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[Axiom-developer] 20070905.01.tpd.patch
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20070905.01.tpd.patch |
|
Date: |
Wed, 5 Sep 2007 07:46:50 -0500 |
This patch
- Adds additional help and regression test files for the algebra:
o Set (SET)
o SegmentBinding (SEGBIND)
o Segment (SEG)
o RomanNumeral.help (ROMAN)
o RegularTriangularSet (REGSET)
o RealClosure (RECLOS)
o RadixExpansion (RADIX)
o Polynomial (POLY)
o Permanent (PERM)
o PartialFraction (PFRAC)
o OrderlyDifferentialPolynomial (ODPOL)
o OrderedVariableList (OVAR)
o Operator (OP)
Tim
================================================================
diff --git a/changelog b/changelog
index eeab935..6ffd80a 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,53 @@
+20070905 tpd src/doc/spadhelp add Set
+20070905 tpd src/algebra/Makefile add Set.help
+20070905 tpd src/algebra/sets.spad add Set.help (SET)
+20070905 tpd src/algebra/sets.spad Set.input
+20070905 tpd src/doc/spadhelp add SegmentBinding
+20070905 tpd src/algebra/Makefile add SegmentBinding.help
+20070905 tpd src/algebra/seg.spad add SegmentBinding.help (SEGBIND)
+20070905 tpd src/algebra/seg.spad SegmentBinding.input
+20070905 tpd src/doc/spadhelp add Segment
+20070905 tpd src/algebra/Makefile add Segment.help
+20070905 tpd src/algebra/seg.spad add Segment.help (SEG)
+20070905 tpd src/algebra/seg.spad Segment.input
+20070905 tpd src/algebra/integer.spad update RomanNumeral.help (ROMAN)
+20070905 tpd src/algebra/integer.spad update RomanNumeral.input
+20070904 tpd src/doc/spadhelp add RegularTriangularSet
+20070904 tpd src/algebra/Makefile add RegularTriangularSet.help
+20070904 tpd src/algebra/regset.spad add RegularTriangularSet.help (REGSET)
+20070904 tpd src/algebra/regset.spad RegularTriangularSet.input
+20070904 tpd src/doc/spadhelp add RealClosure
+20070904 tpd src/algebra/Makefile add RealClosure.help
+20070904 tpd src/algebra/reclos.spad add RealClosure.help (RECLOS)
+20070904 tpd src/algebra/reclos.spad RealClosure.input
+20070904 tpd src/doc/spadhelp add RadixExpansion
+20070904 tpd src/algebra/Makefile add RadixExpansion.help
+20070904 tpd src/algebra/radix.spad add RadixExpansion.help (RADIX)
+20070904 tpd src/algebra/radix.spad RadixExpansion.input
+20070903 tpd src/doc/spadhelp add Polynomial
+20070903 tpd src/algebra/Makefile add Polynomial.help
+20070903 tpd src/algebra/multpoly.spad add Polynomial.help (POLY)
+20070903 tpd src/algebra/multpoly.spad Polynomial.input
+20070903 tpd src/doc/spadhelp add Permanent
+20070903 tpd src/algebra/Makefile add Permanent.help
+20070903 tpd src/algebra/perman.spad add Permanent.help (PERMAN)
+20070903 tpd src/algebra/perman.spad Permanent.input
+20070903 tpd src/doc/spadhelp add PartialFraction
+20070903 tpd src/algebra/Makefile add PartialFraction.help
+20070903 tpd src/algebra/pfr.spad add PartialFraction.help (PFR)
+20070903 tpd src/algebra/pfr.spad PartialFraction.input
+20070903 tpd src/doc/spadhelp add OrderlyDifferentialPolynomial
+20070903 tpd src/algebra/Makefile add OrderlyDifferentialPolynomial.help
+20070903 tpd src/algebra/dpolcat.spad add OrderlyDifferentialPolynomial (ODPOL)
+20070903 tpd src/algebra/dpolcat.spad OrderlyDifferentialPolynomial.input
+20070903 tpd src/doc/spadhelp add OrderedVariableList
+20070903 tpd src/algebra/Makefile add OrderedVariableList.help
+20070903 tpd src/algebra/variable.spad add OrderedVariableList.help (OVAR)
+20070903 tpd src/algebra/variable.spad OrderedVariableList.input
+20070903 tpd src/doc/spadhelp add Operator
+20070903 tpd src/algebra/Makefile add Operator.help
+20070903 tpd src/algebra/opalg.spad add Operator.help (OP)
+20070903 tpd src/algebra/opalg.spad Operator.input
20070903 tpd src/algebra/radix.spad fix typos in help file
20070903 tpd src/algebra/integer.spad fix typos in help file
20070903 tpd src/doc/bookvol4 fix typos
@@ -26,7 +76,7 @@
20070902 tpd src/algebra/Makefile add Matrix.help
20070902 tpd src/algebra/matrix.spad add Matrix.help (MATRIX)
20070902 tpd src/algebra/matrix.spad Matrix.input
-20070902 tpd src/doc/bookvol4 document how to add help files
+20070902 tpd src/doc/Makefile document how to add help files
20070902 tpd src/algebra/Makefile document how to add help files
20070902 tpd src/doc/spadhelp add MappingPackage3
20070902 tpd src/algebra/Makefile add MappingPackage3.help
diff --git a/src/algebra/Makefile.pamphlet b/src/algebra/Makefile.pamphlet
index b011d80..8c50591 100644
--- a/src/algebra/Makefile.pamphlet
+++ b/src/algebra/Makefile.pamphlet
@@ -2098,10 +2098,16 @@ SPADHELP=\
${HELP}/Matrix.help ${HELP}/Multiset.help \
${HELP}/MultivariatePolynomial.help ${HELP}/None.help \
${HELP}/Octonion.help ${HELP}/OneDimensionalArray.help \
+ ${HELP}/Operator.help ${HELP}/OrderedVariableList.help \
+ ${HELP}/OrderlyDifferentialPolynomial.help \
${HELP}/PlaneAlgebraicCurvePlot.help \
- ${HELP}/Permutation.help \
- ${HELP}/RealSolvePackage.help \
- ${HELP}/RomanNumeral.help \
+ ${HELP}/Permutation.help ${HELP}/PartialFraction.help \
+ ${HELP}/Permanent.help ${HELP}/Polynomial.help \
+ ${HELP}/Quaternion.help ${HELP}/RadixExpansion.help \
+ ${HELP}/RealClosure.help ${HELP}/RealSolvePackage.help \
+ ${HELP}/RegularTriangularSet.help \
+ ${HELP}/RomanNumeral.help ${HELP}/Segment.help \
+ ${HELP}/Set.help \
${HELP}/TwoDimensionalViewport.help
@
@@ -2142,10 +2148,15 @@ REGRESS=\
Matrix.regress Multiset.regress \
MultivariatePolynomial.regress None.regress \
Octonion.regress OneDimensionalArray.regress \
- RealSolvePackage.regress \
+ Operator.regress OrderedVariableList.regress \
+ OrderlyDifferentialPolynomial.regress PartialFraction.regress \
+ Permanent.regress Permutation.regress \
PlaneAlgebraicCurvePlot.regress \
- Permutation.regress \
- RomanNumeral.regress
+ Polynomial.regress Quaternion.regress \
+ RadixExpansion.regress RealClosure.regress \
+ RealSolvePackage.regress RegularTriangularSet.regress \
+ RomanNumeral.regress Segment.regress \
+ Set.regress
# this requires graphics
# TwoDimensionalViewport
@@ -2628,8 +2639,60 @@ ${HELP}/OneDimensionalArray.help:
${IN}/array1.spad.pamphlet
@${TANGLE} -R"OneDimensionalArray.input" ${IN}/array1.spad.pamphlet \
>${INPUT}/OneDimensionalArray.input
+${HELP}/Operator.help: ${IN}/opalg.spad.pamphlet
+ @echo 7053 create Operator.help from ${IN}/opalg.spad.pamphlet
+ @${TANGLE} -R"Operator.help" ${IN}/opalg.spad.pamphlet \
+ >${HELP}/Operator.help
+ @cp ${HELP}/Operator.help ${HELP}/OP.help
+ @${TANGLE} -R"Operator.input" ${IN}/opalg.spad.pamphlet \
+ >${INPUT}/Operator.input
+
+${HELP}/OrderedVariableList.help: ${IN}/variable.spad.pamphlet
+ @echo 7054 create OrderedVariableList.help from \
+ ${IN}/variable.spad.pamphlet
+ @${TANGLE} -R"OrderedVariableList.help" ${IN}/variable.spad.pamphlet \
+ >${HELP}/OrderedVariableList.help
+ @cp ${HELP}/OrderedVariableList.help ${HELP}/OVAR.help
+ @${TANGLE} -R"OrderedVariableList.input" ${IN}/variable.spad.pamphlet \
+ >${INPUT}/OrderedVariableList.input
+
+${HELP}/OrderlyDifferentialPolynomial.help: ${IN}/dpolcat.spad.pamphlet
+ @echo 7055 create OrderlyDifferentialPolynomial.help from \
+ ${IN}/dpolcat.spad.pamphlet
+ @${TANGLE} -R"OrderlyDifferentialPolynomial.help" \
+ ${IN}/dpolcat.spad.pamphlet \
+ >${HELP}/OrderlyDifferentialPolynomial.help
+ @cp ${HELP}/OrderlyDifferentialPolynomial.help ${HELP}/ODPOL.help
+ @${TANGLE} -R"OrderlyDifferentialPolynomial.input" \
+ ${IN}/dpolcat.spad.pamphlet \
+ >${INPUT}/OrderlyDifferentialPolynomial.input
+
+${HELP}/PartialFraction.help: ${IN}/pfr.spad.pamphlet
+ @echo 7056 create PartialFraction.help from ${IN}/pfr.spad.pamphlet
+ @${TANGLE} -R"PartialFraction.help" ${IN}/pfr.spad.pamphlet \
+ >${HELP}/PartialFraction.help
+ @cp ${HELP}/PartialFraction.help ${HELP}/PFR.help
+ @${TANGLE} -R"PartialFraction.input" ${IN}/pfr.spad.pamphlet \
+ >${INPUT}/PartialFraction.input
+
+${HELP}/Permanent.help: ${IN}/perman.spad.pamphlet
+ @echo 7057 create Permanent.help from ${IN}/perman.spad.pamphlet
+ @${TANGLE} -R"Permanent.help" ${IN}/perman.spad.pamphlet \
+ >${HELP}/Permanent.help
+ @cp ${HELP}/Permanent.help ${HELP}/PERMAN.help
+ @${TANGLE} -R"Permanent.input" ${IN}/perman.spad.pamphlet \
+ >${INPUT}/Permanent.input
+
+${HELP}/Polynomial.help: ${IN}/multpoly.spad.pamphlet
+ @echo 7058 create Polynomial.help from ${IN}/multpoly.spad.pamphlet
+ @${TANGLE} -R"Polynomial.help" ${IN}/multpoly.spad.pamphlet \
+ >${HELP}/Polynomial.help
+ @cp ${HELP}/Polynomial.help ${HELP}/POLY.help
+ @${TANGLE} -R"Polynomial.input" ${IN}/multpoly.spad.pamphlet \
+ >${INPUT}/Polynomial.input
+
${HELP}/PlaneAlgebraicCurvePlot.help: ${IN}/acplot.spad.pamphlet
- @echo 7053 create PlaneAlgebraicCurvePlot.help from \
+ @echo 7059 create PlaneAlgebraicCurvePlot.help from \
${IN}/acplot.spad.pamphlet
@${TANGLE} -R"PlaneAlgebraicCurvePlot.help" \
${IN}/acplot.spad.pamphlet \
@@ -2640,31 +2703,80 @@ ${HELP}/PlaneAlgebraicCurvePlot.help:
${IN}/acplot.spad.pamphlet
>${INPUT}/PlaneAlgebraicCurvePlot.input
${HELP}/Permutation.help: ${IN}/perm.spad.pamphlet
- @echo 7054 create Permutation.help from ${IN}/perm.spad.pamphlet
+ @echo 7060 create Permutation.help from ${IN}/perm.spad.pamphlet
@${TANGLE} -R"Permutation.help" ${IN}/perm.spad.pamphlet \
>${HELP}/Permutation.help
@cp ${HELP}/Permutation.help ${HELP}/PERM.help
@${TANGLE} -R"Permutation.input" ${IN}/perm.spad.pamphlet \
>${INPUT}/Permutation.input
+${HELP}/Quaternion.help: ${IN}/quat.spad.pamphlet
+ @echo 7061 create Quaternion.help from ${IN}/quat.spad.pamphlet
+ @${TANGLE} -R"Quaternion.help" ${IN}/quat.spad.pamphlet \
+ >${HELP}/Quaternion.help
+ @cp ${HELP}/Quaternion.help ${HELP}/QUAT.help
+ @${TANGLE} -R"Quaternion.input" ${IN}/quat.spad.pamphlet \
+ >${INPUT}/Quaternion.input
+
+${HELP}/RadixExpansion.help: ${IN}/radix.spad.pamphlet
+ @echo 7062 create RadixExpansion.help from ${IN}/radix.spad.pamphlet
+ @${TANGLE} -R"RadixExpansion.help" ${IN}/radix.spad.pamphlet \
+ >${HELP}/RadixExpansion.help
+ @cp ${HELP}/RadixExpansion.help ${HELP}/RADIX.help
+ @${TANGLE} -R"RadixExpansion.input" ${IN}/radix.spad.pamphlet \
+ >${INPUT}/RadixExpansion.input
+
+${HELP}/RealClosure.help: ${IN}/reclos.spad.pamphlet
+ @echo 7063 create RealClosure.help from ${IN}/reclos.spad.pamphlet
+ @${TANGLE} -R"RealClosure.help" ${IN}/reclos.spad.pamphlet \
+ >${HELP}/RealClosure.help
+ @cp ${HELP}/RealClosure.help ${HELP}/RECLOS.help
+ @${TANGLE} -R"RealClosure.input" ${IN}/reclos.spad.pamphlet \
+ >${INPUT}/RealClosure.input
+
${HELP}/RealSolvePackage.help: ${IN}/acplot.spad.pamphlet
- @echo 7055 create RealSolvePackage.help from ${IN}/acplot.spad.pamphlet
+ @echo 7064 create RealSolvePackage.help from ${IN}/acplot.spad.pamphlet
@${TANGLE} -R"RealSolvePackage.help" ${IN}/acplot.spad.pamphlet \
>${HELP}/RealSolvePackage.help
@cp ${HELP}/RealSolvePackage.help ${HELP}/REALSOLV.help
@${TANGLE} -R"RealSolvePackage.input" ${IN}/acplot.spad.pamphlet \
>${INPUT}/RealSolvePackage.input
+${HELP}/RegularTriangularSet.help: ${IN}/regset.spad.pamphlet
+ @echo 7065 create RegularTriangularSet.help from \
+ ${IN}/regset.spad.pamphlet
+ @${TANGLE} -R"RegularTriangularSet.help" ${IN}/regset.spad.pamphlet \
+ >${HELP}/RegularTriangularSet.help
+ @cp ${HELP}/RegularTriangularSet.help ${HELP}/REGSET.help
+ @${TANGLE} -R"RegularTriangularSet.input" ${IN}/regset.spad.pamphlet \
+ >${INPUT}/RegularTriangularSet.input
+
${HELP}/RomanNumeral.help: ${IN}/integer.spad.pamphlet
- @echo 7056 create RomanNumeral.help from ${IN}/integer.spad.pamphlet
+ @echo 7066 create RomanNumeral.help from ${IN}/integer.spad.pamphlet
@${TANGLE} -R"RomanNumeral.help" ${IN}/integer.spad.pamphlet \
>${HELP}/RomanNumeral.help
@cp ${HELP}/RomanNumeral.help ${HELP}/ROMAN.help
@${TANGLE} -R"RomanNumeral.input" ${IN}/integer.spad.pamphlet \
>${INPUT}/RomanNumeral.input
+${HELP}/Segment.help: ${IN}/seg.spad.pamphlet
+ @echo 7067 create Segment.help from ${IN}/seg.spad.pamphlet
+ @${TANGLE} -R"Segment.help" ${IN}/seg.spad.pamphlet \
+ >${HELP}/Segment.help
+ @cp ${HELP}/Segment.help ${HELP}/SEG.help
+ @${TANGLE} -R"Segment.input" ${IN}/seg.spad.pamphlet \
+ >${INPUT}/Segment.input
+
+${HELP}/Set.help: ${IN}/sets.spad.pamphlet
+ @echo 7068 create Set.help from ${IN}/sets.spad.pamphlet
+ @${TANGLE} -R"Set.help" ${IN}/sets.spad.pamphlet \
+ >${HELP}/Set.help
+ @cp ${HELP}/Set.help ${HELP}/SET.help
+ @${TANGLE} -R"Set.input" ${IN}/sets.spad.pamphlet \
+ >${INPUT}/Set.input
+
${HELP}/TwoDimensionalViewport.help: ${IN}/view2d.spad.pamphlet
- @echo 7057 create TwoDimensionalViewport.help from \
+ @echo 7069 create TwoDimensionalViewport.help from \
${IN}/view2d.spad.pamphlet
@${TANGLE} -R"TwoDimensionalViewport.help" \
${IN}/view2d.spad.pamphlet \
diff --git a/src/algebra/dpolcat.spad.pamphlet
b/src/algebra/dpolcat.spad.pamphlet
index 2a17025..f01f7df 100644
--- a/src/algebra/dpolcat.spad.pamphlet
+++ b/src/algebra/dpolcat.spad.pamphlet
@@ -471,6 +471,658 @@ DifferentialSparseMultivariatePolynomial(R, S, V):
@
\section{domain ODPOL OrderlyDifferentialPolynomial}
+<<OrderlyDifferentialPolynomial.input>>=
+-- dpolcat.spad.pamphlet OrderlyDifferentialPolynomial.input
+)spool OrderlyDifferentialPolynomial.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 36
+dpol:= ODPOL(FRAC INT)
+--R
+--R
+--R (1) OrderlyDifferentialPolynomial Fraction Integer
+--R Type:
Domain
+--E 1
+
+--S 2 of 36
+w := makeVariable('w)$dpol
+--R
+--R
+--R (2) theMap(DPOLCAT-;makeVariable;AM;17!0,0)
+--R Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction
Integer)
+--E 2
+
+--S 3 of 36
+z := makeVariable('z)$dpol
+--R
+--R
+--R (3) theMap(DPOLCAT-;makeVariable;AM;17!0,0)
+--R Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction
Integer)
+--E 3
+
+--S 4 of 36
+w.5
+--R
+--R
+--R (4) w
+--R 5
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 4
+
+--S 5 of 36
+w 0
+--R
+--R
+--R (5) w
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 5
+
+--S 6 of 36
+[z.i for i in 1..5]
+--R
+--R
+--R (6) [z ,z ,z ,z ,z ]
+--R 1 2 3 4 5
+--R Type: List OrderlyDifferentialPolynomial Fraction
Integer
+--E 6
+
+--S 7 of 36
+f:= w.4 - w.1 * w.1 * z.3
+--R
+--R
+--R 2
+--R (7) w - w z
+--R 4 1 3
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 7
+
+--S 8 of 36
+g:=(z.1)**3 * (z.2)**2 - w.2
+--R
+--R
+--R 3 2
+--R (8) z z - w
+--R 1 2 2
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 8
+
+--S 9 of 36
+D(f)
+--R
+--R
+--R 2
+--R (9) w - w z - 2w w z
+--R 5 1 4 1 2 3
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 9
+
+--S 10 of 36
+D(f,4)
+--R
+--R
+--R (10)
+--R 2 2
+--R w - w z - 8w w z + (- 12w w - 12w )z - 2w z w
+--R 8 1 7 1 2 6 1 3 2 5 1 3 5
+--R +
+--R 2
+--R (- 8w w - 24w w )z - 8w z w - 6w z
+--R 1 4 2 3 4 2 3 4 3 3
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 10
+
+--S 11 of 36
+df:=makeVariable(f)$dpol
+--R
+--R
+--R (11) theMap(DPOLCAT-;makeVariable;AM;17!0,0)
+--R Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction
Integer)
+--E 11
+
+--S 12 of 36
+df.4
+--R
+--R
+--R (12)
+--R 2 2
+--R w - w z - 8w w z + (- 12w w - 12w )z - 2w z w
+--R 8 1 7 1 2 6 1 3 2 5 1 3 5
+--R +
+--R 2
+--R (- 8w w - 24w w )z - 8w z w - 6w z
+--R 1 4 2 3 4 2 3 4 3 3
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 12
+
+--S 13 of 36
+order(g)
+--R
+--R
+--R (13) 2
+--R Type:
PositiveInteger
+--E 13
+
+--S 14 of 36
+order(g, 'w)
+--R
+--R
+--R (14) 2
+--R Type:
PositiveInteger
+--E 14
+
+--S 15 of 36
+differentialVariables(g)
+--R
+--R
+--R (15) [z,w]
+--R Type: List
Symbol
+--E 15
+
+--S 16 of 36
+degree(g)
+--R
+--R
+--R 2 3
+--R (16) z z
+--R 2 1
+--R Type: IndexedExponents OrderlyDifferentialVariable
Symbol
+--E 16
+
+--S 17 of 36
+degree(g, 'w)
+--R
+--R
+--R (17) 1
+--R Type:
PositiveInteger
+--E 17
+
+--S 18 of 36
+weights(g)
+--R
+--R
+--R (18) [7,2]
+--R Type: List
NonNegativeInteger
+--E 18
+
+--S 19 of 36
+weights(g,'w)
+--R
+--R
+--R (19) [2]
+--R Type: List
NonNegativeInteger
+--E 19
+
+--S 20 of 36
+weight(g)
+--R
+--R
+--R (20) 7
+--R Type:
PositiveInteger
+--E 20
+
+--S 21 of 36
+isobaric?(g)
+--R
+--R
+--R (21) false
+--R Type:
Boolean
+--E 21
+
+--S 22 of 36
+eval(g,['w::Symbol],[f])
+--R
+--R
+--R 2 2 3 2
+--R (22) - w + w z + 4w w z + (2w w + 2w )z + z z
+--R 6 1 5 1 2 4 1 3 2 3 1 2
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 22
+
+--S 23 of 36
+eval(g,variables(w.0),[f])
+--R
+--R
+--R 3 2
+--R (23) z z - w
+--R 1 2 2
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 23
+
+--S 24 of 36
+monomials(g)
+--R
+--R
+--R 3 2
+--R (24) [z z ,- w ]
+--R 1 2 2
+--R Type: List OrderlyDifferentialPolynomial Fraction
Integer
+--E 24
+
+--S 25 of 36
+variables(g)
+--R
+--R
+--R (25) [z ,w ,z ]
+--R 2 2 1
+--R Type: List OrderlyDifferentialVariable
Symbol
+--E 25
+
+--S 26 of 36
+gcd(f,g)
+--R
+--R
+--R (26) 1
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 26
+
+--S 27 of 36
+groebner([f,g])
+--R
+--R
+--R 2 3 2
+--R (27) [w - w z ,z z - w ]
+--R 4 1 3 1 2 2
+--R Type: List OrderlyDifferentialPolynomial Fraction
Integer
+--E 27
+
+--S 28 of 36
+lg:=leader(g)
+--R
+--R
+--R (28) z
+--R 2
+--R Type: OrderlyDifferentialVariable
Symbol
+--E 28
+
+--S 29 of 36
+sg:=separant(g)
+--R
+--R
+--R 3
+--R (29) 2z z
+--R 1 2
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 29
+
+--S 30 of 36
+ig:=initial(g)
+--R
+--R
+--R 3
+--R (30) z
+--R 1
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 30
+
+--S 31 of 36
+g1 := D g
+--R
+--R
+--R 3 2 3
+--R (31) 2z z z - w + 3z z
+--R 1 2 3 3 1 2
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 31
+
+--S 32 of 36
+lg1:= leader g1
+--R
+--R
+--R (32) z
+--R 3
+--R Type: OrderlyDifferentialVariable
Symbol
+--E 32
+
+--S 33 of 36
+pdf:=D(f, lg1)
+--R
+--R
+--R 2
+--R (33) - w
+--R 1
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 33
+
+--S 34 of 36
+prf:=sg * f- pdf * g1
+--R
+--R
+--R 3 2 2 2 3
+--R (34) 2z z w - w w + 3w z z
+--R 1 2 4 1 3 1 1 2
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 34
+
+--S 35 of 36
+lcf:=leadingCoefficient univariate(prf, lg)
+--R
+--R
+--R 2 2
+--R (35) 3w z
+--R 1 1
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 35
+
+--S 36 of 36
+ig * prf - lcf * g * lg
+--R
+--R
+--R 6 2 3 2 2
+--R (36) 2z z w - w z w + 3w z w z
+--R 1 2 4 1 1 3 1 1 2 2
+--R Type: OrderlyDifferentialPolynomial Fraction
Integer
+--E 36
+)spool
+)lisp (bye)
+@
+<<OrderlyDifferentialPolynomial.help>>=
+====================================================================
+OrderlyDifferentialPolynomial examples
+====================================================================
+
+Many systems of differential equations may be transformed to equivalent
+systems of ordinary differential equations where the equations are expressed
+polynomially in terms of the unknown functions. In Axiom, the domain
+constructors OrderlyDifferentialPolynomial (abbreviated ODPOL) and
+SequentialDifferentialPolynomial (abbreviation SDPOL) implement two
+domains of ordinary differential polynomials over any differential
+ring. In the simplest case, this differential ring is usually either
+the ring of integers, or the field of rational numbers. However,
+Axiom can handle ordinary differential polynomials over a field of
+rational functions in a single indeterminate.
+
+The two domains ODPOL and SDPOL are almost identical, the only difference
+being the choice of a different ranking, which is an ordering of the
+derivatives of the indeterminates. The first domain uses an orderly ranking,
+that is, derivatives of higher order are ranked higher, and derivatives of
+the same order are ranked alphabetically. The second domain uses a
+sequential ranking, where derivatives are ordered first alphabetically
+by the differential indeterminates, and then by order. A more general
+domain constructor, DifferentialSparseMultivariatePolynomial (abbreviation
+DSMP) allows both a user-provided list of differential indeterminates
+as well as a user-defined ranking. We shall illustrate ODPOL(FRAC INT),
+which constructs a domain of ordinary differential polynomials in an
+arbitrary number of differential indeterminates with rational numbers
+as coefficients.
