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Diffstat (limited to 'src/input/reclos.input.pamphlet')
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1 files changed, 122 insertions, 0 deletions
diff --git a/src/input/reclos.input.pamphlet b/src/input/reclos.input.pamphlet new file mode 100644 index 00000000..43634594 --- /dev/null +++ b/src/input/reclos.input.pamphlet @@ -0,0 +1,122 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input RECLOS.input} +\author{The Axiom Team} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +This input file was updated by Renaud Rioboo in January 2004 +<<*>>= + +-- Input generated from RealClosureXmpPage +)clear all +Ran := RECLOS(FRAC INT) +-- +-- Some simple signs for square roots, these correspond to an extension +-- of degree 16 of the rational numbers. +-- these examples were given to me by J. Abbot +-- +fourSquares(a:Ran,b:Ran,c:Ran,d:Ran):Ran == sqrt(a)+sqrt(b) - sqrt(c)-sqrt(d) +squareDiff1 := fourSquares(73,548,60,586) +recip(squareDiff1) +sign(squareDiff1) +squareDiff2 := fourSquares(165,778,86,990) +recip(squareDiff2) +sign(squareDiff2) +squareDiff3 := fourSquares(217,708,226,692) +recip(squareDiff3) +sign(squareDiff3) +squareDiff4 := fourSquares(155,836,162,820) +recip(squareDiff4) +sign(squareDiff4) +squareDiff5 := fourSquares(591,772,552,818) +recip(squareDiff5) +sign(squareDiff5) +squareDiff6 := fourSquares(434,1053,412,1088) +recip(squareDiff6) +sign(squareDiff6) +squareDiff7 := fourSquares(514,1049,446,1152) +recip(squareDiff7) +sign(squareDiff7) +squareDiff8 := fourSquares(190,1751,208,1698) +recip(squareDiff8) +sign(squareDiff8) +relativeApprox(squareDiff8,10**(-3))::Float +-- +-- test the Renaud Rioboo fix (Jan 2004) +-- +allRootsOf((x-2)*(x-3)*(x-4))$RECLOS(FRAC INT) +-- +-- check out if the sum of all roots is null +-- example from P.V. Koseleff +-- +l := allRootsOf((x**2-2)**2-2)$Ran +l.1+l.2+l.3+l.4 +removeDuplicates map(mainDefiningPolynomial,l) +map(mainCharacterization,l) +[reduce(+,l),reduce(*,l)-2] +-- +-- a more complicated test that involve an extension of degree 256 +-- example by prof Kahan at ISSAC'92 +-- +)cl prop s2 s5 10 +(s2, s5, s10) := (sqrt(2)$Ran, sqrt(5)$Ran, sqrt(10)$Ran) +eq1:=sqrt(s10+3)*sqrt(s5+2) - sqrt(s10-3)*sqrt(s5-2) = sqrt(10*s2+10) +eq1::Boolean +-- +-- analogous one by [rr] +-- +eq2:=sqrt(s5+2)*sqrt(s2+1) - sqrt(s5-2)*sqrt(s2-1) = sqrt(2*s10+2) +eq2::Boolean +-- +-- these came from J.M. Arnaudies +-- +)cl prop s4 s7 e1 e2 +s3 := sqrt(3)$Ran +s7:= sqrt(7)$Ran +e1 := sqrt(2*s7-3*s3,3) +e2 := sqrt(2*s7+3*s3,3) +-- this should be null +ee1:=e2-e1=s3 +ee1::Boolean +)cl prop pol r1 alpha beta +pol : UP(x,Ran) := x**4+(7/3)*x**2+30*x-(100/3) +r1 := sqrt(7633)$Ran +-- cubic roots +alpha := sqrt(5*r1-436,3)/3 +beta := -sqrt(5*r1+436,3)/3 +-- this should be null +pol.(alpha+beta-1/3) +)cl prop qol r2 alpha beta +r2 := sqrt(153)$Ran +-- roots of order 5 +alpha2 := sqrt(r2-11,5) +beta2 := -sqrt(r2+11,5) +qol : UP(x,Ran) := x**5+10*x**3+20*x+22 +qol(alpha2+beta2) +dst1:=sqrt(9+4*s2)=1+2*s2 +dst1::Boolean +s6:Ran:=sqrt 6 +dst2:=sqrt(5+2*s6)+sqrt(5-2*s6) = 2*s3 +dst2::Boolean +s29:Ran:=sqrt 29 +dst4:=sqrt(16-2*s29+2*sqrt(55-10*s29)) = sqrt(22+2*s5)-sqrt(11+2*s29)+s5 +dst4::Boolean +dst6:=sqrt((112+70*s2)+(46+34*s2)*s5) = (5+4*s2)+(3+s2)*s5 +dst6::Boolean +f3:Ran:=sqrt(3,5) +f25:Ran:=sqrt(1/25,5) +f32:Ran:=sqrt(32/5,5) +f27:Ran:=sqrt(27/5,5) +dst5:=sqrt((f32-f27,3)) = f25*(1+f3-f3**2) +dst5::Boolean +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} |