Cover | Description |
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This repository contains accompanying data for my book Advanced Statistics with Application in R. Please visit my website www.eugened.org for more information about the book. Below is the Table of Contents for the accompanying data provided with the book. You can individually download each piece of data; but my recommendation is to download everything using the zip option in GitHub. |
Code/Data | Chapter | Section | Page | Description |
---|---|---|---|---|
mvsd.r | 1 | 1.5.5 | 19 | "Computation of mean, variance and SD" |
vecomp.r | 1 | 1.5.5 | 20 | Vectorized computation of the double integral |
my1sr.r | 1 | 1.5.6 | 23 | An example for graphics in R |
frlJLET.r | 1 | 1.56 | 24 | Frequency of English letters in a Jack London novel |
Jack_London_Call_of_the_Wild_The_f1.char | 1 | 1.56 | 24 | "Char by char ""Call of the Wild""" |
birthdaysim.r | 1 | 1.6 | 30 | Simulations for the birthday problem |
sampP.r | 1 | 1.8.1 | 38 | Illustration of the sample command |
sudoku.r | 1 | 1.8.2 | 40 | Random sudoku problem |
webhits.r | 2 | 2.1.2 | 45 | Cdf of website hits |
comwebhits.dat | 2 | 2.1.2 | 45 | Data for website hits |
webhitsQ.r | 2 | 2.2.1 | 55 | Quartiles of web hits cdf |
tr.binom.r | 2 | 2.2.2 | 57 | Confidence range for binomial distribution |
longpiece.r | 2 | 2.3 | 60 | Simulations for the broken stick |
simCookie.r | 2 | 2.3 | 61 | Simulations for raisin in the cookie |
truck.turn.r | 2 | 2.4 | 65 | Simulations for safe turn |
gampois.r | 2 | 2.6.1 | 78 | Simulation for approximation Poison by gamma cdf |
pti.gamma.r | 2 | 2.6.3 | 80 | Newton's algorithm for the gamma tight confidence range |
pti.gamma.rome.r | 2 | 2.6.3 | 81 | Modified pti.gamma for Rome example |
varX2.r | 2 | 2.7 | 87 | Simulations for var(X^2) |
mile.r | 2 | 2.9.1 | 97 | Simulations for distance to work (LLN) |
LLNintegral.r | 2 | 2.9.2 | 102 | Integral approximation via Monte Carlo |
LLNIntegral2.r | 2 | 2.9.2 | 103 | Optimal lambda for Monte Carlo approximation |
LLNintegral3.r | 2 | 2.1 | 106 | Normal approximation of the binomial cdf |
clt.binom.r | 2 | 2.1 | 106 | Dynamic convergence of the binomial distribution |
jb.r | 2 | 2.1 | 109 | James Bond chase problem |
cltP.r | 2 | 2.1 | 111 | Violated CLT |
lognDS.r | 2 | 2.11 | 117 | Simulations for mean and variance of the lognormal distribution |
logLnr.r | 2 | 2.11.1 | 119 | Tight confidence range for the lognormal distribution |
expr.r | 2 | 2.13 | 127 | Simulations for logistic distribution |
rugf2.r | 2 | 2.13 | 127 | Newton's algorithm for Gaussian mixture |
genmixN.r | 2 | 2.13 | 128 | Random number generation for Gaussian mixture |
discr.gen.r | 2 | 2.13 | 129 | Random number generation from a discrete distribution |
meanC.r | 2 | 2.13.1 | 130 | Simulations for Cauchy distribution |
pidistr.r | 2 | 2.15 | 136 | Pmf of pi digits |
pidistr1010.r | 2 | 2.15 | 137 | The 10x10 probability matrix for pi digits |
benfordEXP.r | 2 | 2.16 | 142 | Benford's law for Pareto distribution |
sales.RData | 2 | 2.16 | 142 | Sales transactions data (R object) |
benfordFT.r | 2 | 2.16 | 142 | Benford's law analysis of fraudulent transactions |