+
+ dpol:= ODPOL(FRAC INT)
+ OrderlyDifferentialPolynomial Fraction Integer
+ Type: Domain
+
+A differential indeterminate w may be viewed as an infinite sequence of
+algebraic indeterminates, which are the derivatives of w. To facilitate
+referencing these, Axiom provides the operation makeVariable to convert
+an element of type Symbol to a map from the natural numbers to the
+differential polynomial ring.
+
+ w := makeVariable('w)$dpol
+ theMap(DPOLCAT-;makeVariable;AM;17!0,0)
+ Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction Integer)
+
+ z := makeVariable('z)$dpol
+ theMap(DPOLCAT-;makeVariable;AM;17!0,0)
+ Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction Integer)
+
+The fifth derivative of w can be obtained by applying the map w to the
+number 5. Note that the order of differentiation is given as a subscript
+(except when the order is 0).
+
+ w.5
+ w
+ 5
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+ w 0
+ w
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+The first five derivatives of z can be generated by a list.
+
+ [z.i for i in 1..5]
+ [z ,z ,z ,z ,z ]
+ 1 2 3 4 5
+ Type: List OrderlyDifferentialPolynomial Fraction Integer
+
+The usual arithmetic can be used to form a differential polynomial from
+the derivatives.
+
+ f:= w.4 - w.1 * w.1 * z.3
+ 2
+ w - w z
+ 4 1 3
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+ g:=(z.1)**3 * (z.2)**2 - w.2
+ 3 2
+ z z - w
+ 1 2 2
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+The operation D computes the derivative of any differential polynomial.
+
+ D(f)
+ 2
+ w - w z - 2w w z
+ 5 1 4 1 2 3
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+The same operation can compute higher derivatives, like the fourth derivative.
+
+ D(f,4)
+ 2 2
+ w - w z - 8w w z + (- 12w w - 12w )z - 2w z w
+ 8 1 7 1 2 6 1 3 2 5 1 3 5
+ +
+ 2
+ (- 8w w - 24w w )z - 8w z w - 6w z
+ 1 4 2 3 4 2 3 4 3 3
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+The operation makeVariable creates a map to facilitate referencing the
+derivatives of f, similar to the map w.
+
+ df:=makeVariable(f)$dpol
+ theMap(DPOLCAT-;makeVariable;AM;17!0,0)
+ Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction Integer)
+
+The fourth derivative of f may be referenced easily.
+
+ df.4
+ 2 2
+ w - w z - 8w w z + (- 12w w - 12w )z - 2w z w
+ 8 1 7 1 2 6 1 3 2 5 1 3 5
+ +
+ 2
+ (- 8w w - 24w w )z - 8w z w - 6w z
+ 1 4 2 3 4 2 3 4 3 3
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+The operation order returns the order of a differential polynomial,
+or the order in a specified differential indeterminate.
+
+ order(g)
+ 2
+ Type: PositiveInteger
+
+ order(g, 'w)
+ 2
+ Type: PositiveInteger
+
+The operation differentialVariables returns a list of differential
+indeterminates occurring in a differential polynomial.
+
+ differentialVariables(g)
+ [z,w]
+ Type: List Symbol
+
+The operation degree returns the degree, or the degree in the differential
+indeterminate specified.
+
+ degree(g)
+ 2 3
+ z z
+ 2 1
+ Type: IndexedExponents OrderlyDifferentialVariable Symbol
+
+ degree(g, 'w)
+ 1
+ Type: PositiveInteger
+
+The operation weights returns a list of weights of differential monomials
+appearing in differential polynomial, or a list of weights in a specified
+differential indeterminate.
+
+ weights(g)
+ [7,2]
+ Type: List NonNegativeInteger
+
+ weights(g,'w)
+ [2]
+ Type: List NonNegativeInteger
+
+The operation weight returns the maximum weight of all differential
+monomials appearing in the differential polynomial.
+
+ weight(g)
+ 7
+ Type: PositiveInteger
+
+A differential polynomial is isobaric if the weights of all differential
+monomials appearing in it are equal.
+
+ isobaric?(g)
+ false
+ Type: Boolean
+
+To substitute differentially, use eval. Note that we must coerce 'w to
+Symbol, since in ODPOL, differential indeterminates belong to the domain
+Symbol. Compare this result to the next, which substitutes algebraically
+(no substitution is done since w.0 does not appear in g).
+
+ eval(g,['w::Symbol],[f])
+ 2 2 3 2
+ - w + w z + 4w w z + (2w w + 2w )z + z z
+ 6 1 5 1 2 4 1 3 2 3 1 2
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+ eval(g,variables(w.0),[f])
+ 3 2
+ z z - w
+ 1 2 2
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+Since OrderlyDifferentialPolynomial belongs to PolynomialCategory, all
+the operations defined in the latter category, or in packages for the
+latter category, are available.
+
+ monomials(g)
+ 3 2
+ [z z ,- w ]
+ 1 2 2
+ Type: List OrderlyDifferentialPolynomial Fraction Integer
+
+ variables(g)
+ [z ,w ,z ]
+ 2 2 1
+ Type: List OrderlyDifferentialVariable Symbol
+
+ gcd(f,g)
+ 1
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+ groebner([f,g])
+ 2 3 2
+ [w - w z ,z z - w ]
+ 4 1 3 1 2 2
+ Type: List OrderlyDifferentialPolynomial Fraction Integer
+
+The next three operations are essential for elimination procedures in
+differential polynomial rings. The operation leader returns the leader
+of a differential polynomial, which is the highest ranked derivative
+of the differential indeterminates that occurs.
+
+ lg:=leader(g)
+ z
+ 2
+ Type: OrderlyDifferentialVariable Symbol
+
+The operation separant returns the separant of a differential polynomial,
+which is the partial derivative with respect to the leader.
+
+ sg:=separant(g)
+ 3
+ 2z z
+ 1 2
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+The operation initial returns the initial, which is the leading coefficient
+when the given differential polynomial is expressed as a polynomial in
+the leader.
+
+ ig:=initial(g)
+ 3
+ z
+ 1
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+Using these three operations, it is possible to reduce f modulo the
+differential ideal generated by g. The general scheme is to first
+reduce the order, then reduce the degree in the leader. First, eliminate
+z.3 using the derivative of g.
+
+ g1 := D g
+ 3 2 3
+ 2z z z - w + 3z z
+ 1 2 3 3 1 2
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+Find its leader.
+
+ lg1:= leader g1
+ z
+ 3
+ Type: OrderlyDifferentialVariable Symbol
+
+Differentiate f partially with respect to this leader.
+
+ pdf:=D(f, lg1)
+ 2
+ - w
+ 1
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+Compute the partial remainder of f with respect to g.
+
+ prf:=sg * f- pdf * g1
+ 3 2 2 2 3
+ 2z z w - w w + 3w z z
+ 1 2 4 1 3 1 1 2
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+Note that high powers of lg still appear in prf. Compute the leading
+coefficient of prf as a polynomial in the leader of g.
+
+ lcf:=leadingCoefficient univariate(prf, lg)
+ 2 2
+ 3w z
+ 1 1
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+Finally, continue eliminating the high powers of lg appearing in prf to
+obtain the (pseudo) remainder of f modulo g and its derivatives.
+
+ ig * prf - lcf * g * lg
+ 6 2 3 2 2
+ 2z z w - w z w + 3w z w z
+ 1 2 4 1 1 3 1 1 2 2
+ Type: OrderlyDifferentialPolynomial Fraction Integer
+
+See Also:
+o )show OrderlyDifferentialPolynomial
+o $AXIOM/doc/src/algebra/dpolcat.spad.dvi
+
+@
<<domain ODPOL OrderlyDifferentialPolynomial>>=
)abbrev domain ODPOL OrderlyDifferentialPolynomial
++ Author: William Sit
diff --git a/src/algebra/integer.spad.pamphlet
b/src/algebra/integer.spad.pamphlet
index b5e56d2..16ac727 100644
--- a/src/algebra/integer.spad.pamphlet
+++ b/src/algebra/integer.spad.pamphlet
@@ -1443,54 +1443,247 @@ Note that this code is not included in the generated
catdef.spad file.
)set message test on
)set message auto off
)clear all
---S 1 of 5
+--S 1 of 15
+f := operator 'f
+--R
+--R
+--R (1) f
+--R Type:
BasicOperator
+--E 1
+
+--S 2 of 15
+D(f x,x,7)
+--R
+--R
+--R (vii)
+--R (2) f (x)
+--R
+--R Type: Expression
Integer
+--E 2
+
+--S 3 of 15
+a := roman(1978 - 1965)
+--R
+--R
+--R (3) XIII
+--R Type:
RomanNumeral
+--E 3
+
+--S 4 of 15
+x : UTS(ROMAN,'x,0) := x
+--R
+--R
+--R (4) x
+--R Type:
UnivariateTaylorSeries(RomanNumeral,x,0)
+--E 4
+
+--S 5 of 15
+recip(1 - x - x**2)
+--R
+--R
+--R (5)
+--R 2 3 4 5 6 7 8
+--R I + x + II x + III x + V x + VIII x + XIII x + XXI x + XXXIV x
+--R +
+--R 9 10 11
+--R LV x + LXXXIX x + O(x )
+--R Type:
Union(UnivariateTaylorSeries(RomanNumeral,x,0),...)
+--E 5
+
+--S 6 of 15
+m : MATRIX FRAC ROMAN
+--R
+--R Type:
Void
+--E 6
+
+--S 7 of 15
+m := matrix [ [1/(i + j) for i in 1..3] for j in 1..3]
+--R
+--R
+--R + I I I+
+--R |-- --- --|
+--R |II III IV|
+--R | |
+--R | I I I |
+--R (7) |--- -- - |
+--R |III IV V |
+--R | |
+--R | I I I|
+--R |-- - --|
+--R +IV V VI+
+--R Type: Matrix Fraction
RomanNumeral
+--E 7
+
+--S 8 of 15
+inverse m
+--R
+--R
+--R +LXXII - CCXL CLXXX +
+--R | |
+--R (8) |- CCXL CM - DCCXX|
+--R | |
+--R +CLXXX - DCCXX DC +
+--R Type: Union(Matrix Fraction
RomanNumeral,...)
+--E 8
+
+--S 9 of 15
+y := factorial 10
+--R
+--R
+--R (9) 3628800
+--R Type:
PositiveInteger
+--E 9
+
+--S 10 of 15
+roman y
+--R
+--R
+--R (10)
+--R ((((I))))((((I))))((((I)))) (((I)))(((I)))(((I)))(((I)))(((I)))(((I)))
((I))(
+--R (I)) MMMMMMMMDCCC
+--R Type:
RomanNumeral
+--E 10
+
+--S 11 of 15
a := roman(78)
--R
--R
---R (1) LXXVIII
+--R (11) LXXVIII
--R Type:
RomanNumeral
---E 1
+--E 11
---S 2 of 5
+--S 12 of 15
b := roman(87)
--R
--R
---R (2) LXXXVII
+--R (12) LXXXVII
--R Type:
RomanNumeral
---E 2
+--E 12
---S 3 of 5
+--S 13 of 15
a + b
--R
--R
---R (3) CLXV
+--R (13) CLXV
--R Type:
RomanNumeral
---E 3
+--E 13
---S 4 of 5
+--S 14 of 15
a * b
--R
--R
---R (4) MMMMMMDCCLXXXVI
+--R (14) MMMMMMDCCLXXXVI
--R Type:
RomanNumeral
---E 4
+--E 14
---S 5 of 5
+--S 15 of 15
b rem a
--R
--R
---R (5) IX
+--R (15) IX
--R Type:
RomanNumeral
---E 5
+--E 15
)spool
)lisp (bye)
@
-
<<RomanNumeral.help>>=
====================================================================
RomanNumeral Examples
====================================================================
+The Roman numeral package was added to Axiom in MCMLXXXVI for use in
+denoting higher order derivatives.
+
+For example, let f be a symbolic operator.
+
+ f := operator 'f
+ f
+ Type: BasicOperator
+
+This is the seventh derivative of f with respect to x.
+ D(f x,x,7)
+ (vii)
+ f (x)
+ Type: Expression Integer
+
+You can have integers printed as Roman numerals by declaring variables to
+be of type RomanNumeral (abbreviation ROMAN).
+
+ a := roman(1978 - 1965)
+ XIII
+ Type: RomanNumeral
+
+This package now has a small but devoted group of followers that claim
+this domain has shown its efficacy in many other contexts. They claim
+that Roman numerals are every bit as useful as ordinary integers.
+
+In a sense, they are correct, because Roman numerals form a ring and you
+can therefore construct polynomials with Roman numeral coefficients,
+matrices over Roman numerals, etc..
+
+ x : UTS(ROMAN,'x,0) := x
+ x
+ Type: UnivariateTaylorSeries(RomanNumeral,x,0)
+
+Was Fibonacci Italian or ROMAN?
+
+ recip(1 - x - x**2)
+ 2 3 4 5 6 7 8
+ I + x + II x + III x + V x + VIII x + XIII x + XXI x + XXXIV x
+ +
+ 9 10 11
+ LV x + LXXXIX x + O(x )
+ Type: Union(UnivariateTaylorSeries(RomanNumeral,x,0),...)
+
+You can also construct fractions with Roman numeral numerators and
+denominators, as this matrix Hilberticus illustrates.
+
+ m : MATRIX FRAC ROMAN
+ Type: Void
+
+ m := matrix [ [1/(i + j) for i in 1..3] for j in 1..3]
+ + I I I+
+ |-- --- --|
+ |II III IV|
+ | |
+ | I I I |
+ |--- -- - |
+ |III IV V |
+ | |
+ | I I I|
+ |-- - --|
+ +IV V VI+
+ Type: Matrix Fraction RomanNumeral
+
+Note that the inverse of the matrix has integral ROMAN entries.
+
+ inverse m
+ +LXXII - CCXL CLXXX +
+ | |
+ |- CCXL CM - DCCXX|
+ | |
+ +CLXXX - DCCXX DC +
+ Type: Union(Matrix Fraction RomanNumeral,...)
+
+Unfortunately, the spoil-sports say that the fun stops when the
+numbers get big---mostly because the Romans didn't establish
+conventions about representing very large numbers.
+
+ y := factorial 10
+ 3628800
+ Type: PositiveInteger
+
+You work it out!
+
+ roman y
+ ((((I))))((((I))))((((I)))) (((I)))(((I)))(((I)))(((I)))(((I)))(((I))) ((I))(
+ (I)) MMMMMMMMDCCC
+ Type: RomanNumeral
+
+Issue the system command )show RomanNumeral to display the full
+list of operations defined by RomanNumeral.
+
+
a := roman(78)
LXXVIII
Type: RomanNumeral
diff --git a/src/algebra/multpoly.spad.pamphlet
b/src/algebra/multpoly.spad.pamphlet
index 76a5624..68abc3c 100644
--- a/src/algebra/multpoly.spad.pamphlet
+++ b/src/algebra/multpoly.spad.pamphlet
@@ -10,6 +10,798 @@
\tableofcontents
\eject
\section{domain POLY Polynomial}
+<<Polynomial.input>>=
+-- multpoly.spad.pamphlet Polynomial.input
+)spool Polynomial.output
+)set message test on
+)set message auto off
+--S 1 of 46
+x + 1
+--R
+--R
+--R (1) x + 1
+--R Type: Polynomial
Integer
+--E 1
+
+--S 2 of 46
+z - 2.3
+--R
+--R
+--R (2) z - 2.3
+--R Type: Polynomial
Float
+--E 2
+
+--S 3 of 46
+y**2 - z + 3/4
+--R
+--R
+--R 2 3
+--R (3) - z + y + -
+--R 4
+--R Type: Polynomial Fraction
Integer
+--E 3
+
+--S 4 of 46
+y **2 + x*y + y
+--R
+--R
+--R 2
+--R (4) y + (x + 1)y
+--R Type: Polynomial
Integer
+--E 4
+
+--S 5 of 46
+% :: DMP([y,x],INT)
+--R
+--R
+--R 2
+--R (5) y + y x + y
+--R Type:
DistributedMultivariatePolynomial([y,x],Integer)
+--E 5
+
+--S 6 of 46
+p := (y-1)**2 * x * z
+--R
+--R
+--R 2
+--R (6) (x y - 2x y + x)z
+--R Type: Polynomial
Integer
+--E 6
+
+--S 7 of 46
+q := (y-1) * x * (z+5)
+--R
+--R
+--R (7) (x y - x)z + 5x y - 5x
+--R Type: Polynomial
Integer
+--E 7
+
+--S 8 of 46
+factor(q)
+--R
+--R
+--R (8) x(y - 1)(z + 5)
+--R Type: Factored Polynomial
Integer
+--E 8
+
+--S 9 of 46
+p - q**2
+--R
+--R
+--R (9)
+--R 2 2 2 2 2 2 2 2 2
+--R (- x y + 2x y - x )z + ((- 10x + x)y + (20x - 2x)y - 10x + x)z
+--R +
+--R 2 2 2 2
+--R - 25x y + 50x y - 25x
+--R Type: Polynomial
Integer
+--E 9
+
+--S 10 of 46
+gcd(p,q)
+--R
+--R
+--R (10) x y - x
+--R Type: Polynomial
Integer
+--E 10
+
+--S 11 of 46
+factor %
+--R
+--R
+--R (11) x(y - 1)
+--R Type: Factored Polynomial
Integer
+--E 11
+
+--S 12 of 46
+lcm(p,q)
+--R
+--R
+--R 2 2 2
+--R (12) (x y - 2x y + x)z + (5x y - 10x y + 5x)z
+--R Type: Polynomial
Integer
+--E 12
+
+--S 13 of 46
+content p
+--R
+--R
+--R (13) 1
+--R Type:
PositiveInteger
+--E 13
+
+--S 14 of 46
+resultant(p,q,z)
+--R
+--R
+--R 2 3 2 2 2 2
+--R (14) 5x y - 15x y + 15x y - 5x
+--R Type: Polynomial
Integer
+--E 14
+
+--S 15 of 46
+resultant(p,q,x)
+--R
+--R
+--R (15) 0
+--R Type: Polynomial
Integer
+--E 15
+
+--S 16 of 46
+mainVariable p
+--R
+--R
+--R (16) z
+--R Type:
Union(Symbol,...)
+--E 16
+
+--S 17 of 46
+mainVariable(1 :: POLY INT)
+--R
+--R
+--R (17) "failed"
+--R Type:
Union("failed",...)
+--E 17
+
+--S 18 of 46
+ground? p
+--R
+--R
+--R (18) false
+--R Type:
Boolean
+--E 18
+
+--S 19 of 46
+ground?(1 :: POLY INT)
+--R
+--R
+--R (19) true
+--R Type:
Boolean
+--E 19
+
+--S 20 of 46
+variables p
+--R
+--R
+--R (20) [z,y,x]
+--R Type: List
Symbol
+--E 20
+
+--S 21 of 46
+degree(p,x)
+--R
+--R
+--R (21) 1
+--R Type:
PositiveInteger
+--E 21
+
+--S 22 of 46
+degree(p,y)
+--R
+--R
+--R (22) 2
+--R Type:
PositiveInteger
+--E 22
+
+--S 23 of 46
+degree(p,z)
+--R
+--R
+--R (23) 1
+--R Type:
PositiveInteger
+--E 23
+
+--S 24 of 46
+degree(p,[x,y,z])
+--R
+--R
+--R (24) [1,2,1]
+--R Type: List
NonNegativeInteger
+--E 24
+
+--S 25 of 46
+minimumDegree(p,z)
+--R
+--R
+--R (25) 1
+--R Type:
PositiveInteger
+--E 25
+
+--S 26 of 46
+totalDegree p
+--R
+--R
+--R (26) 4
+--R Type:
PositiveInteger
+--E 26
+
+--S 27 of 46
+leadingMonomial p
+--R
+--R
+--R 2
+--R (27) x y z
+--R Type: Polynomial
Integer
+--E 27
+
+--S 28 of 46
+reductum p
+--R
+--R
+--R (28) (- 2x y + x)z
+--R Type: Polynomial
Integer
+--E 28
+
+--S 29 of 46
+p - leadingMonomial p - reductum p
+--R
+--R
+--R (29) 0
+--R Type: Polynomial
Integer
+--E 29
+
+--S 30 of 46
+leadingCoefficient p
+--R
+--R
+--R (30) 1
+--R Type:
PositiveInteger
+--E 30
+
+--S 31 of 46
+p
+--R
+--R
+--R 2
+--R (31) (x y - 2x y + x)z
+--R Type: Polynomial
Integer
+--E 31
+
+--S 32 of 46
+eval(p,x,w)
+--R
+--R
+--R 2
+--R (32) (w y - 2w y + w)z
+--R Type: Polynomial
Integer
+--E 32
+
+--S 33 of 46
+eval(p,x,1)
+--R
+--R
+--R 2
+--R (33) (y - 2y + 1)z
+--R Type: Polynomial
Integer
+--E 33
+
+--S 34 of 46
+eval(p,x,y^2 - 1)
+--R
+--R
+--R 4 3
+--R (34) (y - 2y + 2y - 1)z
+--R Type: Polynomial
Integer
+--E 34
+
+--S 35 of 46
+D(p,x)
+--R
+--R
+--R 2
+--R (35) (y - 2y + 1)z
+--R Type: Polynomial
Integer
+--E 35
+
+--S 36 of 46
+D(p,y)
+--R
+--R
+--R (36) (2x y - 2x)z
+--R Type: Polynomial
Integer
+--E 36
+
+--S 37 of 46
+D(p,z)
+--R
+--R
+--R 2
+--R (37) x y - 2x y + x
+--R Type: Polynomial
Integer
+--E 37
+
+--S 38 of 46
+integrate(p,y)
+--R
+--R
+--R 1 3 2
+--R (38) (- x y - x y + x y)z
+--R 3
+--R Type: Polynomial Fraction
Integer
+--E 38
+
+--S 39 of 46
+qr := monicDivide(p,x+1,x)
+--R
+--R
+--R 2 2
+--R (39) [quotient= (y - 2y + 1)z,remainder= (- y + 2y - 1)z]
+--R Type: Record(quotient: Polynomial Integer,remainder: Polynomial
Integer)
+--E 39
+
+--S 40 of 46
+qr.remainder
+--R
+--R
+--R 2
+--R (40) (- y + 2y - 1)z
+--R Type: Polynomial
Integer
+--E 40
+
+--S 41 of 46
+p - ((x+1) * qr.quotient + qr.remainder)
+--R
+--R
+--R (41) 0
+--R Type: Polynomial
Integer
+--E 41
+
+--S 42 of 46
+p/q
+--R
+--R
+--R (y - 1)z
+--R (42) --------
+--R z + 5
+--R Type: Fraction Polynomial
Integer
+--E 42
+
+--S 43 of 46
+(2/3) * x**2 - y + 4/5
+--R
+--R
+--R 2 2 4
+--R (43) - y + - x + -
+--R 3 5
+--R Type: Polynomial Fraction
Integer
+--E 43
+
+--S 44 of 46
+% :: FRAC POLY INT
+--R
+--R
+--R 2
+--R - 15y + 10x + 12
+--R (44) -----------------
+--R 15
+--R Type: Fraction Polynomial
Integer
+--E 44
+
+--S 45 of 46
+% :: POLY FRAC INT
+--R
+--R
+--R 2 2 4
+--R (45) - y + - x + -
+--R 3 5
+--R Type: Polynomial Fraction
Integer
+--E 45
+
+--S 46 of 46
+map(numeric,%)
+--R
+--R
+--R 2
+--R (46) - 1.0 y + 0.6666666666 6666666667 x + 0.8
+--R Type: Polynomial
Float
+--E 46
+)spool
+)lisp (bye)
+@
+<<Polynomial.help>>=
+====================================================================
+Polynomial examples
+====================================================================
+
+The domain constructor Polynomial (abbreviation: POLY) provides
+polynomials with an arbitrary number of unspecified variables.
+
+It is used to create the default polynomial domains in Axiom. Here
+the coefficients are integers.
+
+ x + 1
+ x + 1
+ Type: Polynomial Integer
+
+Here the coefficients have type Float.
+
+ z - 2.3
+ z - 2.3
+ Type: Polynomial Float
+
+And here we have a polynomial in two variables with coefficients which
+have type Fraction Integer.
+
+ y**2 - z + 3/4
+ 2 3
+ - z + y + -
+ 4
+ Type: Polynomial Fraction Integer
+
+The representation of objects of domains created by Polynomial is that
+of recursive univariate polynomials. The term univariate means "one
+variable". The term multivariate means "possibly more than one variable".
+
+This recursive structure is sometimes obvious from the display of a polynomial.
+
+ y **2 + x*y + y
+ 2
+ y + (x + 1)y
+ Type: Polynomial Integer
+
+In this example, you see that the polynomial is stored as a polynomial
+in y with coefficients that are polynomials in x with integer coefficients.
+In fact, you really don't need to worry about the representation unless you
+are working on an advanced application where it is critical. The polynomial
+types created from DistributedMultivariatePolynomial and
+NewDistributedMultivariatePolynomial are stored and displayed in a
+non-recursive manner.
+
+You see a "flat" display of the above polynomial by converting to
+one of those types.
+
+ % :: DMP([y,x],INT)
+ 2
+ y + y x + y
+ Type: DistributedMultivariatePolynomial([y,x],Integer)
+
+We will demonstrate many of the polynomial facilities by using two
+polynomials with integer coefficients.
+
+By default, the interpreter expands polynomial expressions, even if they
+are written in a factored format.
+
+ p := (y-1)**2 * x * z
+ 2
+ (x y - 2x y + x)z
+ Type: Polynomial Integer
+
+See Factored to see how to create objects in factored form directly.
+
+ q := (y-1) * x * (z+5)
+ (x y - x)z + 5x y - 5x
+ Type: Polynomial Integer
+
+The fully factored form can be recovered by using factor.
+
+ factor(q)
+ x(y - 1)(z + 5)
+ Type: Factored Polynomial Integer
+
+This is the same name used for the operation to factor integers. Such
+reuse of names is called overloading and makes it much easier to think
+of solving problems in general ways. Axiom facilities for factoring
+polynomials created with Polynomial are currently restricted to the
+integer and rational number coefficient cases.
+
+The standard arithmetic operations are available for polynomials.
+
+ p - q**2
+ 2 2 2 2 2 2 2 2 2
+ (- x y + 2x y - x )z + ((- 10x + x)y + (20x - 2x)y - 10x + x)z
+ +
+ 2 2 2 2
+ - 25x y + 50x y - 25x
+ Type: Polynomial Integer
+
+The operation gcd is used to compute the greatest common divisor of
+two polynomials.
