4082483 -0.5826791
f <- function(x){
  return((2-x)*(4-x)*(6-x)+3*5*4+4*3*5-(2-x)*5*5-4*(4-x)*4-3*3*(6-x))
}
f(0)
## [1] 0
curve(f,xlim = c(-5,15))
abline(h=0)

因為函數為三次式，我們將圖形更進一步分割 


curve(f,xlim = c(-1,1))
abline(h=0)

curve(f,xlim = c(12,13))
abline(h=0)

我們選取三組起始點來計算三根


\(a=-1 ,b=-0.2\)作為第一組


\(a=-0.2,b=1\)作為第二組


\(a=12 ,b=13\)作為第三組



第一組(-1,-0.2)


“R_uniroot”

#第一組
# We choose start point a=-1 b=-0.2
uniroot(f,c(-1,-0.2))
## $root
## [1] -0.480742
## 
## $f.root
## [1] 8.279401e-06
## 
## $iter
## [1] 8
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 6.103516e-05


“Bisection Method”

# We choose start point a=-1 b=-0.2
bisection.method(f,-1,-0.2)
## [1] "Bisection Method"
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  20"
## [1] "approximation root : -0.48074"
## 
## 
## ==========  ==========  =========  ==========  ==========  ==========
##          a           b     func.a      func.b           c      func.c
## ==========  ==========  =========  ==========  ==========  ==========
## -1.0000000  -0.2000000  7.0000000  -0.7120000  -0.6000000   0.9360000
## -0.6000000  -0.2000000  0.9360000  -0.7120000  -0.4000000  -0.4160000
## -0.6000000  -0.4000000  0.9360000  -0.4160000  -0.5000000   0.1250000
## -0.5000000  -0.4000000  0.1250000  -0.4160000  -0.4500000  -0.1788750
## -0.5000000  -0.4500000  0.1250000  -0.1788750  -0.4750000  -0.0353281
## -0.5000000  -0.4750000  0.1250000  -0.0353281  -0.4875000   0.0427324
## -0.4875000  -0.4750000  0.0427324  -0.0353281  -0.4812500   0.0031770
## -0.4812500  -0.4750000  0.0031770  -0.0353281  -0.4781250  -0.0162068
## -0.4812500  -0.4781250  0.0031770  -0.0162068  -0.4796875  -0.0065477
## -0.4812500  -0.4796875  0.0031770  -0.0065477  -0.4804687  -0.0016935
## -0.4812500  -0.4804687  0.0031770  -0.0016935  -0.4808594   0.0007397
## -0.4808594  -0.4804687  0.0007397  -0.0016935  -0.4806641  -0.0004774
## -0.4808594  -0.4806641  0.0007397  -0.0004774  -0.4807617   0.0001310
## -0.4807617  -0.4806641  0.0001310  -0.0004774  -0.4807129  -0.0001733
## -0.4807617  -0.4807129  0.0001310  -0.0001733  -0.4807373  -0.0000211
## -0.4807617  -0.4807373  0.0001310  -0.0000211  -0.4807495   0.0000549
## -0.4807495  -0.4807373  0.0000549  -0.0000211  -0.4807434   0.0000169
## -0.4807434  -0.4807373  0.0000169  -0.0000211  -0.4807404  -0.0000021
## -0.4807434  -0.4807404  0.0000169  -0.0000021  -0.4807419   0.0000074
## -0.4807419  -0.4807404  0.0000074  -0.0000021  -0.4807411   0.0000026
## -0.4807411  -0.4807404  0.0000026  -0.0000021  -0.4807407   0.0000002
## ==========  ==========  =========  ==========  ==========  ==========
Bisection Method
a b func.a func.b c func.c
-1.0000000 -0.2000000 7.0000000 -0.7120000 -0.6000000 0.9360000
-0.6000000 -0.2000000 0.9360000 -0.7120000 -0.4000000 -0.4160000
-0.6000000 -0.4000000 0.9360000 -0.4160000 -0.5000000 0.1250000
-0.5000000 -0.4000000 0.1250000 -0.4160000 -0.4500000 -0.1788750
-0.5000000 -0.4500000 0.1250000 -0.1788750 -0.4750000 -0.0353281
-0.5000000 -0.4750000 0.1250000 -0.0353281 -0.4875000 0.0427324
-0.4875000 -0.4750000 0.0427324 -0.0353281 -0.4812500 0.0031770
-0.4812500 -0.4750000 0.0031770 -0.0353281 -0.4781250 -0.0162068
-0.4812500 -0.4781250 0.0031770 -0.0162068 -0.4796875 -0.0065477
-0.4812500 -0.4796875 0.0031770 -0.0065477 -0.4804687 -0.0016935
-0.4812500 -0.4804687 0.0031770 -0.0016935 -0.4808594 0.0007397
-0.4808594 -0.4804687 0.0007397 -0.0016935 -0.4806641 -0.0004774
-0.4808594 -0.4806641 0.0007397 -0.0004774 -0.4807617 0.0001310
-0.4807617 -0.4806641 0.0001310 -0.0004774 -0.4807129 -0.0001733
-0.4807617 -0.4807129 0.0001310 -0.0001733 -0.4807373 -0.0000211
-0.4807617 -0.4807373 0.0001310 -0.0000211 -0.4807495 0.0000549
-0.4807495 -0.4807373 0.0000549 -0.0000211 -0.4807434 0.0000169
-0.4807434 -0.4807373 0.0000169 -0.0000211 -0.4807404 -0.0000021
-0.4807434 -0.4807404 0.0000169 -0.0000021 -0.4807419 0.0000074
-0.4807419 -0.4807404 0.0000074 -0.0000021 -0.4807411 0.0000026
-0.4807411 -0.4807404 0.0000026 -0.0000021 -0.4807407 0.0000002


“False Position Method”