benfordN.r | 2 | 2.16.1 | 143 | Almost-Benford's law distributions |
fracksim.r | 2 | 2.16 | 144 | Benford's law for the t-distribution |
sbmultD.r | 3 | 3.1 | 152 | Density surface viewed at different angle |
CondDensity.mkv | 3 | 3.3 | 169 | Conditional density movie file |
mixedDens.r | 3 | 3.3.2 | 180 | Gaussian mixture distribution of heights |
randS.r | 3 | 3.3.3 | 184 | Random sums |
cancgr.r | 3 | 3.3.4 | 185 | Simulations for cancer growth |
heightweight.r | 3 | 3.5 | 202 | Scatterplot weight versus height |
dn2.r | 3 | 3.5 | 204 | Contours of the bivariate normal density |
oilspill.r | 3 | 3.5 | 206 | Simulations for oil spill in the ocean |
dn3.r | 3 | 3.5.3 | 209 | Generation of random numbers from the bivariate normal distribution |
dint.r | 3 | 3.5.3 | 213 | Double integral approximation using simulations |
copRN.r | 3 | 3.5.4 | 216 | Contours of the copula density |
ell2.r | 3 | 3.5.4 | 216 | Contours of the bivariate normal density |
filelXY.r | 3 | 3.6 | 221 | Simulations for Fieller density |
metpr2.r | 3 | 3.7.1 | 225 | Simulations for meeting problem 2 |
buffon.r | 3 | 3.7.2 | 225 | Buffon's needle problem |
stickSQ.r | 3 | 3.7.2 | 226 | Simulations for random segment example |
sbE.r | 3 | 3.7.2 | 227 | "Simulations for ""Who wins the game?"" example" |
s47.r | 3 | 3.7.2 | 228 | Simulations for random squares |
prmb.r | 3 | 3.8.4 | 235 | Probability bullet |
cdfMED.r | 3 | 3.9 | 238 | Cdf of the median |
pmaxmin.r | 3 | 3.9 | 238 | Distribution of the range |
cov01.r | 3 | 3.9 | 239 | Simulations for random segments |
normQQ.r | 3 | 3.10.2 | 247 | Testing multivariate normal distribution |
intrD3.r | 3 | 3.10.3 | 250 | Simulations for the root of the random cubic equation |
multB.r | 3 | 3.10.4 | 252 | Simulations for the multinomial distribution |
mn3.r | 4 | 4.1.1 | 260 | Genetation and viewing of 3D points |
foold.csv | 4 | 4.1.2 | 263 | Data for temperature in March |
multCLT.r | 4 | 4.1.3 | 269 | Bivariate CLT |
chidf1.r | 4 | 4.2.1 | 277 | The cdf of the chi-square cdf |
salary.r | 5 | 5.1 | 294 | CDF of salary in Connecticut and Vermont |
mortgageROC.csv | 5 | 5.1.1 | 301 | Data on 375 mortgage applicants |
mortgageROC.r | 5 | 5.1.1 | 302 | ROC curve for mortgage applicants |
vomit.r | 5 | 5.1.1 | 304 | Time to vomiting ROC curve |
emesis.txt | 5 | 5.1.1 | 304 | Data for the vomit code |
survROC.r | 5 | 5.1.1 | 307 | ROC curve for cancer patients |
survcanc.csv | 5 | 5.1.1 | 307 | Data for the survROC code |
SCcancer.r | 5 | 5.1.2 | 306 | Survival analysis for cancer patients |
DeathYears.csv | 5 | 5.1.1 | 306 | Survival data for the SCcancer code |
usbflash.csv | 5 | 5.1 | 311 | Data for Problem 29 |
creditpr.r | 5 | 5.1 | 312 | 1996 credit card applicants' analysis |
creditpr.csv | 5 | 5.1 | 312 | Data for the creditpr code |
webhits.hist.r | 5 | 5.2 | 313 | Histogram plot for website hits |
histN.r | 5 | 5.2 | 315 | Histogram and density for normal distribution |
SCcancerQQ.r | 5 | 5.3 | 318 | Q-q plot for cancer patients |
Qqmurder.r | 5 | 5.3 | 318 | Q-q plot to test the uniformity of murders |
wikmurdr.txt | 5 | 5.3 | 318 | Murder rates in 51 states |
