/usr/share/axiom-20170501/src/algebra/COMPCAT.spad is in axiom-source 20170501-3.
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++ Date Last Updated: 18 March 1994
++ Description:
++ This category represents the extension of a ring by a square root of -1.
ComplexCategory(R) : Category == SIG where
R : CommutativeRing
MA ==> MonogenicAlgebra(R, SparseUnivariatePolynomial(R))
FRT ==> FullyRetractableTo(R)
DE ==> DifferentialExtension(R)
FEO ==> FullyEvalableOver(R)
FPM ==> FullyPatternMatchable(R)
P ==> Patternable(R)
FLERO ==> FullyLinearlyExplicitRingOver(R)
CR ==> CommutativeRing
SIG ==> Join(MA,FRT,DE,FEO,FPM,P,FLERO,CR) with
complex
++ indicates that % has sqrt(-1)
imaginary : () -> %
++ imaginary() = sqrt(-1) = %i.
conjugate : % -> %
++ conjugate(x + %i y) returns x - %i y.
complex : (R, R) -> %
++ complex(x,y) constructs x + %i*y.
imag : % -> R
++ imag(x) returns imaginary part of x.
real : % -> R
++ real(x) returns real part of x.
norm : % -> R
++ norm(x) returns x * conjugate(x)
if R has OrderedSet then OrderedSet
if R has IntegralDomain then
IntegralDomain
_exquo : (%,R) -> Union(%,"failed")
++ exquo(x, r) returns the exact quotient of x by r, or
++ "failed" if r does not divide x exactly.
if R has EuclideanDomain then EuclideanDomain
if R has multiplicativeValuation then multiplicativeValuation
if R has additiveValuation then additiveValuation
if R has Field then -- this is a lie; we must know that
Field -- x**2+1 is irreducible in R
if R has ConvertibleTo InputForm then ConvertibleTo InputForm
if R has CharacteristicZero then CharacteristicZero
if R has CharacteristicNonZero then CharacteristicNonZero
if R has RealConstant then
ConvertibleTo Complex DoubleFloat
ConvertibleTo Complex Float
if R has RealNumberSystem then
abs : % -> %
++ abs(x) returns the absolute value of x = sqrt(norm(x)).
if R has TranscendentalFunctionCategory then
TranscendentalFunctionCategory
argument : % -> R
++ argument(x) returns the angle made by (0,1) and (0,x).
if R has RadicalCategory then RadicalCategory
if R has RealNumberSystem then
polarCoordinates : % -> Record(r:R, phi:R)
++ polarCoordinates(x) returns (r, phi) such that
++ x = r * exp(%i * phi).
if R has IntegerNumberSystem then
rational? : % -> Boolean
++ rational?(x) tests if x is a rational number.
rational : % -> Fraction Integer
++ rational(x) returns x as a rational number.
++ Error: if x is not a rational number.
rationalIfCan : % -> Union(Fraction Integer, "failed")
++ rationalIfCan(x) returns x as a rational number, or
++ "failed" if x is not a rational number.
if R has PolynomialFactorizationExplicit and R has EuclideanDomain then
PolynomialFactorizationExplicit
add
import MatrixCategoryFunctions2(%, Vector %, Vector %, Matrix %,
R, Vector R, Vector R, Matrix R)
SUP ==> SparseUnivariatePolynomial
characteristicPolynomial x ==
v := monomial(1,1)$SUP(R)
v**2 - trace(x)*v**1 + norm(x)*v**0
if R has PolynomialFactorizationExplicit and R has EuclideanDomain then
SupR ==> SparseUnivariatePolynomial R
Sup ==> SparseUnivariatePolynomial %
import FactoredFunctionUtilities Sup
import UnivariatePolynomialCategoryFunctions2(R,SupR,%,Sup)
import UnivariatePolynomialCategoryFunctions2(%,Sup,R,SupR)
pp,qq:Sup
if R has IntegerNumberSystem then
myNextPrime: (%,NonNegativeInteger) -> %
myNextPrime(x,n ) == -- prime is actually in R, and = 3(mod 4)
xr:=real(x)-4::R
while not prime? xr repeat
xr:=xr-4::R
complex(xr,0)
--!TT:=InnerModularGcd(%,Sup,32719 :: %,myNextPrime)
--!gcdPolynomial(pp,qq) == modularGcd(pp,qq)$TT
solveLinearPolynomialEquation(lp:List Sup,p:Sup) ==
solveLinearPolynomialEquation(lp,p)$ComplexIntegerSolveLinearPolynomialEquation(R,%)
normPolynomial: Sup -> SupR
normPolynomial pp ==
map(z+->retract(z@%)::R,pp * map(conjugate,pp))
factorPolynomial pp ==
refine(squareFree pp,factorSquareFreePolynomial)
factorSquareFreePolynomial pp ==
pnorm:=normPolynomial pp
k:R:=0
while degree gcd(pnorm,differentiate pnorm)>0 repeat
k:=k+1
pnorm:=normPolynomial
elt(pp,monomial(1,1)-monomial(complex(0,k),0))
fR:=factorSquareFreePolynomial pnorm
numberOfFactors fR = 1 =>
makeFR(1,[["irred",pp,1]])
lF:List Record(flg:Union("nil", "sqfr", "irred", "prime"),
fctr:Sup, xpnt:Integer):=[]