# We choose start point a=-1 b=-0.2
false.position.method(f,-1,-0.2)
## [1] "False Position Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  22"
## [1] "approximation root : -0.480739897910791"
## 
## 
## ===  ==========  ==========  ======  ==========  ==========
##   a           b           c  func.a      func.b      func.c
## ===  ==========  ==========  ======  ==========  ==========
##  -1  -0.2738589  -0.2738589       7  -0.7226300  -0.7226300
##  -1  -0.3418061  -0.3418061       7  -0.6089259  -0.6089259
##  -1  -0.3944800  -0.3944800       7  -0.4381196  -0.4381196
##  -1  -0.4301463  -0.4301463       7  -0.2809796  -0.2809796
##  -1  -0.4521375  -0.4521375       7  -0.1672557  -0.1672557
##  -1  -0.4649224  -0.4649224       7  -0.0952059  -0.0952059
##  -1  -0.4721023  -0.4721023       7  -0.0528245  -0.0528245
##  -1  -0.4760561  -0.4760561       7  -0.0288951  -0.0288951
##  -1  -0.4782100  -0.4782100       7  -0.0156828  -0.0156828
##  -1  -0.4793764  -0.4793764       7  -0.0084759  -0.0084759
##  -1  -0.4800061  -0.4800061       7  -0.0045703  -0.0045703
##  -1  -0.4803453  -0.4803453       7  -0.0024614  -0.0024614
##  -1  -0.4805280  -0.4805280       7  -0.0013247  -0.0013247
##  -1  -0.4806263  -0.4806263       7  -0.0007127  -0.0007127
##  -1  -0.4806792  -0.4806792       7  -0.0003834  -0.0003834
##  -1  -0.4807076  -0.4807076       7  -0.0002062  -0.0002062
##  -1  -0.4807229  -0.4807229       7  -0.0001109  -0.0001109
##  -1  -0.4807311  -0.4807311       7  -0.0000596  -0.0000596
##  -1  -0.4807356  -0.4807356       7  -0.0000321  -0.0000321
##  -1  -0.4807379  -0.4807379       7  -0.0000172  -0.0000172
##  -1  -0.4807392  -0.4807392       7  -0.0000093  -0.0000093
##  -1  -0.4807399  -0.4807399       7  -0.0000050  -0.0000050
## ===  ==========  ==========  ======  ==========  ==========
False Position Method
a b c func.a func.b func.c
-1 -0.2738589 -0.2738589 7 -0.7226300 -0.7226300
-1 -0.3418061 -0.3418061 7 -0.6089259 -0.6089259
-1 -0.3944800 -0.3944800 7 -0.4381196 -0.4381196
-1 -0.4301463 -0.4301463 7 -0.2809796 -0.2809796
-1 -0.4521375 -0.4521375 7 -0.1672557 -0.1672557
-1 -0.4649224 -0.4649224 7 -0.0952059 -0.0952059
-1 -0.4721023 -0.4721023 7 -0.0528245 -0.0528245
-1 -0.4760561 -0.4760561 7 -0.0288951 -0.0288951
-1 -0.4782100 -0.4782100 7 -0.0156828 -0.0156828
-1 -0.4793764 -0.4793764 7 -0.0084759 -0.0084759
-1 -0.4800061 -0.4800061 7 -0.0045703 -0.0045703
-1 -0.4803453 -0.4803453 7 -0.0024614 -0.0024614
-1 -0.4805280 -0.4805280 7 -0.0013247 -0.0013247
-1 -0.4806263 -0.4806263 7 -0.0007127 -0.0007127
-1 -0.4806792 -0.4806792 7 -0.0003834 -0.0003834
-1 -0.4807076 -0.4807076 7 -0.0002062 -0.0002062
-1 -0.4807229 -0.4807229 7 -0.0001109 -0.0001109
-1 -0.4807311 -0.4807311 7 -0.0000596 -0.0000596
-1 -0.4807356 -0.4807356 7 -0.0000321 -0.0000321
-1 -0.4807379 -0.4807379 7 -0.0000172 -0.0000172
-1 -0.4807392 -0.4807392 7 -0.0000093 -0.0000093
-1 -0.4807399 -0.4807399 7 -0.0000050 -0.0000050


“Secant Method”

# We choose start point a=-1 b=-0.2
secant.method(f,-1,-0.2)
## [1] "Secant Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  20"
## [1] "approximation root : -0.480741"
## 
## 
## ===========  ===========  =============  ============  ============  =============
##           a            b              c        func.a        func.b         func.c
## ===========  ===========  =============  ============  ============  =============
##  -1.0000000   -0.2000000   7.000000e+00    -0.7120000    -0.2738589  -7.226300e-01
##  -1.0000000   -0.2000000  -2.670400e-01     7.0000000    -0.7120000  -7.274729e-01
##  -0.2000000   -0.2738589  -3.365808e+02    -0.7120000    -0.7226300   3.948754e+07
##  -0.2738589    4.7471086   4.728345e+00    -0.7226300   191.9268478   1.909443e+02
##   4.7471086   -0.2550252  -2.360635e-01   191.9268478    -0.7331107  -7.345144e-01
##  -0.2550252   -0.2359911   5.292587e+03    -0.7331107    -0.7345022  -1.479170e+11
##  -0.2359911  -10.2826619  -1.027945e+01    -0.7345022  2294.3197520   2.292527e+03
## -10.2826619   -0.2392064  -2.424214e-01  2294.3197520    -0.7349147  -7.350641e-01
##  -0.2392064   -0.2424225  -7.776279e+04    -0.7349147    -0.7350641   4.703081e+14
##  -0.2424225   15.5734410   1.558850e+01    -0.7350641  -773.2333753  -7.784809e+02
##  15.5734410   -0.2574719  -2.724924e-01  -773.2333753    -0.7322618  -7.236959e-01
##  -0.2574719   -0.2724782  -1.089311e+02    -0.7322618    -0.7237068   1.434314e+06
##  -0.2724782   -1.5419319  -1.504303e+00    -0.7237068    22.9450835   2.153346e+01
##  -1.5419319   -0.3112935  -3.454451e-01    22.9450835    -0.6747517  -5.994600e-01
##  -0.3112935   -0.3464494  -2.675412e+00    -0.6747517    -0.5967868   8.899162e+01
##  -0.3464494   -0.6155519  -5.539748e-01    -0.5967868     1.0867730   5.288168e-01
##  -0.6155519   -0.4418406  -4.663161e-01     1.0867730    -0.2221087  -8.708740e-02
##  -0.4418406   -0.4713183  -4.852201e-01    -0.2221087    -0.0575196   2.818160e-02
##  -0.4713183   -0.4816199  -4.808007e-01    -0.0575196     0.0054891   3.738000e-04
##  -0.4816199   -0.4807225  -4.807403e-01     0.0054891    -0.0001134  -2.500000e-06
##  -0.4807225   -0.4807407  -4.807407e-01    -0.0001134    -0.0000002   0.000000e+00
## ===========  ===========  =============  ============  ============  =============
Secant Method
a b c func.a func.b func.c
-1.0000000 -0.2000000 7.000000e+00 -0.7120000 -0.2738589 -7.226300e-01
-1.0000000 -0.2000000 -2.670400e-01 7.0000000 -0.7120000 -7.274729e-01
-0.2000000 -0.2738589 -3.365808e+02 -0.7120000 -0.7226300 3.948754e+07
-0.2738589 4.7471086 4.728345e+00 -0.7226300 191.9268478 1.909443e+02
4.7471086 -0.2550252 -2.360635e-01 191.9268478 -0.7331107 -7.345144e-01
-0.2550252 -0.2359911 5.292587e+03 -0.7331107 -0.7345022 -1.479170e+11
-0.2359911 -10.2826619 -1.027945e+01 -0.7345022 2294.3197520 2.292527e+03
-10.2826619 -0.2392064 -2.424214e-01 2294.3197520 -0.7349147 -7.350641e-01
-0.2392064 -0.2424225 -7.776279e+04 -0.7349147 -0.7350641 4.703081e+14
-0.2424225 15.5734410 1.558850e+01 -0.7350641 -773.2333753 -7.784809e+02
15.5734410 -0.2574719 -2.724924e-01 -773.2333753 -0.7322618 -7.236959e-01
-0.2574719 -0.2724782 -1.089311e+02 -0.7322618 -0.7237068 1.434314e+06
-0.2724782 -1.5419319 -1.504303e+00 -0.7237068 22.9450835 2.153346e+01
-1.5419319 -0.3112935 -3.454451e-01 22.9450835 -0.6747517 -5.994600e-01
-0.3112935 -0.3464494 -2.675412e+00 -0.6747517 -0.5967868 8.899162e+01
-0.3464494 -0.6155519 -5.539748e-01 -0.5967868 1.0867730 5.288168e-01
-0.6155519 -0.4418406 -4.663161e-01 1.0867730 -0.2221087 -8.708740e-02
-0.4418406 -0.4713183 -4.852201e-01 -0.2221087 -0.0575196 2.818160e-02
-0.4713183 -0.4816199 -4.808007e-01 -0.0575196 0.0054891 3.738000e-04
-0.4816199 -0.4807225 -4.807403e-01 0.0054891 -0.0001134 -2.500000e-06
-0.4807225 -0.4807407 -4.807407e-01 -0.0001134 -0.0000002 0.000000e+00