qqband.r | 5 | 5.3.1 | 320 | Q-q plot for uniform distribution |
qqLOGband.r | 5 | 5.3.1 | 321 | Q-q plot for lognormal distribution |
mortgageQQ.r | 5 | 5.3.1 | 322 | Q-q plot for mortgage data |
qqnill.r | 5 | 5.3 | 323 | Problem 5 |
toears.txt | 5 | 5.3 | 323 | Toenail arsenic data |
Goldman.csv | 5 | 5.3 | 323 | Anatomical data |
salaryBAR.r | 5 | 5.4 | 324 | Barplot for Vermont and Connecticut salaries |
kernavN.r | 5 | 5.5 | 326 | Gaussian kernel densities |
kernM.r | 5 | 5.5 | 327 | Gaussian kernel densities with two bw |
n.density.my.r | 5 | 5.5 | 327 | In-house Gaussian kernel density |
eppendorf.r | 5 | 5.5 | 329 | Rat brain oxygen distribution |
eppendorf.txt | 5 | 5.5 | 329 | Data for rat brain oxygen |
toears.r | 5 | 5.5 | 330 | Distribution of toenail arsenic in NH |
toears.txt | 5 | 5.5 | 330 | Data for toenail distribution |
asviol.r | 5 | 5.5 | 330 | Asymmetric violin for salary in VT and CT |
kern.movie.r | 5 | 5.5.1 | 332 | Density movie |
alc3d.r | 5 | 5.5.2 | 333 | 3D alcohol consumption |
alcoholUSA.csv | 5 | 5.5.2 | 333 | Data for 3D plots |
autocrash.csv | 5 | 5.5 | 335 | Automobile accident data |
bvn.density.my.r | 5 | 5.6 | 337 | Bivariate normal density |
bvex.r | 5 | 5.6 | 337 | 2D and 3D bivariate kernel densities |
Eyx.r | 5 | 5.6 | 337 | E(Y|X=x) |
matimage.r | 5 | 5.6.1 | 339 | Matrix image |
R.smooth.r | 5 | 5.6.1 | 340 | Smoothed images of R |
R.pgm | 5 | 5.6.1 | 340 | The pgm image data for R |
salmark.r | 5 | 5.6.2 | 342 | Scatterplot of Forbes data |
Forbes2000.csv | 5 | 5.6.2 | 342 | Forbes data |
nhcancer.r | 5 | 5.6.3 | 342 | NH lung cancer spatial distribution |
NHtowns.csv | 5 | 5.6.3 | 342 | NH town names |
xyNHcancer.csv | 5 | 5.6.3 | 343 | Coordinates of 10439 cancer cases |
xyNHpopulation.csv | 5 | 5.6.3 | 343 | Geographic location of random NH residents |
Lena.pgm | 5 | 5.6 | 346 | Lena image |
IBM_daily.csv | 5 | 5.6 | 346 | IBM stock prices |
robpol.r | 6 | 6.2 | 353 | Police and bank robber |
gMMgamma.r | 6 | 6.2.1 | 356 | Newton's algorithm for the MM estimation of gamma parameters |
pois2est.r | 6 | 6.4.2 | 371 | RMSE for two estimators of lambda |
luR.r | 6 | 6.4.3 | 373 | Simulations for lower and upper bounds of uniform distribution |
arMSE.r | 6 | 6.6.1 | 389 | Simulations for estimation of area of the circle |
robias.r | 6 | 6.6.2 | 393 | Simulations for the bias of c.c. |
cimcorSP.r | 6 | 6.6.2 | 395 | 17x17 stock correlation heatmap |
stocks.zip | 6 | 6.6.2 | 394 | Data on 17 stocks (must be unzipped) |
Rcolor.pdf | 6 | 6.6.2 | 396 | Colors in R (by name) |
olsim.r | 6 | 6.7.2 | 405 | Simulations for simple regression |
lm.trendSP.r | 6 | 6.7.3 | 408 | Prediction of the Google stock price |
truckR.r | 6 | 6.7.4 | 411 | Coefficient of determination for truck driers |
truckR.data.csv | 6 | 6.7.4 | 411 | Data for truck drivers' problem |
betaMM.apply.r | 6 | 6.8.2 | 430 | Simulations for the MM estimator for alpha and beta |
betaMM.r | 6 | 6.8 | 433 | MM estimation of gamma distribution |
gammaInf.r | 6 | 6.9.2 | 449 | ML estimation of alpha and beta of the gamma distribution |
bufprob.r | 6 | 6.10.1 | 456 | Estimation of L/D in the Buffon problem |
regrD.r | 6 | 6.10.1 | 464 | Linear regression with random X |