+
+ gcd(p,q)
+ x y - x
+ Type: Polynomial Integer
+
+In the case of p and q, the gcd is obvious from their definitions. We
+factor the gcd to show this relationship better.
+
+ factor %
+ x(y - 1)
+ Type: Factored Polynomial Integer
+
+The least common multiple is computed by using lcm.
+
+ lcm(p,q)
+ 2 2 2
+ (x y - 2x y + x)z + (5x y - 10x y + 5x)z
+ Type: Polynomial Integer
+
+Use content to compute the greatest common divisor of the coefficients
+of the polynomial.
+
+ content p
+ 1
+ Type: PositiveInteger
+
+Many of the operations on polynomials require you to specify a variable.
+For example, resultant requires you to give the variable in which the
+polynomials should be expressed.
+
+This computes the resultant of the values of p and q, considering them
+as polynomials in the variable z. They do not share a root when thought
+of as polynomials in z.
+
+ resultant(p,q,z)
+ 2 3 2 2 2 2
+ 5x y - 15x y + 15x y - 5x
+ Type: Polynomial Integer
+
+This value is 0 because as polynomials in x the polynomials have a
+common root.
+
+ resultant(p,q,x)
+ 0
+ Type: Polynomial Integer
+
+The data type used for the variables created by Polynomial is Symbol.
+As mentioned above, the representation used by Polynomial is recursive
+and so there is a main variable for nonconstant polynomials.
+
+The operation mainVariable returns this variable. The return type is
+actually a union of Symbol and "failed".
+
+ mainVariable p
+ z
+ Type: Union(Symbol,...)
+
+The latter branch of the union is be used if the polynomial has no
+variables, that is, is a constant.
+
+ mainVariable(1 :: POLY INT)
+ "failed"
+ Type: Union("failed",...)
+
+You can also use the predicate ground? to test whether a polynomial is
+in fact a member of its ground ring.
+
+ ground? p
+ false
+ Type: Boolean
+
+ ground?(1 :: POLY INT)
+ true
+ Type: Boolean
+
+The complete list of variables actually used in a particular polynomial
+is returned by variables. For constant polynomials, this list is empty.
+
+ variables p
+ [z,y,x]
+ Type: List Symbol
+
+The degree operation returns the degree of a polynomial in a specific variable.
+
+ degree(p,x)
+ 1
+ Type: PositiveInteger
+
+ degree(p,y)
+ 2
+ Type: PositiveInteger
+
+ degree(p,z)
+ 1
+ Type: PositiveInteger
+
+If you give a list of variables for the second argument, a list of the
+degrees in those variables is returned.
+
+ degree(p,[x,y,z])
+ [1,2,1]
+ Type: List NonNegativeInteger
+
+The minimum degree of a variable in a polynomial is computed using
+minimumDegree.
+
+ minimumDegree(p,z)
+ 1
+ Type: PositiveInteger
+
+The total degree of a polynomial is returned by totalDegree.
+
+ totalDegree p
+ 4
+ Type: PositiveInteger
+
+It is often convenient to think of a polynomial as a leading monomial plus
+the remaining terms.
+
+ leadingMonomial p
+ 2
+ x y z
+ Type: Polynomial Integer
+
+The reductum operation returns a polynomial consisting of the sum of
+the monomials after the first.
+
+ reductum p
+ (- 2x y + x)z
+ Type: Polynomial Integer
+
+These have the obvious relationship that the original polynomial is
+equal to the leading monomial plus the reductum.
+
+ p - leadingMonomial p - reductum p
+ 0
+ Type: Polynomial Integer
+
+The value returned by leadingMonomial includes the coefficient of that term.
+This is extracted by using leadingCoefficient on the original polynomial.
+
+ leadingCoefficient p
+ 1
+ Type: PositiveInteger
+
+The operation eval is used to substitute a value for a variable in a
+polynomial.
+
+ p
+ 2
+ (x y - 2x y + x)z
+ Type: Polynomial Integer
+
+This value may be another variable, a constant or a polynomial.
+
+ eval(p,x,w)
+ 2
+ (w y - 2w y + w)z
+ Type: Polynomial Integer
+
+ eval(p,x,1)
+ 2
+ (y - 2y + 1)z
+ Type: Polynomial Integer
+
+Actually, all the things being substituted are just polynomials,
+some more trivial than others.
+
+ eval(p,x,y^2 - 1)
+ 4 3
+ (y - 2y + 2y - 1)z
+ Type: Polynomial Integer
+
+Derivatives are computed using the D operation.
+
+ D(p,x)
+ 2
+ (y - 2y + 1)z
+ Type: Polynomial Integer
+
+The first argument is the polynomial and the second is the variable.
+
+ D(p,y)
+ (2x y - 2x)z
+ Type: Polynomial Integer
+
+Even if the polynomial has only one variable, you must specify it.
+
+ D(p,z)
+ 2
+ x y - 2x y + x
+ Type: Polynomial Integer
+
+Integration of polynomials is similar and the integrate operation is used.
+
+Integration requires that the coefficients support division. Axiom
+converts polynomials over the integers to polynomials over the rational
+numbers before integrating them.
+
+ integrate(p,y)
+ 1 3 2
+ (- x y - x y + x y)z
+ 3
+ Type: Polynomial Fraction Integer
+
+It is not possible, in general, to divide two polynomials. In our
+example using polynomials over the integers, the operation monicDivide
+divides a polynomial by a monic polynomial (that is, a polynomial with
+leading coefficient equal to 1). The result is a record of the
+quotient and remainder of the division.
+
+You must specify the variable in which to express the polynomial.
+
+ qr := monicDivide(p,x+1,x)
+ 2 2
+ [quotient= (y - 2y + 1)z,remainder= (- y + 2y - 1)z]
+ Type: Record(quotient: Polynomial Integer,remainder: Polynomial Integer)
+
+The selectors of the components of the record are quotient and remainder.
+Issue this to extract the remainder.
+
+ qr.remainder
+ 2
+ (- y + 2y - 1)z
+ Type: Polynomial Integer
+
+Now that we can extract the components, we can demonstrate the
+relationship among them and the arguments to our original expression
+qr := monicDivide(p,x+1,x).
+
+ p - ((x+1) * qr.quotient + qr.remainder)
+ 0
+ Type: Polynomial Integer
+
+If the / operator is used with polynomials, a fraction object is
+created. In this example, the result is an object of type
+Fraction Polynomial Integer.
+
+ p/q
+ (y - 1)z
+ --------
+ z + 5
+ Type: Fraction Polynomial Integer
+
+If you use rational numbers as polynomial coefficients, the
+resulting object is of type Polynomial Fraction Integer.
+
+ (2/3) * x**2 - y + 4/5
+ 2 2 4
+ - y + - x + -
+ 3 5
+ Type: Polynomial Fraction Integer
+
+This can be converted to a fraction of polynomials and back again, if
+required.
+
+ % :: FRAC POLY INT
+ 2
+ - 15y + 10x + 12
+ -----------------
+ 15
+ Type: Fraction Polynomial Integer
+
+ % :: POLY FRAC INT
+ 2 2 4
+ - y + - x + -
+ 3 5
+ Type: Polynomial Fraction Integer
+
+To convert the coefficients to floating point, map the numeric
+operation on the coefficients of the polynomial.
+
+ map(numeric,%)
+ - 1.0 y + 0.6666666666 6666666667 x + 0.8
+ Type: Polynomial Float
+
+See Also:
+o )help Factored
+o )help UnivariatePolynomial
+o )help MultivariatePolynomial
+o )help DistributedMultivariatePolynomial
+o )help NewDistributedMultivariatePolynomial
+o )show Polynomial
+o $AXIOM/doc/src/algebra/multpoly.spad.dvi
+
+@
<<domain POLY Polynomial>>=
)abbrev domain POLY Polynomial
++ Author: Dave Barton, Barry Trager
diff --git a/src/algebra/opalg.spad.pamphlet b/src/algebra/opalg.spad.pamphlet
index a03c8ce..8eab785 100644
--- a/src/algebra/opalg.spad.pamphlet
+++ b/src/algebra/opalg.spad.pamphlet
@@ -238,6 +238,392 @@ ModuleOperator(R: Ring, M:LeftModule(R)): Exports ==
Implementation where
@
\section{domain OP Operator}
+<<Operator.input>>=
+-- opalg.spad.pamphlet Operator.input
+)spool Operator.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 21
+R := SQMATRIX(2, INT)
+--R
+--R
+--R (1) SquareMatrix(2,Integer)
+--R Type:
Domain
+--E 1
+
+--S 2 of 21
+t := operator("tilde") :: OP(R)
+--R
+--R
+--R (2) tilde
+--R Type: Operator
SquareMatrix(2,Integer)
+--E 2
+
+--S 3 of 21
+)set expose add constructor Operator
+--R
+--R Operator is now explicitly exposed in frame frame0
+--E 3
+
+--S 4 of 21
+evaluate(t, m +-> transpose m)
+--R
+--R
+--R (3) tilde
+--R Type: Operator
SquareMatrix(2,Integer)
+--E 4
+
+--S 5 of 21
+s : R := matrix [ [0, 1], [1, 0] ]
+--R
+--R
+--R +0 1+
+--R (4) | |
+--R +1 0+
+--R Type:
SquareMatrix(2,Integer)
+--E 5
+
+--S 6 of 21
+rho := t * s
+--R
+--R
+--R +0 1+
+--R (5) tilde| |
+--R +1 0+
+--R Type: Operator
SquareMatrix(2,Integer)
+--E 6
+
+--S 7 of 21
+z := rho**4 - 1
+--R
+--R
+--R +0 1+ +0 1+ +0 1+ +0 1+
+--R (6) - 1 + tilde| |tilde| |tilde| |tilde| |
+--R +1 0+ +1 0+ +1 0+ +1 0+
+--R Type: Operator
SquareMatrix(2,Integer)
+--E 7
+
+--S 8 of 21
+m:R := matrix [ [1, 2], [3, 4] ]
+--R
+--R
+--R +1 2+
+--R (7) | |
+--R +3 4+
+--R Type:
SquareMatrix(2,Integer)
+--E 8
+
+--S 9 of 21
+z m
+--R
+--R
+--R +0 0+
+--R (8) | |
+--R +0 0+
+--R Type:
SquareMatrix(2,Integer)
+--E 9
+
+--S 10 of 21
+rho m
+--R
+--R
+--R +3 1+
+--R (9) | |
+--R +4 2+
+--R Type:
SquareMatrix(2,Integer)
+--E 10
+
+--S 11 of 21
+rho rho m
+--R
+--R
+--R +4 3+
+--R (10) | |
+--R +2 1+
+--R Type:
SquareMatrix(2,Integer)
+--E 11
+
+--S 12 of 21
+(rho^3) m
+--R
+--R
+--R +2 4+
+--R (11) | |
+--R +1 3+
+--R Type:
SquareMatrix(2,Integer)
+--E 12
+
+--S 13 of 21
+b := t * s - s * t
+--R
+--R
+--R +0 1+ +0 1+
+--R (12) - | |tilde + tilde| |
+--R +1 0+ +1 0+
+--R Type: Operator
SquareMatrix(2,Integer)
+--E 13
+
+--S 14 of 21
+b m
+--R
+--R
+--R +1 - 3+
+--R (13) | |
+--R +3 - 1+
+--R Type:
SquareMatrix(2,Integer)
+--E 14
+
+--S 15 of 21
+L n ==
+ n = 0 => 1
+ n = 1 => x
+ (2*n-1)/n * x * L(n-1) - (n-1)/n * L(n-2)
+--R
+--R Type:
Void
+--E 15
+
+--S 16 of 21
+dx := operator("D") :: OP(POLY FRAC INT)
+--R
+--R
+--R (15) D
+--R Type: Operator Polynomial Fraction
Integer
+--E 16
+
+--S 17 of 21
+evaluate(dx, p +-> D(p, 'x))
+--R
+--R
+--R (16) D
+--R Type: Operator Polynomial Fraction
Integer
+--E 17
+
+--S 18 of 21
+E n == (1 - x**2) * dx**2 - 2 * x * dx + n*(n+1)
+--R
+--R Type:
Void
+--E 18
+
+--S 19 of 21
+L 15
+--R
+--R Compiling function L with type Integer -> Polynomial Fraction
+--R Integer
+--R Compiling function L as a recurrence relation.
+--R
+--R (18)
+--R 9694845 15 35102025 13 50702925 11 37182145 9 14549535 7
+--R ------- x - -------- x + -------- x - -------- x + -------- x
+--R 2048 2048 2048 2048 2048
+--R +
+--R 2909907 5 255255 3 6435
+--R - ------- x + ------ x - ---- x
+--R 2048 2048 2048
+--R Type: Polynomial Fraction
Integer
+--E 19
+
+--S 20 of 21
+E 15
+--R
+--R Compiling function E with type PositiveInteger -> Operator
+--R Polynomial Fraction Integer
+--R
+--R 2 2
+--R (19) 240 - 2x D - (x - 1)D
+--R Type: Operator Polynomial Fraction
Integer
+--E 20
+
+--S 21 of 21
+(E 15)(L 15)
+--R
+--R
+--R (20) 0
+--R Type: Polynomial Fraction
Integer
+--E 21
+)spool
+)lisp (bye)
+@
+<<Operator.help>>=
+====================================================================
+Operator examples
+====================================================================
+
+Given any ring R, the ring of the Integer-linear operators over R is
+called Operator(R). To create an operator over R, first create a
+basic operator using the operation operator, and then convert it to
+Operator(R) for the R you want.
+
+We choose R to be the two by two matrices over the integers.
+
+ R := SQMATRIX(2, INT)
+ SquareMatrix(2,Integer)
+ Type: Domain
+
+Create the operator tilde on R.
+
+ t := operator("tilde") :: OP(R)
+ tilde
+ Type: Operator SquareMatrix(2,Integer)
+
+Since Operator is unexposed we must either package-call operations
+from it, or expose it explicitly. For convenience we will do the latter.
+
+Expose Operator.
+
+ )set expose add constructor Operator
+
+To attach an evaluation function (from R to R) to an operator over R,
+use evaluate(op, f) where op is an operator over R and f is a function
+R -> R. This needs to be done only once when the operator is defined.
+Note that f must be Integer-linear (that is, f(ax+y) = a f(x) + f(y)
+for any integer a, and any x and y in R).
+
+We now attach the transpose map to the above operator t.
+
+ evaluate(t, m +-> transpose m)
+ tilde
+ Type: Operator SquareMatrix(2,Integer)
+
+Operators can be manipulated formally as in any ring: + is the
+pointwise addition and * is composition. Any element x of R can be
+converted to an operator op(x) over R, and the evaluation function of
+op(x) is left-multiplication by x.
+
+Multiplying on the left by this matrix swaps the two rows.
+
+ s : R := matrix [ [0, 1], [1, 0] ]
+ +0 1+
+ | |
+ +1 0+
+ Type: SquareMatrix(2,Integer)
+
+Can you guess what is the action of the following operator?
+
+ rho := t * s
+ +0 1+
+ tilde| |
+ +1 0+
+ Type: Operator SquareMatrix(2,Integer)
+
+Hint: applying rho four times gives the identity, so rho^4-1 should
+return 0 when applied to any two by two matrix.
+
+ z := rho**4 - 1
+ +0 1+ +0 1+ +0 1+ +0 1+
+ - 1 + tilde| |tilde| |tilde| |tilde| |
+ +1 0+ +1 0+ +1 0+ +1 0+
+ Type: Operator SquareMatrix(2,Integer)
+
+Now check with this matrix.
+
+ m:R := matrix [ [1, 2], [3, 4] ]
+ +1 2+
+ | |
+ +3 4+
+ Type: SquareMatrix(2,Integer)
+
+ z m
+ +0 0+
+ | |
+ +0 0+
+ Type: SquareMatrix(2,Integer)
+
+As you have probably guessed by now, rho acts on matrices by rotating
+the elements clockwise.
+
+ rho m
+ +3 1+
+ | |
+ +4 2+
+ Type: SquareMatrix(2,Integer)
+
+ rho rho m
+ +4 3+
+ | |
+ +2 1+
+ Type: SquareMatrix(2,Integer)
+
+ (rho^3) m
+ +2 4+
+ | |
+ +1 3+
+ Type: SquareMatrix(2,Integer)
+
+Do the swapping of rows and transposition commute? We can check by
+computing their bracket.
+
+ b := t * s - s * t
+ +0 1+ +0 1+
+ - | |tilde + tilde| |
+ +1 0+ +1 0+
+ Type: Operator SquareMatrix(2,Integer)
+
+Now apply it to m.
+
+ b m
+ +1 - 3+
+ | |
+ +3 - 1+
+ Type: SquareMatrix(2,Integer)
+
+Next we demonstrate how to define a differential operator on a polynomial ring.
+
+This is the recursive definition of the n-th Legendre polynomial.
+
+ L n ==
+ n = 0 => 1
+ n = 1 => x
+ (2*n-1)/n * x * L(n-1) - (n-1)/n * L(n-2)
+ Type: Void
+
+Create the differential operator d/dx on polynomials in x over the
+rational numbers.
+
+ dx := operator("D") :: OP(POLY FRAC INT)
+ D
+ Type: Operator Polynomial Fraction Integer
+
+Now attach the map to it.
+
+ evaluate(dx, p +-> D(p, 'x))
+ D
+ Type: Operator Polynomial Fraction Integer
+
+This is the differential equation satisfied by the n-th Legendre polynomial.
+
+ E n == (1 - x**2) * dx**2 - 2 * x * dx + n*(n+1)
+ Type: Void
+
+Now we verify this for n = 15. Here is the polynomial.
+
+ L 15
+ 9694845 15 35102025 13 50702925 11 37182145 9 14549535 7
+ ------- x - -------- x + -------- x - -------- x + -------- x
+ 2048 2048 2048 2048 2048
+ +
+ 2909907 5 255255 3 6435
+ - ------- x + ------ x - ---- x
+ 2048 2048 2048
+ Type: Polynomial Fraction Integer
+
+Here is the operator.
+
+ E 15
+ 240 - 2x D - (x - 1)D
+ Type: Operator Polynomial Fraction Integer
+
+Here is the evaluation.
+
+ (E 15)(L 15)
+ 0
+ Type: Polynomial Fraction Integer
+
+See Also:
+o )show Operator
+o $AXIOM/doc/src/algebra/opalg.spad.dvi
+
+@
<<domain OP Operator>>=
)abbrev domain OP Operator
++ Author: Manuel Bronstein
diff --git a/src/algebra/perman.spad.pamphlet b/src/algebra/perman.spad.pamphlet
index bc2f63f..0e456a2 100644
--- a/src/algebra/perman.spad.pamphlet
+++ b/src/algebra/perman.spad.pamphlet
@@ -86,6 +86,83 @@ GrayCode: public == private where
@
\section{package PERMAN Permanent}
+<<Permanent.input>>=
+-- perman.spad.pamphlet Permanent.input
+)spool Permanent.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 3
+kn n ==
+ r : MATRIX INT := new(n,n,1)
+ for i in 1..n repeat
+ r.i.i := 0
+ r
+--R
+--R Type:
Void
+--E 1
+
+--S 2 of 3
+permanent(kn(5) :: SQMATRIX(5,INT))
+--R
+--R Compiling function kn with type PositiveInteger -> Matrix Integer
+--R
+--R (2) 44
+--R Type:
PositiveInteger
+--E 2
+
+--S 3 of 3
+[permanent(kn(n) :: SQMATRIX(n,INT)) for n in 1..13]
+--R
+--R Cannot compile conversion for types involving local variables. In
+--R particular, could not compile the expression involving ::
+--R SQMATRIX(n,INT)
+--R AXIOM will attempt to step through and interpret the code.
+--R
+--R (3)
+--R [0,1,2,9,44,265,1854,14833,133496,1334961,14684570,176214841,2290792932]
+--R Type: List
NonNegativeInteger
+--E 3
+)spool
+)lisp (bye)
+@
+<<Permanent.help>>=
+====================================================================
+Permanent examples
+====================================================================
+
+The package Permanent provides the function permanent for square
+matrices. The permanent of a square matrix can be computed in the
+same way as the determinant by expansion of minors except that for the
+permanent the sign for each element is 1, rather than being 1 if the
+row plus column indices is positive and -1 otherwise. This function
+is much more difficult to compute efficiently than the determinant.
+An example of the use of permanent is the calculation of the n-th
+derangement number, defined to be the number of different possibilities
+for n couples to dance but never with their own spouse.
+
+Consider an n by n matrix with entries 0 on the diagonal and 1 elsewhere.
+Think of the rows as one-half of each couple (for example, the males) and
+the columns the other half. The permanent of such a matrix gives the
+desired derangement number.
+
+ kn n ==
+ r : MATRIX INT := new(n,n,1)
+ for i in 1..n repeat
+ r.i.i := 0
+ r
+
+Here are some derangement numbers, which you see grow quite fast.
+
+ permanent(kn(5) :: SQMATRIX(5,INT))
+
+ [permanent(kn(n) :: SQMATRIX(n,INT)) for n in 1..13]
+
+See Also:
+o )show Permanent
+o $AXIOM/doc/src/algebra/perman.spad.dvi
+
+@
<<package PERMAN Permanent>>=
)abbrev package PERMAN Permanent
++ Authors: Johannes Grabmeier, Oswald Gschnitzer
diff --git a/src/algebra/pfr.spad.pamphlet b/src/algebra/pfr.spad.pamphlet
index 8286147..ce53bf8 100644
--- a/src/algebra/pfr.spad.pamphlet
+++ b/src/algebra/pfr.spad.pamphlet
@@ -10,6 +10,279 @@
\tableofcontents
\eject
\section{domain PFR PartialFraction}
+<<PartialFraction.input>>=
+-- pfr.spad.pamphlet PartialFraction.input
+)spool PartialFraction.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 10
+partialFraction(1,factorial 10)
+--R
+--R
+--R 159 23 12 1
+--R (1) --- - -- - -- + -
+--R 8 4 2 7
+--R 2 3 5
+--R Type: PartialFraction
Integer
+--E 1
+
+--S 2 of 10
+f := padicFraction(%)
+--R
+--R
+--R 1 1 1 1 1 1 2 1 2 2 2 1
+--R (2) - + -- + -- + -- + -- + -- - -- - -- - -- - - - -- + -
+--R 2 4 5 6 7 8 2 3 4 5 2 7
+--R 2 2 2 2 2 3 3 3 5
+--R Type: PartialFraction
Integer
+--E 2
+
+--S 3 of 10
+compactFraction(f)
+--R
+--R
+--R 159 23 12 1
+--R (3) --- - -- - -- + -
+--R 8 4 2 7
+--R 2 3 5
+--R Type: PartialFraction
Integer
+--E 3
+
+--S 4 of 10
+numberOfFractionalTerms(f)
+--R
+--R
+--R (4) 12
+--R Type:
PositiveInteger
+--E 4
+
+--S 5 of 10
+nthFractionalTerm(f,3)
+--R
+--R
+--R 1
+--R (5) --
+--R 5
+--R 2
+--R Type: PartialFraction
Integer
+--E 5
+
+--S 6 of 10
+partialFraction(1,- 13 + 14 * %i)
+--R
+--R
+--R 1 4
+--R (6) - ------- + -------
+--R 1 + 2%i 3 + 8%i
+--R Type: PartialFraction Complex
Integer
+--E 6
+
+--S 7 of 10
+% :: Fraction Complex Integer
+--R
+--R
+--R %i
+--R (7) - ---------
+--R 14 + 13%i
+--R Type: Fraction Complex
Integer
+--E 7
+
+--S 8 of 10
+u : FR UP(x, FRAC INT) := reduce(*,[primeFactor(x+i,i) for i in 1..4])
+--R
+--R
+--R 2 3 4
+--R (8) (x + 1)(x + 2) (x + 3) (x + 4)
+--R Type: Factored UnivariatePolynomial(x,Fraction
Integer)
+--E 8
+
+--S 9 of 10
+partialFraction(1,u)
+--R
+--R
+--R (9)
+--R 1 1 7 17 2 139 607 3 10115 2 391
44179
+--R --- - x + -- - -- x - 12x - --- --- x + ----- x + --- x +
-----
+--R 648 4 16 8 8 324 432 4 324
+--R ----- + -------- + ------------------- +
---------------------------------
+--R x + 1 2 3 4
+--R (x + 2) (x + 3) (x + 4)
+--R Type: PartialFraction UnivariatePolynomial(x,Fraction
Integer)
+--E 9
+
+--S 10 of 10
+padicFraction %
+--R
+--R
+--R (10)
+--R 1 1 1 17 3 1 607 403
+--R --- - -- -- - - --- ---
+--R 648 4 16 8 4 2 324 432
+--R ----- + ----- - -------- - ----- + -------- - -------- + ----- +
--------
+--R x + 1 x + 2 2 x + 3 2 3 x + 4
2
+--R (x + 2) (x + 3) (x + 3) (x +
4)
+--R +
+--R 13 1
+--R -- --
+--R 36 12
+--R -------- + --------
+--R 3 4
+--R (x + 4) (x + 4)
+--R Type: PartialFraction UnivariatePolynomial(x,Fraction
Integer)
+--E 10
+)spool
+)lisp (bye)
+@
+<<PartialFraction.help>>=
+====================================================================
+PartialFraction examples
+====================================================================
+
+A partial fraction is a decomposition of a quotient into a sum of
+quotients where the denominators of the summands are powers of
+primes. Most people first encounter partial fractions when they are
+learning integral calculus. For a technical discussion of partial
+fractions, see, for example, Lang's Algebra. For example, the rational
+number 1/6 is decomposed into 1/2-1/3. You can compute partial
+fractions of quotients of objects from domains belonging to the
+category EuclideanDomain. For example, Integer, Complex Integer, and
+UnivariatePolynomial(x, Fraction Integer) all belong to EuclideanDomain.
+In the examples following, we demonstrate how to decompose quotients of
+each of these kinds of object into partial fractions.
+
+It is necessary that we know how to factor the denominator when we
+want to compute a partial fraction. Although the interpreter can
+often do this automatically, it may be necessary for you to include a
+call to factor. In these examples, it is not necessary to factor the
+denominators explicitly.
+
+The main operation for computing partial fractions is called partialFraction
+and we use this to compute a decomposition of 1/10!. The first argument to
+partialFraction is the numerator of the quotient and the second argument is
+the factored denominator.
