diff --git a/Makefile.pamphlet b/Makefile.pamphlet index 1f08124..014a659 100644 --- a/Makefile.pamphlet +++ b/Makefile.pamphlet @@ -221,6 +221,7 @@ clean: @ rm -f books/Makefile @ rm -f books/Makefile.dvi @ rm -f books/Makefile.pdf + @ rm -f books/axiom.bib @ rm -f lsp/axiom.sty @ rm -f lsp/Makefile lsp/Makefile.dvi lsp/Makefile.pdf @ rm -rf lsp/gcl* diff --git a/books/Makefile.pamphlet b/books/Makefile.pamphlet index 59dac4e..74df390 100644 --- a/books/Makefile.pamphlet +++ b/books/Makefile.pamphlet @@ -23,6 +23,7 @@ PDF=${AXIOM}/doc IN=${SPD}/books LATEX=latex MAKEINDEX=makeindex +BIBTEX=bibtex DVIPDFM=dvipdfm DVIPS=dvips -Ppdf PS2PDF=ps2pdf @@ -38,6 +39,7 @@ BOOKPDF=${PDF}/bookvol0.pdf ${PDF}/bookvol1.pdf ${PDF}/bookvol2.pdf \ ${PDF}/bookvol11.pdf ${PDF}/bookvol12.pdf ${PDF}/bookvol13.pdf \ ${PDF}/bookvolbib.pdf + OTHER= ${PDF}/refcard.pdf ${PDF}/endpaper.pdf ${PDF}/rosetta.pdf all: announce ${BOOKPDF} ${PDF}/toc.pdf ${OTHER} spadedit @@ -53,7 +55,14 @@ finish: @ echo FINISHED BUILDING PDF FILES books/Makefile @ echo ========================================== -${PDF}/%.pdf: ${IN}/%.pamphlet +${PDF}/axiom.bib: + @ echo =========================================== + @ echo making ${PDF}/axiom.bib from ${IN}/bookvolbib.pamphlet + @ echo =========================================== + @${BOOKS}/tanglec ${BOOKS}/bookvolbib.pamphlet axiom.bib \ + >${PDF}/axiom.bib + +${PDF}/%.pdf: ${IN}/%.pamphlet ${PDF}/axiom.bib @ echo =========================================== @ echo making ${PDF}/$*.pdf from ${IN}/$*.pamphlet @ echo =========================================== @@ -67,6 +76,8 @@ ${PDF}/%.pdf: ${IN}/%.pamphlet ${RM} $*.toc ; \ ${LATEX} $*.pamphlet ; \ ${MAKEINDEX} $*.idx 1>/dev/null 2>/dev/null ; \ + ${BIBTEX} $*.aux ; \ + ${LATEX} $*.pamphlet >/dev/null ; \ ${LATEX} $*.pamphlet >/dev/null ; \ ${DVIPDFM} $*.dvi 2>/dev/null ; \ ${RM} $*.aux $*.dvi $*.log $*.ps $*.idx $*.tex $*.pamphlet ; \ @@ -76,6 +87,8 @@ ${PDF}/%.pdf: ${IN}/%.pamphlet ${RM} $*.toc ; \ ${LATEX} $*.pamphlet >${TMP}/trace ; \ ${MAKEINDEX} $*.idx 1>/dev/null 2>/dev/null ; \ + ${BIBTEX} $*.aux 1>/dev/null 2>/dev/null ; \ + ${LATEX} $*.pamphlet >${TMP}/trace ; \ ${LATEX} $*.pamphlet >${TMP}/trace ; \ ${DVIPDFM} $*.dvi 2>${TMP}/trace ; \ ${RM} $*.aux $*.dvi $*.log $*.ps $*.idx $*.tex $*.pamphlet ; \ diff --git a/books/bookvol0.pamphlet b/books/bookvol0.pamphlet index 874493f..bcb7c0a 100644 --- a/books/bookvol0.pamphlet +++ b/books/bookvol0.pamphlet @@ -9790,7 +9790,7 @@ running on an IBM workstation, for example, issue Axiom can produce \TeX{} output for your \index{output formats!TeX @{\TeX{}}} expressions. \index{TeX output format @{\TeX{}} output format} The output is produced using macros from the \LaTeX{} document -preparation system by Leslie Lamport\cite{1}. The printed version +preparation system by Leslie Lamport\cite{Lamp86}. The printed version of this book was produced using this formatter. To turn on \TeX{} output formatting, issue this. @@ -88398,22 +88398,14 @@ SUCH DAMAGE. \end{verbatim} \eject -\eject -\begin{thebibliography}{99} -\bibitem{1} Lamport, Leslie, -{\it LaTeX: A Document Preparation System,} \\ -Reading, Massachusetts, -Addison-Wesley Publishing Company, Inc., -1986. ISBN 0-201-15790-X -\bibitem{2} Knuth, Donald, {\it The \TeX{}book} \\ -Reading, Massachusetts, -Addison-Wesley Publishing Company, Inc., -1984. ISBN 0-201-13448-9 -\bibitem{3} Jenks, Richard D. and Sutor, Robert S.,\\ -{\it Axiom, The Scientific Computation System} \\ -Springer-Verlag, New York, NY 1992 ISBN 0-387-97855-0 -\bibitem{4} Daly, Timothy, ``The Axiom Literate Documentation''\\ -{\bf http://axiom.axiom-developer.org/axiom-website/documentation.html} -\end{thebibliography} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\chapter{Bibliography} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\bibliographystyle{plain} +\bibliography{axiom} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\chapter{Index} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \printindex \end{document} + diff --git a/books/bookvol1.pamphlet b/books/bookvol1.pamphlet index ca534f1..ddc3fbb 100644 --- a/books/bookvol1.pamphlet +++ b/books/bookvol1.pamphlet @@ -534,7 +534,8 @@ source code for the interpreter, compiler, graphics, browser, and numerics is shipped with the system. There are several websites that host Axiom source code. -Axiom is written using Literate Programming\cite{2} so each file is actually +Axiom is written using Literate Programming\cite{Knut92} +so each file is actually a document rather than just machine source code. The goal is to make the whole system completely literate so people can actually read the system and understand it. This is the first volume in a series of books @@ -924,8 +925,7 @@ interactive ``undo.'' \label{sec:Starting Up and Winding Down} You need to know how to start the Axiom system and how to stop it. We assume that Axiom has been correctly installed on your -machine. Information on how to install Axiom is available on -the wiki website\cite{3}. +machine. To begin using Axiom, issue the command {\bf axiom} to the operating system shell. @@ -6839,7 +6839,7 @@ plotting functions of one or more variables and plotting parametric surfaces. Once the graphics figure appears in a window, move your mouse to the window and click. A control panel appears immediately and allows you to interactively transform the object. Refer to the -original Axiom book\cite{1} and the input files included with Axiom +original Axiom book\cite{Jenk92} and the input files included with Axiom for additional examples. This is an example of Axiom's graphics. From the Control Panel you can @@ -9497,7 +9497,7 @@ domains and their functions and how to write your own functions. \index{Aldor!Spad} \index{Spad} \index{Spad!Aldor} -There is a second language, called {\bf Aldor}\cite{4} that is +There is a second language, called {\bf Aldor}\cite{Watt03} that is compatible with the {\bf Spad} language. They both can create programs than can execute under Axiom. Aldor is a standalone version of the {\bf Spad} language and contains some additional @@ -11867,7 +11867,7 @@ running on an IBM workstation, for example, issue Axiom can produce \TeX{} output for your \index{output formats!TeX @{\TeX{}}} expressions. \index{TeX output format @{\TeX{}} output format} The output is produced using macros from the \LaTeX{} document -preparation system by Leslie Lamport\cite{5}. The printed version +preparation system by Leslie Lamport\cite{Lamp86}. The printed version of this book was produced using this formatter. To turn on \TeX{} output formatting, issue this. @@ -14300,31 +14300,17 @@ The command synonym {\tt )apropos} is equivalent to {\tt )show} \index{)show}. \section{Makefile} -This book is actually a literate program\cite{2} and can contain -executable source code. In particular, the Makefile for this book -is part of the source of the book and is included below. Axiom -uses the ``noweb'' literate programming system by Norman Ramsey\cite{6}. +This book is actually a literate program\cite{Knut92} and can contain +executable source code. \eject -\begin{thebibliography}{99} -\bibitem{1} Jenks, R.J. and Sutor, R.S. -``Axiom -- The Scientific Computation System'' -Springer-Verlag New York (1992) -ISBN 0-387-97855-0 -\bibitem{2} Knuth, Donald E., ``Literate Programming'' -Center for the Study of Language and Information -ISBN 