father.son.csv | 6 | 6.10.1 | 466 | Galton data for father and son heights |
piest.r | 6 | 6.10.1 | 471 | Comparison of four estimators of pi |
gotobed.r | 6 | 6.10.2 | 474 | When students go to bed |
autocrash.r | 6 | 6.10.2 | 475 | Parzen density for autocrash circular data |
autocrash.csv | 6 | 6.10.2 | 475 | Autocrash circular data |
cubMLE.r | 6 | 6.10.5 | 494 | ML estimation of the quadratic model |
mleUNE.r | 6 | 6.10.5 | 497 | MLE for unemployment rate |
cauchy.theta.r | 6 | 6.10.6 | 502 | MLE for the Cauchy distribution |
cauchy.google.r | 6 | 6.10.6 | 503 | MLE for the Google stock price |
mle.gamma.OPT.r | 6 | 6.10.6 | 506 | Comparison of three algorithms for ML estimation |
mle.gamma.CT.r | 6 | 6.10.6 | 506 | Estimation of people in poverty in CT |
bufprobSA.r | 6 | 6.10.6 | 508 | Simulation-based ML for the Buffon problem |
ranlRest.r | 6 | 6.10.6 | 508 | Estimation of the radius of the disk for the random lines problem |
heightweight.r | 6 | 6.1 | 509 | Scatterplot height versus weight of Korean young people |
HeightWeight.csv | 6 | 6.1 | 509 | Height and weight data |
rws.r | 6 | 6.1 | 510 | Random walk on the lattice square |
meanmed.r | 6 | 6.11 | 516 | Simulations for mean and median in Laplace distribution |
robloc.r | 6 | 6.11.1 | 519 | Estimation for noisy data via Gaussian mixture |
gng.r | 6 | 6.11.1 | 521 | Estimation of Google stock price using Gaussian mixture |
AMZN.csv | 6 | 6.11 | 522 | Amazon stock prices |
pvsim.r | 7 | 7.1.1 | 526 | Illustration of p-value using simulations |
boengtr.csv | 7 | 7.1.2 | 530 | Data for Boeing inside trading |
NHBirths2003_2009.csv | 7 | 7.2 | 533 | Data on 33666 NH newborns |
houseprice.txt | 7 | 7.2 | 534 | House prices |
familyincome.r | 7 | 7.2 | 535 | Poverty test |
pvalcost.r | 7 | 7.4 | 550 | Simulations for the p-value living cost example |
powsim.r | 7 | 7.4.2 | 553 | Confirmation of the power of the t-test via simulations |
stucost.r | 7 | 7.4.2 | 554 | Living cost for freshmen and sophomores |
stucost.csv | 7 | 7.4.2 | 554 | Data for living cost |
ttest2pow.r | 7 | 7.4.2 | 557 | Simulations for Welsh and t-test |
sampt2.r | 7 | 7.4.3 | 557 | One- versus two-sided t-test |
salaryMW.r | 7 | 7.4.4 | 559 | Testing the salary for men and women |
salaryMW_paired.csv | 7 | 7.4.4 | 559 | Salary data |
nonpN.r | 7 | 7.4.4 | 560 | Comparison of power functions for parametric and nonparametric test |
varq.r | 7 | 7.5.1 | 563 | Two-sided variance test |
vartest.r | 7 | 7.5.1 | 563 | Equal-tail probabilities and unbiased test for variance |
smallF.r | 7 | 7.6.2 | 572 | Newton's algorithm for unbiased test for variances |
vartestSP.r | 7 | 7.6.2 | 573 | Testing the volatility of two stocks |
binprop1.r | 7 | 7.6.3 | 575 | Simulations for the power function of the binomial test |
binprop2.r | 7 | 7.6.3 | 577 | Simulations for the test of two binomial proportions |
poistest.r | 7 | 7.6.4 | 578 | Power function for the Poison distribution |
poissamn.r | 7 | 7.6.4 | 579 | Sample size for the Poisson distribution |
vartestSP2.r | 7 | 7.6 | 580 | Variance test for GOOGLE and AMAZON |
GOOG.csv | 7 | 7.6 | 580 | Data for GOOGLE |
AMZN.csv | 7 | 7.6 | 580 | Data for AMZN |
