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last edited 11 years ago by test1 |
Edit detail for SandBoxComplexManifold revision 5 of 7
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Editor: Bill Page
Time: 2009/06/21 16:14:16 GMT-7 |
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| Note: clean and add a little documentation | ||
added: Complex domain constructor done differently. DirectProduct lifts many operations from the underlying domain automatically. Complex values are represented as conjugate pairs. removed: - elt:(%,"pos") -> R - elt:(%,"neg") -> R changed: - elt(x,"pos") == rep(x).1 - elt(x,"neg") == rep(x).2 - - imaginary():% == per directProduct vector [1,-1] pair(x:R,y:R):Rep == directProduct vector [x,y] dup(x:R):Rep == directProduct vector [x,x] import List R imaginary():% == per pair(1,-1) changed: - conj(x:%):% == per directProduct vector [rep(x).2,rep(x).1] conj(x:%):% == per pair(rep(x).2,rep(x).1) changed: - (x:% * y:%):% == per directProduct vector [ _ - real(x)*rep(y).1 + imag(x)*rep(y).2, _ - real(x)*rep(y).2 - imag(x)*rep(y).1] (x:% * y:%):% == per pair( _ real(x)*rep(y).1 + imag(x)*rep(y).2, _ real(x)*rep(y).2 - imag(x)*rep(y).1) added: iabs(x:%):R == sqrt norm x abs(x:%):% == per dup iabs x changed: - (2::R)*atan(imag(x)/(sqrt(norm(x))+real(x))) - sqrt(x:%):% == per( sqrt(sqrt(norm(x)))*directProduct(vector [ _ (2::R)*atan(imag(x)/(iabs(x)+real(x))) sqrt(x:%):% == per( sqrt(iabs(x))*pair( _ changed: - cos(arg(x)/(2::R)) - sin(arg(x)/(2::R))] ) ) - - -- [sqrt(abs real x)+sqrt(abs imag x). - -- sign(real x)*sqrt(abs real x)-sign(imag x)*sqrt(abs imag x)] - -- too complicated? - --sqrt(x:%):% == per directProduct vector [ _ - -- (if real(x)>=0 then sqrt real(x) else sqrt(-real(x))) + _ - -- (if imag(x)<=0 then sqrt(-imag(x)) else sqrt imag(x)), _ - -- (if real(x)>=0 then sqrt real(x) else -sqrt(-real(x))) + _ - -- (if imag(x)<=0 then sqrt(-imag(x)) else -sqrt imag(x)) ] cos(arg(x)/(2::R)) - sin(arg(x)/(2::R)) ) ) added: \begin{axiom} )show CM EXPR INT \end{axiom} Compare: \begin{axiom} )show COMPLEX EXPR INT \end{axiom} Tests:
Complex domain constructor done differently.DirectProduct lifts many operations from the underlying domain automatically.
Complex values are represented as conjugate pairs. \begin{spad} )abbrev domain CM ComplexManifold ComplexManifold(R:Join(Field,RadicalCategory,TranscendentalFunctionCategory,OrderedSet)): Join(RadicalCategory,DirectProductCategory(2,R)) with imaginary: () -> % real: % -> R imag: % -> R conj: % -> % norm: % -> R arg: % -> R coerce: % -> Complex R == DirectProduct(2,R) add -- represent as conjugate pair Rep == DirectProduct(2,R) pair(x:R,y:R):Rep == directProduct vector [x,y] dup(x:R):Rep == directProduct vector [x,x] import List R
imaginary():% == per pair(1,-1) real(x:%):R == (rep(x).1 + rep(x).2)/(2::R) imag(x:%):R == (rep(x).1 - rep(x).2)/(2::R) -- just swap conj(x:%):% == per pair(rep(x).2,rep(x).1) -- multiplication is interesting (x:% y:%):% == per pair( _ real(x)rep(y).1 + imag(x)rep(y).2, _ real(x)rep(y).2 - imag(x)rep(y).1) norm(x:%):R == retract(xconj(x)) iabs(x:%):R == sqrt norm x abs(x:%):% == per dup iabs x arg(x:%):R == real(x)<0 and imag(x)=0 => pi() (2::R)atan(imag(x)/(iabs(x)+real(x))) sqrt(x:%):% == per( sqrt(iabs(x))pair( cos(arg(x)/(2::R)) + sin(arg(x)/(2::R)), cos(arg(x)/(2::R)) - sin(arg(x)/(2::R)) ) )
coerce(x:%):OutputForm == complex(real x,imag x)$Complex(R)::OutputForm coerce(x:%):Complex(R) == complex(real x,imag x) \end{spad}
\begin{axiom} )show CM EXPR INT \end{axiom} Compare: \begin{axiom} )show COMPLEX EXPR INT \end{axiom}
Tests: \begin{axiom} a:CM(EXPR INT) := 3 b:CM(EXPR INT) := -5 real(a) norm a norm b ab:=ab real ab imag ab s1:=sqrt(a) real s1 imag s1 s2:=sqrt(b) real s2 imag s2 I:CM(EXPR INT) := imaginary() real I norm I imag I imag conj I s3:=sqrt(I) s3s3 c1:=conj(a+bI) real c1 imag c1 norm c1 c1::Complex(EXPR INT) sqrt % s4:=sqrt(c1) real s4 imag s4 s4s4 %::Complex(EXPR INT) normalize % %::Complex(EXPR INT)::Complex(INT) \end{axiom}
Some or all expressions may not have rendered properly, because Axiom returned the following error:Error: export AXIOM=/usr/local/lib/open-axiom/x86_64-unknown-linux/1.3.0-2009-01-05; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/open-axiom/x86_64-unknown-linux/1.3.0-2009-01-05/lib; LANG=en_US.UTF-8 $AXIOM/bin/AXIOMsys < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1256844308733328586-25px.axm Segmentation fault
Some or all expressions may not have rendered properly, because Latex returned the following error:
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[2] (./6273261374623983092-16.0px.aux) ) (see the transcript file for additional information) Output written on 6273261374623983092-16.0px.dvi (2 pages, 2852 bytes). Transcript written on 6273261374623983092-16.0px.log.