0-937073-81-4 -Stanford CA (1992) -\bibitem{3} Daly, Timothy, ``The Axiom Wiki Website''\\ -{\bf http://axiom.axiom-developer.org} -\bibitem{4} Watt, Stephen, ``Aldor'',\\ -{\bf http://www.aldor.org} -\bibitem{5} Lamport, Leslie, ``Latex -- A Document Preparation System'', -Addison-Wesley, New York ISBN 0-201-52983-1 -\bibitem{6} Ramsey, Norman ``Noweb -- A Simple, Extensible Tool for -Literate Programming''\\ -{\bf http://www.eecs.harvard.edu/ $\tilde{}$nr/noweb} -\bibitem{7} Daly, Timothy, "The Axiom Literate Documentation"\\ -{\bf http://axiom.axiom-developer.org/axiom-website/documentation.html} -\end{thebibliography} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\chapter{Bibliography} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\bibliographystyle{plain} +\bibliography{axiom} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\chapter{Index} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \printindex \end{document} + diff --git a/books/bookvol10.1.pamphlet b/books/bookvol10.1.pamphlet index 05c94b7..f8fee1e 100644 --- a/books/bookvol10.1.pamphlet +++ b/books/bookvol10.1.pamphlet @@ -4,7 +4,7 @@ \mainmatter \setcounter{chapter}{0} % Chapter 1 \chapter{Interval Arithmetic} -Lambov \cite{Lambov06} defines a set of useful formulas for +Lambov \cite{Lamb06} defines a set of useful formulas for computing intervals using the IEEE-754 floating-point standard. The first thing to note is that IEEE floating point defaults to @@ -256,9 +256,9 @@ an exception. If it contains a negative part, the implementation will crop it to only its non-negative part to allow that computations such as $\sqrt{0}$ ca be carried out in exact real arithmetic. -\chapter{Integration \cite{Bro98b}} +\chapter{Integration} -An {\sl elementary function} +An {\sl elementary function}\cite{Bro98b} \index{elementary function} of a variable $x$ is a function that can be obtained from the rational functions in $x$ by repeatedly adjoining @@ -289,7 +289,7 @@ last century, the difficulties posed by algebraic functions caused Hardy (1916) to state that ``there is reason to suppose that no such method can be given''. This conjecture was eventually disproved by Risch (1970), who described an algorithm for this problem in a series -of reports \cite{Ost1845,Ris68,Ris69a,Ris69b}. +of reports \cite{Ostr1845,Risc68,Risc69a,Risc69b,Risc70}. In the past 30 years, this procedure has been repeatedly improved, extended and refined, yielding practical algorithms that are now becoming standard and are implemented in most @@ -413,7 +413,7 @@ approach is the need to factor polynomials over $\mathbb{R}$, $\mathbb{C}$, or $\overline{K}$, thereby introducing algebraic numbers even if the integrand and its integral are both in $\mathbb{Q}(x)$. On the other hand, introducing algebraic numbers may be necessary, for -example it is proven in \cite{Ris69a} that any field containing an +example it is proven in \cite{Risc69a} that any field containing an integral of $1/(x^2+2)$ must also contain $\sqrt{2}$. Modern research has yielded so-called ``rational'' algorithms that \begin{itemize} @@ -423,8 +423,8 @@ calculations being done in $K(x)$, and express the integral \end{itemize} The first rational algorithms for integration date back to the -$19^{{\rm th}}$ century, when both Hermite \cite{Her1872} and -Ostrogradsky \cite{Ost1845} invented methods for +$19^{{\rm th}}$ century, when both Hermite \cite{Herm1872} and +Ostrogradsky \cite{Ostr1845} invented methods for computing the $v$ of \ref{Int4} entirely within $K(x)$. We describe here only Hermite's method, since it is the one that has been generalized to arbitrary elementary @@ -445,7 +445,7 @@ finally that D_1=\frac{D/R}{{\rm gcd}(R,D/R)} \] Computing recursively a squarefree factorization of $R$ completes the -one for $D$. Note that \cite{Yu76} presents a more efficient method for +one for $D$. Note that \cite{Yun76} presents a more efficient method for this decomposition. Let now $f \in K(x)$ be our integrand, and write $f=P+A/D$ where $P,A,D \in K[x]$, gcd$(A,D)=1$, and\\ ${\rm deg}(A)<{\rm deg}(D)$. @@ -487,7 +487,7 @@ follows from \ref{Int2} that where the $\alpha_i$'s are the zeros of $D$ in $\overline{K}$, and the $a_i$'s are the residues of $f$ at the $\alpha_i$'s. The problem is then to compute those residues without splitting $D$. Rothstein -\cite{Ro77} and Trager \cite{Tr76} independently proved that the +\cite{Roth77} and Trager \cite{Trag76} independently proved that the $\alpha_i$'s are exactly the zeros of \begin{equation}\label{Int5} R={\rm resultant}_x(D,A-tD^{\prime}) \in K[t] @@ -502,7 +502,7 @@ where $R=\prod_{i=1}^m R_i^{e_i}$ is the irreducible factorization of $R$ over $K$. Note that this algorithm requires factoring $R$ into irreducibles over $K$, and computing greatest common divisors in $(K[t]/(R_i))[x]$, hence computing with algebraic numbers. Trager and -Lazard \& Rioboo \cite{LR90} independently discovered that those +Lazard \& Rioboo \cite{Laza90} independently discovered that those computations can be avoided, if one uses the subresultant PRS algorithm to compute the resultant of \ref{Int5}: let $(R_0,R_1,\ldots R_k\ne 0,0,\ldots)$ be the subresultant PRS with @@ -528,7 +528,7 @@ extension $K[t]/(Q_i)$. Even this step can be avoided: it is in fact sufficient to ensure that $Q_i$ and the leading coefficient with respect to $x$ of $R_{k_i}$ do not have a nontrivial common factor, which implies then that the remainder by $Q_i$ is nonzero, see -\cite{Mul97} for details and other alternatives for computing +\cite{Muld97} for details and other alternatives for computing ${\rm pp}_x(R_{k_i})(a,x)$ \section{Algebraic Functions} @@ -718,7 +718,7 @@ and $F=27x^4+108x^3+418x^2+108x+27$. The system \ref{Int10} admits a unique solution $f_1=f_2=0, f_3=-2$ and $f_4=(x+1)/x$, whose denominator is not coprime with $V$, so the Hermite reduction is not applicable. -The above problem was first solved by Trager \cite{Tr84}, who proved +The above problem was first solved by Trager \cite{Trag84}, who proved that if $w$ is an {\sl integral basis, i.e.} its elements generate ${\bf O}_{K[x]}$ over $K[x]$, then the system \ref{Int8} always has a unique solution in $K(x)$ when $m > 1$, and that solution always has a @@ -728,9 +728,9 @@ a factor of $FUV^{m-1}$ where $F \in K[x]$ is squarefree and coprime with $UV$. He also described an algorithm for computing an integral basis, a necessary preprocessing for his Hermite reduction. The main problem with that approach is that computing the integral basis, -whether by the method of \cite{Tr84} or the local alternative \cite{vH94}, +whether by the method of \cite{Trag84} or the local alternative \cite{Hoei94}, can be in general more expansive than the rest of the reduction -process. We describe here the lazy Hermite reduction \cite{REF-Bro98}, which +process. We describe here the lazy Hermite reduction \cite{Bron98}, which avoids the precomputation of an integral basis. It is based on the observation that if $m > 1$ and \ref{Int8} does not have a solution allowing us to perform the reduction, then either @@ -745,7 +745,7 @@ also made up of integral elements, so that that $K[x]$-module generated by the new basis strictly contains the one generated by $w$: \noindent -{\bf Theorem 1 (\cite{REF-Bro98})} {\sl Suppose that $m \ge 2$ and that +{\bf Theorem 1 (\cite{Bron98})} {\sl Suppose that $m \ge 2$ and that $\{S_1,\ldots,S_n\}$ as given by \ref{Int9} are linearly dependent over $K(x)$, and let $T_1,\ldots,T_n \in K[x]$ be not all 0 and such that $\sum_{i=1}^n T_iS_i=0$. Then, @@ -756,7 +756,7 @@ Furthermore, if $\gcd(T_1,\ldots,T_n)=1$ then $w_0 \notin K[x]w_1+\cdots+K[x]w_n$.