+
+ partialFraction(1,factorial 10)
+ 159 23 12 1
+ --- - -- - -- + -
+ 8 4 2 7
+ 2 3 5
+ Type: PartialFraction Integer
+
+Since the denominators are powers of primes, it may be possible to
+expand the numerators further with respect to those primes. Use the
+operation padicFraction to do this.
+
+ f := padicFraction(%)
+ 1 1 1 1 1 1 2 1 2 2 2 1
+ - + -- + -- + -- + -- + -- - -- - -- - -- - - - -- + -
+ 2 4 5 6 7 8 2 3 4 5 2 7
+ 2 2 2 2 2 3 3 3 5
+ Type: PartialFraction Integer
+
+The operation compactFraction returns an expanded fraction into the usual
+form. The compacted version is used internally for computational efficiency.
+
+ compactFraction(f)
+ 159 23 12 1
+ --- - -- - -- + -
+ 8 4 2 7
+ 2 3 5
+ Type: PartialFraction Integer
+
+You can add, subtract, multiply and divide partial fractions. In addition,
+you can extract the parts of the decomposition. numberOfFractionalTerms
+computes the number of terms in the fractional part. This does not include
+the whole part of the fraction, which you get by calling wholePart. In
+this example, the whole part is just 0.
+
+ numberOfFractionalTerms(f)
+ 12
+ Type: PositiveInteger
+
+The operation nthFractionalTerm returns the individual terms in the
+decomposition. Notice that the object returned is a partial fraction
+itself. firstNumer and firstDenom extract the numerator and denominator
+of the first term of the fraction.
+
+ nthFractionalTerm(f,3)
+ 1
+ --
+ 5
+ 2
+ Type: PartialFraction Integer
+
+Given two gaussian integers, you can decompose their quotient into a
+partial fraction.
+
+ partialFraction(1,- 13 + 14 * %i)
+ 1 4
+ - ------- + -------
+ 1 + 2%i 3 + 8%i
+ Type: PartialFraction Complex Integer
+
+To convert back to a quotient, simply use a conversion.
+
+ % :: Fraction Complex Integer
+ %i
+ - ---------
+ 14 + 13%i
+ Type: Fraction Complex Integer
+
+To conclude this section, we compute the decomposition of
+
+ 1
+ -------------------------------
+ 2 3 4
+ (x + 1)(x + 2) (x + 3) (x + 4)
+
+
+The polynomials in this object have type
+UnivariatePolynomial(x, Fraction Integer).
+
+We use the primeFactor operation to create the denominator in factored
+form directly.
+
+ u : FR UP(x, FRAC INT) := reduce(*,[primeFactor(x+i,i) for i in 1..4])
+ 2 3 4
+ (x + 1)(x + 2) (x + 3) (x + 4)
+ Type: Factored UnivariatePolynomial(x,Fraction Integer)
+
+These are the compact and expanded partial fractions for the quotient.
+
+ partialFraction(1,u)
+ 1 1 7 17 2 139 607 3 10115 2 391 44179
+ --- - x + -- - -- x - 12x - --- --- x + ----- x + --- x + -----
+ 648 4 16 8 8 324 432 4 324
+ ----- + -------- + ------------------- + ---------------------------------
+ x + 1 2 3 4
+ (x + 2) (x + 3) (x + 4)
+ Type: PartialFraction UnivariatePolynomial(x,Fraction Integer)
+
+ padicFraction %
+ 1 1 1 17 3 1 607 403
+ --- - -- -- - - --- ---
+ 648 4 16 8 4 2 324 432
+ ----- + ----- - -------- - ----- + -------- - -------- + ----- + --------
+ x + 1 x + 2 2 x + 3 2 3 x + 4 2
+ (x + 2) (x + 3) (x + 3) (x + 4)
+ +
+ 13 1
+ -- --
+ 36 12
+ -------- + --------
+ 3 4
+ (x + 4) (x + 4)
+ Type: PartialFraction UnivariatePolynomial(x,Fraction Integer)
+
+See Also:
+o )help Factored
+o )help Complex
+o )help FullPartialFractionExpansionXmpPage
+o )show PartialFraction
+o $AXIOM/doc/src/algebra/pfr.spad.dvi
+
+@
<<domain PFR PartialFraction>>=
)abbrev domain PFR PartialFraction
++ Author: Robert S. Sutor
diff --git a/src/algebra/quat.spad.pamphlet b/src/algebra/quat.spad.pamphlet
index ca56315..ebb8941 100644
--- a/src/algebra/quat.spad.pamphlet
+++ b/src/algebra/quat.spad.pamphlet
@@ -5233,6 +5233,245 @@ QuaternionCategory(R: CommutativeRing): Category ==
@
\section{domain QUAT Quaternion}
+<<Quaternion.input>>=
+-- quat.spad.pamphlet Quaternion.input
+)spool Quaternion.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 13
+q := quatern(2/11,-8,3/4,1)
+--R
+--R
+--R 2 3
+--R (1) -- - 8i + - j + k
+--R 11 4
+--R Type: Quaternion Fraction
Integer
+--E 1
+
+--S 2 of 13
+[real q, imagI q, imagJ q, imagK q]
+--R
+--R
+--R 2 3
+--R (2) [--,- 8,-,1]
+--R 11 4
+--R Type: List Fraction
Integer
+--E 2
+
+--S 3 of 13
+inv q
+--R
+--R
+--R 352 15488 484 1936
+--R (3) ------ + ------ i - ----- j - ------ k
+--R 126993 126993 42331 126993
+--R Type: Quaternion Fraction
Integer
+--E 3
+
+--S 4 of 13
+q^6
+--R
+--R
+--R 2029490709319345 48251690851 144755072553 48251690851
+--R (4) - ---------------- - ----------- i + ------------ j + ----------- k
+--R 7256313856 1288408 41229056 10307264
+--R Type: Quaternion Fraction
Integer
+--E 4
+
+--S 5 of 13
+r := quatern(-2,3,23/9,-89); q + r
+--R
+--R
+--R 20 119
+--R (5) - -- - 5i + --- j - 88k
+--R 11 36
+--R Type: Quaternion Fraction
Integer
+--E 5
+
+--S 6 of 13
+q * r - r * q
+--R
+--R
+--R 2495 817
+--R (6) - ---- i - 1418j - --- k
+--R 18 18
+--R Type: Quaternion Fraction
Integer
+--E 6
+
+--S 7 of 13
+i:=quatern(0,1,0,0)
+--R
+--R
+--R (7) i
+--R Type: Quaternion
Integer
+--E 7
+
+--S 8 of 13
+j:=quatern(0,0,1,0)
+--R
+--R
+--R (8) j
+--R Type: Quaternion
Integer
+--E 8
+
+--S 9 of 13
+k:=quatern(0,0,0,1)
+--R
+--R
+--R (9) k
+--R Type: Quaternion
Integer
+--E 9
+
+--S 10 of 13
+[i*i, j*j, k*k, i*j, j*k, k*i, q*i]
+--R
+--R
+--R 2 3
+--R (10) [- 1,- 1,- 1,k,i,j,8 + -- i + j - - k]
+--R 11 4
+--R Type: List Quaternion Fraction
Integer
+--E 10
+
+--S 11 of 13
+norm q
+--R
+--R
+--R 126993
+--R (11) ------
+--R 1936
+--R Type: Fraction
Integer
+--E 11
+
+--S 12 of 13
+conjugate q
+--R
+--R
+--R 2 3
+--R (12) -- + 8i - - j - k
+--R 11 4
+--R Type: Quaternion Fraction
Integer
+--E 12
+
+--S 13 of 13
+q * %
+--R
+--R
+--R 126993
+--R (13) ------
+--R 1936
+--R Type: Quaternion Fraction
Integer
+--E 13
+)spool
+)lisp (bye)
+@
+<<Quaternion.help>>=
+====================================================================
+Quaternion examples
+====================================================================
+
+The domain constructor Quaternion implements quaternions over
+commutative rings.
+
+The basic operation for creating quaternions is quatern. This is a
+quaternion over the rational numbers.
+
+ q := quatern(2/11,-8,3/4,1)
+ 2 3
+ -- - 8i + - j + k
+ 11 4
+ Type: Quaternion Fraction Integer
+
+The four arguments are the real part, the i imaginary part, the j
+imaginary part, and the k imaginary part, respectively.
+
+ [real q, imagI q, imagJ q, imagK q]
+ 2 3
+ [--,- 8,-,1]
+ 11 4
+ Type: List Fraction Integer
+
+Because q is over the rationals (and nonzero), you can invert it.
+
+ inv q
+ 352 15488 484 1936
+ ------ + ------ i - ----- j - ------ k
+ 126993 126993 42331 126993
+ Type: Quaternion Fraction Integer
+
+The usual arithmetic (ring) operations are available
+
+ q^6
+ 2029490709319345 48251690851 144755072553 48251690851
+ - ---------------- - ----------- i + ------------ j + ----------- k
+ 7256313856 1288408 41229056 10307264
+ Type: Quaternion Fraction Integer
+
+ r := quatern(-2,3,23/9,-89); q + r
+ 20 119
+ - -- - 5i + --- j - 88k
+ 11 36
+ Type: Quaternion Fraction Integer
+
+In general, multiplication is not commutative.
+
+ q * r - r * q
+ 2495 817
+ - ---- i - 1418j - --- k
+ 18 18
+ Type: Quaternion Fraction Integer
+
+There are no predefined constants for the imaginary i, j, and k parts,
+but you can easily define them.
+
+ i:=quatern(0,1,0,0)
+ i
+ Type: Quaternion Integer
+
+ j:=quatern(0,0,1,0)
+ j
+ Type: Quaternion Integer
+
+ k:=quatern(0,0,0,1)
+ k
+ Type: Quaternion Integer
+
+These satisfy the normal identities.
+
+ [i*i, j*j, k*k, i*j, j*k, k*i, q*i]
+ 2 3
+ [- 1,- 1,- 1,k,i,j,8 + -- i + j - - k]
+ 11 4
+ Type: List Quaternion Fraction Integer
+
+The norm is the quaternion times its conjugate.
+
+ norm q
+ 126993
+ ------
+ 1936
+ Type: Fraction Integer
+
+ conjugate q
+ 2 3
+ -- + 8i - - j - k
+ 11 4
+ Type: Quaternion Fraction Integer
+
+ q * %
+ 126993
+ ------
+ 1936
+ Type: Quaternion Fraction Integer
+
+See Also:
+o )help Octonion
+o )help Complex
+o )help CliffordAlgebra
+o )show Quaternion
+o $AXIOM/doc/src/algebra/quat.spad.dvi
+
+@
<<domain QUAT Quaternion>>=
)abbrev domain QUAT Quaternion
++ Author: Robert S. Sutor
diff --git a/src/algebra/radix.spad.pamphlet b/src/algebra/radix.spad.pamphlet
index 3e536be..98364b1 100644
--- a/src/algebra/radix.spad.pamphlet
+++ b/src/algebra/radix.spad.pamphlet
@@ -10,6 +10,286 @@
\tableofcontents
\eject
\section{domain RADIX RadixExpansion}
+<<RadixExpansion.input>>=
+-- radix.spad.pamphlet RadixExpansion.input
+)spool RadixExpansion.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 17
+111::RadixExpansion(5)
+--R
+--R
+--R (1) 421
+--R Type: RadixExpansion
5
+--E 1
+
+--S 2 of 17
+(5/24)::RadixExpansion(2)
+--R
+--R
+--R __
+--R (2) 0.00110
+--R Type: RadixExpansion
2
+--E 2
+
+--S 3 of 17
+(5/24)::RadixExpansion(3)
+--R
+--R
+--R __
+--R (3) 0.012
+--R Type: RadixExpansion
3
+--E 3
+
+--S 4 of 17
+(5/24)::RadixExpansion(8)
+--R
+--R
+--R __
+--R (4) 0.152
+--R Type: RadixExpansion
8
+--E 4
+
+--S 5 of 17
+(5/24)::RadixExpansion(10)
+--R
+--R
+--R _
+--R (5) 0.2083
+--R Type: RadixExpansion
10
+--E 5
+
+--S 6 of 17
+(5/24)::RadixExpansion(12)
+--R
+--R
+--R (6) 0.26
+--R Type: RadixExpansion
12
+--E 6
+
+--S 7 of 17
+(5/24)::RadixExpansion(16)
+--R
+--R
+--R _
+--R (7) 0.35
+--R Type: RadixExpansion
16
+--E 7
+
+--S 8 of 17
+(5/24)::RadixExpansion(36)
+--R
+--R
+--R (8) 0.7I
+--R Type: RadixExpansion
36
+--E 8
+
+--S 9 of 17
+(5/24)::RadixExpansion(38)
+--R
+--R
+--R _____
+--R (9) 0 . 7 34 31 25 12
+--R Type: RadixExpansion
38
+--E 9
+
+--S 10 of 17
+a := (76543/210)::RadixExpansion(8)
+--R
+--R
+--R ____
+--R (10) 554.37307
+--R Type: RadixExpansion
8
+--E 10
+
+--S 11 of 17
+w := wholeRagits a
+--R
+--R
+--R (11) [5,5,4]
+--R Type: List
Integer
+--E 11
+
+--S 12 of 17
+f0 := prefixRagits a
+--R
+--R
+--R (12) [3]
+--R Type: List
Integer
+--E 12
+
+--S 13 of 17
+f1 := cycleRagits a
+--R
+--R
+--R (13) [7,3,0,7]
+--R Type: List
Integer
+--E 13
+
+--S 14 of 17
+u:RadixExpansion(8):=wholeRadix(w)+fractRadix(f0,f1)
+--R
+--R
+--R ____
+--R (14) 554.37307
+--R Type: RadixExpansion
8
+--E 14
+
+--S 15 of 17
+v: RadixExpansion(12) := fractRadix([1,2,3,11], [0])
+--R
+--R
+--R _
+--R (15) 0.123B0
+--R Type: RadixExpansion
12
+--E 15
+
+--S 16 of 17
+fractRagits(u)
+--R
+--R
+--R _______
+--R (16) [3,7,3,0,7,7]
+--R Type: Stream
Integer
+--E 16
+
+--S 17 of 17
+a :: Fraction(Integer)
+--R
+--R
+--R 76543
+--R (17) -----
+--R 210
+--R Type: Fraction
Integer
+--E 17
+)spool
+)lisp (bye)
+@
+<<RadixExpansion.help>>=
+====================================================================
+RadixExpansion examples
+====================================================================
+
+It possible to expand numbers in general bases.
+
+Here we expand 111 in base 5. This means
+ 10^2+10^1+10^0 = 4 * 5^2+2 * 5^1 + 5^0
+
+ 111::RadixExpansion(5)
+ 421
+ Type: RadixExpansion 5
+
+You can expand fractions to form repeating expansions.
+
+ (5/24)::RadixExpansion(2)
+ __
+ 0.00110
+ Type: RadixExpansion 2
+
+ (5/24)::RadixExpansion(3)
+ __
+ 0.012
+ Type: RadixExpansion 3
+
+ (5/24)::RadixExpansion(8)
+ __
+ 0.152
+ Type: RadixExpansion 8
+
+ (5/24)::RadixExpansion(10)
+ _
+ 0.2083
+ Type: RadixExpansion 10
+
+For bases from 11 to 36 the letters A through Z are used.
+
+ (5/24)::RadixExpansion(12)
+ 0.26
+ Type: RadixExpansion 12
+
+ (5/24)::RadixExpansion(16)
+ _
+ 0.35
+ Type: RadixExpansion 16
+
+ (5/24)::RadixExpansion(36)
+ 0.7I
+ Type: RadixExpansion 36
+
+For bases greater than 36, the ragits are separated by blanks.
+
+ (5/24)::RadixExpansion(38)
+ _____
+ 0 . 7 34 31 25 12
+ Type: RadixExpansion 38
+
+The RadixExpansion type provides operations to obtain the individual
+ragits. Here is a rational number in base 8.
+
+ a := (76543/210)::RadixExpansion(8)
+ ____
+ 554.37307
+ Type: RadixExpansion 8
+
+The operation wholeRagits returns a list of the ragits for the
+integral part of the number.
+
+ w := wholeRagits a
+ [5,5,4]
+ Type: List Integer
+
+The operations prefixRagits and cycleRagits return lists of the initial
+and repeating ragits in the fractional part of the number.
+
+ f0 := prefixRagits a
+ [3]
+ Type: List Integer
+
+ f1 := cycleRagits a
+ [7,3,0,7]
+ Type: List Integer
+
+You can construct any radix expansion by giving the whole, prefix and
+cycle parts. The declaration is necessary to let Axiom know the base
+of the ragits.
+
+ u:RadixExpansion(8):=wholeRadix(w)+fractRadix(f0,f1)
+ ____
+ 554.37307
+ Type: RadixExpansion 8
+
+If there is no repeating part, then the list [0] should be used.
+
+ v: RadixExpansion(12) := fractRadix([1,2,3,11], [0])
+ _
+ 0.123B0
+ Type: RadixExpansion 12
+
+If you are not interested in the repeating nature of the expansion, an
+infinite stream of ragits can be obtained using fractRagits.
+
+ fractRagits(u)
+ _______
+ [3,7,3,0,7,7]
+ Type: Stream Integer
+
+Of course, it's possible to recover the fraction representation:
+
+ a :: Fraction(Integer)
+ 76543
+ -----
+ 210
+ Type: Fraction Integer
+
+See Also:
+o )help DecimalExpansion
+o )help BinaryExpansion
+o )help HexadecimalExpansion
+o )show RadixExpansion
+o $AXIOM/doc/src/algebra/radix.spad.dvi
+
+@
<<domain RADIX RadixExpansion>>=
)abbrev domain RADIX RadixExpansion
++ Author: Stephen M. Watt
diff --git a/src/algebra/reclos.spad.pamphlet b/src/algebra/reclos.spad.pamphlet
index b0400ad..85889cc 100644
--- a/src/algebra/reclos.spad.pamphlet
+++ b/src/algebra/reclos.spad.pamphlet
@@ -901,6 +901,1192 @@ The implementation is based on interval arithmetic.
Moreover, the
design of this constructor and its related packages allows an easy
use of other codings for real algebraic numbers.
ordered field
+<<RealClosure.input>>=
+-- reclos.spad.pamphlet RealClosure.input
+)spool RealClosure.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 67
+Ran := RECLOS(FRAC INT)
+--R
+--R
+--R (1) RealClosure Fraction Integer
+--R Type:
Domain
+--E 1
+
+--S 2 of 67
+fourSquares(a:Ran,b:Ran,c:Ran,d:Ran):Ran==sqrt(a)+sqrt(b)-sqrt(c)-sqrt(d)
+--R
+--R Function declaration fourSquares : (RealClosure Fraction Integer,
+--R RealClosure Fraction Integer,RealClosure Fraction Integer,
+--R RealClosure Fraction Integer) -> RealClosure Fraction Integer has
+--R been added to workspace.
+--R Type:
Void
+--E 2
+
+--S 3 of 67
+squareDiff1 := fourSquares(73,548,60,586)
+--R
+--R Compiling function fourSquares with type (RealClosure Fraction
+--R Integer,RealClosure Fraction Integer,RealClosure Fraction Integer
+--R ,RealClosure Fraction Integer) -> RealClosure Fraction Integer
+--R
+--R +---+ +--+ +---+ +--+
+--R (3) - \|586 - \|60 + \|548 + \|73
+--R Type: RealClosure Fraction
Integer
+--E 3
+
+--S 4 of 67
+recip(squareDiff1)
+--R
+--R
+--R (4)
+--R +---+ +--+ +--+ +--+ +---+ +---+
+--R ((54602\|548 + 149602\|73 )\|60 + 49502\|73 \|548 + 9900895)\|586
+--R +
+--R +--+ +---+ +--+ +---+ +--+
+--R (154702\|73 \|548 + 30941947)\|60 + 10238421\|548 + 28051871\|73
+--R Type: Union(RealClosure Fraction
Integer,...)
+--E 4
+
+--S 5 of 67
+sign(squareDiff1)
+--R
+--R
+--R (5) 1
+--R Type:
PositiveInteger
+--E 5
+
+--S 6 of 67
+squareDiff2 := fourSquares(165,778,86,990)
+--R
+--R
+--R +---+ +--+ +---+ +---+
+--R (6) - \|990 - \|86 + \|778 + \|165
+--R Type: RealClosure Fraction
Integer
+--E 6
+
+--S 7 of 67
+recip(squareDiff2)
+--R
+--R
+--R (7)
+--R +---+ +---+ +--+ +---+ +---+
+--R ((556778\|778 + 1209010\|165 )\|86 + 401966\|165 \|778 +
144019431)
+--R *
+--R +---+
+--R \|990
+--R +
+--R +---+ +---+ +--+ +---+
+---+
+--R (1363822\|165 \|778 + 488640503)\|86 + 162460913\|778 +
352774119\|165
+--R Type: Union(RealClosure Fraction
Integer,...)
+--E 7
+
+--S 8 of 67
+sign(squareDiff2)
+--R
+--R
+--R (8) 1
+--R Type:
PositiveInteger
+--E 8
+
+--S 9 of 67
+squareDiff3 := fourSquares(217,708,226,692)
+--R
+--R
+--R +---+ +---+ +---+ +---+
+--R (9) - \|692 - \|226 + \|708 + \|217
+--R Type: RealClosure Fraction
Integer
+--E 9
+
+--S 10 of 67
+recip(squareDiff3)
+--R
+--R
+--R (10)
+--R +---+ +---+ +---+ +---+ +---+
+---+
+--R ((- 34102\|708 - 61598\|217 )\|226 - 34802\|217 \|708 -
13641141)\|692
+--R +
+--R +---+ +---+ +---+ +---+
+---+
+--R (- 60898\|217 \|708 - 23869841)\|226 - 13486123\|708 - 24359809\|217
+--R Type: Union(RealClosure Fraction
Integer,...)
+--E 10
+
+--S 11 of 67
+sign(squareDiff3)
+--R
+--R
+--R (11) - 1
+--R Type:
Integer
+--E 11
+
+--S 12 of 67
+squareDiff4 := fourSquares(155,836,162,820)
+--R
+--R
+--R +---+ +---+ +---+ +---+
+--R (12) - \|820 - \|162 + \|836 + \|155
+--R Type: RealClosure Fraction
Integer
+--E 12
+
+--S 13 of 67
+recip(squareDiff4)
+--R
+--R
+--R (13)
+--R +---+ +---+ +---+ +---+ +---+
+---+
+--R ((- 37078\|836 - 86110\|155 )\|162 - 37906\|155 \|836 -
13645107)\|820
+--R +
+--R +---+ +---+ +---+ +---+
+---+
+--R (- 85282\|155 \|836 - 30699151)\|162 - 13513901\|836 - 31384703\|155
+--R Type: Union(RealClosure Fraction
Integer,...)
+--E 13
+
+--S 14 of 67
+sign(squareDiff4)
+--R
+--R
+--R (14) - 1
+--R Type:
Integer
+--E 14
+
+--S 15 of 67
+squareDiff5 := fourSquares(591,772,552,818)
+--R
+--R
+--R +---+ +---+ +---+ +---+
+--R (15) - \|818 - \|552 + \|772 + \|591
+--R Type: RealClosure Fraction
Integer
+--E 15
+
+--S 16 of 67
+recip(squareDiff5)
+--R
+--R
+--R (16)
+--R +---+ +---+ +---+ +---+ +---+
+---+
+--R ((70922\|772 + 81058\|591 )\|552 + 68542\|591 \|772 + 46297673)\|818
+--R +
+--R +---+ +---+ +---+ +---+ +---+
+--R (83438\|591 \|772 + 56359389)\|552 + 47657051\|772 + 54468081\|591
+--R Type: Union(RealClosure Fraction
Integer,...)
+--E 16
+
+--S 17 of 67
+sign(squareDiff5)
+--R
+--R
+--R (17) 1
+--R Type:
PositiveInteger
+--E 17
+
+--S 18 of 67
+squareDiff6 := fourSquares(434,1053,412,1088)
+--R
+--R
+--R +----+ +---+ +----+ +---+
+--R (18) - \|1088 - \|412 + \|1053 + \|434
+--R Type: RealClosure Fraction
Integer
+--E 18
+
+--S 19 of 67
+recip(squareDiff6)
+--R
+--R
+--R (19)
+--R +----+ +---+ +---+ +---+ +----+
+--R ((115442\|1053 + 179818\|434 )\|412 + 112478\|434 \|1053 +
76037291)
+--R *
+--R +----+
+--R \|1088
+--R +
+--R +---+ +----+ +---+ +----+
+---+
+--R (182782\|434 \|1053 + 123564147)\|412 + 77290639\|1053 +
120391609\|434
+--R Type: Union(RealClosure Fraction
Integer,...)
+--E 19
+
+--S 20 of 67
+sign(squareDiff6)
+--R
+--R
+--R (20) 1
+--R Type:
PositiveInteger
+--E 20
+
+--S 21 of 67
+squareDiff7 := fourSquares(514,1049,446,1152)
+--R
+--R
+--R +----+ +---+ +----+ +---+
+--R (21) - \|1152 - \|446 + \|1049 + \|514
+--R Type: RealClosure Fraction
Integer
+--E 21
+
+--S 22 of 67
+recip(squareDiff7)
+--R
+--R
+--R (22)
+--R +----+ +---+ +---+ +---+ +----+
+--R ((349522\|1049 + 499322\|514 )\|446 + 325582\|514 \|1049 +
239072537)
+--R *
+--R +----+
+--R \|1152
+--R +
+--R +---+ +----+ +---+ +----+
+---+
+--R (523262\|514 \|1049 + 384227549)\|446 + 250534873\|1049 +
357910443\|514
+--R Type: Union(RealClosure Fraction
Integer,...)
+--E 22
+
+--S 23 of 67
+sign(squareDiff7)
+--R
+--R
+--R (23) 1
+--R Type:
PositiveInteger
+--E 23
+
+--S 24 of 67
+squareDiff8 := fourSquares(190,1751,208,1698)
+--R
+--R
+--R +----+ +---+ +----+ +---+
+--R (24) - \|1698 - \|208 + \|1751 + \|190
+--R Type: RealClosure Fraction
Integer
+--E 24
+
+--S 25 of 67
+recip(squareDiff8)
+--R
+--R
+--R (25)
+--R +----+ +---+ +---+ +---+ +----+
+--R (- 214702\|1751 - 651782\|190 )\|208 - 224642\|190 \|1751
+--R +
+--R - 129571901
+--R *
+--R +----+
+--R \|1698
+--R +
+--R +---+ +----+ +---+ +----+
+--R (- 641842\|190 \|1751 - 370209881)\|208 - 127595865\|1751
+--R +
+--R +---+
+--R - 387349387\|190
+--R Type: Union(RealClosure Fraction
Integer,...)
