\documentclass[12pt]{article}
\usepackage{fullpage}
\usepackage{noweb}
\usepackage{amsmath}

\title{A Literate Programming Example}
\begin{document}
\maketitle

% Weave with
% noweb::noweave("leukopt.Rnw")
% write(noweb::noweb.sty, file="noweb.sty")
% system("pdflatex leukopt")

% Tangle with
% noweb::notangle("leukopt.Rnw", target = "leukopt.R", out = "leukopt.R")

\section*{The Log-Likelihood}
The log-likelihood for the leukemia data is

\begin{equation*}
  \ell(\beta_0, \beta_1, \beta_2, \delta) = -n \log\delta + \sum (z_i - e^{z_i})
\end{equation*}
with $\delta = 1/\gamma$ and
$z_i = (\log t_i - \beta_0 - \beta_1 x_i - \beta_2 u_i)/\delta$.

The log likelihood is computed by the expression
<<compute log-likelihood>>=
-n * log(delta) + sum(z - exp(z))
@
after computing \texttt{n} and \texttt{z} as
<<compute n and z>>=
n <- length(t)
z <- (log(t) - beta[1] - beta[2] * x - beta[3] * u) / delta
@ 
A function to compute the log-likelihood is therefore
<<log-likelihood>>=
ll <- function(beta, delta, t, x, u) {
    <<compute n and z>>
    <<compute log-likelihood>>
}
@ 

\section*{The Gradient of the Log-Likelihood}
The values of \texttt{exp(z)} and \texttt{exp(z) - 1} are needed
several times in computing the gradient, so it is useful to compute
these once and save them in variables:
<<compute exp_z and exp_z_m_1>>=
exp_z <- exp(z)
exp_z_m_1 <- exp_z - 1
@ 
The gradient is then computed by the expression
<<compute gradient>>=
c(sum(exp_z_m_1) / delta,
  sum(exp_z_m_1 * x) / delta,
  sum(exp_z_m_1 * u) / delta,
  (sum(exp_z * z - z) - n) / delta)
@ 
A function to compute the gradient of the log-likelihood is then
<<gradient>>=
llg <- function(beta, delta, t, x, u) {
    <<compute n and z>>
    <<compute exp_z and exp_z_m_1>>
    <<compute gradient>>
}
@ 
\end{document}

<<leukopt.R>>=
<<log-likelihood>>

<<gradient>>
@