} \noindent -{\bf Theorem 2 (\cite{REF-Bro98})} {\sl Suppose that $m \ge 2$ and that +{\bf Theorem 2 (\cite{Bron98})} {\sl Suppose that $m \ge 2$ and that $\{S_1,\ldots,S_n\}$ as given by \ref{Int9} are linearly independent over $K(x)$, and let $Q,T_1,\ldots,T_n \in K[x]$ be such that \[ @@ -771,7 +771,7 @@ Furthermore, if $\gcd(Q,T_1,\ldots,T_n)=1$ and $\deg(\gcd(V,Q)) \ge 1$, then $w_0 \notin K[x]w_1+\cdots+K[x]w_n$.} -{\bf Theorem 3 (\cite{REF-Bro98})} {\sl Suppose that the denominator $F$ of +{\bf Theorem 3 (\cite{Bron98})} {\sl Suppose that the denominator $F$ of some $w_i$ is not squarefree, and let $F=F_1F_2^2\cdots F_k^k$ be its squarefree factorization. Then,} \[ @@ -951,7 +951,7 @@ integration problem by allowing only new logarithms to appear linearly in the integral, all the other terms appearing in the integral being already in the integrand. -{\bf Theorem 4 (Liouville \cite{Lio1833a,Lio1833b})} {\sl +{\bf Theorem 4 (Liouville \cite{Liou1833a,Liou1833b})} {\sl Let $E$ be an algebraic extension of the rational function field $K(x)$, and $f \in E$. If $f$ has an elementary integral, then there exist $v \in E$, constants $c_1,\ldots,c_n \in \overline{K}$ and @@ -960,9 +960,10 @@ $u_1,\ldots,u_k \in E(c_1,\ldots,c_k)^{*}$ such that} f=v^{\prime}+c_1\frac{u_1^{\prime}}{u_1}+\cdots+c_k\frac{u_k^{\prime}}{u_k} \end{equation} The above is a restriction to algebraic functions of the strong -Liouville Theorem, whose proof can be found in \cite{Bro97,Ris69b}. An elegant +Liouville Theorem, whose proof can be found in \cite{Bron97,Risc69b}. +An elegant and elementary algebraic proof of a slightly weaker version can be -found in \cite{Ro72}. As a consequence, we can look for an integral of +found in \cite{Rose72}. As a consequence, we can look for an integral of the form \ref{Int4}, Liouville's Theorem guaranteeing that there is no elementary integral if we cannot find one in that form. Note that the above theorem does not say that every integral must have the above @@ -983,7 +984,7 @@ $c_1,\ldots,c_k$. Since $D$ is squarefree, it can be shown that $v \in {\bf O}_{K[x]}$ for any solution, and in fact $v$ corresponds to the polynomial part of the integral of rational functions. It is however more difficult to compute than the integral -of polynomials, so Trager \cite{Tr84} gave a change of variable that +of polynomials, so Trager \cite{Trag84} gave a change of variable that guarantees that either $v^{\prime}=0$ or $f$ has no elementary integral. In order to describe it, we need to define the analogue for algebraic functions of having a nontrivial polynomial part: we say that @@ -1007,7 +1008,7 @@ and ${\rm deg}(C) \ge {\rm deg}(B_i)$ for each $i$. We say that the differential $\alpha{}dx$ is integral at infinity if $\alpha x^{1+1/r} \in {\bf O}_\infty$ where $r$ is the smallest -ramification index at infinity. Trager \cite{Tr84} described an +ramification index at infinity. Trager \cite{Trag84} described an algorithm that converts an arbitrary integral basis $w_1,\ldots,w_n$ into one that is also normal at infinity, so the first part of his integration algorithm is as follows: @@ -1071,7 +1072,7 @@ $K(z)$, and $w$ is normal at infinity \end{itemize} A primitive element can be computed by considering linear combinations of the generators of $E$ over $K(x)$ with random coefficients in -$K(x)$, and Trager \cite{Tr84} describes an absolute factorization +$K(x)$, and Trager \cite{Trag84} describes an absolute factorization algorithm, so the above assumptions can be ensured, although those steps can be computationally very expensive, except in the case of simple radical extensions. Before describing the second part of @@ -1131,7 +1132,8 @@ elementary, with the smallest possible number of logarithms. Steps 3 to 6 requires computing in the splitting field $K_0$ of $R$ over $K$, but it can be proven that, as in the case of rational functions, $K_0$ is the minimal algebraic extension of $K$ necessary to express the -integral in the form \ref{Int4}. Trager \cite{Tr84} describes a representation +integral in the form \ref{Int4}. Trager \cite{Trag84} +describes a representation of divisors as fractional ideals and gives algorithms for the arithmetic of divisors and for testing whether a given divisor is principal. In order to determine whether there exists an integer $N$ @@ -1141,7 +1143,7 @@ extension to one over a finite field $\mathbb{F}_{p^q}$ for some known that for every divisor $\delta=\sum{n_PP}$ such that $\sum{n_P}=0$, $M\delta$ is principal for some integer $1 \le M \le (1+\sqrt{p^q})^{2g}$, where $g$ is the genus of the curve -\cite{We71}, so we compute such an $M$ by testing $M=1,2,3,\ldots$ until +\cite{Weil71}, so we compute such an $M$ by testing $M=1,2,3,\ldots$ until we find it. It can then be shown that for almost all primes $p$, if $M\delta$ is not principal in characteristic 0, the $N\delta$ is not principal for any integer $N \ne 0$. Since we can test whether the @@ -1149,7 +1151,7 @@ prime $p$ is ``good'' by testing whether the image in $\mathbb{F}_{p^q}$ of the discriminant of the discriminant of the minimal polynomial for $y$ over $K[z]$ is 0, this yields a complete algorithm. In the special case of hyperelliptic extensions, {\sl i.e.} -simple radical extensions of degree 2, Bertrand \cite{Ber95} describes a +simple radical extensions of degree 2, Bertrand \cite{Bert95} describes a simpler representation of divisors for which the arithmetic and principality tests are more efficient than the general methods. @@ -1287,7 +1289,7 @@ new constant, and an exponential could in fact be algebraic, for example $\mathbb{Q}(x)(log(x),log(2x))=\mathbb{Q}(log(2))(x)(log(x))$ and $\mathbb{Q}(x)(e^{log(x)/2})=\mathbb{Q}(x)(\sqrt{x})$. There are however algorithms that detect all such occurences and modify the -tower accordingly \cite{Ris79}, so we can assume that all the logarithms +tower accordingly \cite{Risc79}, so we can assume that all the logarithms and exponentials appearing in $E$ are monomials, and that ${\rm Const}(E)=C$. Let now $k_0$ be the largest index such that $t_{k_0}$ is transcendental over $K=C(x)(t_1,\ldots,t_{k_0-1})$ and @@ -1430,7 +1432,7 @@ $r_0,\ldots,r_{d-1} \in K$. Again, it is easy to verify that for any R=\frac{1}{{\rm deg}_t(S)}\frac{r_{d-1}}{c_d}\frac{S'}{S}+\overline{R} \] where $\overline{R} \in K[t]$ is such that $\overline{R}=0$ or -${\rm deg}_t(\overline{R}) < e-1$. Furthermore, it can be proven \cite{Bro97} +${\rm deg}_t(\overline{R}) < e-1$. Furthermore, it can be proven \cite{Bron97} that if $R+A/D$ has an elementary integral over $K(t)$, then $r_{d-1}/{c_d}$ is a constant, which implies that \[ @@ -1480,7 +1482,7 @@ g=\sum_{i=1}^k\sum_{a|Q_i(a)=0} a\log(\gcd{}_t(D,A-aD')) Note that the roots of each $Q_i$ must all be constants, and that the arguments of the logarithms can be obtained directly from the subresultant PRS of $D$ and $A-zD'$ as in the rational function -case. It can then be proven \cite{Bro97} that +case. It can then be proven \cite{Bron97} that \begin{itemize} \item $f-g'$ is always ``simpler'' than $f$ \item the splitting field of $Q_1\cdots Q_k$ over $K$ is the minimal @@ -1555,7 +1557,7 @@ $z$ be a new indeterminante and compute \begin{equation}\label{Int16} R(z)={\rm resultant_t}({\rm pp_z}({\rm resultant_y}(G-tHD',F)),D) \in K[t] \end{equation} -It can then be proven \cite{Bro90c} that if $f$ has an elementary integral +It can then be proven \cite{Bron90c} that if $f$ has an elementary integral over $E$, then $R|\kappa(R)$ in $K[z]$. {\bf Example 12} {\sl @@ -1607,7 +1609,7 @@ to $f_d$, either proving that \ref{Int18} has no solution, in which case $f$ has no elementary integral, or obtaining the constant $v_{d+1}$, and $v_d$ up to an additive constant (in fact, we apply recursively a specialized version of the integration algorithm to equations of the -form \ref{Int18}, see \cite{Bro97} for details). Write then +form \ref{Int18}, see \cite{Bron97} for details). Write then $v_d=\overline{v_d}+c_d$ where $\overline{v_d} \in K$ is known and $c_d \in {\rm Const}(K)$ is undetermined. Equating the coefficients of $t^{d-1}$ yields @@ -1654,8 +1656,8 @@ The above problem is called a {\sl Risch differential equation over K}. Although solving it seems more complicated than solving $g'=f$, it is actually simpler than an integration problem because we look for the solutions $v_i$ in $K$ only rather than in an extension of -$K$. Bronstein \cite{Bro90c,Bro91a,Bro97} and Risch -\cite{Ris68,Ris69a,Ris69b} describe algorithms for solving this type +$K$. Bronstein \cite{Bron90c,Bron91a,Bron97} and Risch +\cite{Risc68,Risc69a,Risc69b} describe algorithms for solving this type of equation when $K$ is an elementary extension of the rational function field. @@ -1708,7 +1710,7 @@ b where $at+b$ and $ct+d$ are the remainders module $t^2+1$ of $A$ and $V$ respectively. The above is a coupled differential system, which can be solved by methods similar to the ones used for Risch -differential equations \cite{Bro97}. If it has no solution, then the +differential equations \cite{Bron97}. If it has no solution, then the integral is not elementary, otherwise we reduce the integrand to $h \in K[t]$, at which point the polynomial reduction either proves that its integral is not elementary, or reduce the integrand to an @@ -1898,7 +1900,7 @@ whose solution is $v_2=2$, implying that $h=2y'$, hence that In the general case when $E$ is not a radical extension of $K(t)$, \ref{Int21} is solved by bounding ${\rm deg}_t(v_i)$ and comparing the Puiseux expansions at infinity of $\sum_{i=1}^n v_iw_i$ with those of the form -\ref{Int20} of $h$, see \cite{Bro90c,Ris68} for details. +\ref{Int20} of $h$, see \cite{Bron90c,Risc68} for details. \subsection{The algebraic exponential case} The transcendental exponential case method also generalizes to the @@ -2022,7 +2024,7 @@ $v=\sum_{i=1}^n v_iw_i/t^m$ where $v_1,\ldots,v_m \in K[t]$. We can compute $v$ by bounding ${\rm deg}_t(v_i)$ and comparing the Puiseux expansions at $t=0$ and at infinity of $\sum_{i=1}^n v_iw_i/t^m$ with those of the form \ref{Int20} of the integrand, -see \cite{Bro90c,Ris68} for details. +see \cite{Bron90c,Risc68} for details. Once we are reduced to solving \ref{Int13} for $v \in K$, constants $c_1,\ldots,c_k \in \overline{K}$ and @@ -2032,13 +2034,14 @@ places above $t=0$ and at infinity in a manner similar to the algebraic logarithmic case, at which point the algorithm proceeds by constructing the divisors $\delta_j$ and the $u_j$'s as in that case. Again, the details are quite technical and can be found in -\cite{Bro90c,Ris68,Ris69a}. +\cite{Bron90c,Risc68,Risc69a}. -\chapter{Singular Value Decomposition \cite{Pu09}} +\chapter{Singular Value Decomposition} \section{Singular Value Decomposition Tutorial} When you browse standard web sources like Wikipedia to learn about -Singular Value Decomposition or SVD you find many equations, but +Singular Value Decomposition \cite{Puff09} +or SVD you find many equations, but not an intuitive explanation of what it is or how it works. SVD is a way of factoring matrices into a series of linear approximations that expose the underlying structure of the matrix. Two important @@ -2445,7 +2448,7 @@ are the same. We are trying to predict patterns of how words occur in documents instead of trying to predict patterns of how players score on holes. \chapter{Quaternions} -from \cite{Alt05}: +from \cite{Altm05}: \begin{quotation} Quaternions are inextricably linked to rotations. Rotations, however, are an accident of three-dimensional space. @@ -2467,8 +2470,8 @@ The Theory of Quaternions is due to Sir William Rowan Hamilton, Royal Astronomer of Ireland, who presented his first paper on the subject to the Royal Irish Academy in 1843. His Lectures on Quaternions were published in 1853, and his Elements, in 1866, -shortly after his death. The Elements of Quaternions by Tait \cite{Ta1890} is -the accepted text-book for advanced students. +shortly after his death. The Elements of Quaternions by Tait \cite{Tait1890} +is the accepted text-book for advanced students. Large portions of this file are derived from a public domain version of Tait's book combined with the algebra available in Axiom. @@ -7651,13 +7654,13 @@ i = \right] $$ -\chapter{Clifford Algebra \cite{Fl09}} +\chapter{Clifford Algebra} -This is quoted from John Fletcher's web page \cite{Fl09} (with permission). +This is quoted from John Fletcher's web page \cite{Flet09} (with permission). The theory of Clifford Algebra includes a statement that each Clifford Algebra is isomorphic to a matrix representation. Several authors -discuss this and in particular Ablamowicz \cite{Ab98} gives examples of +discuss this and in particular Ablamowicz \cite{Abla98} gives examples of derivation of the matrix representation. A matrix will itself satisfy the characteristic polynomial equation obeyed by its own eigenvalues. This relationship can be used to calculate the inverse of @@ -7672,7 +7675,8 @@ Clifford(2), Clifford(3) and Clifford(2,2). Introductory texts on Clifford algebra state that for any chosen Clifford Algebra there is a matrix representation which is equivalent. Several authors discuss this in more detail and in particular, -Ablamowicz \cite{Ab98} shows that the matrices can be derived for each algebra +Ablamowicz \cite{Abla98} +shows that the matrices can be derived for each algebra from a choice of idempotent, a member of the algebra which when squared gives itself. The idea of this paper is that any matrix obeys the characteristic equation of its own eigenvalues, and that therefore @@ -7687,7 +7691,7 @@ implementation. This knowledge is not believed to be new, but the theory is distributed in the literature and the purpose of this paper is to make it clear. The examples have been first developed using a system of symbolic algebra described in another paper by this -author \cite{Fl01}. +author \cite{Flet01}. \section{Clifford Basis Matrix Theory} @@ -8129,7 +8133,7 @@ simple cases of wide usefulness. \subsection{Example 3: Clifford (2,2)} -The following basis matrices are given by Ablamowicz \cite{Ab98} +The following basis matrices are given by Ablamowicz \cite{Abla98} \[ \begin{array}{cc} @@ -8379,7 +8383,7 @@ and \[n^{-1}_2 = \frac{n^3_2- 4n^2_2 + 8n_2 - 8}{4}\] This expression can be evaluated easily using a computer algebra -system for Clifford algebra such as described in Fletcher \cite{Fl01}. +system for Clifford algebra such as described in Fletcher \cite{Flet01}. The result is \[ @@ -8423,15 +8427,16 @@ It is well known that the most difficult part in constructing AG-code is the computation of a basis of the vector space ``L(D)'' where D is a divisor of the function field of an irreducible curve. To compute such a basis, PAFF used the Brill-Noether algorithm which was generalized -to any plane curve by D. LeBrigand and J.J. Risler \cite{LR88}. In -\cite{Ha96} +to any plane curve by D. LeBrigand and J.J. Risler \cite{LeBr88}. In +\cite{Hach96} you will find more details about the algorithmic aspect of the Brill-Noether algorithm. Also, if you prefer, as I do, a strictly -algebraic approach, see \cite{Ha95}. This is the approach I used in my thesis -(\cite{Ha96}) and of course this is where you will find complete details about +algebraic approach, see \cite{Hach95}. This is the approach I used in my thesis +(\cite{Hach96}) +and of course this is where you will find complete details about the implementation of the algorithm. The algebraic approach use the theory of algebraic function field in one variable : you will find in -\cite{St93} a very good introduction to this theory and AG-codes. +\cite{Stic93} a very good introduction to this theory and AG-codes. It is important to notice that PAFF can be used for most computation related to the function field of an irreducible plane curve. For @@ -8444,7 +8449,7 @@ There is also the package PAFFFF which is especially designed to be used over finite fields. This package is essentially the same as PAFF, except that the computation are done over ``dynamic extensions'' of the ground field. For this, I used a simplify version of the notion of -dynamic algebraic closure as proposed by D. Duval \cite{Du95}. +dynamic algebraic closure as proposed by D. Duval \cite{Duva95}. Example 1 @@ -8484,7 +8489,7 @@ notation for the binomial coefficients There are $n$ factors in the numerator and $n$ in the denominator. Viewed as a function of $u$, $C(u+k,n)$ is a polynomial of degree $n$. -The figure above, Hamming \cite{Ham62} +The figure above, Hamming \cite{Hamm62} calls a lozenge diagram. A line starting at a point on the left edge and following some path across the page defines an interpolation formula if the following rules are used. @@ -8560,182 +8565,13 @@ Gaussian Elimination \chapter{Diophantine Equations} Diophantine Equations -\begin{thebibliography}{99} - -\bibitem[Ablamowicz 98]{Ab98} Ablamowicz, Rafal\\ -``Spinor Representations of Clifford Algebras: A Symbolic Approach''\\ -Computer Physics Communications -Vol. 115, No. 2-3, December 11, 1998, pages 510-535. - -\bibitem[Altmann 05]{Alt05} Altmann, Simon L.\\ -``Rotations, Quaternions, and Double Groups''\\ -Dover Publications, Inc. 2005 ISBN 0-486-44518-6 - -\bibitem[Bertrand 95]{Ber95} Bertrand, Laurent\\ -``Computing a hyperelliptic integral using arithmetic in the jacobian -of the curve''\\ -{\sl Applicable Algebra in Engineering, Communication and Computing}, -6:275-298, 1995 - -\bibitem[Bronstein 90c]{Bro90c} Bronstein, M.\\ -``On the integration of elementary functions''\\ -{\sl Journal of Symbolic Computation} 9(2):117-173, February 1990 - -\bibitem[Bronstein 91a]{Bro91a} Bronstein, M.\\ -``The Risch differential equation on an algebraic curve''\\ -in Watt [Wat91], pp241-246 ISBN 0-89791-437-6 LCCN QA76.95.I59 1991 - -\bibitem[Bronstein 97]{Bro97} Bronstein, M.\\ -``Symbolic Integration I--Transcendental Functions.''\\ -Springer, Heidelberg, 1997 ISBN 3-540-21493-3 -\verb|evil-wire.org/arrrXiv/Mathematics/Bronstein,_Symbolic_Integration_I,1997.pdf| - -\bibitem[Bronstein 98b]{Bro98b} Bronstein, Manuel\\ -``Symbolic Integration Tutorial''\\ -INRIA Sophia Antipolis ISSAC 1998 Rostock - -\bibitem[Bronstein 98]{REF-Bro98} Bronstein, M.\\ -``The lazy hermite reduction''\\ -Rapport de Recherche RR-3562, INRIA, 1998 - -\bibitem[Duval 95]{Du95} Duval, D.\\ -``Evaluation dynamique et cl\^oture alg\'ebrique en Axiom''.\\ -Journal of Pure and Applied Algebra, no99, 1995, pp. 267--295. - -\bibitem[Fletcher 01]{Fl01} Fletcher, John P.\\ -``Symbolic processing of Clifford Numbers in C++''\\ -Paper 25, AGACSE 2001. - -\bibitem[Fletcher 09]{Fl09} Fletcher, John P.\\ -``Clifford Numbers and their inverses calculated using the matrix -representation.''\\ -Chemical Engineering and -Applied Chemistry, School of Engineering and Applied Science, Aston -University, Aston Triangle, Birmingham B4 7 ET, U. K. \\ -\verb|www.ceac.aston.ac.uk/research/staff/jpf/papers/paper24/index.php| - -\bibitem[Hathway 1896]{Ha1896} Hathway, Arthur S.\\ -``A Primer Of Quaternions''\\ -(1896) - -\bibitem[Hache 95a]{Ha95} Hach\'e, G.\\ -``Computation in algebraic function fields for effective -construction of algebraic-geometric codes''\\ -Lecture Notes in Computer Science, vol. 948, 1995, pp. 262--278. - -\bibitem[Hache 96]{Ha96} Hach\'e, G.\\ -``Construction effective des codes g\'eom\'etriques''\\ -Th\'ese de doctorat de l'Universit\'e Pierre et Marie Curie (Paris 6), -Septembre 1996. - -\bibitem[Hamming 62]{Ham62} Hamming R W.\\ -``Numerical Methods for Scientists and Engineers''\\ -Dover (1973) ISBN 0-486-65241-6 - -\bibitem[Hermite 1872]{Her1872} Hermite, E.\\ -``Sur l'int\'{e}gration des fractions rationelles.''\\ -{\sl Nouvelles Annales de Math\'{e}matiques} -($2^{eme}$ s\'{e}rie), 11:145-148, 1872 - -\bibitem[van Hoeij 94]{vH94} van Hoeij, M.\\ -``An algorithm for computing an integral basis in an algebraic -function field''\\ -Journal of Symbolic Computation, 18(4) pp353-363 Oct. 1994 -CODEN JSYCEH ISSN 0747-7171 - -\bibitem[Le Brigand 88]{LR88} Le Brigand, D.; Risler, J.J.\\ -``Algorithme de Brill-Noether et codes de Goppa''\\ -Bull. Soc. Math. France, vol. 116, 1988, pp. 231--253. - -\bibitem[Lazard 90]{LR90} Lazard, Daniel; Rioboo, Renaud\\ -``Integration of rational functions: Rational computation of the -logarithmic part''\\ -{\sl Journal of Symbolic Computation}, 9:113-116:1990 - -\bibitem[Liouville 1833a]{Lio1833a} Liouville, Joseph\\ -``Premier m\'{e}moire sur la -d\'{e}termination des int\'{e}grales dont la valeur est -alg\'{e}brique''\\ -{\sl Journal de l'Ecole Polytechnique}, 14:124-148, 1833 - -\bibitem[Liouville 1833b]{Lio1833b} Liouville, Joseph\\ -``Second m\'{e}moire sur la d\'{e}termination des int\'{e}grales -dont la valeur est alg\'{e}brique''\\ -{\sl Journal de l'Ecole Polytechnique}, 14:149-193, 1833 - -\bibitem[Mulders 97]{Mul97} Mulders. Thom\\ -``A note on subresultants and a correction to the lazard/rioboo/trager -formula in rational function integration''\\ -{\sl Journal of Symbolic Computation}, 24(1):45-50, 1997 - -\bibitem[Ostrogradsky 1845]{Ost1845} Ostrogradsky. M.W.\\ -``De l'int\'{e}gration des fractions rationelles.''\\ -{\sl Bulletin de la Classe Physico-Math\'{e}matiques de -l'Acae\'{e}mie Imp\'{e}riale des Sciences de St. P\'{e}tersbourg,} -IV:145-167,286-300, 1845 - -\bibitem[Puffinware 09]{Pu09} Puffinware LLC.