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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/herm.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 herm.as}
-\author{Michael Richardson}
-\maketitle
-\begin{abstract}
-\end{abstract}
-\eject
-\tableofcontents
-\eject
-\begin{verbatim}
--- N.B. ndftip.as inlines this, must be recompiled if this is.
-
--- To test:
--- sed -ne '1,/^#if NeverAssertThis/d;/#endif/d;p' < herm.as > herm.input
--- axiom
--- )set nag host <some machine running nagd>
--- )r herm.input
-\end{verbatim}
-\section{PackedHermitianSequence}
-<<PackedHermitianSequence>>=
-#include "axiom.as"
-
-INT ==> Integer ;
-NNI ==> NonNegativeInteger ;
-PHS ==> PackedHermitianSequence ;
-
-+++ Author: M.G. Richardson
-+++ Date Created: 1995 Nov. 24
-+++ Date Last Updated:
-+++ Basic Functions:
-+++ Related Constructors: Vector
-+++ Also See:
-+++ AMS Classifications:
-+++ Keywords:
-+++ References:
-+++ Description:
-+++ This type represents packed Hermitian sequences - that is, complex
-+++ sequences s, whose tails ("rest s", in Axiom terms) are conjugate to
-+++ themselves reversed - by real sequences, in the "standard" manner;
-+++ in this, the real parts of the elements of the first "half" of the
-+++ tail are stored there and the imaginary parts are stored in reverse
-+++ order in the second "half" of the tail.
-+++ (If the tail has an odd number of elements, its middle element is
-+++ real and is stored unchanged. The "halves" mentioned above then
-+++ refer to the elements before and after the middle, respectively.)
-
-PackedHermitianSequence(R : CommutativeRing) : LinearAggregate(R) with{
-
- pHS : List R -> % ;
-++ pHS(l) converts the list l to a packedHermitianSequence.
-
- expand : % -> Vector Complex R ;
-++ expand(h) converts the packedHermitianSequence h to a Hermitian
-++ sequence (a complex vector).
-
- packHS : Vector Complex R -> % ;
-++ packHS(v) checks that the complex vector v represents a Hermitian
-++ sequences and, if so, converts it to a packedHermitianSequence;
-++ otherwise an error message is printed.
-
- conjHerm : % -> % ;
-++ conjHerm(h) returns the packedHermitianSequence which represents the
-++ Hermitian sequence conjugate to that represented by h.
-
- coerce : % -> OutputForm -- shouldn't need this, should be inherited
- -- from Vector.
-
-} == Vector(R) add {
- Rep ==> Vector R;
- import from Rep;
- import from INT ;
- import from R ;
- import from Vector R ;
- import from Complex R ;
- import from Vector Complex R ;
- import from ErrorFunctions ;
- import from String ;
- import from List String ;
-
- local (..)(a:INT,b:INT):Generator INT == {
- generate {
- t := a ;
- while (t <= b) repeat {
- yield t ;
- t := t + 1 ;
- }
- }
- }
-
- pHS(l : List R) : % == (vector l) pretend PHS R ;
-
- expand(h : %) : Vector Complex R == {
-
- local len : NNI ;
- local nvals, npairs, n1, n2 : INT ;
- local fullh : Vector Complex R ;
-
- {
- len := # h ;
- nvals := len pretend INT ; -- pretend since :: fails
- npairs := (nvals - 1) quo 2 ;
- fullh := new(len, 0) ;
- (nvals = 0) => () ;
- fullh.1 := complex(h.1,0) ;
- (nvals = 1) => () ;
- fullh.(npairs+2) := complex(h.(npairs+2),0) ; -- need if even length
- -- (not worth testing)
- for j in 1 .. npairs repeat {
- n1 := j + 1 ;
- n2 := nvals - j + 1 ;
- fullh.n1 := complex(h.n1, h.n2) ;
- fullh.n2 := complex(h.n1, -(h.n2)) ;
- }
-
- }
-
- fullh
-
- }
-
- packHS(v : Vector Complex R) : % == {
-
- local len : NNI ;
- local nonhs : String == "The argument of packHS is not Hermitian" ;
- local nvals, testprs, n1, n2 : INT ;
- local hpacked : Vector R ;
- local v1, v2 : Complex R ;
- local r1, i1, r2, i2 : R ;
-
- {
- len := # v ;
- nvals := len pretend INT ; -- pretend since :: fails
- testprs := nvals quo 2 ;
- hpacked := new(len, 0) ;
- (nvals = 0) => () ;
- if imag(v.1) ~= 0
- then error [nonhs, " - the first element must be real."]
