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authordos-reis <gdr@axiomatics.org>2011-09-15 18:48:07 +0000
committerdos-reis <gdr@axiomatics.org>2011-09-15 18:48:07 +0000
commit12c856f9901ef3d6d82fb99855ecdf3e0b91484b (patch)
tree96fc5a4bf51d9777f8d46ac1e4023c0d8e545f8a /src/algebra/ffrac.as.pamphlet
parent9025c74466dcd1c38dde5e4c63934ff0b2e4f18d (diff)
downloadopen-axiom-12c856f9901ef3d6d82fb99855ecdf3e0b91484b.tar.gz
* algebra/axtimer.as.pamphlet: Remove.
* algebra/ffrac.as.pamphlet: Likewise. * algebra/herm.as.pamphlet: Likewise. * algebra/interval.as.pamphlet: Likewise. * algebra/invnode.as.pamphlet: Likewise. * algebra/invrender.as.pamphlet: Likewise. * algebra/invtypes.as.pamphlet: Likewise. * algebra/invutils.as.pamphlet: Likewise. * algebra/iviews.as.pamphlet: Likewise. * algebra/ndftip.as.pamphlet: Likewise. * algebra/nepip.as.pamphlet: Likewise. * algebra/noptip.as.pamphlet: Likewise. * algebra/nqip.as.pamphlet: Likewise. * algebra/nrc.as.pamphlet: Likewise. * algebra/nsfip.as.pamphlet: Likewise.
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-\documentclass{article}
-\usepackage{open-axiom}
-\begin{document}
-\title{\$SPAD/src/algebra ffrac.as}
-\author{Michael Richardson}
-\maketitle
-\begin{abstract}
-\end{abstract}
-\eject
-\tableofcontents
-\eject
-\begin{verbatim}
-
--- FormalFraction
-
--- N.B. ndftip.as inlines this, must be recompiled if this is.
-
--- To test:
--- sed '1,/^#if NeverAssertThis/d;/#endif/d' < ffrac.as > ffrac.input
--- axiom
--- )set nag host <some machine running nagd>
--- )r ffrac.input
-
-\end{verbatim}
-\section{FormalFraction}
-<<FormalFraction>>=
-
-#include "axiom.as"
-
-FFRAC ==> FormalFraction ;
-
-OF ==> OutputForm ;
-SC ==> SetCategory ;
-FRAC ==> Fraction ;
-ID ==> IntegralDomain ;
-
-+++ Author: M.G. Richardson
-+++ Date Created: 1996 Jan. 23
-+++ Date Last Updated:
-+++ Basic Functions:
-+++ Related Constructors: Fraction
-+++ Also See:
-+++ AMS Classifications:
-+++ Keywords:
-+++ References:
-+++ Description:
-+++ This type represents formal fractions - that is, pairs displayed as
-+++ fractions with no simplification.
-+++
-+++ If the elements of the pair have a type X which is an integral
-+++ domain, a FFRAC X can be coerced to a FRAC X, provided that this
-+++ is a valid type. A FRAC X can always be coerced to a FFRAC X.
-+++ If the type of the elements is a Field, a FFRAC X can be coerced
-+++ to X.
-+++
-+++ Formal fractions are used to return results from numerical methods
-+++ which determine numerator and denominator separately, to enable
-+++ users to inspect these components and recognise, for example,
-+++ ratios of very small numbers as potentially indeterminate.
-
-FormalFraction(X : SC) : SC with {
-
--- Could generalise further to allow numerator and denominator to be of
--- different types, X and Y, both SCs. "Left as an exercise."
-
- / : (X,X) -> % ;
-++ / forms the formal quotient of two items.
-
- numer : % -> X ;
-++ numer returns the numerator of a FormalFraction.
-
- denom : % -> X ;
-++ denom returns the denominator of a FormalFraction.
-
- if X has ID then {
-
- coerce : % -> FRAC(X pretend ID) ;
-++ coerce x converts a FormalFraction over an IntegralDomain to a
-++ Fraction over that IntegralDomain.
-
- coerce : FRAC(X pretend ID) -> % ;
-++ coerce converts a Fraction to a FormalFraction.
-
- }
-
- if X has Field then coerce : % -> (X pretend Field) ;
-
-} == add {
-
- import from Record(num : X, den : X) ;
-
- Rep == Record(num : X, den : X) ; -- representation
-
- ((x : %) = (y : %)) : Boolean ==
- ((rep(x).num = rep(y).num) and (rep(x).den = rep(y).den)) ;
-
- ((n : X)/(d : X)) : % == per(record(n,d)) ;
-
- coerce(r : %) : OF == (rep(r).num :: OF) / (rep(r).den :: OF) ;
-
- numer(r : %) : X == rep(r).num ;
-
- denom(r : %) : X == rep(r).den ;
-
- if X has ID then {
-
- coerce(r : %) : FRAC(X pretend ID)
- == ((rep(r).num)/(rep(r).den)) @ (FRAC(X pretend ID)) ;
-
- coerce(x : FRAC(X pretend ID)) : % == x pretend % ;
-
- }
-
- if X has Field then coerce(r : %) : (X pretend Field)
- == ((rep(r).num)/(rep(r).den)) $ (X pretend Field) ;
-
-}
-
-#if NeverAssertThis
-
-)lib ffrac
-
-f1 : FormalFraction Integer
-f1 := 6/3
-
--- 6
--- -
--- 3
-
-f2 := (3.6/2.4)$FormalFraction Float
-
--- 3.6
--- ---
--- 2.4
-
-numer f1
-
--- 6
-
-denom f2
-
--- 2.4
-
-f1 :: FRAC INT
-
--- 2
-
-% :: FormalFraction Integer
-
--- 2
--- -
--- 1
-
-f2 :: Float
-
--- 1.5
-
-output "End of tests"
-
-#endif
-@
-\section{License}
-<<license>>=
---Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
---All rights reserved.
---
---Redistribution and use in source and binary forms, with or without
---modification, are permitted provided that the following conditions are
---met:
---
--- - Redistributions of source code must retain the above copyright
--- notice, this list of conditions and the following disclaimer.
---
--- - Redistributions in binary form must reproduce the above copyright
--- notice, this list of conditions and the following disclaimer in
--- the documentation and/or other materials provided with the
--- distribution.
---
--- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--- names of its contributors may be used to endorse or promote products
--- derived from this software without specific prior written permission.
---
---THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
---IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
---TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
---PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
---OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
---EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
---PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
---PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
---LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
---NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
---SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
-@
-<<*>>=
-<<license>>
-
-<<FormalFraction>>
-@
-\eject
-\begin{thebibliography}{99}
-\bibitem{1} nothing
-\end{thebibliography}
-\end{document}