\\ -``Singular Value Decomposition (SVD) Tutorial''\\ -\verb|www.puffinwarellc.com/p3a.htm| - -\bibitem[Risch 68]{Ris68} Risch, Robert\\ -``On the integration of elementary functions -which are built up using algebraic operations''\\ -Research Report -SP-2801/002/00, System Development Corporation, Santa Monica, CA, USA, 1968 - -\bibitem[Risch 69a]{Ris69a} Risch, Robert\\ -``Further results on elementary functions''\\ -Research Report RC-2042, IBM Research, Yorktown Heights, NY, USA, 1969 - -\bibitem[Risch 69b]{Ris69b} Risch, Robert\\ -``The problem of integration in finite terms''\\ -{\sl Transactions of the American Mathematical Society} 139:167-189, 1969 - -\bibitem[Risch 79]{Ris79} Risch, Robert\\ -``Algebraic properties of the elementary functions of analysis''\\ -{\sl American Journal of Mathematics}, 101:743-759, 1979 - -\bibitem[Rosenlicht 72]{Ro72} Rosenlicht, Maxwell\\ -``Integration in finite terms''\\ -{\sl American Mathematical Monthly}, 79:963-972, 1972 - -\bibitem[Rothstein 77]{Ro77} Rothstein, Michael\\ -``A new algorithm for the integration of -exponential and logarithmic functions''\\ -In {\sl Proceedings of the 1977 MACSYMA Users Conference}, -pages 263-274. NASA Pub CP-2012, 1977 - -\bibitem[Stichtenoth 93]{St93} Stichtenoth, H.\\ -``Algebraic function fields and codes''\\ -Springer-Verlag, 1993, University Text. - -\bibitem[Tait 1890]{Ta1890} Tait, P.G.\\ -``An Elementary Treatise on Quaternions''\\ -C.J. Clay and Sons, Cambridge University Press Warehouse, Ave Maria Lane 1890 - -\bibitem[Trager 76]{Tr76} Trager, Barry\\ -``Algebraic factoring and rational function integration''\\ -In {Proceedings of SYMSAC'76} pages 219-226, 1976 - -\bibitem[Trager 84]{Tr84} Trager, Barry\\ -``On the integration of algebraic functions''\\ -PhD thesis, MIT, Computer Science, 1984 - -\bibitem[Lambov 06]{Lambov06} Lambov, Branimir\\ -``Interval Arithmetic Using SSE-2''\\ -in Lecture Notes in Computer Science, Springer ISBN 978-3-540-85520-0 -(2006) pp102-113 - -\bibitem[Weil 71]{We71} Weil, Andr\'{e}\\ -``Courbes alg\'{e}briques et vari\'{e}t\'{e}s Abeliennes''\\ -Hermann, Paris, 1971 - -\bibitem[Yun 76]{Yu76} Yun, D.Y.Y.\\ -``On square-free decomposition algorithms''\\ -{\sl Proceedings of SYMSAC'76} pages 26-35, 1976 - -\end{thebibliography} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\chapter{Bibliography} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\bibliographystyle{plain} +\bibliography{axiom} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \chapter{Index} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \printindex \end{document} diff --git a/books/bookvol10.3.pamphlet b/books/bookvol10.3.pamphlet index 0015f59..4cad1e5 100644 --- a/books/bookvol10.3.pamphlet +++ b/books/bookvol10.3.pamphlet @@ -18303,9 +18303,10 @@ CharacterClass: Join(SetCategory, ConvertibleTo String, \end{chunk} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{domain CLIF CliffordAlgebra\cite{7,12}} +\section{domain CLIF CliffordAlgebra} \subsection{Vector (linear) spaces} -This information is originally from Paul Leopardi's presentation on +This information is originally from Paul Leopardi's \cite{Leop03} +presentation on the {\sl Introduction to Clifford Algebras} and is included here as an outline with his permission. Further details are based on the book by Doran and Lasenby called {\sl Geometric Algebra for Physicists}. @@ -18372,7 +18373,7 @@ for $A$. {\bf Definition: Dimension} The dimension of a vector space is the number of basis elements, which is unique since all bases of a vector space have the same number of elements. -\subsection{Quadratic Forms\cite{1}} +\subsection{Quadratic Forms} For vector space $\mathbb{V}$ over field $\mathbb{F}$, characteristic $\ne 2$: \begin{list}{} @@ -18383,7 +18384,7 @@ $$b:\mathbb{V}{\rm\ x\ }\mathbb{V} \rightarrow \mathbb{F}{\rm\ ,given\ by\ }$$ $$b(x,y):=\frac{1}{2}(f(x+y)-f(x)=f(y))$$ is a symmetric bilinear form \end{list} -\subsection{Quadratic spaces, Clifford Maps\cite{1,2}} +\subsection{Quadratic spaces, Clifford Maps} \begin{list}{} \item A quadratic space is the pair($\mathbb{V}$,$f$), where $f$ is a quadratic form on $\mathbb{V}$ @@ -18392,7 +18393,7 @@ $$\rho : \mathbb{V} \rightarrow \mathbb{A}$$ where $\mathbb{A}$ is an associated algebra, and $$(\rho v)^2 = f(v),{\rm\ \ \ } \forall v \in \mathbb{V}$$ \end{list} -\subsection{Universal Clifford algebras\cite{1}} +\subsection{Universal Clifford algebras} \begin{list}{} \item The {\sl universal Clifford algebra} $Cl(f)$ for the quadratic space $(\mathbb{V},f)$ is the algebra generated by the image of the Clifford @@ -18402,7 +18403,7 @@ $\phi_{\mathbb{A}} \exists$ a homomorphism $$P_\mathbb{A}:Cl(f) \rightarrow \mathbb{A}$$ $$\rho_\mathbb{A} = P_\mathbb{A}\circ\rho_f$$ \end{list} -\subsection{Real Clifford algebras $\mathbb{R}_{p,q}$\cite{2}} +\subsection{Real Clifford algebras $\mathbb{R}_{p,q}$} \begin{list}{} \item The real quadratic space $\mathbb{R}^{p,q}$ is $\mathbb{R}^{p+q}$ with $$\phi(x):=-\sum_{k:=-q}^{-1}{x_k^2}+\sum_{k=1}^p{x_k^2}$$ @@ -18422,7 +18423,7 @@ $$\mathbb{P}(S):={\rm\ the\ }\ power\ set\ {\rm\ of\ }S$$ \item For $m \le n \in \mathbb{Z}$, define $$\zeta(m,n):=\{m,m+1,\ldots,n-1,n\}\backslash\{0\}$$ \end{list} -\subsection{Frames for Clifford algebras\cite{9,10,11}} +\subsection{Frames for Clifford algebras} \begin{list}{} \item A {\sl frame} is an ordered basis $(\gamma_{-q},\ldots,\gamma_p)$ for $\mathbb{R}^{p,q}$ which puts a quadratic form into the canonical @@ -18433,7 +18434,7 @@ $$\gamma:\zeta(-q,p) \rightarrow \mathbb{R}^{p,q}$$ $$\rho:\mathbb{R}^{p,q} \rightarrow \mathbb{R}_{p,q}$$ $$(\rho\gamma k)^2 = \phi\gamma k = {\rm\ sgn\ }k$$ \end{list} -\subsection{Real frame groups\cite{5,6}} +\subsection{Real frame groups} \begin{list}{} \item For $p,q \in \mathbb{N}$, define the real {\sl frame group} $\mathbb{G}_{p,q}$ via the map @@ -18449,7 +18450,7 @@ $$(g_k)^2 = \right.$$ $$g_kg_m = \mu g_mg_k{\rm\ \ \ }\forall k \ne m\rangle$$ \end{list} -\subsection{Canonical products\cite{1,3,4}} +\subsection{Canonical products} \begin{list}{} \item The real frame group $\mathbb{G}_{p,q}$ has order $2^{p+q+1}$ \item Each member $w$ can be expressed as the canonically ordered product @@ -18457,7 +18458,7 @@ $$w=\mu^a\prod_{k \in T}{g_k}$$ $$\ =\mu^a\prod_{k=-q,k\ne0}^p{g_k^{b_k}}$$ where $T \subseteq \zeta(-q,p),a,b_k \in \{0,1\}$ \end{list} -\subsection{Clifford algebra of frame group\cite{1,4,5,6}} +\subsection{Clifford algebra of frame group} \begin{list}{} \item For $p,q \in \mathbb{N}$ embed $\mathbb{G}_{p,q}$ into $\mathbb{R}_{p,q}$ via the map @@ -18471,7 +18472,7 @@ $$e:\mathbb{P}\zeta(-q,p) \rightarrow \mathbb{R}_{p,q}, \item Each $a \in \mathbb{R}_{p,q}$ can be expressed as $$a = \sum_{T \subseteq \zeta(-q,p)}{a_T e_T}$$ \end{list} -\subsection{Neutral matrix representations\cite{1,2,8}} +\subsection{Neutral matrix representations} The {\sl representation map} $P_m$ and {\sl representation matrix} $R_m$ make the following diagram commute: \begin{tabular}{ccc} @@ -28603,7 +28604,7 @@ from /home/greg/Axiom/DFLOAT.nrlib/code So it is clear that he has added a new function called {\tt doubleFloatFormat} which takes a string argument that specifies the common lisp format control string (\"{}\~{},4,,F\"{}). -For reference we quote from the common lisp manual \cite{1}. +For reference we quote from the common lisp manual. On page 582 we find: \begin{quote} @@ -156871,6 +156872,11 @@ Note that this code is not included in the generated catdef.spad file. \getchunk{domain XRPOLY XRecursivePolynomial} \end{chunk} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\chapter{Bibliography} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\bibliographystyle{plain} +\bibliography{axiom} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \chapter{Index} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \printindex diff --git a/books/bookvol10.4.pamphlet