+--E 25
+
+--S 26 of 67
+sign(squareDiff8)
+--R
+--R
+--R (26) - 1
+--R Type:
Integer
+--E 26
+
+--S 27 of 67
+relativeApprox(squareDiff8,10**(-3))::Float
+--R
+--R
+--R (27) - 0.2340527771 5937700123 E -10
+--R Type:
Float
+--E 27
+
+--S 28 of 67
+l := allRootsOf((x**2-2)**2-2)$Ran
+--R
+--R
+--R (28) [%A33,%A34,%A35,%A36]
+--R Type: List RealClosure Fraction
Integer
+--E 28
+
+--S 29 of 67
+removeDuplicates map(mainDefiningPolynomial,l)
+--R
+--R
+--R 4 2
+--R (29) [? - 4? + 2]
+--RType: List Union(SparseUnivariatePolynomial RealClosure Fraction
Integer,"failed")
+--E 29
+
+--S 30 of 67
+map(mainCharacterization,l)
+--R
+--R
+--R (30) [[- 2,- 1[,[- 1,0[,[0,1[,[1,2[]
+--RType: List Union(RightOpenIntervalRootCharacterization(RealClosure Fraction
Integer,SparseUnivariatePolynomial RealClosure Fraction Integer),"failed")
+--E 30
+
+--S 31 of 67
+[reduce(+,l),reduce(*,l)-2]
+--R
+--R
+--R (31) [0,0]
+--R Type: List RealClosure Fraction
Integer
+--E 31
+
+--S 32 of 67
+(s2, s5, s10) := (sqrt(2)$Ran, sqrt(5)$Ran, sqrt(10)$Ran)
+--R
+--R
+--R +--+
+--R (32) \|10
+--R Type: RealClosure Fraction
Integer
+--E 32
+
+--S 33 of 67
+eq1:=sqrt(s10+3)*sqrt(s5+2) - sqrt(s10-3)*sqrt(s5-2) = sqrt(10*s2+10)
+--R
+--R
+--R +---------+ +--------+ +---------+ +--------+ +-----------+
+--R | +--+ | +-+ | +--+ | +-+ | +-+
+--R (33) - \|\|10 - 3 \|\|5 - 2 + \|\|10 + 3 \|\|5 + 2 = \|10\|2 + 10
+--R Type: Equation RealClosure Fraction
Integer
+--E 33
+
+--S 34 of 67
+eq1::Boolean
+--R
+--R
+--R (34) true
+--R Type:
Boolean
+--E 34
+
+--S 35 of 67
+eq2:=sqrt(s5+2)*sqrt(s2+1) - sqrt(s5-2)*sqrt(s2-1) = sqrt(2*s10+2)
+--R
+--R
+--R +--------+ +--------+ +--------+ +--------+ +----------+
+--R | +-+ | +-+ | +-+ | +-+ | +--+
+--R (35) - \|\|5 - 2 \|\|2 - 1 + \|\|5 + 2 \|\|2 + 1 = \|2\|10 + 2
+--R Type: Equation RealClosure Fraction
Integer
+--E 35
+
+--S 36 of 67
+eq2::Boolean
+--R
+--R
+--R (36) true
+--R Type:
Boolean
+--E 36
+
+--S 37 of 67
+s3 := sqrt(3)$Ran
+--R
+--R
+--R +-+
+--R (37) \|3
+--R Type: RealClosure Fraction
Integer
+--E 37
+
+--S 38 of 67
+s7:= sqrt(7)$Ran
+--R
+--R
+--R +-+
+--R (38) \|7
+--R Type: RealClosure Fraction
Integer
+--E 38
+
+--S 39 of 67
+e1 := sqrt(2*s7-3*s3,3)
+--R
+--R
+--R +-------------+
+--R 3| +-+ +-+
+--R (39) \|2\|7 - 3\|3
+--R Type: RealClosure Fraction
Integer
+--E 39
+
+--S 40 of 67
+e2 := sqrt(2*s7+3*s3,3)
+--R
+--R
+--R +-------------+
+--R 3| +-+ +-+
+--R (40) \|2\|7 + 3\|3
+--R Type: RealClosure Fraction
Integer
+--E 40
+
+--S 41 of 67
+e2-e1-s3
+--R
+--R
+--R (41) 0
+--R Type: RealClosure Fraction
Integer
+--E 41
+
+--S 42 of 67
+pol : UP(x,Ran) := x**4+(7/3)*x**2+30*x-(100/3)
+--R
+--R
+--R 4 7 2 100
+--R (42) x + - x + 30x - ---
+--R 3 3
+--R Type: UnivariatePolynomial(x,RealClosure Fraction
Integer)
+--E 42
+
+--S 43 of 67
+r1 := sqrt(7633)$Ran
+--R
+--R
+--R +----+
+--R (43) \|7633
+--R Type: RealClosure Fraction
Integer
+--E 43
+
+--S 44 of 67
+alpha := sqrt(5*r1-436,3)/3
+--R
+--R
+--R +--------------+
+--R 1 3| +----+
+--R (44) - \|5\|7633 - 436
+--R 3
+--R Type: RealClosure Fraction
Integer
+--E 44
+
+--S 45 of 67
+beta := -sqrt(5*r1+436,3)/3
+--R
+--R
+--R +--------------+
+--R 1 3| +----+
+--R (45) - - \|5\|7633 + 436
+--R 3
+--R Type: RealClosure Fraction
Integer
+--E 45
+
+--S 46 of 67
+pol.(alpha+beta-1/3)
+--R
+--R
+--R (46) 0
+--R Type: RealClosure Fraction
Integer
+--E 46
+
+--S 47 of 67
+qol : UP(x,Ran) := x**5+10*x**3+20*x+22
+--R
+--R
+--R 5 3
+--R (47) x + 10x + 20x + 22
+--R Type: UnivariatePolynomial(x,RealClosure Fraction
Integer)
+--E 47
+
+--S 48 of 67
+r2 := sqrt(153)$Ran
+--R
+--R
+--R +---+
+--R (48) \|153
+--R Type: RealClosure Fraction
Integer
+--E 48
+
+--S 49 of 67
+alpha2 := sqrt(r2-11,5)
+--R
+--R
+--R +-----------+
+--R 5| +---+
+--R (49) \|\|153 - 11
+--R Type: RealClosure Fraction
Integer
+--E 49
+
+--S 50 of 67
+beta2 := -sqrt(r2+11,5)
+--R
+--R
+--R +-----------+
+--R 5| +---+
+--R (50) - \|\|153 + 11
+--R Type: RealClosure Fraction
Integer
+--E 50
+
+--S 51 of 67
+qol(alpha2+beta2)
+--R
+--R
+--R (51) 0
+--R Type: RealClosure Fraction
Integer
+--E 51
+
+--S 52 of 67
+dst1:=sqrt(9+4*s2)=1+2*s2
+--R
+--R
+--R +---------+
+--R | +-+ +-+
+--R (52) \|4\|2 + 9 = 2\|2 + 1
+--R Type: Equation RealClosure Fraction
Integer
+--E 52
+
+--S 53 of 67
+dst1::Boolean
+--R
+--R
+--R (53) true
+--R Type:
Boolean
+--E 53
+
+--S 54 of 67
+s6:Ran:=sqrt 6
+--R
+--R
+--R +-+
+--R (54) \|6
+--R Type: RealClosure Fraction
Integer
+--E 54
+
+--S 55 of 67
+dst2:=sqrt(5+2*s6)+sqrt(5-2*s6) = 2*s3
+--R
+--R
+--R +-----------+ +---------+
+--R | +-+ | +-+ +-+
+--R (55) \|- 2\|6 + 5 + \|2\|6 + 5 = 2\|3
+--R Type: Equation RealClosure Fraction
Integer
+--E 55
+
+--S 56 of 67
+dst2::Boolean
+--R
+--R
+--R (56) true
+--R Type:
Boolean
+--E 56
+
+--S 57 of 67
+s29:Ran:=sqrt 29
+--R
+--R
+--R +--+
+--R (57) \|29
+--R Type: RealClosure Fraction
Integer
+--E 57
+
+--S 58 of 67
+dst4:=sqrt(16-2*s29+2*sqrt(55-10*s29)) = sqrt(22+2*s5)-sqrt(11+2*s29)+s5
+--R
+--R
+--R (58)
+--R +--------------------------------+
+--R | +--------------+ +-----------+ +----------+
+--R | | +--+ +--+ | +--+ | +-+
+-+
+--R \|2\|- 10\|29 + 55 - 2\|29 + 16 = - \|2\|29 + 11 + \|2\|5 + 22 +
\|5
+--R Type: Equation RealClosure Fraction
Integer
+--E 58
+
+--S 59 of 67
+dst4::Boolean
+--R
+--R
+--R (59) true
+--R Type:
Boolean
+--E 59
+
+--S 60 of 67
+dst6:=sqrt((112+70*s2)+(46+34*s2)*s5) = (5+4*s2)+(3+s2)*s5
+--R
+--R
+--R +--------------------------------+
+--R | +-+ +-+ +-+ +-+ +-+ +-+
+--R (60) \|(34\|2 + 46)\|5 + 70\|2 + 112 = (\|2 + 3)\|5 + 4\|2 + 5
+--R Type: Equation RealClosure Fraction
Integer
+--E 60
+
+--S 61 of 67
+dst6::Boolean
+--R
+--R
+--R (61) true
+--R Type:
Boolean
+--E 61
+
+--S 62 of 67
+f3:Ran:=sqrt(3,5)
+--R
+--R
+--R 5+-+
+--R (62) \|3
+--R Type: RealClosure Fraction
Integer
+--E 62
+
+--S 63 of 67
+f25:Ran:=sqrt(1/25,5)
+--R
+--R
+--R +--+
+--R | 1
+--R (63) 5|--
+--R \|25
+--R Type: RealClosure Fraction
Integer
+--E 63
+
+--S 64 of 67
+f32:Ran:=sqrt(32/5,5)
+--R
+--R
+--R +--+
+--R |32
+--R (64) 5|--
+--R \| 5
+--R Type: RealClosure Fraction
Integer
+--E 64
+
+--S 65 of 67
+f27:Ran:=sqrt(27/5,5)
+--R
+--R
+--R +--+
+--R |27
+--R (65) 5|--
+--R \| 5
+--R Type: RealClosure Fraction
Integer
+--E 65
+
+--S 66 of 67
+dst5:=sqrt((f32-f27,3)) = f25*(1+f3-f3**2)
+--R
+--R
+--R +---------------+
+--R | +--+ +--+ +--+
+--R | |27 |32 5+-+2 5+-+ | 1
+--R (66) 3|- 5|-- + 5|-- = (- \|3 + \|3 + 1) 5|--
+--R \| \| 5 \| 5 \|25
+--R Type: Equation RealClosure Fraction
Integer
+--E 66
+
+--S 67 of 67
+dst5::Boolean
+--R
+--R
+--R (67) true
+--R Type:
Boolean
+--E 67
+)spool
+)lisp (bye)
+@
+<<RealClosure.help>>=
+====================================================================
+RealClosure examples
+====================================================================
+
+The Real Closure 1.0 package provided by Renaud Rioboo consists of
+different packages, categories and domains :
+
+The package RealPolynomialUtilitiesPackage which needs a Field F and a
+UnivariatePolynomialCategory domain with coefficients in F. It computes
+some simple functions such as Sturm and Sylvester sequences sturmSequence,
+sylvesterSequence.
+
+The category RealRootCharacterizationCategory provides abstract
+functions to work with "real roots" of univariate polynomials. These
+resemble variables with some functionality needed to compute important
+operations.
+
+The category RealClosedField provides common operations available over
+real closed fiels. These include finding all the roots of a univariate
+polynomial, taking square (and higher) roots, ...
+
+The domain RightOpenIntervalRootCharacterization is the main code that
+provides the functionality of RealRootCharacterizationCategory for the
+case of archimedean fields. Abstract roots are encoded with a left
+closed right open interval containing the root together with a
+defining polynomial for the root.
+
+The RealClosure domain is the end-user code. It provides usual arithmetic
+with real algebraic numbers, along with the functionality of a real closed
+field. It also provides functions to approximate a real algebraic number
+by an element of the base field. This approximation may either be absolute,
+approximate or relative (relativeApprox).
+
+====================================================================
+CAVEATS
+====================================================================
+
+Since real algebraic expressions are stored as depending on "real
+roots" which are managed like variables, there is an ordering on
+these. This ordering is dynamical in the sense that any new algebraic
+takes precedence over older ones. In particular every creation
+function raises a new "real root". This has the effect that when you
+type something like sqrt(2) + sqrt(2) you have two new variables
+which happen to be equal. To avoid this name the expression such as in
+s2 := sqrt(2) ; s2 + s2
+
+Also note that computing times depend strongly on the ordering you
+implicitly provide. Please provide algebraics in the order which seems
+most natural to you.
+
+====================================================================
+LIMITATIONS
+====================================================================
+
+This packages uses algorithms which are published in [1] and [2] which
+are based on field arithmetics, in particular for polynomial gcd
+related algorithms. This can be quite slow for high degree polynomials
+and subresultants methods usually work best. Beta versions of the
+package try to use these techniques in a better way and work
+significantly faster. These are mostly based on unpublished algorithms
+and cannot be distributed. Please contact the author if you have a
+particular problem to solve or want to use these versions.
+
+Be aware that approximations behave as post-processing and that all
+computations are done exactly. They can thus be quite time consuming when
+depending on several ``real roots''.
+
+====================================================================
+REFERENCES
+====================================================================
+
+
+[1] R. Rioboo : Real Algebraic Closure of an ordered Field : Implementation
+ in Axiom.
+ In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206-215.
+
+[2] Z. Ligatsikas, R. Rioboo, M. F. Roy : Generic computation of the real
+ closure of an ordered field.
+ In Mathematics and Computers in Simulation Volume 42, Issue 4-6,
+ November 1996.
+
+====================================================================
+EXAMPLES
+====================================================================
+
+We shall work with the real closure of the ordered field of
+rational numbers.
+
+ Ran := RECLOS(FRAC INT)
+ RealClosure Fraction Integer
+ Type: Domain
+
+Some simple signs for square roots, these correspond to an extension
+of degree 16 of the rational numbers. Examples provided by J. Abbot.
+
+ fourSquares(a:Ran,b:Ran,c:Ran,d:Ran):Ran==sqrt(a)+sqrt(b)-sqrt(c)-sqrt(d)
+ Type: Void
+
+These produce values very close to zero.
+
+ squareDiff1 := fourSquares(73,548,60,586)
+ +---+ +--+ +---+ +--+
+ - \|586 - \|60 + \|548 + \|73
+ Type: RealClosure Fraction Integer
+
+ recip(squareDiff1)
+ +---+ +--+ +--+ +--+ +---+ +---+
+ ((54602\|548 + 149602\|73 )\|60 + 49502\|73 \|548 + 9900895)\|586
+ +
+ +--+ +---+ +--+ +---+ +--+
+ (154702\|73 \|548 + 30941947)\|60 + 10238421\|548 + 28051871\|73
+ Type: Union(RealClosure Fraction Integer,...)
+
+ sign(squareDiff1)
+ 1
+ Type: PositiveInteger
+
+ squareDiff2 := fourSquares(165,778,86,990)
+ +---+ +--+ +---+ +---+
+ - \|990 - \|86 + \|778 + \|165
+ Type: RealClosure Fraction Integer
+
+ recip(squareDiff2)
+ +---+ +---+ +--+ +---+ +---+
+ ((556778\|778 + 1209010\|165 )\|86 + 401966\|165 \|778 + 144019431)
+ *
+ +---+
+ \|990
+ +
+ +---+ +---+ +--+ +---+ +---+
+ (1363822\|165 \|778 + 488640503)\|86 + 162460913\|778 + 352774119\|165
+ Type: Union(RealClosure Fraction Integer,...)
+
+ sign(squareDiff2)
+ 1
+ Type: PositiveInteger
+
+ squareDiff3 := fourSquares(217,708,226,692)
+ +---+ +---+ +---+ +---+
+ - \|692 - \|226 + \|708 + \|217
+ Type: RealClosure Fraction Integer
+
+ recip(squareDiff3)
+ +---+ +---+ +---+ +---+ +---+ +---+
+ ((- 34102\|708 - 61598\|217 )\|226 - 34802\|217 \|708 - 13641141)\|692
+ +
+ +---+ +---+ +---+ +---+ +---+
+ (- 60898\|217 \|708 - 23869841)\|226 - 13486123\|708 - 24359809\|217
+ Type: Union(RealClosure Fraction Integer,...)
+
+ sign(squareDiff3)
+ - 1
+ Type: Integer
+
+ squareDiff4 := fourSquares(155,836,162,820)
+ +---+ +---+ +---+ +---+
+ - \|820 - \|162 + \|836 + \|155
+ Type: RealClosure Fraction Integer
+
+ recip(squareDiff4)
+ +---+ +---+ +---+ +---+ +---+ +---+
+ ((- 37078\|836 - 86110\|155 )\|162 - 37906\|155 \|836 - 13645107)\|820
+ +
+ +---+ +---+ +---+ +---+ +---+
+ (- 85282\|155 \|836 - 30699151)\|162 - 13513901\|836 - 31384703\|155
+ Type: Union(RealClosure Fraction Integer,...)
+
+ sign(squareDiff4)
+ - 1
+ Type: Integer
+
+ squareDiff5 := fourSquares(591,772,552,818)
+ +---+ +---+ +---+ +---+
+ - \|818 - \|552 + \|772 + \|591
+ Type: RealClosure Fraction Integer
+
+ recip(squareDiff5)
+ +---+ +---+ +---+ +---+ +---+ +---+
+ ((70922\|772 + 81058\|591 )\|552 + 68542\|591 \|772 + 46297673)\|818
+ +
+ +---+ +---+ +---+ +---+ +---+
+ (83438\|591 \|772 + 56359389)\|552 + 47657051\|772 + 54468081\|591
+ Type: Union(RealClosure Fraction Integer,...)
+
+ sign(squareDiff5)
+ 1
+ Type: PositiveInteger
+
+ squareDiff6 := fourSquares(434,1053,412,1088)
+ +----+ +---+ +----+ +---+
+ - \|1088 - \|412 + \|1053 + \|434
+ Type: RealClosure Fraction Integer
+
+ recip(squareDiff6)
+ +----+ +---+ +---+ +---+ +----+
+ ((115442\|1053 + 179818\|434 )\|412 + 112478\|434 \|1053 + 76037291)
+ *
+ +----+
+ \|1088
+ +
+ +---+ +----+ +---+ +----+ +---+
+ (182782\|434 \|1053 + 123564147)\|412 + 77290639\|1053 + 120391609\|434
+ Type: Union(RealClosure Fraction Integer,...)
+
+ sign(squareDiff6)
+ 1
+ Type: PositiveInteger
+
+ squareDiff7 := fourSquares(514,1049,446,1152)
+ +----+ +---+ +----+ +---+
+ - \|1152 - \|446 + \|1049 + \|514
+ Type: RealClosure Fraction Integer
+
+ recip(squareDiff7)
+ +----+ +---+ +---+ +---+ +----+
+ ((349522\|1049 + 499322\|514 )\|446 + 325582\|514 \|1049 + 239072537)
+ *
+ +----+
+ \|1152
+ +
+ +---+ +----+ +---+ +----+ +---+
+ (523262\|514 \|1049 + 384227549)\|446 + 250534873\|1049 + 357910443\|514
+ Type: Union(RealClosure Fraction Integer,...)
+
+ sign(squareDiff7)
+ 1
+ Type: PositiveInteger
+
+ squareDiff8 := fourSquares(190,1751,208,1698)
+ +----+ +---+ +----+ +---+
+ - \|1698 - \|208 + \|1751 + \|190
+ Type: RealClosure Fraction Integer
+
+ recip(squareDiff8)
+ +----+ +---+ +---+ +---+ +----+
+ (- 214702\|1751 - 651782\|190 )\|208 - 224642\|190 \|1751
+ +
+ - 129571901
+ *
+ +----+
+ \|1698
+ +
+ +---+ +----+ +---+ +----+
+ (- 641842\|190 \|1751 - 370209881)\|208 - 127595865\|1751
+ +
+ +---+
+ - 387349387\|190
+ Type: Union(RealClosure Fraction Integer,...)
+
+ sign(squareDiff8)
+ - 1
+ Type: Integer
+
+This should give three digits of precision
+
+ relativeApprox(squareDiff8,10**(-3))::Float
+ - 0.2340527771 5937700123 E -10
+ Type: Float
+
+The sum of these 4 roots is 0
+
+ l := allRootsOf((x**2-2)**2-2)$Ran
+ [%A33,%A34,%A35,%A36]
+ Type: List RealClosure Fraction Integer
+
+Check that they are all roots of the same polynomial
+
+ removeDuplicates map(mainDefiningPolynomial,l)
+ 4 2
+ [? - 4? + 2]
+ Type: List Union(
+ SparseUnivariatePolynomial RealClosure Fraction Integer,
+ "failed")
+
+We can see at a glance that they are separate roots
+
+ map(mainCharacterization,l)
+ [[- 2,- 1[,[- 1,0[,[0,1[,[1,2[]
+ Type: List Union(
+ RightOpenIntervalRootCharacterization(
+ RealClosure Fraction Integer,
+ SparseUnivariatePolynomial RealClosure Fraction Integer),
+ "failed")
+
+Check the sum and product
+
+ [reduce(+,l),reduce(*,l)-2]
+ [0,0]
+ Type: List RealClosure Fraction Integer
+
+A more complicated test that involve an extension of degree 256.
+This is a way of checking nested radical identities.
+
+ (s2, s5, s10) := (sqrt(2)$Ran, sqrt(5)$Ran, sqrt(10)$Ran)
+ +--+
+ \|10
+ Type: RealClosure Fraction Integer
+
+ eq1:=sqrt(s10+3)*sqrt(s5+2) - sqrt(s10-3)*sqrt(s5-2) = sqrt(10*s2+10)
+ +---------+ +--------+ +---------+ +--------+ +-----------+
+ | +--+ | +-+ | +--+ | +-+ | +-+
+ - \|\|10 - 3 \|\|5 - 2 + \|\|10 + 3 \|\|5 + 2 = \|10\|2 + 10
+ Type: Equation RealClosure Fraction Integer
+
+ eq1::Boolean
+ true
+ Type: Boolean
+
+ eq2:=sqrt(s5+2)*sqrt(s2+1) - sqrt(s5-2)*sqrt(s2-1) = sqrt(2*s10+2)
+ +--------+ +--------+ +--------+ +--------+ +----------+
+ | +-+ | +-+ | +-+ | +-+ | +--+
+ - \|\|5 - 2 \|\|2 - 1 + \|\|5 + 2 \|\|2 + 1 = \|2\|10 + 2
+ Type: Equation RealClosure Fraction Integer
+
+ eq2::Boolean
+ true
+ Type: Boolean
+
+Some more examples from J. M. Arnaudies
+
+ s3 := sqrt(3)$Ran
+ +-+
+ \|3
+ Type: RealClosure Fraction Integer
+
+ s7:= sqrt(7)$Ran
+ +-+
+ \|7
+ Type: RealClosure Fraction Integer
+
+ e1 := sqrt(2*s7-3*s3,3)
+ +-------------+
+ 3| +-+ +-+
+ \|2\|7 - 3\|3
+ Type: RealClosure Fraction Integer
+
+ e2 := sqrt(2*s7+3*s3,3)
+ +-------------+
+ 3| +-+ +-+
+ \|2\|7 + 3\|3
+ Type: RealClosure Fraction Integer
+
+This should be null
+
+ e2-e1-s3
+ 0
+ Type: RealClosure Fraction Integer
+
+A quartic polynomial
+
+ pol : UP(x,Ran) := x**4+(7/3)*x**2+30*x-(100/3)
+ 4 7 2 100
+ x + - x + 30x - ---
+ 3 3
+ Type: UnivariatePolynomial(x,RealClosure Fraction Integer)
+
+Add some cubic roots
+
+ r1 := sqrt(7633)$Ran
+ +----+
+ \|7633
+ Type: RealClosure Fraction Integer
+
+ alpha := sqrt(5*r1-436,3)/3
+ +--------------+
+ 1 3| +----+
+ - \|5\|7633 - 436
+ 3
+ Type: RealClosure Fraction Integer
+
+ beta := -sqrt(5*r1+436,3)/3
+ +--------------+
+ 1 3| +----+
+ - - \|5\|7633 + 436
+ 3
+ Type: RealClosure Fraction Integer
+
+this should be null
+
+ pol.(alpha+beta-1/3)
+ 0
+ Type: RealClosure Fraction Integer
+
+A quintic polynomial
+
+ qol : UP(x,Ran) := x**5+10*x**3+20*x+22
+ 5 3
+ x + 10x + 20x + 22
+ Type: UnivariatePolynomial(x,RealClosure Fraction Integer)
+
+Add some cubic roots
+
+ r2 := sqrt(153)$Ran
+ +---+
+ \|153
+ Type: RealClosure Fraction Integer
+
+ alpha2 := sqrt(r2-11,5)
+ +-----------+
+ 5| +---+
+ \|\|153 - 11
+ Type: RealClosure Fraction Integer
+
+ beta2 := -sqrt(r2+11,5)
+ +-----------+
+ 5| +---+
+ - \|\|153 + 11
+ Type: RealClosure Fraction Integer
+
+this should be null
+
+ qol(alpha2+beta2)
+ 0
+ Type: RealClosure Fraction Integer
+
+Finally, some examples from the book Computer Algebra by Davenport,
+Siret and Tournier (page 77). The last one is due to Ramanujan.