corn0.r | 7 | 7.7 | 581 | Power functions for correlation coefficient |
cor0.r | 7 | 7.7 | 582 | Simulations for testing correlation coefficient |
ciumax.r | 7 | 7.8 | 585 | CI for uniform distribution |
Cimovie.r | 7 | 7.8 | 586 | CI animation |
houprCI.r | 7 | 7.8 | 588 | CI and confidence range for the house price |
civar.r | 7 | 7.8.3 | 592 | Shortest CI for variance and SD |
ci.binpr.r | 7 | 7.8.4 | 593 | Shortest CI for the binomial probability |
cfmus.r | 7 | 7.8.5 | 596 | "Simulations for confidence region for (mu,sigma)" |
thastest.r | 7 | 7.9 | 600 | Three asymptotic tests for binomial probability |
powlinmod.r | 7 | 7.9 | 602 | Simulations for testing the regression coefficient by three tests |
thlinmod.r | 7 | 7.9 | 602 | Simulation-derived cdfs for four tests |
powlinmodC.r | 7 | 7.9 | 604 | Type I adjustments for three tests |
chismult.r | 7 | 7.9.1 | 607 | Simulations for chi-square and Wald tests |
frlJL.r | 7 | 7.9.1 | 607 | Wald test for English letter analysis |
Mark_Twain_The_Adventures_of_Tom_Sawyer_f1.txt.char | 7 | 7.9.1 | 607 | Mark Twain novel char data |
mnist_train.csv.zip | 7 | 7.9.2 | 608 | Handwritten digit train set |
mnist_test.csv | 7 | 7.9.2 | 608 | Handwritten digit test set |
dig.mnist.r | 7 | 7.9.2 | 609 | Plotting and classification of handwritten digits |
wtext.r | 7 | 7.9 | 611 | Analysis of English letters in the novel by Jane Austen |
Jane_Austen_Pride_and_Prejudice.char | 7 | 7.9 | 611 | "Char by char ""Pride and Prejudice""" |
saldisc.r | 7 | 7.1 | 623 | Drug or not to drug example |
saldisc.csv | 7 | 7.1 | 623 | Data for the saldisc code |
dvalREG.r | 7 | 7.1 | 626 | d-value for linear regression |
simLM.r | 8 | 8.3.1 | 647 | Simulations for statistical properties of the quadratic regression |
roblinreg2.r | 8 | 8.3.1 | 648 | Simulations to study violation of the normal assumption |
roblinreg.r | 8 | 8.3.1 | 649 | Simulations with the apply function |
CDpf.r | 8 | 8.4.1 | 653 | Estimation of the Cobb-Douglas production function |
CDpf.csv | 8 | 8.4.1 | 653 | Data for pf |
qtrpow.r | 8 | 8.4.3 | 660 | Sample size determination for the regression coefficient |
linpower.r | 8 | 8.4.3 | 660 | Simulations for the F-test in quadratic regression |
simCB.r | 8 | 8.4.3 | 664 | Simultaneous confidence band for quadratic regression |
olsnormT.r | 8 | 8.4.5 | 668 | Simulations for linear regression with random predictor |
pfx123.csv | 8 | 8.4 | 670 | Data for pf from Problem 8 |
dvalPMED.r | 8 | 8.5.2 | 675 | D-value for personalized medicine |
dvalPMED.csv | 8 | 8.5.2 | 675 | Data for dvalPMED |
kidsdrink.r | 8 | 8.6.1 | 677 | Kids drinking alcohol example |
kidsdrink.csv | 8 | 8.6.1 | 677 | Data for kids drinking |
leftright.r | 8 | 8.6.2 | 680 | False discovery example |
leftright.csv | 8 | 8.6.2 | 680 | Data for false discovery example |
hfn.r | 8 | 8.6.3 | 682 | "Heigh, foot, and nose example" |
HeightFootNose.csv | 8 | 8.6.3 | 682 | "Data for height, foot, and nose example" |
amzn.r | 8 | 8.6.5 | 692 | Autoregression for AMZN stock price |
AMZN_weekly.csv | 8 | 8.6.5 | 691 | Data for AMZN autoregression example |
salMW.r | 8 | 8.7.1 | 697 | Gender difference in salary |