b/books/bookvol10.4.pamphlet index 65a295f..0847029 100644 --- a/books/bookvol10.4.pamphlet +++ b/books/bookvol10.4.pamphlet @@ -14767,7 +14767,7 @@ DistinctDegreeFactorize(F,FP): C == T The special functions in this section are developed as special cases but can all be expressed in terms of generalized hypergeomentric functions ${}_pF_q$ or its generalization, the Meijer G function. -\cite{Luk169,Luk269} +\cite{Luke69a,Luke69b} The long term plan is to reimplement these functions using the generalized version. \begin{chunk}{DoubleFloatSpecialFunctions.input} @@ -15716,7 +15716,7 @@ DoubleFloatSpecialFunctions(): Exports == Impl where \end{chunk} \subsection{The Exponential Integral} \subsubsection{The E1 function} -(Quoted from Segletes\cite{2}): +(Quoted from Segletes\cite{Segl98}): A number of useful integrals exist for which no exact solutions have been found. In other cases, an exact solution, if found, may be @@ -15775,7 +15775,7 @@ exponential integral family may be analytically related. However, this technique only allows for the transformation of one integral into another. There remains the problem of evaluating $E_1(x)$. There is an exact solution to the integral of $(e^{-t}/t)$, appearing in a number -of mathematical references \cite{4,5} which is obtainable by +of mathematical references which is obtainable by expanding the exponential into a power series and integrating term by term. That exact solution, which is convergent, may be used to specify $E_1(x)$ as @@ -15789,7 +15789,7 @@ E_1(x)=-\gamma-ln(x) Euler's constant, $\gamma$, equal to $0.57721\ldots$, arises when the power series expansion for $(e^{-t}/t)$ is integrated and evaluated at -its upper limit, as $x\rightarrow\infty$\cite{6}. +its upper limit, as $x\rightarrow\infty$. Employing eqn (5), however, to evaluate $E_1(x)$ is problematic for finite $x$ significantly larger than unity. One may well ask of the @@ -15857,8 +15857,8 @@ fit. While some steps are taken to make the fits intelligent ({\sl e.g.}, transformation of variables), the fits are all piecewise over the domain of the integral. -Cody and Thatcher \cite{7} performed what is perhaps the definitive -work, with the use of Chebyshev\cite{18,19} approximations to the exponential +Cody and Thatcher performed what is perhaps the definitive +work, with the use of Chebyshev approximations to the exponential integral $E_1$. Like others, they fit the integral over a piecewise series of subdomains (three in their case) and provide the fitting parameters necessary to evaluate the function to various required @@ -16041,7 +16041,7 @@ $$E_{n+1}(z)=\frac{1}{n}\left(e^{-z}-zE_n(z)\right)\ \ \ (n=1,2,3,\ldots)$$ The base case of the recursion depends on E1 above. -The formula is 5.1.14 in Abramowitz and Stegun, 1965, p229\cite{4}. +The formula is 5.1.14 in Abramowitz and Stegun, 1965, p229 \begin{chunk}{package DFSFUN DoubleFloatSpecialFunctions} En(n:PI,x:R):OPR == n=1 => E1(x) @@ -16051,12 +16051,12 @@ The formula is 5.1.14 in Abramowitz and Stegun, 1965, p229\cite{4}. \end{chunk} \subsection{The Ei Function} -This function is based on Kin L. Lee's work\cite{8}. See also \cite{21}. +This function is based on Kin L. Lee's work. \subsubsection{Abstract} The exponential integral Ei(x) is evaluated via Chebyshev series expansion of its associated functions to achieve high relative accuracy throughout the entire real line. The Chebyshev coefficients -for these functions are given to 30 significant digits. Clenshaw's\cite{20} +for these functions are given to 30 significant digits. Clenshaw's method is modified to furnish an efficient procedure for the accurate solution of linear systems having near-triangular coefficient matrices. @@ -16067,13 +16067,13 @@ Ei(x)=\int_{-\infty}^{X}{\frac{e^u}{u}}\ du=-E_1(-x), x \ne 0 \end{equation} is usually based on the value of its associated functions, for example, $xe^{-x}Ei(x)$. High accuracy tabulations of integral (1) by -means of Taylor series techniques are given by Harris \cite{9} and -Miller and Hurst \cite{10}. The evaluation of $Ei(x)$ for +means of Taylor series techniques are given by Harris and +Miller and Hurst. The evaluation of $Ei(x)$ for $-4 \le x \le \infty$ by means of Chebyshev series is provided by -Clenshaw \cite{11} to have the absolute accuracy of 20 decimal +Clenshaw to have the absolute accuracy of 20 decimal places. The evaluation of the same integral (1) by rational approximation of its associated functions is furnished by Cody and -Thacher \cite{12,13} for $-\infty < x < \infty$, and has the relative +Thacher for $-\infty < x < \infty$, and has the relative accuracy of 17 significant figures. The approximation of Cody and Thacher from the point of view of @@ -16089,7 +16089,7 @@ functions that are accurate to 30 significant figures by a modification of Clenshaw's procedure. To verify the accuracy of the several Chebyshev series, values of the associated functions were checked against those computed by Taylor series and those of Murnaghan -and Wrench \cite{14} (see Remarks on Convergence and Accuracy). +and Wrench (see Remarks on Convergence and Accuracy). Although for most purposes fewer than 30 figures of accuracy are required, such high accuracy is desirable for the following @@ -16106,7 +16106,7 @@ approximated. To take account of the errors commited by these routines, the function values must have an accuracy higher than the approximation to be determined. Consequently, high-precision results are useful as a master function for finding approximations for (or -involving) $Ei(x)$ (e.g. \cite{12,13}) where prescribed accuracy is +involving) $Ei(x)$ where prescribed accuracy is less than 30 figures. \subsubsection{Discussion} @@ -16199,10 +16199,10 @@ coefficients $A_k^{(0)}$ and $A_k^{(1)}$ can be obtained analytically (if possible) or by numerical quadrature. However, since each function in table 1 satisfies a linear differential equation with polynomial coefficients, the Chebyshev coefficients can be more readily evaluated -by the method of Clenshaw \cite{16}. +by the method of Clenshaw. -There are several variations of Clenshaw's procedure (see, -e.g. \cite{17}), but for high-precision computation, where multiple +There are several variations of Clenshaw's procedure, +but for high-precision computation, where multiple precision arithmetic is employed, we find his original procedure easiest to implement. However, straightforward application of it may result in a loss of accuracy if the trial solutions selected are not @@ -16248,7 +16248,7 @@ p(pt+q) \end{equation} It can be demonstrated that if $B_k$ are the Chebyshev coefficients of a function $\Psi(t)$, then $C_k$, the Chebyshev coefficients of -$t^r\Psi(t)$ for positive integers r, are given by \cite{16} +$t^r\Psi(t)$ for positive integers r, are given by \begin{equation} C_k=2^{-r}\sum_{i=0}^r\binom{r}{i}B_{\vert k-r+2i\vert} \end{equation} @@ -16277,7 +16277,7 @@ p^2A_{k+1}^{(0)}\\ \end{array} \right\} \end{equation} -The relation \cite{16} +The relation \begin{equation} 2kA_k^{(0)}=A_{k-1}^{(1)}-A_{k+1}^{(1)} \end{equation} @@ -16304,7 +16304,7 @@ p^2A_{k-1}+2p(2k+q-2)A_k+8q(k+1)A_{k+1}+2p(2k-q+6)A_{k+2}-p^2A_{k+3}\\ \right\} \end{equation} The superscript of $A_k^{(0)}$ is dropped for simplicity. In order to -solve the infinite system 20, Clenshaw \cite{11} essentially +solve the infinite system 20, Clenshaw essentially considered