+
+ dst1:=sqrt(9+4*s2)=1+2*s2
+ +---------+
+ | +-+ +-+
+ \|4\|2 + 9 = 2\|2 + 1
+ Type: Equation RealClosure Fraction Integer
+
+ dst1::Boolean
+ true
+ Type: Boolean
+
+ s6:Ran:=sqrt 6
+ +-+
+ \|6
+ Type: RealClosure Fraction Integer
+
+ dst2:=sqrt(5+2*s6)+sqrt(5-2*s6) = 2*s3
+ +-----------+ +---------+
+ | +-+ | +-+ +-+
+ \|- 2\|6 + 5 + \|2\|6 + 5 = 2\|3
+ Type: Equation RealClosure Fraction Integer
+
+ dst2::Boolean
+ true
+ Type: Boolean
+
+ s29:Ran:=sqrt 29
+ +--+
+ \|29
+ Type: RealClosure Fraction Integer
+
+ dst4:=sqrt(16-2*s29+2*sqrt(55-10*s29)) = sqrt(22+2*s5)-sqrt(11+2*s29)+s5
+ +--------------------------------+
+ | +--------------+ +-----------+ +----------+
+ | | +--+ +--+ | +--+ | +-+ +-+
+ \|2\|- 10\|29 + 55 - 2\|29 + 16 = - \|2\|29 + 11 + \|2\|5 + 22 + \|5
+ Type: Equation RealClosure Fraction Integer
+
+ dst4::Boolean
+ true
+ Type: Boolean
+
+ dst6:=sqrt((112+70*s2)+(46+34*s2)*s5) = (5+4*s2)+(3+s2)*s5
+ +--------------------------------+
+ | +-+ +-+ +-+ +-+ +-+ +-+
+ \|(34\|2 + 46)\|5 + 70\|2 + 112 = (\|2 + 3)\|5 + 4\|2 + 5
+ Type: Equation RealClosure Fraction Integer
+
+ dst6::Boolean
+ true
+ Type: Boolean
+
+ f3:Ran:=sqrt(3,5)
+ 5+-+
+ \|3
+ Type: RealClosure Fraction Integer
+
+ f25:Ran:=sqrt(1/25,5)
+ +--+
+ | 1
+ 5|--
+ \|25
+ Type: RealClosure Fraction Integer
+
+ f32:Ran:=sqrt(32/5,5)
+ +--+
+ |32
+ 5|--
+ \| 5
+ Type: RealClosure Fraction Integer
+
+ f27:Ran:=sqrt(27/5,5)
+ +--+
+ |27
+ 5|--
+ \| 5
+ Type: RealClosure Fraction Integer
+
+ dst5:=sqrt((f32-f27,3)) = f25*(1+f3-f3**2)
+ +---------------+
+ | +--+ +--+ +--+
+ | |27 |32 5+-+2 5+-+ | 1
+ 3|- 5|-- + 5|-- = (- \|3 + \|3 + 1) 5|--
+ \| \| 5 \| 5 \|25
+ Type: Equation RealClosure Fraction Integer
+
+ dst5::Boolean
+ true
+ Type: Boolean
+
+See Also:
+o )help RightOpenIntervalRootCharacterization
+o )help RealClosedField
+o )help RealRootCharacterizationCategory
+o )help UnivariatePolynomialCategory
+o )help Field
+o )help RealPolynomialUtilitiesPackage
+o )show RealClosure
+o $AXIOM/doc/src/algebra/reclos.spad.dvi
+
+@
<<domain RECLOS RealClosure>>=
)abbrev domain RECLOS RealClosure
++ Author: Renaud Rioboo
diff --git a/src/algebra/regset.spad.pamphlet b/src/algebra/regset.spad.pamphlet
index 99d0747..2a1e80c 100644
--- a/src/algebra/regset.spad.pamphlet
+++ b/src/algebra/regset.spad.pamphlet
@@ -1369,6 +1369,1376 @@ chains). Among them {\bf RegularTriangularSet} and
by Marc Moreno Maza for computing triangular decompositions of polynomial
systems. This method is refined in the package {\bf LazardSetSolvingPackage}
in order to produce decompositions by means of Lazard triangular sets.
+<<RegularTriangularSet.input>>=
+-- regset.spad.pamphlet RegularTriangularSet.input
+)spool RegularTriangularSet.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 34
+R := Integer
+--R
+--R
+--R (1) Integer
+--R Type:
Domain
+--E 1
+
+--S 2 of 34
+ls : List Symbol := [x,y,z,t]
+--R
+--R
+--R (2) [x,y,z,t]
+--R Type: List
Symbol
+--E 2
+
+--S 3 of 34
+V := OVAR(ls)
+--R
+--R
+--R (3) OrderedVariableList [x,y,z,t]
+--R Type:
Domain
+--E 3
+
+--S 4 of 34
+E := IndexedExponents V
+--R
+--R
+--R (4) IndexedExponents OrderedVariableList [x,y,z,t]
+--R Type:
Domain
+--E 4
+
+--S 5 of 34
+P := NSMP(R, V)
+--R
+--R
+--R (5) NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--R Type:
Domain
+--E 5
+
+--S 6 of 34
+x: P := 'x
+--R
+--R
+--R (6) x
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 6
+
+--S 7 of 34
+y: P := 'y
+--R
+--R
+--R (7) y
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 7
+
+--S 8 of 34
+z: P := 'z
+--R
+--R
+--R (8) z
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 8
+
+--S 9 of 34
+t: P := 't
+--R
+--R
+--R (9) t
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 9
+
+--S 10 of 34
+T := REGSET(R,E,V,P)
+--R
+--R
+--R (10)
+--R RegularTriangularSet(Integer,IndexedExponents OrderedVariableList
[x,y,z,t],O
+--R rderedVariableList
[x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedV
+--R ariableList [x,y,z,t]))
+--R Type:
Domain
+--E 10
+
+--S 11 of 34
+p1 := x ** 31 - x ** 6 - x - y
+--R
+--R
+--R 31 6
+--R (11) x - x - x - y
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 11
+
+--S 12 of 34
+p2 := x ** 8 - z
+--R
+--R
+--R 8
+--R (12) x - z
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 12
+
+--S 13 of 34
+p3 := x ** 10 - t
+--R
+--R
+--R 10
+--R (13) x - t
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 13
+
+--S 14 of 34
+lp := [p1, p2, p3]
+--R
+--R
+--R 31 6 8 10
+--R (14) [x - x - x - y,x - z,x - t]
+--RType: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 14
+
+--S 15 of 34
+zeroSetSplit(lp)$T
+--R
+--R
+--R 5 4 2 3 8 5 3 2 4 2
+--R (15) [{z - t ,t z y + 2z y - t + 2t + t - t ,(t - t)x - t y - z }]
+--RType: List RegularTriangularSet(Integer,IndexedExponents
OrderedVariableList [x,y,z,t],OrderedVariableList
[x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t]))
+--E 15
+
+--S 16 of 34
+lts := zeroSetSplit(lp,false)$T
+--R
+--R
+--R (16)
+--R 5 4 2 3 8 5 3 2 4 2
+--R [{z - t ,t z y + 2z y - t + 2t + t - t ,(t - t)x - t y - z },
+--R 3 5 2 3 2
+--R {t - 1,z - t,t z y + 2z y + 1,z x - t}, {t,z,y,x}]
+--RType: List RegularTriangularSet(Integer,IndexedExponents
OrderedVariableList [x,y,z,t],OrderedVariableList
[x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t]))
+--E 16
+
+--S 17 of 34
+[coHeight(ts) for ts in lts]
+--R
+--R
+--R (17) [1,0,0]
+--R Type: List
NonNegativeInteger
+--E 17
+
+--S 18 of 34
+f1 := y**2*z+2*x*y*t-2*x-z
+--R
+--R
+--R 2
+--R (18) (2t y - 2)x + z y - z
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 18
+
+--S 19 of 34
+f2:=-x**3*z+ 4*x*y**2*z+4*x**2*y*t+2*y**3*t+4*x**2-10*y**2+4*x*z-10*y*t+2
+--R
+--R
+--R 3 2 2 3 2
+--R (19) - z x + (4t y + 4)x + (4z y + 4z)x + 2t y - 10y - 10t y + 2
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 19
+
+--S 20 of 34
+f3 := 2*y*z*t+x*t**2-x-2*z
+--R
+--R
+--R 2
+--R (20) (t - 1)x + 2t z y - 2z
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 20
+
+--S 21 of 34
+f4:=-x*z**3+4*y*z**2*t+4*x*z*t**2+2*y*t**3+4*x*z+4*z**2-10*y*t- 10*t**2+2
+--R
+--R
+--R 3 2 2 3 2 2
+--R (21) (- z + (4t + 4)z)x + (4t z + 2t - 10t)y + 4z - 10t + 2
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 21
+
+--S 22 of 34
+lf := [f1, f2, f3, f4]
+--R
+--R
+--R (22)
+--R 2
+--R [(2t y - 2)x + z y - z,
+--R 3 2 2 3 2
+--R - z x + (4t y + 4)x + (4z y + 4z)x + 2t y - 10y - 10t y + 2,
+--R 2
+--R (t - 1)x + 2t z y - 2z,
+--R 3 2 2 3 2 2
+--R (- z + (4t + 4)z)x + (4t z + 2t - 10t)y + 4z - 10t + 2]
+--RType: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 22
+
+--S 23 of 34
+zeroSetSplit(lf)$T
+--R
+--R
+--R (23)
+--R 2 8 6 2 3 2
+--R [{t - 1,z - 16z + 256z - 256,t y - 1,(z - 8z)x - 8z + 16},
+--R 2 2 2
+--R {3t + 1,z - 7t - 1,y + t,x + z},
+--R 8 6 2 3 2
+--R {t - 10t + 10t - 1,z,(t - 5t)y - 5t + 1,x},
+--R 2 2
+--R {t + 3,z - 4,y + t,x - z}]
+--RType: List RegularTriangularSet(Integer,IndexedExponents
OrderedVariableList [x,y,z,t],OrderedVariableList
[x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t]))
+--E 23
+
+--S 24 of 34
+lts2 := zeroSetSplit(lf,false)$T
+--R
+--R
+--R (24)
+--R 8 6 2 3 2
+--R [{t - 10t + 10t - 1,z,(t - 5t)y - 5t + 1,x},
+--R 2 8 6 2 3 2
+--R {t - 1,z - 16z + 256z - 256,t y - 1,(z - 8z)x - 8z + 16},
+--R 2 2 2 2 2
+--R {3t + 1,z - 7t - 1,y + t,x + z}, {t + 3,z - 4,y + t,x - z}]
+--RType: List RegularTriangularSet(Integer,IndexedExponents
OrderedVariableList [x,y,z,t],OrderedVariableList
[x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t]))
+--E 24
+
+--S 25 of 34
+[coHeight(ts) for ts in lts2]
+--R
+--R
+--R (25) [0,0,0,0]
+--R Type: List
NonNegativeInteger
+--E 25
+
+--S 26 of 34
+degrees := [degree(ts) for ts in lts2]
+--R
+--R
+--R (26) [8,16,4,4]
+--R Type: List
NonNegativeInteger
+--E 26
+
+--S 27 of 34
+reduce(+,degrees)
+--R
+--R
+--R (27) 32
+--R Type:
PositiveInteger
+--E 27
+
+--S 28 of 34
+u : R := 2
+--R
+--R
+--R (28) 2
+--R Type:
Integer
+--E 28
+
+--S 29 of 34
+q1 := 2*(u-1)**2+ 2*(x-z*x+z**2)+ y**2*(x-1)**2- 2*u*x+ 2*y*t*(1-x)*(x-z)+_
+ 2*u*z*t*(t-y)+ u**2*t**2*(1-2*z)+ 2*u*t**2*(z-x)+ 2*u*t*y*(z-1)+_
+ 2*u*z*x*(y+1)+ (u**2-2*u)*z**2*t**2+ 2*u**2*z**2+ 4*u*(1-u)*z+_
+ t**2*(z-x)**2
+--R
+--R
+--R (29)
+--R 2 2 2 2 2 2
+--R (y - 2t y + t )x + (- 2y + ((2t + 4)z + 2t)y + (- 2t + 2)z - 4t -
2)x
+--R +
+--R 2 2 2 2
+--R y + (- 2t z - 4t)y + (t + 10)z - 8z + 4t + 2
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 29
+
+--S 30 of 34
+q2 := t*(2*z+1)*(x-z)+ y*(z+2)*(1-x)+ u*(u-2)*t+ u*(1-2*u)*z*t+_
+ u*y*(x+u-z*x-1)+ u*(u+1)*z**2*t
+--R
+--R
+--R 2
+--R (30) (- 3z y + 2t z + t)x + (z + 4)y + 4t z - 7t z
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 30
+
+--S 31 of 34
+q3 := -u**2*(z-1)**2+ 2*z*(z-x)-2*(x-1)
+--R
+--R
+--R 2
+--R (31) (- 2z - 2)x - 2z + 8z - 2
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 31
+
+--S 32 of 34
+q4 := u**2+4*(z-x**2)+3*y**2*(x-1)**2- 3*t**2*(z-x)**2+_
+ 3*u**2*t**2*(z-1)**2+u**2*z*(z-2)+6*u*t*y*(z+x+z*x-1)
+--R
+--R
+--R (32)
+--R 2 2 2 2 2 2
+--R (3y - 3t - 4)x + (- 6y + (12t z + 12t)y + 6t z)x + 3y + (12t z -
12t)y
+--R +
+--R 2 2 2 2
+--R (9t + 4)z + (- 24t - 4)z + 12t + 4
+--R Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 32
+
+--S 33 of 34
+lq := [q1, q2, q3, q4]
+--R
+--R
+--R (33)
+--R [
+--R 2 2 2
+--R (y - 2t y + t )x
+--R +
+--R 2 2 2 2
+--R (- 2y + ((2t + 4)z + 2t)y + (- 2t + 2)z - 4t - 2)x + y
+--R +
+--R 2 2 2
+--R (- 2t z - 4t)y + (t + 10)z - 8z + 4t + 2
+--R ,
+--R 2 2
+--R (- 3z y + 2t z + t)x + (z + 4)y + 4t z - 7t z, (- 2z - 2)x - 2z + 8z
- 2,
+--R
+--R 2 2 2 2 2 2
+--R (3y - 3t - 4)x + (- 6y + (12t z + 12t)y + 6t z)x + 3y
+--R +
+--R 2 2 2 2
+--R (12t z - 12t)y + (9t + 4)z + (- 24t - 4)z + 12t + 4
+--R ]
+--RType: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t])
+--E 33
+
+--S 34 of 34
+zeroSetSplit(lq,true,true)$T
+--R
+--R[1 <4,0> -> |4|; {0}]W[2 <5,0>,<3,1> -> |8|; {0}][2 <4,1>,<3,1> -> |7|;
{0}][1 <3,1> -> |3|; {0}]G[2 <4,1>,<4,1> -> |8|; {0}]W[3 <5,1>,<4,1>,<3,2> ->
|12|; {0}]GI[3 <4,2>,<4,1>,<3,2> -> |11|; {0}]GWw[3 <4,1>,<3,2>,<5,2> -> |12|;
{0}][3 <3,2>,<3,2>,<5,2> -> |11|; {0}]GIwWWWw[4 <3,2>,<4,2>,<5,2>,<2,3> ->
|14|; {0}][4 <2,2>,<4,2>,<5,2>,<2,3> -> |13|; {0}]Gwww[5
<3,2>,<3,2>,<4,2>,<5,2>,<2,3> -> |17|; {0}]Gwwwwww[8
<3,2>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |30|; {0}]Gwwwwww[8
<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |31|; {0}][8
<3,3>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |30|; {0}][8
<2,3>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |29|; {0}][8
<1,3>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |28|; {0}][7
<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |27|; {0}][6
<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |23|; {0}][5
<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |19|; {0}]GIGIWwww[6
<5,2>,<4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |23|; {0}][6
<4,3>,<4,2>,<4,2>,<5,2>,<3,3!
>,<2,3> -> |22|; {0}]GIGI[6 <3,4>,<4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |21|; {0}][6
><2,4>,<4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |20|; {0}]GGG[5
><4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |18|; {0}]GIGIWwwwW[6
><5,2>,<4,2>,<5,2>,<3,3>,<3,3>,<2,3> -> |22|; {0}][6
><4,3>,<4,2>,<5,2>,<3,3>,<3,3>,<2,3> -> |21|; {0}]GIwwWwWWWWWWWwWWWWwwwww[8
><4,2>,<5,2>,<3,3>,<3,3>,<4,3>,<2,3>,<3,4>,<3,4> -> |27|; {0}][8
><3,3>,<5,2>,<3,3>,<3,3>,<4,3>,<2,3>,<3,4>,<3,4> -> |26|; {0}][8
><2,3>,<5,2>,<3,3>,<3,3>,<4,3>,<2,3>,<3,4>,<3,4> -> |25|;
>{0}]Gwwwwwwwwwwwwwwwwwwww[9
><5,2>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |29|; {0}]GI[9
><4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |28|; {0}][9
><3,3>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |27|; {0}][9
><2,3>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |26|;
>{0}]GGwwwwwwwwwwwwWWwwwwwwww[11
><3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |33|;
>{0}][11 <2,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> !
|32|; {0}][11 <1,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>!
,<3,3>,<3,4>,<3,4> -> |31|; {0}]GGGwwwwwwwwwwwww[12
<2,3>,<2,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> ->
|34|; {0}]GGwwwwwwwwwwwww[13
<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |38|; {0}]Gwwwwwwwwwwwww[13
<2,3>,<3,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |39|; {0}]GGGwwwwwwwwwwwww[15
<3,3>,<4,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |46|; {0}][14
<4,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |43|; {0}]GIGGGGIGGI[14
<3,4>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |42|; {0}]GGG[14
<2,4>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |41|; {0}][14
<1,4>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |40|; {0}]GGG[13
<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |39|;!
{0}]Gwwwwwwwwwwwww[15
<3,3>,<3,3>,<4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |48|; {0}]Gwwwwwwwwwwwww[15
<4,3>,<4,3>,<3,3>,<4,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |49|; {0}]GIGI[15
<3,4>,<4,3>,<3,3>,<4,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |48|; {0}]G[14
<4,3>,<3,3>,<4,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |45|; {0}][13
<3,3>,<4,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |41|; {0}]Gwwwwwwwwwwwww[13
<4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |42|; {0}]GIGGGGIGGI[13
<3,4>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |41|; {0}]GGGGGGGG[13
<2,4>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |40|; {0}][13
<1,4>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |39|; {0}][13 <0,4>,<4,3>,<4,3>,!
<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -!
> |38|; {0}][12
> <4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> ->
> |38|; {1}][11
> <4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |34|;
> {1}][10 <3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |30|;
> {1}][10 <2,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |29|;
> {1}]GGGwwwwwwwwwwwww[11
> <3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |33|;
> {1}]GGGwwwwwwwwwwwww[12
> <4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> ->
> |38|; {1}]Gwwwwwwwwwwwww[12
> <3,3>,<4,3>,<5,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> ->
> |39|; {1}]GGwwwwwwwwwwwww[13
> <5,3>,<4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
> -> |44|; {1}]GIGGGGIGGIW[13
> <4,4>,<4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
> -> |43|; {1}]GGW[13
> <3,4>,<4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
> -> |42|; {1}]GGG[12 <4,3>,<4,3>,<4,3>,<3,3>!
,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |39|; {1}]Gwwwwwwwwwwwww[12
<4,3>,<4,3>,<5,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> ->
|40|; {1}]Gwwwwwwwwwwwww[13
<5,3>,<5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |46|; {1}]GIGIW[13
<4,4>,<5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |45|; {1}][13
<3,4>,<5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |44|; {1}][13
<2,4>,<5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
-> |43|; {1}]GG[12
<5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> ->
|41|; {1}]GIGGGGIGGIW[12
<4,4>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> ->
|40|; {1}]GGGGGGW[12
<3,4>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> ->
|39|; {1}][12
<2,4>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> ->
|38|; {1}][12 <1,4>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,!
3>,<3,3>,<3,3>,<3,4>,<3,4> -> |37|; {1}]GGG[11 <4,3>,<5,3>,<3!
,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |36|; {1}][10
<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |32|; {1}][9
<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |27|; {1}]W[9
<2,4>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |26|; {1}][9
<1,4>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |25|; {1}][8
<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |24|; {1}]W[8
<2,4>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |23|; {1}][8
<1,4>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |22|; {1}][7
<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |21|; {1}]w[7
<3,4>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |20|; {1}][7
<2,4>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |19|; {1}][7
<1,4>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |18|; {1}][6
<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |17|; {1}]GGwwwwww[7
<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |21|; {1}]GIW[7
<2,4>,<3,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |20|; {1}]GG[6
<3,3>,<3,3>,<3,3>,<3,3>,<3!
,4>,<3,4> -> |18|; {1}]Gwwwwww[7 <4,3>,<4,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> ->
|23|; {1}]GIW[7 <3,4>,<4,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |22|; {1}][6
<4,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |19|; {1}]GIW[6
<3,4>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |18|; {1}]GGW[6
<2,4>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |17|; {1}][6
<1,4>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |16|; {1}]GGG[5
<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |15|; {1}]GIW[5 <2,4>,<3,3>,<3,3>,<3,4>,<3,4>
-> |14|; {1}]GG[4 <3,3>,<3,3>,<3,4>,<3,4> -> |12|; {1}][3 <3,3>,<3,4>,<3,4> ->
|9|; {1}]W[3 <2,4>,<3,4>,<3,4> -> |8|; {1}][3 <1,4>,<3,4>,<3,4> -> |7|; {1}]G[2
<3,4>,<3,4> -> |6|; {1}]G[1 <3,4> -> |3|; {1}][1 <2,4> -> |2|; {1}][1 <1,4> ->
|1|; {1}]
+--R *** QCMPACK Statistics ***
+--R Table size: 36
+--R Entries reused: 255
+--R
+--R *** REGSETGCD: Gcd Statistics ***
+--R Table size: 125
+--R Entries reused: 0
+--R
+--R *** REGSETGCD: Inv Set Statistics ***
+--R Table size: 30
+--R Entries reused: 0
+--R
+--R (34)
+--R [
+--R {
+--R 24 23 22
+--R 960725655771966t + 386820897948702t + 8906817198608181t
+--R +
+--R 21 20 19
+--R 2704966893949428t + 37304033340228264t + 7924782817170207t
+--R +
+--R 18 17 16
+--R 93126799040354990t + 13101273653130910t + 156146250424711858t
+--R +
+--R 15 14 13
+--R 16626490957259119t + 190699288479805763t + 24339173367625275t
+--R +
+--R 12 11
10
+--R 180532313014960135t + 35288089030975378t + 135054975747656285t
+--R +
+--R 9 8 7
+--R 34733736952488540t + 75947600354493972t + 19772555692457088t
+--R +
+--R 6 5 4
+--R 28871558573755428t + 5576152439081664t + 6321711820352976t
+--R +
+--R 3 2
+--R 438314209312320t + 581105748367008t - 60254467992576t +
1449115951104
+--R ,
+--R
+--R 23
+--R 26604210869491302385515265737052082361668474181372891857784t
+--R +
+--R 22
+--R 443104378424686086067294899528296664238693556855017735265295t
+--R +
+--R 21
+--R 279078393286701234679141342358988327155321305829547090310242t
+--R +
+--R
20
+--R 3390276361413232465107617176615543054620626391823613392185226t
+--R +
+--R 19
+--R 941478179503540575554198645220352803719793196473813837434129t
+--R +
+--R
18
+--R 11547855194679475242211696749673949352585747674184320988144390t
+--R +
+--R
17
+--R 1343609566765597789881701656699413216467215660333356417241432t
+--R +
+--R
16
+--R 23233813868147873503933551617175640859899102987800663566699334t
+--R +
+--R 15
+--R 869574020537672336950845440508790740850931336484983573386433t
+--R +
+--R
14
+--R 31561554305876934875419461486969926554241750065103460820476969t
+--R +
+--R
13
+--R 1271400990287717487442065952547731879554823889855386072264931t
+--R +
+--R
12
+--R 31945089913863736044802526964079540198337049550503295825160523t
+--R +
+--R
11
+--R 3738735704288144509871371560232845884439102270778010470931960t
+--R +
+--R
10
+--R 25293997512391412026144601435771131587561905532992045692885927t
+--R +
+--R 9
+--R 5210239009846067123469262799870052773410471135950175008046524t
+--R +
+--R
8
+--R 15083887986930297166259870568608270427403187606238713491129188t
+--R +
+--R 7
+--R 3522087234692930126383686270775779553481769125670839075109000t
+--R +
+--R 6
+--R 6079945200395681013086533792568886491101244247440034969288588t
+--R +
+--R 5
+--R 1090634852433900888199913756247986023196987723469934933603680t
+--R +
+--R 4
+--R 1405819430871907102294432537538335402102838994019667487458352t
+--R +
+--R 3
+--R 88071527950320450072536671265507748878347828884933605202432t
+--R +
+--R 2
+--R 135882489433640933229781177155977768016065765482378657129440t
+--R +
+--R - 13957283442882262230559894607400314082516690749975646520320t
+--R +
+--R 334637692973189299277258325709308472592117112855749713920
+--R *
+--R z
+--R +
+--R 23
+--R 8567175484043952879756725964506833932149637101090521164936t
+--R +
+--R 22
+--R 149792392864201791845708374032728942498797519251667250945721t
+--R +
+--R 21
+--R 77258371783645822157410861582159764138123003074190374021550t
+--R +
+--R 20
+--R 1108862254126854214498918940708612211184560556764334742191654t
+--R +
+--R 19
+--R 213250494460678865219774480106826053783815789621501732672327t
+--R +
+--R 18
+--R 3668929075160666195729177894178343514501987898410131431699882t
+--R +
+--R 17
+--R 171388906471001872879490124368748236314765459039567820048872t
+--R +
+--R 16
+--R 7192430746914602166660233477331022483144921771645523139658986t
+--R +
+--R 15
+--R - 128798674689690072812879965633090291959663143108437362453385t
+--R +
+--R 14
+--R 9553010858341425909306423132921134040856028790803526430270671t
+--R +
+--R 13
+--R - 13296096245675492874538687646300437824658458709144441096603t
+--R +
+--R 12
+--R 9475806805814145326383085518325333106881690568644274964864413t
+--R +
+--R 11
+--R 803234687925133458861659855664084927606298794799856265539336t
+--R +
+--R 10
+--R 7338202759292865165994622349207516400662174302614595173333825t
+--R +
+--R 9
+--R 1308004628480367351164369613111971668880538855640917200187108t
+--R +
+--R 8
+--R 4268059455741255498880229598973705747098216067697754352634748t
+--R +
+--R 7
+--R 892893526858514095791318775904093300103045601514470613580600t
+--R +
+--R 6
+--R 1679152575460683956631925852181341501981598137465328797013652t
+--R +
+--R 5
+--R 269757415767922980378967154143357835544113158280591408043936t
+--R +
+--R 4
+--R 380951527864657529033580829801282724081345372680202920198224t
+--R +
+--R 3
+--R 19785545294228495032998826937601341132725035339452913286656t
+--R +
+--R 2
+--R 36477412057384782942366635303396637763303928174935079178528t
+--R +
+--R - 3722212879279038648713080422224976273210890229485838670848t
+--R +
+--R 89079724853114348361230634484013862024728599906874105856
+--R ,
+--R 3 2 3 2
+--R (3z - 11z + 8z + 4)y + 2t z + 4t z - 5t z - t,
+--R 2
+--R (z + 1)x + z - 4z + 1}
+--R ]
+--RType: List RegularTriangularSet(Integer,IndexedExponents
OrderedVariableList [x,y,z,t],OrderedVariableList
[x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList
[x,y,z,t]))
+--E 34
+)spool
+)lisp (bye)
+@
+<<RegularTriangularSet.help>>=
+====================================================================
+RegularTriangularSet examples
+====================================================================
+
+The RegularTriangularSet domain constructor implements regular
+triangular sets. These particular triangular sets were introduced by
+M. Kalkbrener (1991) in his PhD Thesis under the name regular chains.