第二組(-0.2,0)


“R_uniroot”

#第二組
# We choose start point a=-0.2 b=1
uniroot(f,c(-0.2,1))
## $root
## [1] -5.977228e-07
## 
## $f.root
## [1] -3.586333e-06
## 
## $iter
## [1] 9
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 6.103516e-05


“Bisection Method”

# We choose start point a=-0.2 b=1
bisection.method(f,-0.2,1)
## [1] "Bisection Method"
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  21"
## [1] "approximation root : 0"
## 
## 
## ==========  =========  ==========  ==========  ==========  ==========
##          a          b      func.a      func.b           c      func.c
## ==========  =========  ==========  ==========  ==========  ==========
## -0.2000000  1.0000000  -0.7120000  17.0000000   0.4000000   4.2560000
## -0.2000000  0.4000000  -0.7120000   4.2560000   0.1000000   0.7190000
## -0.2000000  0.1000000  -0.7120000   0.7190000  -0.0500000  -0.2698750
## -0.0500000  0.1000000  -0.2698750   0.7190000   0.0250000   0.1574844
## -0.0500000  0.0250000  -0.2698750   0.1574844  -0.0125000  -0.0731230
## -0.0125000  0.0250000  -0.0731230   0.1574844   0.0062500   0.0379685
## -0.0125000  0.0062500  -0.0731230   0.0379685  -0.0031250  -0.0186328
## -0.0031250  0.0062500  -0.0186328   0.0379685   0.0015625   0.0094043
## -0.0031250  0.0015625  -0.0186328   0.0094043  -0.0007812  -0.0046802
## -0.0007812  0.0015625  -0.0046802   0.0094043   0.0003906   0.0023456
## -0.0007812  0.0003906  -0.0046802   0.0023456  -0.0001953  -0.0011714
## -0.0001953  0.0003906  -0.0011714   0.0023456   0.0000977   0.0005861
## -0.0001953  0.0000977  -0.0011714   0.0005861  -0.0000488  -0.0002929
## -0.0000488  0.0000977  -0.0002929   0.0005861   0.0000244   0.0001465
## -0.0000488  0.0000244  -0.0002929   0.0001465  -0.0000122  -0.0000732
## -0.0000122  0.0000244  -0.0000732   0.0001465   0.0000061   0.0000366
## -0.0000122  0.0000061  -0.0000732   0.0000366  -0.0000031  -0.0000183
## -0.0000031  0.0000061  -0.0000183   0.0000366   0.0000015   0.0000092
## -0.0000031  0.0000015  -0.0000183   0.0000092  -0.0000008  -0.0000046
## -0.0000008  0.0000015  -0.0000046   0.0000092   0.0000004   0.0000023
## -0.0000008  0.0000004  -0.0000046   0.0000023  -0.0000002  -0.0000011
## -0.0000002  0.0000004  -0.0000011   0.0000023   0.0000001   0.0000006
## ==========  =========  ==========  ==========  ==========  ==========
Bisection Method
a b func.a func.b c func.c
-0.2000000 1.0000000 -0.7120000 17.0000000 0.4000000 4.2560000
-0.2000000 0.4000000 -0.7120000 4.2560000 0.1000000 0.7190000
-0.2000000 0.1000000 -0.7120000 0.7190000 -0.0500000 -0.2698750
-0.0500000 0.1000000 -0.2698750 0.7190000 0.0250000 0.1574844
-0.0500000 0.0250000 -0.2698750 0.1574844 -0.0125000 -0.0731230
-0.0125000 0.0250000 -0.0731230 0.1574844 0.0062500 0.0379685
-0.0125000 0.0062500 -0.0731230 0.0379685 -0.0031250 -0.0186328
-0.0031250 0.0062500 -0.0186328 0.0379685 0.0015625 0.0094043
-0.0031250 0.0015625 -0.0186328 0.0094043 -0.0007812 -0.0046802
-0.0007812 0.0015625 -0.0046802 0.0094043 0.0003906 0.0023456
-0.0007812 0.0003906 -0.0046802 0.0023456 -0.0001953 -0.0011714
-0.0001953 0.0003906 -0.0011714 0.0023456 0.0000977 0.0005861
-0.0001953 0.0000977 -0.0011714 0.0005861 -0.0000488 -0.0002929
-0.0000488 0.0000977 -0.0002929 0.0005861 0.0000244 0.0001465
-0.0000488 0.0000244 -0.0002929 0.0001465 -0.0000122 -0.0000732
-0.0000122 0.0000244 -0.0000732 0.0001465 0.0000061 0.0000366
-0.0000122 0.0000061 -0.0000732 0.0000366 -0.0000031 -0.0000183
-0.0000031 0.0000061 -0.0000183 0.0000366 0.0000015 0.0000092
-0.0000031 0.0000015 -0.0000183 0.0000092 -0.0000008 -0.0000046
-0.0000008 0.0000015 -0.0000046 0.0000092 0.0000004 0.0000023
-0.0000008 0.0000004 -0.0000046 0.0000023 -0.0000002 -0.0000011
-0.0000002 0.0000004 -0.0000011 0.0000023 0.0000001 0.0000006


“False Position Method”