Salary.csv | 8 | 8.7.1 | 696 | Salary data for men and women |
nile.r | 8 | 8.7.1 | 700 | Nile river example |
NileFlow.csv | 8 | 8.7.1 | 700 | Data for Nile flow |
housepr.r | 8 | 8.7.1 | 701 | Regressions for house price in two areas |
houseprice.csv | 8 | 8.7.1 | 701 | Data for house prices in two areas |
obesegene.r | 8 | 8.7.1 | 702 | BMI-gene interaction example |
obesegene.csv | 8 | 8.7.1 | 702 | Data for the BMI-gene regression |
simpson.r | 8 | 8.7.1 | 704 | Simpson paradox |
simpson.csv | 8 | 8.7.2 | 704 | Data for Simpson paradox example |
movrat.r | 8 | 8.7.2 | 706 | Movie rating example |
movrat.csv | 8 | 8.7.2 | 706 | Data for movie rating |
BPlong.r | 8 | 8.7.3 | 708 | Blood pressure treatment |
BPdata.csv | 8 | 8.7.3 | 708 | Data for blood pressure |
QoL.csv | 8 | 8.7.3 | 710 | Quality of life data |
qolS.r | 8 | 8.7.3 | 710 | Quality of life example |
flu.r | 8 | 8.7.4 | 714 | Flu incidence ANOVA example |
linrep.r | 8 | 8.7.4 | 716 | Regression on averages and ANOVA |
consIR.r | 8 | 8.7.5 | 721 | Internet radio example |
consIR.csv | 8 | 8.7.5 | 721 | Data for internet radio example |
CollegeSalary.csv | 8 | 8.7 | 772 | Salary data on 36 college employees |
walD.r | 8 | 8.8 | 724 | Black Friday shopping example |
blackfriday.csv | 8 | 8.8 | 724 | Data for Black Friday shoppers |
lungsm.r | 8 | 8.8.2 | 733 | Two-by-two table via logistic regression |
geodrink.r | 8 | 8.8.2 | 735 | Binge drinking among kids |
kidsdrinkDAT.csv | 8 | 8.8.2 | 735 | Data for binge drinking |
poisR.r | 8 | 8.8.3 | 737 | Poisson regression for traffic tickets |
Traffic.Viol.csv | 8 | 8.8.3 | 737 | Data for traffic violations |
cloglogV.r | 8 | 8.8.3 | 738 | Poisson and log-log regression |
amazshop.csv | 8 | 8.8 | 740 | Data for Problem 10 |
marathon.r | 9 | 9.5 | 770 | Marathon nonlinear regression example |
marathonWR2.txt | 9 | 9.5 | 770 | Data for marathon records |
dnaRAD.csv | 9 | 9.5 | 773 | Cell survival data |
dnaRAD.r | 9 | 9.5 | 774 | Change-point nonlinear regression for cell survival |
twocph.r | 9 | 9.5 | 775 | Two-compartment pharmacokinetics model |
twocph.csv | 9 | 9.5 | 775 | Data for the two-compartment example |
twocphCI.r | 9 | 9.5 | 778 | Three CIs for the two-compartment model |
ces.r | 9 | 9.5 | 780 | CES and Cobb-Douglas production functions |
CES.csv | 9 | 9.5 | 780 | Data for the ces R code |
FallingHat.csv | 9 | 9.5 | 781 | Data for the falling example |
freefall.r | 9 | 9.5 | 781 | The R code for the falling hat example |
gammaNLS.r | 9 | 9.5 | 784 | Simulations for the gamma distribution |
SCcancerQQ2.r | 9 | 9.5 | 784 | Mixture exponential distribution |
nen.r | 9 | 9.6.1 | 787 | Comparison of three distribution approximations |
power.nls.r | 9 | 9.6.2 | 789 | Comparison of three power functions |
expgr.r | 9 | 9.6.3 | 791 | Confidence region for the two-parameter regression |
ci.nls.r | 9 | 9.6.4 | 792 | Three CIs |
q2pr.r | 9 | 9.7.2 | 798 | Probability of two local minima |
michm.r | 9 | 9.9.1 | 806 | Michaelis-Menten nonlinear regression |
berndet.r | 10 | 10.2 | 818 | Bernoulli matrix |
block.inv.r | 10 | 10.2 | 818 | Block inverse |
nLINEXP.r | 10 | 10.6.3 | 836 | Newton's algorithm for a nonlinear equation |