+Regular chains and their related concepts are presented in the paper
+"On the Theories of Triangular sets" By P. Aubry, D. Lazard and
+M. Moreno Maza (to appear in the Journal of Symbolic Computation).
+The RegularTriangularSet constructor also provides a new method
+(by the third author) for solving polynomial system by means of
+regular chains. This method has two ways of solving. One has the
+same specifications as Kalkbrener's algorithm (1991) and the other is
+closer to Lazard's method (Discr. App. Math, 1991). Moreover, this
+new method removes redundant component from the decompositions when
+this is not too expensive. This is always the case with
+square-free regular chains. So if you want to obtain decompositions
+without redundant components just use the SquareFreeRegularTriangularSet
+domain constructor or the LazardSetSolvingPackage package constructor.
+See also the LexTriangularPackage and ZeroDimensionalSolvePackage for the
+case of algebraic systems with a finite number of (complex) solutions.
+
+One of the main features of regular triangular sets is that they
+naturally define towers of simple extensions of a field. This allows
+to perform with multivariate polynomials the same kind of operations
+as one can do in an EuclideanDomain.
+
+The RegularTriangularSet constructor takes four arguments. The first
+one, R, is the coefficient ring of the polynomials; it must belong to
+the category GcdDomain. The second one, E, is the exponent monoid of
+the polynomials; it must belong to the category OrderedAbelianMonoidSup.
+The third one, V, is the ordered set of variables; it must belong to the
+category OrderedSet. The last one is the polynomial ring; it must belong
+to the category RecursivePolynomialCategory(R,E,V). The abbreviation for
+RegularTriangularSet is REGSET. See also the constructor RegularChain
+which only takes two arguments, the coefficient ring and the ordered
+set of variables; in that case, polynomials are necessarily built with
+the NewSparseMultivariatePolynomial domain constructor.
+
+We shall explain now how to use the constructor REGSET and how to read
+the decomposition of a polynomial system by means of regular sets.
+
+Let us give some examples. We start with an easy one
+(Donati-Traverso) in order to understand the two ways of solving
+polynomial systems provided by the REGSET constructor.
+
+Define the coefficient ring.
+
+ R := Integer
+ Integer
+ Type: Domain
+
+Define the list of variables,
+
+ ls : List Symbol := [x,y,z,t]
+ [x,y,z,t]
+ Type: List Symbol
+
+and make it an ordered set;
+
+ V := OVAR(ls)
+ OrderedVariableList [x,y,z,t]
+ Type: Domain
+
+then define the exponent monoid.
+
+ E := IndexedExponents V
+ IndexedExponents OrderedVariableList [x,y,z,t]
+ Type: Domain
+
+Define the polynomial ring.
+
+ P := NSMP(R, V)
+ NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+ Type: Domain
+
+Let the variables be polynomial.
+
+ x: P := 'x
+ x
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+ y: P := 'y
+ y
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ z: P := 'z
+ z
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ t: P := 't
+ t
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+Now call the RegularTriangularSet domain constructor.
+
+ T := REGSET(R,E,V,P)
+ RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],O
+ rderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedV
+ ariableList [x,y,z,t]))
+ Type: Domain
+
+Define a polynomial system.
+
+ p1 := x ** 31 - x ** 6 - x - y
+ 31 6
+ x - x - x - y
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ p2 := x ** 8 - z
+ 8
+ x - z
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ p3 := x ** 10 - t
+ 10
+ x - t
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ lp := [p1, p2, p3]
+ 31 6 8 10
+ [x - x - x - y,x - z,x - t]
+ Type: List NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+First of all, let us solve this system in the sense of Kalkbrener.
+
+ zeroSetSplit(lp)$T
+ 5 4 2 3 8 5 3 2 4 2
+ [{z - t ,t z y + 2z y - t + 2t + t - t ,(t - t)x - t y - z }]
+ Type: List RegularTriangularSet(Integer,
+ IndexedExponents OrderedVariableList [x,y,z,t],
+ OrderedVariableList [x,y,z,t],
+ NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t]))
+
+And now in the sense of Lazard (or Wu and other authors).
+
+ lts := zeroSetSplit(lp,false)$T
+ 5 4 2 3 8 5 3 2 4 2
+ [{z - t ,t z y + 2z y - t + 2t + t - t ,(t - t)x - t y - z },
+ 3 5 2 3 2
+ {t - 1,z - t,t z y + 2z y + 1,z x - t}, {t,z,y,x}]
+ Type: List RegularTriangularSet(Integer,
+ IndexedExponents OrderedVariableList [x,y,z,t],
+ OrderedVariableList [x,y,z,t],
+ NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t]))
+
+We can see that the first decomposition is a subset of the second.
+So how can both be correct ?
+
+Recall first that polynomials from a domain of the category
+RecursivePolynomialCategory are regarded as univariate polynomials in
+their main variable. For instance the second polynomial in the first
+set of each decomposition has main variable y and its initial
+(i.e. its leading coefficient w.r.t. its main variable) is t z.
+
+Now let us explain how to read the second decomposition. Note that
+the non-constant initials of the first set are t^4-t and t z. Then
+the solutions described by this first set are the common zeros of
+its polynomials that do not cancel the polynomials t^4-t and ty z.
+Now the solutions of the input system lp satisfying these equations
+are described by the second and the third sets of the decomposition.
+Thus, in some sense, they can be considered as degenerated solutions.
+The solutions given by the first set are called the generic points of
+the system; they give the general form of the solutions. The first
+decomposition only provides these generic points. This latter
+decomposition is useful when they are many degenerated solutions
+(which is sometimes hard to compute) and when one is only interested
+in general informations, like the dimension of the input system.
+
+We can get the dimensions of each component of a decomposition as follows.
+
+ [coHeight(ts) for ts in lts]
+ [1,0,0]
+ Type: List NonNegativeInteger
+
+Thus the first set has dimension one. Indeed t can take any value,
+except 0 or any third root of 1, whereas z is completely determined
+from t, y is given by z and t, and finally x is given by the other
+three variables. In the second and the third sets of the second
+decomposition the four variables are completely determined and thus
+these sets have dimension zero.
+
+We give now the precise specifications of each decomposition. This
+assume some mathematical knowledge. However, for the non-expert user,
+the above explanations will be sufficient to understand the other
+features of the RSEGSET constructor.
+
+The input system lp is decomposed in the sense of Kalkbrener as
+finitely many regular sets T1,...,Ts such that the radical ideal
+generated by lp is the intersection of the radicals of the saturated
+ideals of T1,...,Ts. In other words, the affine variety associated
+with lp is the union of the closures (w.r.t. Zarisky topology) of the
+regular-zeros sets of T1,...,Ts.
+
+N. B. The prime ideals associated with the radical of the saturated
+ideal of a regular triangular set have all the same dimension;
+moreover these prime ideals can be given by characteristic sets with
+the same main variables. Thus a decomposition in the sense of
+Kalkbrener is unmixed dimensional. Then it can be viewed as a lazy
+decomposition into prime ideals (some of these prime ideals being
+merged into unmixed dimensional ideals).
+
+Now we explain the other way of solving by means of regular triangular
+sets. The input system lp is decomposed in the sense of Lazard as
+finitely many regular triangular sets T1,...,Ts such that the affine
+variety associated with lp is the union of the regular-zeros sets of
+T1,...,Ts. Thus a decomposition in the sense of Lazard is also a
+decomposition in the sense of Kalkbrener; the converse is false as we
+have seen before.
+
+When the input system has a finite number of solutions, both ways of
+solving provide similar decompositions as we shall see with this
+second example (Caprasse).
+
+Define a polynomial system.
+
+ f1 := y**2*z+2*x*y*t-2*x-z
+ 2
+ (2t y - 2)x + z y - z
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ f2:=-x**3*z+ 4*x*y**2*z+4*x**2*y*t+2*y**3*t+4*x**2-10*y**2+4*x*z-10*y*t+2
+ 3 2 2 3 2
+ - z x + (4t y + 4)x + (4z y + 4z)x + 2t y - 10y - 10t y + 2
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ f3 := 2*y*z*t+x*t**2-x-2*z
+ 2
+ (t - 1)x + 2t z y - 2z
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ f4:=-x*z**3+4*y*z**2*t+4*x*z*t**2+2*y*t**3+4*x*z+4*z**2-10*y*t- 10*t**2+2
+ 3 2 2 3 2 2
+ (- z + (4t + 4)z)x + (4t z + 2t - 10t)y + 4z - 10t + 2
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ lf := [f1, f2, f3, f4]
+ 2
+ [(2t y - 2)x + z y - z,
+ 3 2 2 3 2
+ - z x + (4t y + 4)x + (4z y + 4z)x + 2t y - 10y - 10t y + 2,
+ 2
+ (t - 1)x + 2t z y - 2z,
+ 3 2 2 3 2 2
+ (- z + (4t + 4)z)x + (4t z + 2t - 10t)y + 4z - 10t + 2]
+ Type: List NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+First of all, let us solve this system in the sense of Kalkbrener.
+
+ zeroSetSplit(lf)$T
+ 2 8 6 2 3 2
+ [{t - 1,z - 16z + 256z - 256,t y - 1,(z - 8z)x - 8z + 16},
+ 2 2 2
+ {3t + 1,z - 7t - 1,y + t,x + z},
+ 8 6 2 3 2
+ {t - 10t + 10t - 1,z,(t - 5t)y - 5t + 1,x},
+ 2 2
+ {t + 3,z - 4,y + t,x - z}]
+ Type: List RegularTriangularSet(Integer,
+ IndexedExponents OrderedVariableList [x,y,z,t],
+ OrderedVariableList [x,y,z,t],
+ NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t]))
+
+And now in the sense of Lazard (or Wu and other authors).
+
+ lts2 := zeroSetSplit(lf,false)$T
+ 8 6 2 3 2
+ [{t - 10t + 10t - 1,z,(t - 5t)y - 5t + 1,x},
+ 2 8 6 2 3 2
+ {t - 1,z - 16z + 256z - 256,t y - 1,(z - 8z)x - 8z + 16},
+ 2 2 2 2 2
+ {3t + 1,z - 7t - 1,y + t,x + z}, {t + 3,z - 4,y + t,x - z}]
+ Type: List RegularTriangularSet(Integer,
+ IndexedExponents OrderedVariableList [x,y,z,t],
+ OrderedVariableList [x,y,z,t],
+ NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t]))
+
+Up to the ordering of the components, both decompositions are identical.
+
+Let us check that each component has a finite number of solutions.
+
+ [coHeight(ts) for ts in lts2]
+ [0,0,0,0]
+ Type: List NonNegativeInteger
+
+Let us count the degrees of each component,
+
+ degrees := [degree(ts) for ts in lts2]
+ [8,16,4,4]
+ Type: List NonNegativeInteger
+
+and compute their sum.
+
+ reduce(+,degrees)
+ 32
+ Type: PositiveInteger
+
+We study now the options of the zeroSetSplit operation. As we have seen
+yet, there is an optional second argument which is a boolean value. If this
+value is true (this is the default) then the decomposition is computed in
+the sense of Kalkbrener, otherwise it is computed in the sense of Lazard.
+
+There is a second boolean optional argument that can be used (in that
+case the first optional argument must be present). This second option
+allows you to get some information during the computations.
+
+Therefore, we need to understand a little what is going on during the
+computations. An important feature of the algorithm is that the
+intermediate computations are managed in some sense like the processes
+of a Unix system. Indeed, each intermediate computation may generate
+other intermediate computations and the management of all these
+computations is a crucial task for the efficiency. Thus any
+intermediate computation may be suspended, killed or resumed,
+depending on algebraic considerations that determine priorities for
+these processes. The goal is of course to go as fast as possible
+towards the final decomposition which means to avoid as much as
+possible unnecessary computations.
+
+To follow the computations, one needs to set to true the second
+argument. Then a lot of numbers and letters are displayed. Between a
+[ and a ] one has the state of the processes at a given time. Just
+after [ one can see the number of processes. Then each process is
+represented by two numbers between < and >. A process consists of a
+list of polynomial ps and a triangular set ts; its goal is to compute
+the common zeros of ps that belong to the regular-zeros set of ts.
+After the processes, the number between pipes gives the total number
+of polynomials in all the sets ps. Finally, the number between braces
+gives the number of components of a decomposition that are already
+computed. This number may decrease.
+
+Let us take a third example (Czapor-Geddes-Wang) to see how this
+information is displayed.
+
+Define a polynomial system.
+
+ u : R := 2
+ 2
+ Type: Integer
+
+ q1 := 2*(u-1)**2+ 2*(x-z*x+z**2)+ y**2*(x-1)**2- 2*u*x+ 2*y*t*(1-x)*(x-z)+_
+ 2*u*z*t*(t-y)+ u**2*t**2*(1-2*z)+ 2*u*t**2*(z-x)+ 2*u*t*y*(z-1)+_
+ 2*u*z*x*(y+1)+ (u**2-2*u)*z**2*t**2+ 2*u**2*z**2+ 4*u*(1-u)*z+_
+ t**2*(z-x)**2}
+ 2 2 2 2 2 2
+ (y - 2t y + t )x + (- 2y + ((2t + 4)z + 2t)y + (- 2t + 2)z - 4t - 2)x
+ +
+ 2 2 2 2
+ y + (- 2t z - 4t)y + (t + 10)z - 8z + 4t + 2
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ q2 := t*(2*z+1)*(x-z)+ y*(z+2)*(1-x)+ u*(u-2)*t+ u*(1-2*u)*z*t+_
+ u*y*(x+u-z*x-1)+ u*(u+1)*z**2*t}
+ (- 3z y + 2t z + t)x + (z + 4)y + 4t z - 7t z
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ q3 := -u**2*(z-1)**2+ 2*z*(z-x)-2*(x-1)
+ (- 2z - 2)x - 2z + 8z - 2
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ q4 := u**2+4*(z-x**2)+3*y**2*(x-1)**2- 3*t**2*(z-x)**2+_
+ 3*u**2*t**2*(z-1)**2+u**2*z*(z-2)+6*u*t*y*(z+x+z*x-1)}
+ 2 2 2 2 2 2
+ (3y - 3t - 4)x + (- 6y + (12t z + 12t)y + 6t z)x + 3y + (12t z - 12t)y
+ +
+ 2 2 2 2
+ (9t + 4)z + (- 24t - 4)z + 12t + 4
+ Type: NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+ lq := [q1, q2, q3, q4]
+ [
+ 2 2 2
+ (y - 2t y + t )x
+ +
+ 2 2 2 2
+ (- 2y + ((2t + 4)z + 2t)y + (- 2t + 2)z - 4t - 2)x + y
+ +
+ 2 2 2
+ (- 2t z - 4t)y + (t + 10)z - 8z + 4t + 2
+ ,
+ 2 2
+ (- 3z y + 2t z + t)x + (z + 4)y + 4t z - 7t z, (- 2z - 2)x - 2z + 8z - 2,
+
+ 2 2 2 2 2 2
+ (3y - 3t - 4)x + (- 6y + (12t z + 12t)y + 6t z)x + 3y
+ +
+ 2 2 2 2
+ (12t z - 12t)y + (9t + 4)z + (- 24t - 4)z + 12t + 4
+ ]
+ Type: List NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t])
+
+Let us try the information option. N.B. The timing should be between
+1 and 10 minutes, depending on your machine.
+
+ zeroSetSplit(lq,true,true)$T
+ [1 <4,0> -> |4|; {0}]W[2 <5,0>,<3,1> -> |8|; {0}]
+ [2 <4,1>,<3,1> -> |7|; {0}]
+ [1 <3,1> -> |3|; {0}]G
+ [2 <4,1>,<4,1> -> |8|; {0}]W
+ [3 <5,1>,<4,1>,<3,2> -> |12|; {0}]GI
+ [3 <4,2>,<4,1>,<3,2> -> |11|; {0}]GWw
+ [3 <4,1>,<3,2>,<5,2> -> |12|; {0}]
+ [3 <3,2>,<3,2>,<5,2> -> |11|; {0}]GIwWWWw
+ [4 <3,2>,<4,2>,<5,2>,<2,3> -> |14|; {0}]
+ [4 <2,2>,<4,2>,<5,2>,<2,3> -> |13|; {0}]Gwww
+ [5 <3,2>,<3,2>,<4,2>,<5,2>,<2,3> -> |17|; {0}]Gwwwwww
+ [8 <3,2>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |30|; {0}]Gwwwwww
+ [8 <4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |31|; {0}]
+ [8 <3,3>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |30|; {0}]
+ [8 <2,3>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |29|; {0}]
+ [8 <1,3>,<4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |28|; {0}]
+ [7 <4,2>,<4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |27|; {0}]
+ [6 <4,2>,<4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |23|; {0}]
+ [5 <4,2>,<4,2>,<4,2>,<5,2>,<2,3> -> |19|; {0}]GIGIWwww
+ [6 <5,2>,<4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |23|; {0}]
+ [6 <4,3>,<4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |22|; {0}]GIGI
+ [6 <3,4>,<4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |21|; {0}]
+ [6 <2,4>,<4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |20|; {0}]GGG
+ [5 <4,2>,<4,2>,<5,2>,<3,3>,<2,3> -> |18|; {0}]GIGIWwwwW
+ [6 <5,2>,<4,2>,<5,2>,<3,3>,<3,3>,<2,3> -> |22|; {0}]
+ [6 <4,3>,<4,2>,<5,2>,<3,3>,<3,3>,<2,3> -> |21|; {0}]
+ GIwwWwWWWWWWWwWWWWwwwww
+ [8 <4,2>,<5,2>,<3,3>,<3,3>,<4,3>,<2,3>,<3,4>,<3,4> -> |27|; {0}]
+ [8 <3,3>,<5,2>,<3,3>,<3,3>,<4,3>,<2,3>,<3,4>,<3,4> -> |26|; {0}]
+ [8 <2,3>,<5,2>,<3,3>,<3,3>,<4,3>,<2,3>,<3,4>,<3,4> -> |25|; {0}]
+ Gwwwwwwwwwwwwwwwwwwww
+ [9 <5,2>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |29|; {0}]GI
+ [9 <4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |28|; {0}]
+ [9 <3,3>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |27|; {0}]
+ [9 <2,3>,<3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,4>,<3,4> -> |26|; {0}]
+ GGwwwwwwwwwwwwWWwwwwwwww
+ [11 <3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
+ -> |33|; {0}]
+ [11 <2,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
+ -> |32|; {0}]
+ [11 <1,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
+ -> |31|; {0}]GGGwwwwwwwwwwwww
+ [12 <2,3>,<2,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,
+ <3,4> -> |34|; {0}]GGwwwwwwwwwwwww
+ [13 <3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |38|; {0}]Gwwwwwwwwwwwww
+ [13 <2,3>,<3,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |39|; {0}]GGGwwwwwwwwwwwww
+ [15 <3,3>,<4,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,
+ <3,3>,<3,3>,<3,4>,<3,4> -> |46|; {0}]
+ [14 <4,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,
+ <3,3>,<3,4>,<3,4> -> |43|; {0}]GIGGGGIGGI
+ [14 <3,4>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,
+ <3,3>,<3,4>,<3,4> -> |42|; {0}]GGG
+ [14 <2,4>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,
+ <3,3>,<3,4>,<3,4> -> |41|; {0}]
+ [14 <1,4>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,
+ <3,3>,<3,4>,<3,4> -> |40|; {0}]GGG
+ [13 <3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |39|; {0}]Gwwwwwwwwwwwww
+ [15 <3,3>,<3,3>,<4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,
+ <3,3>,<3,3>,<3,4>,<3,4> -> |48|; {0}]Gwwwwwwwwwwwww
+ [15 <4,3>,<4,3>,<3,3>,<4,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,
+ <3,3>,<3,3>,<3,4>,<3,4> -> |49|; {0}]GIGI
+ [15 <3,4>,<4,3>,<3,3>,<4,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,
+ <3,3>,<3,3>,<3,4>,<3,4> -> |48|; {0}]G
+ [14 <4,3>,<3,3>,<4,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,
+ <3,3>,<3,4>,<3,4> -> |45|; {0}]
+ [13 <3,3>,<4,3>,<4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |41|; {0}]Gwwwwwwwwwwwww
+ [13 <4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |42|; {0}]GIGGGGIGGI
+ [13 <3,4>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |41|; {0}]GGGGGGGG
+ [13 <2,4>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |40|; {0}]
+ [13 <1,4>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |39|; {0}]
+ [13 <0,4>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |38|; {0}]
+ [12 <4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,
+ <3,4> -> |38|; {1}]
+ [11 <4,3>,<3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
+ -> |34|; {1}]
+ [10 <3,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
+ -> |30|; {1}]
+ [10 <2,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
+ -> |29|; {1}]GGGwwwwwwwwwwwww
+ [11 <3,3>,<3,3>,<4,3>,<3,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
+ -> |33|; {1}]GGGwwwwwwwwwwwww
+ [12 <4,3>,<3,3>,<4,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,
+ <3,4> -> |38|; {1}]Gwwwwwwwwwwwww
+ [12 <3,3>,<4,3>,<5,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,
+ <3,4> -> |39|; {1}]GGwwwwwwwwwwwww
+ [13 <5,3>,<4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |44|; {1}]GIGGGGIGGIW
+ [13 <4,4>,<4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |43|; {1}]GGW
+ [13 <3,4>,<4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |42|; {1}]GGG
+ [12 <4,3>,<4,3>,<4,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,
+ <3,4> -> |39|; {1}]Gwwwwwwwwwwwww
+ [12 <4,3>,<4,3>,<5,3>,<3,3>,<4,3>,<3,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,
+ <3,4> -> |40|; {1}]Gwwwwwwwwwwwww
+ [13 <5,3>,<5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |46|; {1}]GIGIW
+ [13 <4,4>,<5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |45|; {1}]
+ [13 <3,4>,<5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |44|; {1}]
+ [13 <2,4>,<5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,
+ <3,4>,<3,4> -> |43|; {1}]GG
+ [12 <5,3>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,
+ <3,4> -> |41|; {1}]GIGGGGIGGIW
+ [12 <4,4>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,
+ <3,4> -> |40|; {1}]GGGGGGW
+ [12 <3,4>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,
+ <3,4> -> |39|; {1}]
+ [12 <2,4>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,
+ <3,4> -> |38|; {1}]
+ [12 <1,4>,<4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,
+ <3,4> -> |37|; {1}]GGG
+ [11 <4,3>,<5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
+ -> |36|; {1}]
+ [10 <5,3>,<3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4>
+ -> |32|; {1}]
+ [9 <3,3>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |27|; {1}]W
+ [9 <2,4>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |26|; {1}]
+ [9 <1,4>,<3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |25|; {1}]
+ [8 <3,3>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |24|; {1}]W
+ [8 <2,4>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |23|; {1}]
+ [8 <1,4>,<4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |22|; {1}]
+ [7 <4,3>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |21|; {1}]w
+ [7 <3,4>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |20|; {1}]
+ [7 <2,4>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |19|; {1}]
+ [7 <1,4>,<2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |18|; {1}]
+ [6 <2,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |17|; {1}]GGwwwwww
+ [7 <3,3>,<3,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |21|; {1}]GIW
+ [7 <2,4>,<3,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |20|; {1}]GG
+ [6 <3,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |18|; {1}]Gwwwwww
+ [7 <4,3>,<4,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |23|; {1}]GIW
+ [7 <3,4>,<4,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |22|; {1}]
+ [6 <4,3>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |19|; {1}]GIW
+ [6 <3,4>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |18|; {1}]GGW
+ [6 <2,4>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |17|; {1}]
+ [6 <1,4>,<3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |16|; {1}]GGG
+ [5 <3,3>,<3,3>,<3,3>,<3,4>,<3,4> -> |15|; {1}]GIW
+ [5 <2,4>,<3,3>,<3,3>,<3,4>,<3,4> -> |14|; {1}]GG
+ [4 <3,3>,<3,3>,<3,4>,<3,4> -> |12|; {1}]
+ [3 <3,3>,<3,4>,<3,4> -> |9|; {1}]W
+ [3 <2,4>,<3,4>,<3,4> -> |8|; {1}]
+ [3 <1,4>,<3,4>,<3,4> -> |7|; {1}]G
+ [2 <3,4>,<3,4> -> |6|; {1}]G
+ [1 <3,4> -> |3|; {1}]
+ [1 <2,4> -> |2|; {1}]
+ [1 <1,4> -> |1|; {1}]
+ *** QCMPACK Statistics ***
+ Table size: 36
+ Entries reused: 255
+
+ *** REGSETGCD: Gcd Statistics ***
+ Table size: 125
+ Entries reused: 0
+
+ *** REGSETGCD: Inv Set Statistics ***
+ Table size: 30
+ Entries reused: 0
+
+ [
+ {
+ 24 23 22
+ 960725655771966t + 386820897948702t + 8906817198608181t
+ +
+ 21 20 19
+ 2704966893949428t + 37304033340228264t + 7924782817170207t
+ +
+ 18 17 16
+ 93126799040354990t + 13101273653130910t + 156146250424711858t
+ +
+ 15 14 13
+ 16626490957259119t + 190699288479805763t + 24339173367625275t
+ +
+ 12 11 10
+ 180532313014960135t + 35288089030975378t + 135054975747656285t
+ +
+ 9 8 7
+ 34733736952488540t + 75947600354493972t + 19772555692457088t
+ +
+ 6 5 4
+ 28871558573755428t + 5576152439081664t + 6321711820352976t
+ +
+ 3 2
+ 438314209312320t + 581105748367008t - 60254467992576t + 1449115951104
+ ,
+
+ 23
+ 26604210869491302385515265737052082361668474181372891857784t
+ +
+ 22
+ 443104378424686086067294899528296664238693556855017735265295t
+ +
+ 21
+ 279078393286701234679141342358988327155321305829547090310242t
+ +
+ 20
+ 3390276361413232465107617176615543054620626391823613392185226t
+ +
+ 19
+ 941478179503540575554198645220352803719793196473813837434129t
+ +
+ 18
+ 11547855194679475242211696749673949352585747674184320988144390t
+ +
+ 17
+ 1343609566765597789881701656699413216467215660333356417241432t
+ +
+ 16
+ 23233813868147873503933551617175640859899102987800663566699334t
+ +
+ 15
+ 869574020537672336950845440508790740850931336484983573386433t
+ +
+ 14
+ 31561554305876934875419461486969926554241750065103460820476969t
+ +
+ 13
+ 1271400990287717487442065952547731879554823889855386072264931t
+ +
+ 12
+ 31945089913863736044802526964079540198337049550503295825160523t
+ +
+ 11
+ 3738735704288144509871371560232845884439102270778010470931960t
+ +
+ 10
+ 25293997512391412026144601435771131587561905532992045692885927t
+ +
+ 9
+ 5210239009846067123469262799870052773410471135950175008046524t
+ +
+ 8
+ 15083887986930297166259870568608270427403187606238713491129188t
+ +
+ 7
+ 3522087234692930126383686270775779553481769125670839075109000t
+ +
+ 6
+ 6079945200395681013086533792568886491101244247440034969288588t
+ +
+ 5
+ 1090634852433900888199913756247986023196987723469934933603680t
+ +
+ 4
+ 1405819430871907102294432537538335402102838994019667487458352t
+ +
+ 3
+ 88071527950320450072536671265507748878347828884933605202432t
+ +
+ 2
+ 135882489433640933229781177155977768016065765482378657129440t
+ +
+ - 13957283442882262230559894607400314082516690749975646520320t
+ +
+ 334637692973189299277258325709308472592117112855749713920
+ *
+ z
+ +
+ 23
+ 8567175484043952879756725964506833932149637101090521164936t
+ +
+ 22
+ 149792392864201791845708374032728942498797519251667250945721t
+ +
+ 21
+ 77258371783645822157410861582159764138123003074190374021550t
+ +
+ 20
+ 1108862254126854214498918940708612211184560556764334742191654t
+ +
+ 19
+ 213250494460678865219774480106826053783815789621501732672327t
+ +
+ 18
+ 3668929075160666195729177894178343514501987898410131431699882t
+ +
+ 17
+ 171388906471001872879490124368748236314765459039567820048872t
+ +
+ 16
+ 7192430746914602166660233477331022483144921771645523139658986t
+ +
+ 15
+ - 128798674689690072812879965633090291959663143108437362453385t
+ +
+ 14
+ 9553010858341425909306423132921134040856028790803526430270671t
+ +
+ 13
+ - 13296096245675492874538687646300437824658458709144441096603t
+ +
+ 12
+ 9475806805814145326383085518325333106881690568644274964864413t
+ +
+ 11
+ 803234687925133458861659855664084927606298794799856265539336t
+ +
+ 10
+ 7338202759292865165994622349207516400662174302614595173333825t
+ +
+ 9
+ 1308004628480367351164369613111971668880538855640917200187108t
+ +
+ 8
+ 4268059455741255498880229598973705747098216067697754352634748t
+ +
+ 7
+ 892893526858514095791318775904093300103045601514470613580600t
+ +
+ 6
+ 1679152575460683956631925852181341501981598137465328797013652t
+ +
+ 5
+ 269757415767922980378967154143357835544113158280591408043936t
+ +
+ 4
+ 380951527864657529033580829801282724081345372680202920198224t
+ +
+ 3
+ 19785545294228495032998826937601341132725035339452913286656t
+ +
+ 2
+ 36477412057384782942366635303396637763303928174935079178528t
+ +
+ - 3722212879279038648713080422224976273210890229485838670848t
+ +
+ 89079724853114348361230634484013862024728599906874105856
+ ,
+ 3 2 3 2
+ (3z - 11z + 8z + 4)y + 2t z + 4t z - 5t z - t,
+ 2
+ (z + 1)x + z - 4z + 1}
+ ]
+ Type: List RegularTriangularSet(Integer,
+ IndexedExponents OrderedVariableList [x,y,z,t],
+ OrderedVariableList [x,y,z,t],
+ NewSparseMultivariatePolynomial(Integer,
+ OrderedVariableList [x,y,z,t]))
+
+Between a sequence of processes, thus between a ] and a [ you can see
+capital letters W, G, I and lower case letters i, w. Each time a
+capital letter appears a non-trivial computation has be performed and
+its result is put in a hash-table. Each time a lower case letter
+appears a needed result has been found in an hash-table. The use of
+these hash-tables generally speed up the computations. However, on
+very large systems, it may happen that these hash-tables become too
+big to be handle by your AXIOM configuration. Then in these
+exceptional cases, you may prefer getting a result (even if it takes a
+long time) than getting nothing. Hence you need to know how to
+prevent the RSEGSET constructor from using these hash-tables. In that
+case you will be using the zeroSetSplit with five arguments. The
+first one is the input system lp as above. The second one is a
+boolean value hash? which is true iff you want to use hash-tables.