# We choose start point a=-0.2 b=1
false.position.method(f,-0.2,1)
## [1] "False Position Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  28"
## [1] "approximation root : -1.74663519647567e-06"
## 
## 
## ==========  ===  ==========  ==========  ======  ==========
##          a    b           c      func.a  func.b      func.c
## ==========  ===  ==========  ==========  ======  ==========
## -0.1517615    1  -0.1517615  -0.6306951      17  -0.6306951
## -0.1105601    1  -0.1105601  -0.5153266      17  -0.5153266
## -0.0778858    1  -0.0778858  -0.3940478      17  -0.3940478
## -0.0534671    1  -0.0534671  -0.2863452      17  -0.2863452
## -0.0360167    1  -0.0360167  -0.2004868      17  -0.2004868
## -0.0239410    1  -0.0239410  -0.1367540      17  -0.1367540
## -0.0157698    1  -0.0157698  -0.0916304      17  -0.0916304
## -0.0103241    1  -0.0103241  -0.0606644      17  -0.0606644
## -0.0067316    1  -0.0067316  -0.0398453      17  -0.0398453
## -0.0043775    1  -0.0043775  -0.0260348      17  -0.0260348
## -0.0028417    1  -0.0028417  -0.0169530      17  -0.0169530
## -0.0018426    1  -0.0018426  -0.0110148      17  -0.0110148
## -0.0011939    1  -0.0011939  -0.0071462      17  -0.0071462
## -0.0007732    1  -0.0007732  -0.0046320      17  -0.0046320
## -0.0005006    1  -0.0005006  -0.0030005      17  -0.0030005
## -0.0003240    1  -0.0003240  -0.0019429      17  -0.0019429
## -0.0002097    1  -0.0002097  -0.0012578      17  -0.0012578
## -0.0001357    1  -0.0001357  -0.0008141      17  -0.0008141
## -0.0000878    1  -0.0000878  -0.0005269      17  -0.0005269
## -0.0000568    1  -0.0000568  -0.0003410      17  -0.0003410
## -0.0000368    1  -0.0000368  -0.0002206      17  -0.0002206
## -0.0000238    1  -0.0000238  -0.0001428      17  -0.0001428
## -0.0000154    1  -0.0000154  -0.0000924      17  -0.0000924
## -0.0000100    1  -0.0000100  -0.0000598      17  -0.0000598
## -0.0000064    1  -0.0000064  -0.0000387      17  -0.0000387
## -0.0000042    1  -0.0000042  -0.0000250      17  -0.0000250
## -0.0000027    1  -0.0000027  -0.0000162      17  -0.0000162
## -0.0000017    1  -0.0000017  -0.0000105      17  -0.0000105
## ==========  ===  ==========  ==========  ======  ==========
False Position Method
a b c func.a func.b func.c
-0.1517615 1 -0.1517615 -0.6306951 17 -0.6306951
-0.1105601 1 -0.1105601 -0.5153266 17 -0.5153266
-0.0778858 1 -0.0778858 -0.3940478 17 -0.3940478
-0.0534671 1 -0.0534671 -0.2863452 17 -0.2863452
-0.0360167 1 -0.0360167 -0.2004868 17 -0.2004868
-0.0239410 1 -0.0239410 -0.1367540 17 -0.1367540
-0.0157698 1 -0.0157698 -0.0916304 17 -0.0916304
-0.0103241 1 -0.0103241 -0.0606644 17 -0.0606644
-0.0067316 1 -0.0067316 -0.0398453 17 -0.0398453
-0.0043775 1 -0.0043775 -0.0260348 17 -0.0260348
-0.0028417 1 -0.0028417 -0.0169530 17 -0.0169530
-0.0018426 1 -0.0018426 -0.0110148 17 -0.0110148
-0.0011939 1 -0.0011939 -0.0071462 17 -0.0071462
-0.0007732 1 -0.0007732 -0.0046320 17 -0.0046320
-0.0005006 1 -0.0005006 -0.0030005 17 -0.0030005
-0.0003240 1 -0.0003240 -0.0019429 17 -0.0019429
-0.0002097 1 -0.0002097 -0.0012578 17 -0.0012578
-0.0001357 1 -0.0001357 -0.0008141 17 -0.0008141
-0.0000878 1 -0.0000878 -0.0005269 17 -0.0005269
-0.0000568 1 -0.0000568 -0.0003410 17 -0.0003410
-0.0000368 1 -0.0000368 -0.0002206 17 -0.0002206
-0.0000238 1 -0.0000238 -0.0001428 17 -0.0001428
-0.0000154 1 -0.0000154 -0.0000924 17 -0.0000924
-0.0000100 1 -0.0000100 -0.0000598 17 -0.0000598
-0.0000064 1 -0.0000064 -0.0000387 17 -0.0000387
-0.0000042 1 -0.0000042 -0.0000250 17 -0.0000250
-0.0000027 1 -0.0000027 -0.0000162 17 -0.0000162
-0.0000017 1 -0.0000017 -0.0000105 17 -0.0000105


“Secant Method”

# We choose start point a=-0.2 b=1
secant.method(f,-0.2,1)
## [1] "Secant Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  8"
## [1] "approximation root : 0"
## 
## 
## ==========  ==========  ==========  ==========  ==========  ==========
##          a           b           c      func.a      func.b      func.c
## ==========  ==========  ==========  ==========  ==========  ==========
## -0.2000000   1.0000000  -0.7120000  17.0000000  -0.1517615  -0.6306951
## -0.2000000   1.0000000   0.9537006  -0.7120000  17.0000000  15.7693093
##  1.0000000  -0.1517615  -0.1120339  17.0000000  -0.6306951  -0.5201782
## -0.1517615  -0.1105601   0.8955375  -0.6306951  -0.5153266  14.2788648
## -0.1105601   0.0734781   0.0274730  -0.5153266   0.5052603   0.1738747
##  0.0734781  -0.0176334  -0.0048949   0.5052603  -0.1020635  -0.0290816
## -0.0176334  -0.0023217   0.0004638  -0.1020635  -0.0138654   0.0027856
## -0.0023217   0.0000854   0.0000027  -0.0138654   0.0005127   0.0000160
##  0.0000854  -0.0000004   0.0000000   0.0005127  -0.0000024   0.0000000
## ==========  ==========  ==========  ==========  ==========  ==========
Secant Method
a b c func.a func.b func.c
-0.2000000 1.0000000 -0.7120000 17.0000000 -0.1517615 -0.6306951
-0.2000000 1.0000000 0.9537006 -0.7120000 17.0000000 15.7693093
1.0000000 -0.1517615 -0.1120339 17.0000000 -0.6306951 -0.5201782
-0.1517615 -0.1105601 0.8955375 -0.6306951 -0.5153266 14.2788648
-0.1105601 0.0734781 0.0274730 -0.5153266 0.5052603 0.1738747
0.0734781 -0.0176334 -0.0048949 0.5052603 -0.1020635 -0.0290816
-0.0176334 -0.0023217 0.0004638 -0.1020635 -0.0138654 0.0027856
-0.0023217 0.0000854 0.0000027 -0.0138654 0.0005127 0.0000160
0.0000854 -0.0000004 0.0000000 0.0005127 -0.0000024 0.0000000


第三組(12,13)


“R_uniroot”

#第三組
# We choose start point a=12 b=13
uniroot(f,c(12,13))
## $root
## [1] 12.48074
## 
## $f.root
## [1] 0.0004214783
## 
## $iter
## [1] 4
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 6.103516e-05


“Bisection Method”

