commit 67ebaf88644f0bf47103af79fee76d015d43ce00
parent 7ad81d682ee7957bb87889e24138c37ed16db3d9
Author: Mattias Andrée <maandree@kth.se>
Date: Sat, 23 Jul 2016 17:49:40 +0200
Add two exercises to the manual
[M10] Convergence of the Lucas Number ratios
[M12+] Factorisation of factorials
Diffstat:
3 files changed, 100 insertions(+), 1 deletion(-)
diff --git a/Makefile b/Makefile
@@ -82,7 +82,8 @@ TEXSRC =\
doc/bit-operations.tex\
doc/number-theory.tex\
doc/random-numbers.tex\
- doc/not-implemented.tex
+ doc/not-implemented.tex\
+ doc/exercises.tex
HDR_PUBLIC = zahl.h $(HDR_SEMIPUBLIC)
HDR = $(HDR_PUBLIC) $(HDR_PRIVATE)
diff --git a/doc/exercises.tex b/doc/exercises.tex
@@ -0,0 +1,96 @@
+\chapter{Exercises}
+\label{chap:Exercises}
+
+% TODO
+%
+% All exercises should be group with a chapter
+%
+% ▶ Recommended
+% M Matematically oriented
+% HM Higher matematical education required
+%
+% 00 Immediate
+% 10 Simple
+% 20 Medium
+% 30 Moderately hard
+% 40 Term project
+% 50 Research project
+%
+% ⁺ High risk of undershoot difficulty
+
+
+\begin{enumerate}[label=\textbf{\arabic*}.]
+
+
+\item {[\textit{M10}]} \textbf{Convergence of the Lucas Number ratios}
+
+Find an approximation for $\displaystyle{ \lim_{n \to \infty} \frac{L_{n + 1}}{L_n}}$,
+where $L_n$ is the $n^{\text{th}}$ Lucas Number \psecref{sec:Lucas numbers}.
+
+\( \displaystyle{
+ L_n \stackrel{\text{\tiny{def}}}{\text{=}} \left \{ \begin{array}{ll}
+ 2 & \text{if} ~ n = 0 \\
+ 1 & \text{if} ~ n = 1 \\
+ L_{n - 1} + L_{n + 1} & \text{otherwise}
+ \end{array} \right .
+}\)
+
+
+\item {[\textit{M12${}^+$}]} \textbf{Factorisation of factorials}
+
+Implement the function
+
+\vspace{-1em}
+\begin{alltt}
+ void factor_fact(z_t n);
+\end{alltt}
+\vspace{-1em}
+
+\noindent
+which prints the prime factorisation of $n!$ (the $n^{\text{th}}$ factorial).
+The function shall be efficient for all $n$ where all primes $p \le n$ can
+be found efficiently. You can assume that $n \ge 2$. You should not evaluate $n!$.
+
+
+\end{enumerate}
+
+
+
+\chapter{Solutions}
+\label{chap:Solutions}
+
+
+\begin{enumerate}[label=\textbf{\arabic*}.]
+
+\item \textbf{Convergence of the Lucas Number ratios}
+
+It would be a mistake to use bignum, and bigint in particular,
+to solve this problem. Good old mathematics is a much better solution.
+
+$$ \lim_{n \to \infty} \frac{L_{n + 1}}{L_n} = \lim_{n \to \infty} \frac{L_{n}}{L_{n - 1}} = \lim_{n \to \infty} \frac{L_{n - 1}}{L_{n - 2}} $$
+
+$$ \frac{L_{n}}{L_{n - 1}} = \frac{L_{n - 1}}{L_{n - 2}} $$
+
+$$ \frac{L_{n - 1} + L_{n - 2}}{L_{n - 1}} = \frac{L_{n - 1}}{L_{n - 2}} $$
+
+$$ 1 + \frac{L_{n - 2}}{L_{n - 1}} = \frac{L_{n - 1}}{L_{n - 2}} $$
+
+$$ 1 + \varphi = \frac{1}{\varphi} $$
+
+So the ratio tends toward the golden ration.
+
+
+\item \textbf{Factorisation of factorials}
+
+Base your implementation on
+
+\( \displaystyle{
+ n! = \prod_{p~\in~\textbf{P}}^{n} p^{k_p}, ~\text{where}~
+ k_p = \sum_{a = 1}^{\lfloor \log_p n \rfloor} \lfloor np^{-a} \rfloor.
+}\)
+
+There is no need to calculate $\lfloor \log_p n \rfloor$,
+you will see when $p^a > n$.
+
+
+\end{enumerate}
diff --git a/doc/libzahl.tex b/doc/libzahl.tex
@@ -6,6 +6,7 @@
\usepackage{amsmath, amssymb, mathtools, MnSymbol, mathrsfs, esvect}
\usepackage{tipa, color, graphicx}
\usepackage{shorttoc, minitoc}
+\usepackage{enumitem}
\usepackage[english]{babel}
\selectlanguage{english}
\definecolor{linkcolour}{RGB}{112, 0, 0}
@@ -103,6 +104,7 @@ copyright notice and this permission notice appear in all copies.}
\input doc/number-theory.tex
\input doc/random-numbers.tex
\input doc/not-implemented.tex
+\input doc/exercises.tex
\appendix