- else {
- hpacked.1 := real(v.1) ;
- (nvals = 1) => () ;
- for j in 1 .. testprs repeat {
- n1 := j + 1 ;
- n2 := nvals - j + 1 ;
- v1 := v.n1 ;
- v2 := v.n2 ;
- r1 := real v1 ;
- i1 := imag v1 ;
- r2 := real v2 ;
- i2 := imag v2 ;
- if r1 ~= r2 or i1 ~= -i2
- then if n1 = n2
- then error [nonhs,
- " - element ",
- string(n1),
- " must be real to be self-conjugate."]
- else error [nonhs,
- " - elements ",
- string(n1),
- " and ",
- string(n2),
- " are not conjugate."]
- else {
- hpacked.n2 := i1 ; -- This order means that when the tail of v
- hpacked.n1 := r1 ; -- has odd length, the (real part) of its
- -- middle element ends up in that position.
- }
- }
- }
- }
-
- hpacked pretend %
-
- }
-
- local set!(x: %, i: INT, v: R): () == {
- (rep x).i := v;
- }
- conjHerm(h : %) : % == {
-
- local len : NNI ;
- local nvals, npairs : INT ;
- local ch : % ;
-
- ch := copy h ;
- len := # h ;
- (len < 3) => ch ; -- these Hermitian sequences are self-conjugate.
- nvals := len pretend INT ; -- pretend since :: fails
- npairs := (nvals - 1) quo 2 ;
- for j in (nvals - npairs + 1) .. nvals repeat ch.j := - h.j ;
- ch
-
- }
-
- import from List OutputForm ;
-
- coerce(h : %) : OutputForm ==
- bracket commaSeparate [
- qelt(h, k) :: OutputForm for k in minIndex h .. maxIndex h]
-
-}
-
-#if NeverAssertThis
-
-)lib herm
-
-h0 := pHS([] :: List INT)
-
--- []
-
-h1 := pHS [1]
-
--- [1]
-
-h2 := pHS [1,2]
-
--- [1,2]
-
-h3 := pHS [1,2,3]
-
--- [1,2,3]
-
-h4 := pHS [1,2,3,4]
-
--- [1,2,3,4]
-
-h5 := pHS [1,2,3,4,5]
-
--- [1,2,3,4,5]
-
-
-f0 := expand h0
-
--- []
-
-f1 := expand h1
-
--- [1]
-
-f2 := expand h2
-
--- [1,2]
-
-f3 := expand h3
-
--- [1,2 + 3%i,2 - 3%i]
-
-f4 := expand h4
-
--- [1,2 + 4%i,3,2 - 4%i]
-
-f5 := expand h5
-
--- [1,2 + 5%i,3 + 4%i,3 - 4%i,2 - 5%i]
-
-packHS f0
-
--- []
-
-packHS f1
-
--- [1]
-
-packHS f2
-
--- [1,2]
-
-packHS f3
-
--- [1,2,3]
-
-packHS f4
-
--- [1,2,3,4]
-
-packHS f5
-
--- [1,2,3,4,5]
-
-packHS vector[%i,3,3,3]
-
--- Error signalled from user code:
--- The argument of packHS is not Hermitian - the first element must
--- be real.
-
-packHS vector [1, 3, 5, 7]
-
--- Error signalled from user code:
--- The argument of packHS is not Hermitian - elements 2 and 4 are
--- not conjugate.
-
-packHS [1, 3, %i, 3]
-
--- Error signalled from user code:
--- The argument of packHS is not Hermitian - element 3 must be real
--- to be self-conjugate.
-
-conjHerm h0
-
--- []
-
-conjHerm h1
-
--- [1]
-
-conjHerm h2
-
--- [1,2]
-
-conjHerm h3
-
--- [1,2,- 3]
-
-conjHerm h4
-
--- [1,2,3,- 4]
-
-conjHerm h5
-
--- [1,2,3,- 4,- 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>>
-
-<<PackedHermitianSequence>>
-@
-\eject
-\begin{thebibliography}{99}
-\bibitem{1} nothing
-\end{thebibliography}
-\end{document}