# We choose start point a=12 b=12
bisection.method(f,12,13)
## [1] "Bisection Method"
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  20"
## [1] "approximation root : 12.480742"
## 
## 
## ========  ========  ==========  ===========  ========  ==========
##        a         b      func.a       func.b         c      func.c
## ========  ========  ==========  ===========  ========  ==========
## 12.00000  13.00000  72.0000000  -91.0000000  12.50000  -3.1250000
## 12.00000  12.50000  72.0000000   -3.1250000  12.25000  35.9843750
## 12.25000  12.50000  35.9843750   -3.1250000  12.37500  16.8222656
## 12.37500  12.50000  16.8222656   -3.1250000  12.43750   6.9475098
## 12.43750  12.50000   6.9475098   -3.1250000  12.46875   1.9360657
## 12.46875  12.50000   1.9360657   -3.1250000  12.48438  -0.5882530
## 12.46875  12.48438   1.9360657   -0.5882530  12.47656   0.6754584
## 12.47656  12.48438   0.6754584   -0.5882530  12.48047   0.0439909
## 12.48047  12.48438   0.0439909   -0.5882530  12.48242  -0.2720340
## 12.48047  12.48242   0.0439909   -0.2720340  12.48145  -0.1139973
## 12.48047  12.48145   0.0439909   -0.1139973  12.48096  -0.0349971
## 12.48047  12.48096   0.0439909   -0.0349971  12.48071   0.0044984
## 12.48071  12.48096   0.0044984   -0.0349971  12.48084  -0.0152490
## 12.48071  12.48084   0.0044984   -0.0152490  12.48077  -0.0053752
## 12.48071  12.48077   0.0044984   -0.0053752  12.48074  -0.0004384
## 12.48071  12.48074   0.0044984   -0.0004384  12.48073   0.0020300
## 12.48073  12.48074   0.0020300   -0.0004384  12.48074   0.0007958
## 12.48074  12.48074   0.0007958   -0.0004384  12.48074   0.0001787
## 12.48074  12.48074   0.0001787   -0.0004384  12.48074  -0.0001298
## 12.48074  12.48074   0.0001787   -0.0001298  12.48074   0.0000245
## 12.48074  12.48074   0.0000245   -0.0001298  12.48074  -0.0000527
## ========  ========  ==========  ===========  ========  ==========
Bisection Method
a b func.a func.b c func.c
12.00000 13.00000 72.0000000 -91.0000000 12.50000 -3.1250000
12.00000 12.50000 72.0000000 -3.1250000 12.25000 35.9843750
12.25000 12.50000 35.9843750 -3.1250000 12.37500 16.8222656
12.37500 12.50000 16.8222656 -3.1250000 12.43750 6.9475098
12.43750 12.50000 6.9475098 -3.1250000 12.46875 1.9360657
12.46875 12.50000 1.9360657 -3.1250000 12.48438 -0.5882530
12.46875 12.48438 1.9360657 -0.5882530 12.47656 0.6754584
12.47656 12.48438 0.6754584 -0.5882530 12.48047 0.0439909
12.48047 12.48438 0.0439909 -0.5882530 12.48242 -0.2720340
12.48047 12.48242 0.0439909 -0.2720340 12.48145 -0.1139973
12.48047 12.48145 0.0439909 -0.1139973 12.48096 -0.0349971
12.48047 12.48096 0.0439909 -0.0349971 12.48071 0.0044984
12.48071 12.48096 0.0044984 -0.0349971 12.48084 -0.0152490
12.48071 12.48084 0.0044984 -0.0152490 12.48077 -0.0053752
12.48071 12.48077 0.0044984 -0.0053752 12.48074 -0.0004384
12.48071 12.48074 0.0044984 -0.0004384 12.48073 0.0020300
12.48073 12.48074 0.0020300 -0.0004384 12.48074 0.0007958
12.48074 12.48074 0.0007958 -0.0004384 12.48074 0.0001787
12.48074 12.48074 0.0001787 -0.0004384 12.48074 -0.0001298
12.48074 12.48074 0.0001787 -0.0001298 12.48074 0.0000245
12.48074 12.48074 0.0000245 -0.0001298 12.48074 -0.0000527


“False Position Method”

# We choose start point a=12 b=12
false.position.method(f,12,13)
## [1] "False Position Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  7"
## [1] "approximation root : 12.4807406902856"
## 
## 
## ========  ===  ========  =========  ======  =========
##        a    b         c     func.a  func.b     func.c
## ========  ===  ========  =========  ======  =========
## 12.44172   13  12.44172  6.2740086     -91  6.2740086
## 12.47773   13  12.47773  0.4874456     -91  0.4874456
## 12.48051   13  12.48051  0.0375246     -91  0.0375246
## 12.48072   13  12.48072  0.0028867     -91  0.0028867
## 12.48074   13  12.48074  0.0002221     -91  0.0002221
## 12.48074   13  12.48074  0.0000171     -91  0.0000171
## 12.48074   13  12.48074  0.0000013     -91  0.0000013
## ========  ===  ========  =========  ======  =========
False Position Method
a b c func.a func.b func.c
12.44172 13 12.44172 6.2740086 -91 6.2740086
12.47773 13 12.47773 0.4874456 -91 0.4874456
12.48051 13 12.48051 0.0375246 -91 0.0375246
12.48072 13 12.48072 0.0028867 -91 0.0028867
12.48074 13 12.48074 0.0002221 -91 0.0002221
12.48074 13 12.48074 0.0000171 -91 0.0000171
12.48074 13 12.48074 0.0000013 -91 0.0000013


“Secant Method”

# We choose start point a=12 b=13
secant.method(f,12,13)
## [1] "Secant Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  5"
## [1] "approximation root : 12.480741"
## 
## 
## ========  ========  ========  ===========  ===========  ===========
##        a         b         c       func.a       func.b       func.c
## ========  ========  ========  ===========  ===========  ===========
## 12.00000  13.00000  72.00000  -91.0000000   12.4417178    6.2740086
## 12.00000  13.00000  12.75340   72.0000000  -91.0000000  -46.0189560
## 13.00000  12.44172  12.47540  -91.0000000    6.2740086    0.8626558
## 12.44172  12.47773  12.48101    6.2740086    0.4874456   -0.0443439
## 12.47773  12.48076  12.48074    0.4874456   -0.0030080   -0.0000170
## 12.48076  12.48074  12.48074   -0.0030080    0.0000014    0.0000000
## ========  ========  ========  ===========  ===========  ===========
Secant Method
a b c func.a func.b func.c
12.00000 13.00000 72.00000 -91.0000000 12.4417178 6.2740086
12.00000 13.00000 12.75340 72.0000000 -91.0000000 -46.0189560
13.00000 12.44172 12.47540 -91.0000000 6.2740086 0.8626558
12.44172 12.47773 12.48101 6.2740086 0.4874456 -0.0443439
12.47773 12.48076 12.48074 0.4874456 -0.0030080 -0.0000170
12.48076 12.48074 12.48074 -0.0030080 0.0000014 0.0000000

結論與比較

Question6.

  • Consider a multinomial observation \(X =(x1,x2,x3,x4)\)with class probabilities given by \((p_1,p_2,p_3,p_4)=(\frac{2+{\theta} }{_4},\frac{1+{\theta}}{4},\frac{1+{\theta}}{4},\frac{{\theta}}{4})\), where \(0<{\theta}<1\). The sample size is \(n={\sum x_i}\) and the parameter θ is to be estimated from the observedfrequencies \((1997, 906, 904, 32)\), i.e., sample size 3839. Use the secant, Ridder’s (or Brent’s), and Newton-Raphosn methods to find the MLE (via \(l’(θ)\)). Youmay choose your own starting points and convergence criterion (preferred \(10^{-6}\) or smaller).s
library(pracma)
## 
## Attaching package: 'pracma'
## The following objects are masked from 'package:numDeriv':
## 
##     grad, hessian, jacobian
g <- function(x){1997/(2+x)-1810/(1-x)+32/x}
curve(g,xlim = c(0,0.1))
abline(h=0)


“Newton-Raphosn Method”

#Newton
Newton <- function(f,S,tol){
i=1
N_table <- matrix(c(0,S,f(S)),1,3)
colnames(N_table) <- c('Iterative','x','f(x)')
tr = tol
while(tr>=tol){
  ss <- N_table[i,2] - f(N_table[i,2])/grad(f,N_table[i,2])
  tr <- abs(ss-N_table[i,2])
  i = i+1
  N_table <- rbind(N_table,c(i-1,ss,f(ss)))
}
  print("Newton's Method ")
  print('covegence criterion : tolerence <1e-6')
  print(paste('number of iterations : ',i))
  print(paste('approximation root :', ss))
  print(knitr::kable(N_table,"rst",digits = 7))
  kable(N_table, "html") %>% kable_styling(bootstrap_options = "striped", full_width = F) %>% add_header_above(c("Newton-Raphosn Method"= 3 ))
}


“Ridder’s Method”