the required solution as the limiting solution of the sequence of truncated systems consisting of the first $M+1$ equations of the same system, that is, the solution of the system @@ -16380,7 +16380,7 @@ $S(\alpha)$ are equal, respectively, to the left members of equations designation holds for $R(\beta)$ and $S(\beta)$.) The quantities $\alpha_k$ and $\beta_k$ are known as trial solutions -in reference \cite{12}. Clenshaw has pointed out that if $\alpha_k$ +in reference. Clenshaw has pointed out that if $\alpha_k$ and $\beta_k$ are not sufficiently independent, loss of significance will occur in the formation of the linear combination 24, with consequent loss of accuracy. Clenshaw suggested the Gauss-Seidel @@ -16728,7 +16728,7 @@ evaluation also checks with that of the function values of table 4 (computed with 30-digit floating-point arithmetic using the coefficients of table 3) for at least 28-1/2 significant digits. Evaluation of Ei(x) using the coefficients of table 3 also -checked with Murnaghan and Wrench \cite{14} for 28-1/2 significant +checked with Murnaghan and Wrench for 28-1/2 significant figures. {\vbox{\vskip 1cm}} @@ -17555,7 +17555,7 @@ $\infty$ & -1.000 & 0.100000000 0000000000 00000000001 E 01\\ 32 & 1.000 & 0.103341356 4216241049 43493552567 E 01\\ \end{tabular} -\subsection{The Fresnel Integral\cite{PEA56,LOS60}} +\subsection{The Fresnel Integral\cite{Pear56,Losc60}} The Fresnel function is \[C(x) - iS(x) = \int_0^x{i^{-t^2}}~dt = \int_0^x{\exp(-i\pi{}t^2/2)}~dt\] @@ -17587,7 +17587,7 @@ $|\rm{arc\ }z| \le \pi-\epsilon$, ($\epsilon > 0$), for $|z| \gg 1$ is given by \left(1-\frac{1\cdot{}3}{(2z)^2}+\frac{1\cdot{}3\cdot{}5\cdot{}7}{(2z)^4}- \cdots\right)-\frac{\cos z}{\sqrt{2\pi{}z}}\left(\frac{1}{(2z)}- \frac{1\cdot{}3\cdot{}5}{(2z)^3}+\cdots\right)\] -(Note: Pearcey has a sign error for the second term (\cite{PEA56},p7) +(Note: Pearcey has a sign error for the second term (\cite{Pear56},p7) The first approximation is \[C(z) \approx \frac{1}{2} + \frac{\sin z}{\sqrt{2\pi{}z}}\] @@ -52980,7 +52980,7 @@ IntegerFactorizationPackage(I): Exports == Implementation where \end{chunk} \subsection{PollardSmallFactor} -This is Brent's\cite{1} optimization of Pollard's\cite{2} rho factoring. +This is Brent's optimization of Pollard's rho factoring. Brent's algorithm is about 24 percent faster than Pollard's. Pollard;s algorithm has complexity $O(p^{1/2})$ where $p$ is the smallest prime factor of the composite number $N$. @@ -140129,7 +140129,7 @@ PolynomialGcdPackage(E,OV,R,P):C == T where if degree gcd(uf,differentiate uf)=0 then return [uf,ltry] \end{chunk} -In Gathen \cite{GG99} we find a discussion of applying the Euclidean +In Gathen \cite{Gath99} we find a discussion of applying the Euclidean algorithm to elements of a field. In a field every nonzero rational number is a unit. If we want to define a single element such that \[gcd(f,g) \in {\bf Q}[x]\] we choose a monic polynomial, that is, the @@ -175476,6 +175476,11 @@ ZeroDimensionalSolvePackage(R,ls,ls2): Exports == Implementation where \getchunk{package ZDSOLVE ZeroDimensionalSolvePackage} \end{chunk} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\chapter{Bibliography} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\bibliographystyle{plain} +\bibliography{axiom} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \chapter{Index} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \printindex diff --git a/books/bookvol10.5.pamphlet b/books/bookvol10.5.pamphlet index 09485ba..5a36774 100644 --- a/books/bookvol10.5.pamphlet +++ b/books/bookvol10.5.pamphlet @@ -4,10 +4,10 @@ \mainmatter \setcounter{secnumdepth}{0} % override the one in bookheader.tex \setcounter{chapter}{0} % Chapter 1 -\chapter{Numerical Analysis \cite{4}} +\chapter{Numerical Analysis} We can describe each number as $x^{*}$ which has a machine-representable form which differs from the number $x$ it is intended to represent. -Quoting Householder we get: +Quoting Householder \cite{Hous81} we get: \[x^{*}=\pm(x_1\beta^{-1} + x_2\beta^{-2}+\cdots+x_\lambda\beta^\lambda) \beta^\sigma\] where $\beta$ is the base, usually 2 or 10, $\lambda$ is a positive @@ -128,7 +128,7 @@ For real matrices, TRANSx=T and TRANSx=C have the same meaning. For Hermitian matrices, TRANSx=T is not allowed. For complex symmetric matrices, TRANSx=H is not allowed. -There were 38 BLAS Level 1 routines defined in \cite{REF-LAW79}. They are +There were 38 BLAS Level 1 routines defined in \cite{Laws79}. They are \begin{itemize} \item Dot product SDSDOT, DSDOT, DQ-IDOT DQ-ADOT C-UDOT C-CDOT DDOT SDOT \item Constant times a vector plus a vector CAXPY DAXPY SAXPY diff --git a/books/bookvol10.pamphlet b/books/bookvol10.pamphlet index c8f1684..6f77d64 100644 --- a/books/bookvol10.pamphlet +++ b/books/bookvol10.pamphlet @@ -19266,7 +19266,7 @@ clean: \chapter{Implementation} -\section{Elementary Functions\cite{4}} +\section{Elementary Functions} \subsection{Rationale for Branch Cuts and Identities} Perhaps one of the most vexing problems to be addressed when @@ -19279,7 +19279,7 @@ issue facing the mathematical library developer is the plethora of possibilities, and while some choices are demonstrably inferior, there is rarely a choice which is clearly best. -Following Kahan [1], we will refer to the mathematical formula we use +Following Kahan\cite{Kaha86}, we will refer to the mathematical formula we use to define the principal branch of each such function as its principal expression. For the inverse trigonometric and inverse hyperbolic functions, this principal expression is given in terms of the @@ -19469,18 +19469,13 @@ $\begin{array}{l} \end{tabular} \eject -\begin{thebibliography}{99} -\bibitem{1} Kahan, W., “Branch cuts for complex elementary functions, or, -Much ado about nothing's sign bit”, Proceedings of the joint IMA/SIAM -conference on The State of the Art in Numerical Analysis, University of -Birmingham, A. Iserles and M.J.D. Powell, eds, Clarendon Press, -Oxford,1987, 165-210. -\bibitem{2} IEEE standard 754-1985 for binary floating-point arithmetic, -reprinted in ACM SIGPLAN Notices 22 \#2 (1987), 9-25. -\bibitem{3} IEEE standard 754-2008 -\bibitem{4} Numerical Mathematics Consortium -Technical Specification 1.0 (Draft) -\verb|http://www.nmconstorium.org| -\end{thebibliography} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\chapter{Bibliography} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\bibliographystyle{plain} +\bibliography{axiom} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\chapter{Index} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \printindex \end{document} diff --git a/books/bookvol11.pamphlet b/books/bookvol11.pamphlet index 01dbce1..5cfa64e 100644 --- a/books/bookvol11.pamphlet +++ b/books/bookvol11.pamphlet @@ -1,6 +1,9 @@ \documentclass[dvipdfm]{book} \newcommand{\VolumeName}{Volume 11: Axiom Browser} \input{bookheader.tex} +\mainmatter +\setcounter{chapter}{0} % Chapter 1 +\setcounter{secnumdepth}{0} % override the one in bookheader.tex \chapter{Overview} This book contains the Firefox browser AJAX routines. @@ -840,8 +843,6 @@ result sent from the server. This is the standard CSS style section that gets included with every page. We do this here but it could be a separate style sheet. It hardly matters either way as the style sheet is trivial. -\begin{verbatim} -\end{verbatim} \begin{chunk}{style}