+The third one is boolean value clos? which is true iff you want to
+solve your system in the sense of Kalkbrener, the other way remaining
+that of Lazard. The fourth argument is boolean value info? which is
+true iff you want to display information during the computations. The
+last one is boolean value prep? which is true iff you want to use some
+heuristics that are performed on the input system before starting the
+real algorithm. The value of this flag is true when you are using
+zeroSetSplit with less than five arguments. Note that there is no
+available signature for zeroSetSplit with four arguments.
+
+We finish this section by some remarks about both ways of solving, in
+the sense of Kalkbrener or in the sense of Lazard. For problems with
+a finite number of solutions, there are theoretically equivalent and
+the resulting decompositions are identical, up to the ordering of the
+components. However, when solving in the sense of Lazard, the
+algorithm behaves differently. In that case, it becomes more
+incremental than in the sense of Kalkbrener. That means the
+polynomials of the input system are considered one after another
+whereas in the sense of Kalkbrener the input system is treated more
+globally.
+
+This makes an important difference in positive dimension. Indeed when
+solving in the sense of Kalkbrener, the Primeidealkettensatz of Krull
+is used. That means any regular triangular containing more
+polynomials than the input system can be deleted. This is not
+possible when solving in the sense of Lazard. This explains why
+Kalkbrener's decompositions usually contain less components than those
+of Lazard. However, it may happen with some examples that the
+incremental process (that cannot be used when solving in the sense of
+Kalkbrener) provide a more efficient way of solving than the global
+one even if the Primeidealkettensatz is used. Thus just try both,
+with the various options, before concluding that you cannot solve your
+favorite system with zeroSetSplit. There exist more options at the
+development level that are not currently available in this public
+version.
+
+See Also:
+o )help GcdDomain
+o )help OrderedAbelianMonoidSup
+o )help OrderedSet
+o )help RecursivePolynomialCategory
+o )help RegularChain
+o )help NewSparseMultivariatePolynomial
+o )help ZeroDimensionalSolvePackage
+o )help LexTriangularPackage
+o )help LazardSetSolvingPackage
+o )help SquareFreeRegularTriangularSet
+o )show RegularTriangularSet
+o $AXIOM/doc/src/algebra/regset.spad.dvi
+
+@
<<domain REGSET RegularTriangularSet>>=
)abbrev domain REGSET RegularTriangularSet
++ Author: Marc Moreno Maza
diff --git a/src/algebra/seg.spad.pamphlet b/src/algebra/seg.spad.pamphlet
index 63036d6..7d6026b 100644
--- a/src/algebra/seg.spad.pamphlet
+++ b/src/algebra/seg.spad.pamphlet
@@ -88,6 +88,176 @@ SegmentExpansionCategory(S: OrderedRing, L:
StreamAggregate(S)): Category ==
@
\section{domain SEG Segment}
+<<Segment.input>>=
+-- seg.spad.pamphlet Segment.input
+)spool Segment.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 10
+s := 3..10
+--R
+--R
+--R (1) 3..10
+--R Type: Segment
PositiveInteger
+--E 1
+
+--S 2 of 10
+lo s
+--R
+--R
+--R (2) 3
+--R Type:
PositiveInteger
+--E 2
+
+--S 3 of 10
+hi s
+--R
+--R
+--R (3) 10
+--R Type:
PositiveInteger
+--E 3
+
+--S 4 of 10
+t := 10..3 by -2
+--R
+--R
+--R (4) 10..3 by - 2
+--R Type: Segment
PositiveInteger
+--E 4
+
+--S 5 of 10
+incr s
+--R
+--R
+--R (5) 1
+--R Type:
PositiveInteger
+--E 5
+
+--S 6 of 10
+incr t
+--R
+--R
+--R (6) - 2
+--R Type:
Integer
+--E 6
+
+--S 7 of 10
+l := [1..3, 5, 9, 15..11 by -1]
+--R
+--R
+--R (7) [1..3,5..5,9..9,15..11 by - 1]
+--R Type: List Segment
PositiveInteger
+--E 7
+
+--S 8 of 10
+expand s
+--R
+--R
+--R (8) [3,4,5,6,7,8,9,10]
+--R Type: List
Integer
+--E 8
+
+--S 9 of 10
+expand t
+--R
+--R
+--R (9) [10,8,6,4]
+--R Type: List
Integer
+--E 9
+
+--S 10 of 10
+expand l
+--R
+--R
+--R (10) [1,2,3,5,9,15,14,13,12,11]
+--R Type: List
Integer
+--E 10
+)spool
+)lisp (bye)
+@
+<<Segment.help>>=
+====================================================================
+Segment examples
+====================================================================
+
+The Segment domain provides a generalized interval type.
+
+Segments are created using the .. construct by indicating the
+(included) end points.
+
+ s := 3..10
+ 3..10
+ Type: Segment PositiveInteger
+
+The first end point is called the lo and the second is called hi.
+
+ lo s
+ 3
+ Type: PositiveInteger
+
+These names are used even though the end points might belong to an
+unordered set.
+
+ hi s
+ 10
+ Type: PositiveInteger
+
+In addition to the end points, each segment has an integer "increment".
+An increment can be specified using the "by" construct.
+
+ t := 10..3 by -2
+ 10..3 by - 2
+ Type: Segment PositiveInteger
+
+This part can be obtained using the incr function.
+
+ incr s
+ 1
+ Type: PositiveInteger
+
+Unless otherwise specified, the increment is 1.
+
+ incr t
+ - 2
+ Type: Integer
+
+A single value can be converted to a segment with equal end points.
+This happens if segments and single values are mixed in a list.
+
+ l := [1..3, 5, 9, 15..11 by -1]
+ [1..3,5..5,9..9,15..11 by - 1]
+ Type: List Segment PositiveInteger
+
+If the underlying type is an ordered ring, it is possible to perform
+additional operations. The expand operation creates a list of points
+in a segment.
+
+ expand s
+ [3,4,5,6,7,8,9,10]
+ Type: List Integer
+
+If k > 0, then expand(l..h by k) creates the list [l, l+k, ..., lN]
+where lN <= h < lN+k. If k < 0, then lN >= h > lN+k.
+
+ expand t
+ [10,8,6,4]
+ Type: List Integer
+
+It is also possible to expand a list of segments. This is equivalent
+to appending lists obtained by expanding each segment individually.
+
+ expand l
+ [1,2,3,5,9,15,14,13,12,11]
+ Type: List Integer
+
+See Also:
+o )help UniversalSegment
+o )help SegmentBinding
+o )show Segment
+o $AXIOM/doc/src/algebra/seg.spad.dvi
+
+@
<<domain SEG Segment>>=
)abbrev domain SEG Segment
++ Author: Stephen M. Watt
@@ -216,6 +386,118 @@ SegmentFunctions2(R:Type, S:Type): public == private where
@
\section{domain SEGBIND SegmentBinding}
+<<SegmentBinding.input>>=
+-- seg.spad.pamphlet SegmentBinding.input
+)spool SegmentBinding.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 5
+x = a..b
+--R
+--R
+--R (1) x= a..b
+--R Type: SegmentBinding
Symbol
+--E 1
+
+--S 2 of 5
+sum(i**2, i = 0..n)
+--R
+--R
+--R 3 2
+--R 2n + 3n + n
+--R (2) -------------
+--R 6
+--R Type: Fraction Polynomial
Integer
+--E 2
+
+--S 3 of 5
+sb := y = 1/2..3/2
+--R
+--R
+--R 1 3
+--R (3) y= (-)..(-)
+--R 2 2
+--R Type: SegmentBinding Fraction
Integer
+--E 3
+
+--S 4 of 5
+variable(sb)
+--R
+--R
+--R (4) y
+--R Type:
Symbol
+--E 4
+
+--S 5 of 5
+segment(sb)
+--R
+--R
+--R 1 3
+--R (5) (-)..(-)
+--R 2 2
+--R Type: Segment Fraction
Integer
+--E 5
+)spool
+)lisp (bye)
+@
+<<SegmentBinding.help>>=
+====================================================================
+SegmentBinding examples
+====================================================================
+
+The SegmentBinding type is used to indicate a range for a named symbol.
+
+First give the symbol, then an = and finally a segment of values.
+
+ x = a..b
+ x= a..b
+ Type: SegmentBinding Symbol
+
+This is used to provide a convenient syntax for arguments to certain
+operations.
+
+ sum(i**2, i = 0..n)
+ 3 2
+ 2n + 3n + n
+ -------------
+ 6
+ Type: Fraction Polynomial Integer
+
+ draw(x**2, x = -2..2)
+ TwoDimensionalViewport: "x*x"
+ Type: TwoDimensionalViewport
+
+
+The left-hand side must be of type Symbol but the right-hand side can
+be a segment over any type.
+
+ sb := y = 1/2..3/2
+ 1 3
+ y= (-)..(-)
+ 2 2
+ Type: SegmentBinding Fraction Integer
+
+The left- and right-hand sides can be obtained using the variable and
+segment operations.
+
+ variable(sb)
+ y
+ Type: Symbol
+
+ segment(sb)
+ 1 3
+ (-)..(-)
+ 2 2
+ Type: Segment Fraction Integer
+
+See Also:
+o )help Segment
+o )help UniversalSegment
+o )show SegmentBinding
+o $AXIOM/doc/src/algebra/seg.spad.dvi
+
+@
<<domain SEGBIND SegmentBinding>>=
)abbrev domain SEGBIND SegmentBinding
++ Author:
diff --git a/src/algebra/sets.spad.pamphlet b/src/algebra/sets.spad.pamphlet
index 5f373d4..0a348b4 100644
--- a/src/algebra/sets.spad.pamphlet
+++ b/src/algebra/sets.spad.pamphlet
@@ -10,6 +10,314 @@
\tableofcontents
\eject
\section{domain SET Set}
+<<Set.input>>=
+-- sets.spad.pamphlet Set.input
+)spool Set.output
+)set message test on
+)set message auto off
+)clear all
+--S 1 of 20
+s := set [x**2-1, y**2-1, z**2-1]
+--R
+--R
+--R 2 2 2
+--R (1) {x - 1,y - 1,z - 1}
+--R Type: Set Polynomial
Integer
+--E 1
+
+--S 2 of 20
+t := set [x**i - i+1 for i in 2..10 | prime? i]
+--R
+--R
+--R 2 3 5 7
+--R (2) {x - 1,x - 2,x - 4,x - 6}
+--R Type: Set Polynomial
Integer
+--E 2
+
+--S 3 of 20
+i := intersect(s,t)
+--R
+--R
+--R 2
+--R (3) {x - 1}
+--R Type: Set Polynomial
Integer
+--E 3
+
+--S 4 of 20
+u := union(s,t)
+--R
+--R
+--R 2 3 5 7 2 2
+--R (4) {x - 1,x - 2,x - 4,x - 6,y - 1,z - 1}
+--R Type: Set Polynomial
Integer
+--E 4
+
+--S 5 of 20
+difference(s,t)
+--R
+--R
+--R 2 2
+--R (5) {y - 1,z - 1}
+--R Type: Set Polynomial
Integer
+--E 5
+
+--S 6 of 20
+symmetricDifference(s,t)
+--R
+--R
+--R 3 5 7 2 2
+--R (6) {x - 2,x - 4,x - 6,y - 1,z - 1}
+--R Type: Set Polynomial
Integer
+--E 6
+
+--S 7 of 20
+member?(y, s)
+--R
+--R
+--R (7) false
+--R Type:
Boolean
+--E 7
+
+--S 8 of 20
+member?((y+1)*(y-1), s)
+--R
+--R
+--R (8) true
+--R Type:
Boolean
+--E 8
+
+--S 9 of 20
+subset?(i, s)
+--R
+--R
+--R (9) true
+--R Type:
Boolean
+--E 9
+
+--S 10 of 20
+subset?(u, s)
+--R
+--R
+--R (10) false
+--R Type:
Boolean
+--E 10
+
+--S 11 of 20
+gs := set [g for i in 1..11 | primitive?(g := i::PF 11)]
+--R
+--R
+--R (11) {2,6,7,8}
+--R Type: Set PrimeField
11
+--E 11
+
+--S 12 of 20
+complement gs
+--R
+--R
+--R (12) {1,3,4,5,9,10,0}
+--R Type: Set PrimeField
11
+--E 12
+
+--S 13 of 20
+a := set [i**2 for i in 1..5]
+--R
+--R
+--R (13) {1,4,9,16,25}
+--R Type: Set
PositiveInteger
+--E 13
+
+--S 14 of 20
+insert!(32, a)
+--R
+--R
+--R (14) {1,4,9,16,25,32}
+--R Type: Set
PositiveInteger
+--E 14
+
+--S 15 of 20
+remove!(25, a)
+--R
+--R
+--R (15) {1,4,9,16,32}
+--R Type: Set
PositiveInteger
+--E 15
+
+--S 16 of 20
+a
+--R
+--R
+--R (16) {1,4,9,16,32}
+--R Type: Set
PositiveInteger
+--E 16
+
+--S 17 of 20
+b := b0 := set [i**2 for i in 1..5]
+--R
+--R
+--R (17) {1,4,9,16,25}
+--R Type: Set
PositiveInteger
+--E 17
+
+--S 18 of 20
+b := union(b, {32})
+--R
+--R
+--R (18) {1,4,9,16,25,32}
+--R Type: Set
PositiveInteger
+--E 18
+
+--S 19 of 20
+b := difference(b, {25})
+--R
+--R
+--R (19) {1,4,9,16,32}
+--R Type: Set
PositiveInteger
+--E 19
+
+--S 20 of 20
+b0
+--R
+--R
+--R (20) {1,4,9,16,25}
+--R Type: Set
PositiveInteger
+--E 20
+)spool
+)lisp (bye)
+@
+<<Set.help>>=
+====================================================================
+Set examples
+====================================================================
+
+The Set domain allows one to represent explicit finite sets of values.
+These are similar to lists, but duplicate elements are not allowed.
+
+Sets can be created by giving a fixed set of values ...
+
+ s := set [x**2-1, y**2-1, z**2-1]
+ 2 2 2
+ {x - 1,y - 1,z - 1}
+ Type: Set Polynomial Integer
+
+or by using a collect form, just as for lists. In either case, the
+set is formed from a finite collection of values.
+
+ t := set [x**i - i+1 for i in 2..10 | prime? i]
+ 2 3 5 7
+ {x - 1,x - 2,x - 4,x - 6}
+ Type: Set Polynomial Integer
+
+The basic operations on sets are intersect, union, difference, and
+symmetricDifference.
+
+ i := intersect(s,t)
+ 2
+ {x - 1}
+ Type: Set Polynomial Integer
+
+ u := union(s,t)
+ 2 3 5 7 2 2
+ {x - 1,x - 2,x - 4,x - 6,y - 1,z - 1}
+ Type: Set Polynomial Integer
+
+The set difference(s,t) contains those members of s which are not in t.
+
+ difference(s,t)
+ 2 2
+ {y - 1,z - 1}
+ Type: Set Polynomial Integer
+
+The set symmetricDifference(s,t) contains those elements which are
+in s or t but not in both.
+
+ symmetricDifference(s,t)
+ 3 5 7 2 2
+ {x - 2,x - 4,x - 6,y - 1,z - 1}
+ Type: Set Polynomial Integer
+
+Set membership is tested using the member? operation.
+
+ member?(y, s)
+ false
+ Type: Boolean
+
+ member?((y+1)*(y-1), s)
+ true
+ Type: Boolean
+
+The subset? function determines whether one set is a subset of another.
+
+ subset?(i, s)
+ true
+ Type: Boolean
+
+ subset?(u, s)
+ false
+ Type: Boolean
+
+When the base type is finite, the absolute complement of a set is
+defined. This finds the set of all multiplicative generators of
+PrimeField 11---the integers mod 11.
+
+ gs := set [g for i in 1..11 | primitive?(g := i::PF 11)]
+ {2,6,7,8}
+ Type: Set PrimeField 11
+
+The following values are not generators.
+
+ complement gs
+ {1,3,4,5,9,10,0}
+ Type: Set PrimeField 11
+
+Often the members of a set are computed individually; in addition,
+values can be inserted or removed from a set over the course of a
+computation.
+
+There are two ways to do this:
+
+ a := set [i**2 for i in 1..5]
+ {1,4,9,16,25}
+ Type: Set PositiveInteger
+
+One is to view a set as a data structure and to apply updating operations.
+
+ insert!(32, a)
+ {1,4,9,16,25,32}
+ Type: Set PositiveInteger
+
+ remove!(25, a)
+ {1,4,9,16,32}
+ Type: Set PositiveInteger
+
+ a
+ {1,4,9,16,32}
+ Type: Set PositiveInteger
+
+The other way is to view a set as a mathematical entity and to
+create new sets from old.
+
+ b := b0 := set [i**2 for i in 1..5]
+ {1,4,9,16,25}
+ Type: Set PositiveInteger
+
+ b := union(b, {32})
+ {1,4,9,16,25,32}
+ Type: Set PositiveInteger
+
+ b := difference(b, {25})
+ {1,4,9,16,32}
+ Type: Set PositiveInteger
+
+ b0
+ {1,4,9,16,25}
+ Type: Set PositiveInteger
+
+See Also:
+o )help List
+o )show Set
+o $AXIOM/doc/src/algebra/sets.spad.dvi
+
+@
<<domain SET Set>>=
)abbrev domain SET Set
++ Author: Michael Monagan; revised by Richard Jenks
diff --git a/src/algebra/variable.spad.pamphlet
b/src/algebra/variable.spad.pamphlet
index 0f50171..994aa3b 100644
--- a/src/algebra/variable.spad.pamphlet
+++ b/src/algebra/variable.spad.pamphlet
@@ -10,6 +10,101 @@
\tableofcontents
\eject
\section{domain OVAR OrderedVariableList}
+<<OrderedVariableList.input>>=
+-- variable.spad.pamphlet OrderedVariableList.input
+)spool OrderedVariableList.output
+)set message test on
+)set message auto off
+)clear all
+--S 1
+ls:List Symbol:=['x,'a,'z]
+--R
+--R
+--R (1) [x,a,z]
+--R Type: List
Symbol
+--E 1
+
+--S 2
+Z:=OVAR ls
+--R
+--R
+--R (2) OrderedVariableList [x,a,z]
+--R Type:
Domain
+--E 2
+
+--S 3
+size()$Z
+--R
+--R
+--R (3) 3
+--R Type:
NonNegativeInteger
+--E 3
+
+--S 4
+lv:=[index(i::PI)$Z for i in 1..size()$Z]
+--R
+--R Compiling function G1408 with type Integer -> Boolean
+--R Compiling function G1572 with type NonNegativeInteger -> Boolean
+--R
+--R (4) [x,a,z]
+--R Type: List OrderedVariableList
[x,a,z]
+--E 4
+
+--S 5
+sorted?(>,lv)
+--R
+--R
+--R (5) true
+--R Type:
Boolean
+--E 5
+)spool
+)lisp (bye)
+@
+<<OrderedVariableList.help>>=
+====================================================================
+OrderedVariableList examples
+====================================================================
+
+The domain OrderedVariableList provides symbols which are restricted
+to a particular list and have a definite ordering. Those two features
+are specified by a List Symbol object that is the argument to the
+domain.
+
+This is a sample ordering of three symbols.
+
+ ls:List Symbol:=['x,'a,'z]
+ [x,a,z]
+ Type: List Symbol
+
+Let's build the domain
+
+ Z:=OVAR ls
+ OrderedVariableList [x,a,z]
+ Type: Domain
+
+How many variables does it have?
+
+ size()$Z
+ 3
+ Type: NonNegativeInteger
+
+They are (in the imposed order)
+
+ lv:=[index(i::PI)$Z for i in 1..size()$Z]
+ [x,a,z]
+ Type: List OrderedVariableList [x,a,z]
+
+Check that the ordering is right
+
+ sorted?(>,lv)
+ true
+ Type: Boolean
+
+See Also:
+o )show OrderedVariableList
+o $AXIOM/doc/src/algebra/variable.spad.dvi
+
+@
<<domain OVAR OrderedVariableList>>=
)abbrev domain OVAR OrderedVariableList
++ Description:
diff --git a/src/doc/spadhelp.pamphlet b/src/doc/spadhelp.pamphlet
index 024164b..3275622 100644
--- a/src/doc/spadhelp.pamphlet
+++ b/src/doc/spadhelp.pamphlet
@@ -1513,9 +1513,15 @@ MappingPackage1 (MAPPKG1) MappingPackage2
(MAPPKG2)
MappingPackage3 (MAPPKG3) Matrix (MATRIX)
Multiset (MSET) MultivariatePolynomial (MPOLY)
None (NONE) Octonion (OCT)
-OneDimensionalArray (ARRAY1)
+OneDimensionalArray (ARRAY1) Operator (OP)
+OrderedVariableList (OVAR) OrderlyDifferentialPolynomial (ODPOL)
+PartialFraction (PFR) Permanent (PERMAN)
+Polynomial (POLY) Quaternion (QUAT)
+RadixExpansion (RADIX) RealClosure (RECLOS)
PlaneAlgebraicCurvePlot (ACPLOT) RealSolvePackage (REALSOLV)
-RomanNumeral (ROMAN) TwoDimensionalViewport (VIEW2D)
+RegularTriangularSet (REGSET) RomanNumeral (ROMAN)
+Segment (SEG) SegmentBinding (SEGBIND)
+Set (SET) TwoDimensionalViewport (VIEW2D)
@
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