#Ridders method function
ridders.method<-function(f,a,b){
  fa <- f(a) ; fb <- f(b) ; 
  c <- (a+b)/2 ; fc <- f(c)
  e_Q <- (fc+sign(fb)*(fc^2-fa*fb)^(1/2)/fb)
  d <- c-(c-a)*sign(fa-fb)*(fc/(fc^2-fa*fb)^(1/2))
  fd <- f(d)
  na=c();nb=c();nd=c();nfa=c();nfb=c();nfd=c()
  if((fa*fb)>0){
    print("Wrong Input")
  }
  i <- 0
  W1 <- c(a,b,fa,fb,d,fd)
  while (abs(b-a) > 0.000001){
    i<-i+1
    fa <- f(a) ; fb <- f(b) ; fd <- f(d)
    
    if((fa*fd)<0){ 
      a = a ; fa = fa
      b = d ; fb = fd
      na = c(na,a) ; nb = c(nb,b)
      nfa = c(nfa,fa) ; nfb = c(nfb,fb)
    }
    else{ 
      a = d ; fa = fd
      b = b ; fb = fb
      na = c(na,a) ; nb = c(nb,b) ; nfa = c(nfa,fa) ; nfb = c(nfb,fb)
    }
    c = (a+b)/2 ; fc <- f(c)
    d <- c-(c-a)*sign(fa-fb)*(fc/(fc^2-fa*fb)^(1/2)) ; fd <- f(d)
    nd = c(nd,d) ; nfd = c(nfd,fd)
  }
  W2 <- c(na,nb,nfa,nfb,nd,nfd)
  W2 <- matrix(data = W2,ncol = 6,byrow = F)
  W2 <- rbind(W1,W2)
  W <- data.frame("a"=W2[,1],"b"=W2[,2],"func.a"=W2[,3],"func.b"=W2[,4],"d"=W2[,5],"func.d"=W2[,6])
  d=round(b,digits = 6)
  print("Ridder's Method ")
  print('covegence criterion : |b-a|<1e-6')
  print(paste('number of iterations : ',i))
  print(paste('approximation root :', d))
  print(knitr::kable(W,"rst",digits = 7))
  kable(W,"html") %>% kable_styling(bootstrap_options = "striped", full_width = F) %>% add_header_above(c(" Secant Method" = 6))
}

“uniroot”

g <- function(x){1997/(2+x)-1810/(1-x)+32/x}
uniroot(g,c(0.02,0.05))
## $root
## [1] 0.03570009
## 
## $f.root
## [1] 0.3362926
## 
## $iter
## [1] 5
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 6.103516e-05

“newton”

g <- function(x){1997/(2+x)-1810/(1-x)+32/x}
Newton(g,0.05,0.000001)
## [1] "Newton's Method "
## [1] "covegence criterion : tolerence <1e-6"
## [1] "number of iterations :  6"
## [1] "approximation root : 0.035712302240628"
## 
## 
## =========  =========  ============
## Iterative          x          f(x)
## =========  =========  ============
##         0  0.0500000  -291.1168164
##         1  0.0309488   149.4448545
##         2  0.0351208    16.5268914
##         3  0.0357034     0.2459974
##         4  0.0357123     0.0000560
##         5  0.0357123     0.0000000
## =========  =========  ============
Newton-Raphosn Method
Iterative x f(x)
0 0.0500000 -291.1168164
1 0.0309488 149.4448545
2 0.0351208 16.5268914
3 0.0357034 0.2459974
4 0.0357123 0.0000560
5 0.0357123 0.0000000

“ridder”

g <- function(x){1997/(2+x)-1810/(1-x)+32/x}
ridders.method(g,0.02,0.05)
## [1] "Ridder's Method "
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  51"
## [1] "approximation root : 0.035713"
## 
## 
## =========  =========  ===========  ============  =========  ============
##         a          b       func.a        func.b          d        func.d
## =========  =========  ===========  ============  =========  ============
## 0.0200000  0.0500000  741.6750859  -291.1168164  0.0343561    38.6615034
## 0.0343561  0.0500000   38.6615034  -291.1168164  0.0486078  -269.3362458
## 0.0343561  0.0486078   38.6615034  -269.3362458  0.0472218  -246.5854603
## 0.0343561  0.0472218   38.6615034  -246.5854603  0.0458414  -222.7740624
## 0.0343561  0.0458414   38.6615034  -222.7740624  0.0444663  -197.7999955
## 0.0343561  0.0444663   38.6615034  -197.7999955  0.0430959  -171.5475223
## 0.0343561  0.0430959   38.6615034  -171.5475223  0.0417296  -143.8847583
## 0.0343561  0.0417296   38.6615034  -143.8847583  0.0403668  -114.6606400
## 0.0343561  0.0403668   38.6615034  -114.6606400  0.0390069   -83.7011661
## 0.0343561  0.0390069   38.6615034   -83.7011661  0.0376491   -50.8046995
## 0.0343561  0.0376491   38.6615034   -50.8046995  0.0362926   -15.7360362
## 0.0343561  0.0362926   38.6615034   -15.7360362  0.0349365    21.7811582
## 0.0349365  0.0362926   21.7811582   -15.7360362  0.0355168     5.4082161
## 0.0355168  0.0362926    5.4082161   -15.7360362  0.0360970   -10.4841009
## 0.0355168  0.0360970    5.4082161   -10.4841009  0.0359015    -5.1804194
## 0.0355168  0.0359015    5.4082161    -5.1804194  0.0357059     0.1759373
## 0.0357059  0.0359015    0.1759373    -5.1804194  0.0358951    -5.0061627
## 0.0357059  0.0358951    0.1759373    -5.0061627  0.0358887    -4.8318496
## 0.0357059  0.0358887    0.1759373    -4.8318496  0.0358823    -4.6574801
## 0.0357059  0.0358823    0.1759373    -4.6574801  0.0358759    -4.4830543
## 0.0357059  0.0358759    0.1759373    -4.4830543  0.0358695    -4.3085720
## 0.0357059  0.0358695    0.1759373    -4.3085720  0.0358631    -4.1340333
## 0.0357059  0.0358631    0.1759373    -4.1340333  0.0358567    -3.9594382
## 0.0357059  0.0358567    0.1759373    -3.9594382  0.0358503    -3.7847865
## 0.0357059  0.0358503    0.1759373    -3.7847865  0.0358439    -3.6100782
## 0.0357059  0.0358439    0.1759373    -3.6100782  0.0358375    -3.4353134
## 0.0357059  0.0358375    0.1759373    -3.4353134  0.0358311    -3.2604919
## 0.0357059  0.0358311    0.1759373    -3.2604919  0.0358247    -3.0856138
## 0.0357059  0.0358247    0.1759373    -3.0856138  0.0358184    -2.9106790
## 0.0357059  0.0358184    0.1759373    -2.9106790  0.0358120    -2.7356875
## 0.0357059  0.0358120    0.1759373    -2.7356875  0.0358056    -2.5606393
## 0.0357059  0.0358056    0.1759373    -2.5606393  0.0357992    -2.3855343
## 0.0357059  0.0357992    0.1759373    -2.3855343  0.0357928    -2.2103724
## 0.0357059  0.0357928    0.1759373    -2.2103724  0.0357864    -2.0351537
## 0.0357059  0.0357864    0.1759373    -2.0351537  0.0357800    -1.8598782
## 0.0357059  0.0357800    0.1759373    -1.8598782  0.0357736    -1.6845457
## 0.0357059  0.0357736    0.1759373    -1.6845457  0.0357672    -1.5091562
## 0.0357059  0.0357672    0.1759373    -1.5091562  0.0357608    -1.3337098
## 0.0357059  0.0357608    0.1759373    -1.3337098  0.0357544    -1.1582064
## 0.0357059  0.0357544    0.1759373    -1.1582064  0.0357480    -0.9826459
## 0.0357059  0.0357480    0.1759373    -0.9826459  0.0357417    -0.8070283
## 0.0357059  0.0357417    0.1759373    -0.8070283  0.0357353    -0.6313537
## 0.0357059  0.0357353    0.1759373    -0.6313537  0.0357289    -0.4556218
## 0.0357059  0.0357289    0.1759373    -0.4556218  0.0357225    -0.2798328
## 0.0357059  0.0357225    0.1759373    -0.2798328  0.0357161    -0.1039866
## 0.0357059  0.0357161    0.1759373    -0.1039866  0.0357097     0.0719169
## 0.0357097  0.0357161    0.0719169    -0.1039866  0.0357135    -0.0320836
## 0.0357097  0.0357135    0.0719169    -0.0320836  0.0357109     0.0398291
## 0.0357109  0.0357135    0.0398291    -0.0320836  0.0357120     0.0077431
## 0.0357120  0.0357135    0.0077431    -0.0320836  0.0357132    -0.0243409
## 0.0357120  0.0357132    0.0077431    -0.0243409  0.0357129    -0.0165981
## 0.0357120  0.0357129    0.0077431    -0.0165981  0.0357126    -0.0088552
## =========  =========  ===========  ============  =========  ============
Secant Method
a b func.a func.b d func.d
0.0200000 0.0500000 741.6750859 -291.1168164 0.0343561 38.6615034
0.0343561 0.0500000 38.6615034 -291.1168164 0.0486078 -269.3362458
0.0343561 0.0486078 38.6615034 -269.3362458 0.0472218 -246.5854603
0.0343561 0.0472218 38.6615034 -246.5854603 0.0458414 -222.7740624
0.0343561 0.0458414 38.6615034 -222.7740624 0.0444663 -197.7999955
0.0343561 0.0444663 38.6615034 -197.7999955 0.0430959 -171.5475223
0.0343561 0.0430959 38.6615034 -171.5475223 0.0417296 -143.8847583
0.0343561 0.0417296 38.6615034 -143.8847583 0.0403668 -114.6606400
0.0343561 0.0403668 38.6615034 -114.6606400 0.0390069 -83.7011661
0.0343561 0.0390069 38.6615034 -83.7011661 0.0376491 -50.8046995
0.0343561 0.0376491 38.6615034 -50.8046995 0.0362926 -15.7360362
0.0343561 0.0362926 38.6615034 -15.7360362 0.0349365 21.7811582
0.0349365 0.0362926 21.7811582 -15.7360362 0.0355168 5.4082161
0.0355168 0.0362926 5.4082161 -15.7360362 0.0360970 -10.4841009
0.0355168 0.0360970 5.4082161 -10.4841009 0.0359015 -5.1804194
0.0355168 0.0359015 5.4082161 -5.1804194 0.0357059 0.1759373
0.0357059 0.0359015 0.1759373 -5.1804194 0.0358951 -5.0061627
0.0357059 0.0358951 0.1759373 -5.0061627 0.0358887 -4.8318496
0.0357059 0.0358887 0.1759373 -4.8318496 0.0358823 -4.6574801
0.0357059 0.0358823 0.1759373 -4.6574801 0.0358759 -4.4830543
0.0357059 0.0358759 0.1759373 -4.4830543 0.0358695 -4.3085720
0.0357059 0.0358695 0.1759373 -4.3085720 0.0358631 -4.1340333
0.0357059 0.0358631 0.1759373 -4.1340333 0.0358567 -3.9594382
0.0357059 0.0358567 0.1759373 -3.9594382 0.0358503 -3.7847865
0.0357059 0.0358503 0.1759373 -3.7847865 0.0358439 -3.6100782
0.0357059 0.0358439 0.1759373 -3.6100782 0.0358375 -3.4353134
0.0357059 0.0358375 0.1759373 -3.4353134 0.0358311 -3.2604919
0.0357059 0.0358311 0.1759373 -3.2604919 0.0358247 -3.0856138
0.0357059 0.0358247 0.1759373 -3.0856138 0.0358184 -2.9106790
0.0357059 0.0358184 0.1759373 -2.9106790 0.0358120 -2.7356875
0.0357059 0.0358120 0.1759373 -2.7356875 0.0358056 -2.5606393
0.0357059 0.0358056 0.1759373 -2.5606393 0.0357992 -2.3855343
0.0357059 0.0357992 0.1759373 -2.3855343 0.0357928 -2.2103724
0.0357059 0.0357928 0.1759373 -2.2103724 0.0357864 -2.0351537
0.0357059 0.0357864 0.1759373 -2.0351537 0.0357800 -1.8598782
0.0357059 0.0357800 0.1759373 -1.8598782 0.0357736 -1.6845457
0.0357059 0.0357736 0.1759373 -1.6845457 0.0357672 -1.5091562
0.0357059 0.0357672 0.1759373 -1.5091562 0.0357608 -1.3337098
0.0357059 0.0357608 0.1759373 -1.3337098 0.0357544 -1.1582064
0.0357059 0.0357544 0.1759373 -1.1582064 0.0357480 -0.9826459
0.0357059 0.0357480 0.1759373 -0.9826459 0.0357417 -0.8070283
0.0357059 0.0357417 0.1759373 -0.8070283 0.0357353 -0.6313537
0.0357059 0.0357353 0.1759373 -0.6313537 0.0357289 -0.4556218
0.0357059 0.0357289 0.1759373 -0.4556218 0.0357225 -0.2798328
0.0357059 0.0357225 0.1759373 -0.2798328 0.0357161 -0.1039866
0.0357059 0.0357161 0.1759373 -0.1039866 0.0357097 0.0719169
0.0357097 0.0357161 0.0719169 -0.1039866 0.0357135 -0.0320836
0.0357097 0.0357135 0.0719169 -0.0320836 0.0357109 0.0398291
0.0357109 0.0357135 0.0398291 -0.0320836 0.0357120 0.0077431
0.0357120 0.0357135 0.0077431 -0.0320836 0.0357132 -0.0243409
0.0357120 0.0357132 0.0077431 -0.0243409 0.0357129 -0.0165981
0.0357120 0.0357129 0.0077431 -0.0165981 0.0357126 -0.0088552

“ridder 2”

ridders(g,0.02,0.1,tol = 10^-6)
## $root
## [1] 0.0357123
## 
## $f.root
## [1] 2.163141e-07
## 
## $niter
## [1] 10
## 
## $estim.prec
## [1] 7.586792e-07

“secant”

g <- function(x){1997/(2+x)-1810/(1-x)+32/x}
secant.method(g,0.02,0.05)
## [1] "Secant Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  6"
## [1] "approximation root : 0.035712"
## 
## 
## =========  =========  ===========  ============  ============  ============
##         a          b            c        func.a        func.b        func.c
## =========  =========  ===========  ============  ============  ============
## 0.0200000  0.0500000  741.6750859  -291.1168164     0.0415438  -140.0006490
## 0.0200000  0.0500000    0.0439274   741.6750859  -291.1168164  -187.6458756
## 0.0500000  0.0415438    0.0264516  -291.1168164  -140.0006490   336.0430612
## 0.0415438  0.0337096    0.0353333  -140.0006490    58.0914055    10.5322785
## 0.0337096  0.0360070    0.0357614    58.0914055    -8.0495846    -1.3502346
## 0.0360070  0.0357274    0.0357114    -8.0495846    -0.4153208     0.0258887
## 0.0357274  0.0357122    0.0357123    -0.4153208     0.0031156     0.0000220
## =========  =========  ===========  ============  ============  ============
Secant Method
a b c func.a func.b func.c
0.0200000 0.0500000 741.6750859 -291.1168164 0.0415438 -140.0006490
0.0200000 0.0500000 0.0439274 741.6750859 -291.1168164 -187.6458756
0.0500000 0.0415438 0.0264516 -291.1168164 -140.0006490 336.0430612
0.0415438 0.0337096 0.0353333 -140.0006490 58.0914055 10.5322785
0.0337096 0.0360070 0.0357614 58.0914055 -8.0495846 -1.3502346
0.0360070 0.0357274 0.0357114 -8.0495846 -0.4153208 0.0258887
0.0357274 0.0357122 0.0357123 -0.4153208 0.0031156 0.0000220

“bisection”

g <- function(x){1997/(2+x)-1810/(1-x)+32/x}
bisection.method(g,0.02,0.05)
## [1] "Bisection Method"
## [1] "covegence criterion : |b-a|<1e-6"
## [1] "number of iterations :  15"
## [1] "approximation root : 0.035713"
## 
## 
## =========  =========  ===========  ============  =========  ============
##         a          b       func.a        func.b          c        func.c
## =========  =========  ===========  ============  =========  ============
## 0.0200000  0.0500000  741.6750859  -291.1168164  0.0350000    19.9648272
## 0.0350000  0.0500000   19.9648272  -291.1168164  0.0425000  -159.6748709
## 0.0350000  0.0425000   19.9648272  -159.6748709  0.0387500   -77.6366721
## 0.0350000  0.0387500   19.9648272   -77.6366721  0.0368750   -31.0791034
## 0.0350000  0.0368750   19.9648272   -31.0791034  0.0359375    -6.1619503
## 0.0350000  0.0359375   19.9648272    -6.1619503  0.0354688     6.7442244
## 0.0354688  0.0359375    6.7442244    -6.1619503  0.0357031     0.2526096
## 0.0357031  0.0359375    0.2526096    -6.1619503  0.0358203    -2.9642074
## 0.0357031  0.0358203    0.2526096    -2.9642074  0.0357617    -1.3581950
## 0.0357031  0.0357617    0.2526096    -1.3581950  0.0357324    -0.5533932
## 0.0357031  0.0357324    0.2526096    -0.5533932  0.0357178    -0.1505421
## 0.0357031  0.0357178    0.2526096    -0.1505421  0.0357104     0.0509961
## 0.0357104  0.0357178    0.0509961    -0.1505421  0.0357141    -0.0497824
## 0.0357104  0.0357141    0.0509961    -0.0497824  0.0357123     0.0006045
## 0.0357123  0.0357141    0.0006045    -0.0497824  0.0357132    -0.0245895
## 0.0357123  0.0357132    0.0006045    -0.0245895  0.0357127    -0.0119926
## =========  =========  ===========  ============  =========  ============
Bisection Method
a b func.a func.b c func.c
0.0200000 0.0500000 741.6750859 -291.1168164 0.0350000 19.9648272
0.0350000 0.0500000 19.9648272 -291.1168164 0.0425000 -159.6748709
0.0350000 0.0425000 19.9648272 -159.6748709 0.0387500 -77.6366721
0.0350000 0.0387500 19.9648272 -77.6366721 0.0368750 -31.0791034
0.0350000 0.0368750 19.9648272 -31.0791034 0.0359375 -6.1619503
0.0350000 0.0359375 19.9648272 -6.1619503 0.0354688 6.7442244
0.0354688 0.0359375 6.7442244 -6.1619503 0.0357031 0.2526096
0.0357031 0.0359375 0.2526096 -6.1619503 0.0358203 -2.9642074
0.0357031 0.0358203 0.2526096 -2.9642074 0.0357617 -1.3581950
0.0357031 0.0357617 0.2526096 -1.3581950 0.0357324 -0.5533932
0.0357031 0.0357324 0.2526096 -0.5533932 0.0357178 -0.1505421
0.0357031 0.0357178 0.2526096 -0.1505421 0.0357104 0.0509961
0.0357104 0.0357178 0.0509961 -0.1505421 0.0357141 -0.0497824
0.0357104 0.0357141 0.0509961 -0.0497824 0.0357123 0.0006045
0.0357123 0.0357141 0.0006045 -0.0497824 0.0357132 -0.0245895
0.0357123 0.0357132 0.0006045 -0.0245895 0.0357127 -0.0119926

“false position”

g <- function(x){1997/(2+x)-1810/(1-x)+32/x}
false.position.method(g,0.02,0.05)
## [1] "False Position Method"
## [1] "covegence criterion : |Cn-Cn-1|<1e-6"
## [1] "number of iterations :  12"
## [1] "approximation root : 0.035712683196537"
## 
## 
## ====  =========  =========  ========  ============  ============
##    a          b          c    func.a        func.b        func.c
## ====  =========  =========  ========  ============  ============
## 0.02  0.0415438  0.0415438  741.6751  -140.0006490  -140.0006490
## 0.02  0.0381229  0.0381229  741.6751   -62.5228480   -62.5228480
## 0.02  0.0367139  0.0367139  741.6751   -26.8792427   -26.8792427
## 0.02  0.0361293  0.0361293  741.6751   -11.3562642   -11.3562642
## 0.02  0.0358861  0.0358861  741.6751    -4.7618337    -4.7618337
## 0.02  0.0357848  0.0357848  741.6751    -1.9903170    -1.9903170
## 0.02  0.0357425  0.0357425  741.6751    -0.8307805    -0.8307805
## 0.02  0.0357249  0.0357249  741.6751    -0.3465820    -0.3465820
## 0.02  0.0357176  0.0357176  741.6751    -0.1445519    -0.1445519
## 0.02  0.0357145  0.0357145  741.6751    -0.0602836    -0.0602836
## 0.02  0.0357132  0.0357132  741.6751    -0.0251395    -0.0251395
## 0.02  0.0357127  0.0357127  741.6751    -0.0104835    -0.0104835
## ====  =========  =========  ========  ============  ============
False Position Method
a b c func.a func.b func.c
0.02 0.0415438 0.0415438 741.6751 -140.0006490 -140.0006490
0.02 0.0381229 0.0381229 741.6751 -62.5228480 -62.5228480
0.02 0.0367139 0.0367139 741.6751 -26.8792427 -26.8792427
0.02 0.0361293 0.0361293 741.6751 -11.3562642 -11.3562642
0.02 0.0358861 0.0358861 741.6751 -4.7618337 -4.7618337
0.02 0.0357848 0.0357848 741.6751 -1.9903170 -1.9903170
0.02 0.0357425 0.0357425 741.6751 -0.8307805 -0.8307805
0.02 0.0357249 0.0357249 741.6751 -0.3465820 -0.3465820
0.02 0.0357176 0.0357176 741.6751 -0.1445519 -0.1445519
0.02 0.0357145 0.0357145 741.6751 -0.0602836 -0.0602836
0.02 0.0357132 0.0357132 741.6751 -0.0251395 -0.0251395
0.02 0.0357127 0.0357127 741.6751 -0.0104835 -0.0104835
Do you ever shine?