for u in factorList fR repeat
p1:=map((z:R):%+->z::%,u.fctr)
if not zero? k then
p1:=elt(p1,monomial(1,1)+monomial(complex(0,k),0))
p2:=gcd(p1,pp)
lF:=cons(["irred",p2,1],lF)
pp:=(pp exquo p2)::Sup
makeFR(pp,lF)
rank() == 2
discriminant() == -4 :: R
norm x == real(x)**2 + imag(x)**2
trace x == 2 * real x
imaginary() == complex(0, 1)
conjugate x == complex(real x, - imag x)
characteristic() == characteristic()$R
map(fn, x) == complex(fn real x, fn imag x)
x = y == real(x) = real(y) and imag(x) = imag(y)
x + y == complex(real x + real y, imag x + imag y)
- x == complex(- real x, - imag x)
r:R * x:% == complex(r * real x, r * imag x)
coordinates(x:%) == [real x, imag x]
n:Integer * x:% == complex(n * real x, n * imag x)
differentiate(x:%, d:R -> R) == complex(d real x, d imag x)
definingPolynomial() ==
monomial(1,2)$(SUP R) + monomial(1,0)$(SUP R)
reduce(pol:SUP R) ==
part:= (monicDivide(pol,definingPolynomial())).remainder
complex(coefficient(part,0),coefficient(part,1))
lift(x) == monomial(real x,0)$(SUP R)+monomial(imag x,1)$(SUP R)
minimalPolynomial x ==
zero? imag x =>
monomial(1, 1)$(SUP R) - monomial(real x, 0)$(SUP R)
monomial(1, 2)$(SUP R) - monomial(trace x, 1)$(SUP R)
+ monomial(norm x, 0)$(SUP R)
coordinates(x:%, v:Vector %):Vector(R) ==
ra := real(a := v(minIndex v))
rb := real(b := v(maxIndex v))
(#v ^= 2) or
((d := recip(ra * (ib := imag b) - (ia := imag a) * rb))
case "failed") =>error "coordinates: vector is not a basis"
rx := real x
ix := imag x
[d::R * (rx * ib - ix * rb), d::R * (ra * ix - ia * rx)]
coerce(x:%):OutputForm ==
re := (r := real x)::OutputForm
ie := (i := imag x)::OutputForm
zero? i => re
outi := "%i"::Symbol::OutputForm
ip :=
(i = 1) => outi
((-i) = 1) => -outi
ie * outi
zero? r => ip
re + ip
retract(x:%):R ==
not zero?(imag x) =>
error "Imaginary part is nonzero. Cannot retract."
real x
retractIfCan(x:%):Union(R, "failed") ==
not zero?(imag x) => "failed"
real x
x:% * y:% ==
complex(real x * real y - imag x * imag y,
imag x * real y + imag y * real x)
reducedSystem(m:Matrix %):Matrix R ==
vertConcat(map(real, m), map(imag, m))
reducedSystem(m:Matrix %, v:Vector %):
Record(mat:Matrix R, vec:Vector R) ==
rh := reducedSystem(v::Matrix %)@Matrix(R)
[reducedSystem(m)@Matrix(R), column(rh, minColIndex rh)]
if R has RealNumberSystem then
abs(x:%):% == (sqrt norm x)::%
if R has RealConstant then
convert(x:%):Complex(DoubleFloat) ==
complex(convert(real x)@DoubleFloat,convert(imag x)@DoubleFloat)
convert(x:%):Complex(Float) ==
complex(convert(real x)@Float, convert(imag x)@Float)
if R has ConvertibleTo InputForm then
convert(x:%):InputForm ==
convert([convert("complex"::Symbol), convert real x,
convert imag x]$List(InputForm))@InputForm
if R has ConvertibleTo Pattern Integer then
convert(x:%):Pattern Integer ==
convert(x)$ComplexPattern(Integer, R, %)
if R has ConvertibleTo Pattern Float then
convert(x:%):Pattern Float ==
convert(x)$ComplexPattern(Float, R, %)
if R has PatternMatchable Integer then
patternMatch(x:%, p:Pattern Integer,
l:PatternMatchResult(Integer, %)) ==
patternMatch(x, p, l)$ComplexPatternMatch(Integer, R, %)
if R has PatternMatchable Float then
patternMatch(x:%, p:Pattern Float,
l:PatternMatchResult(Float, %)) ==
patternMatch(x, p, l)$ComplexPatternMatch(Float, R, %)
if R has OrderedSet then
x < y ==
real x = real y => imag x < imag y
real x < real y
if R has IntegerNumberSystem then
rational? x == zero? imag x
rational x ==
zero? imag x => rational real x
error "Not a rational number"
rationalIfCan x ==
zero? imag x => rational real x
"failed"
if R has Field then
inv x ==
zero? imag x => (inv real x)::%
r := norm x
complex(real(x) / r, - imag(x) / r)
if R has IntegralDomain then
_exquo(x:%, r:R) ==
(r = 1) => x
(r1 := real(x) exquo r) case "failed" => "failed"
(r2 := imag(x) exquo r) case "failed" => "failed"
complex(r1, r2)
_exquo(x:%, y:%) ==
zero? imag y => x exquo real y
x * conjugate(y) exquo norm(y)
recip(x:%) == 1 exquo x
if R has OrderedRing then
unitNormal x ==
zero? x => [1,x,1]
(u := recip x) case % => [x, 1, u]
zero? real x =>
c := unitNormal imag x
[complex(0, c.unit), (c.associate * imag x)::%,
complex(0, - c.associate)]
c := unitNormal real x
x := c.associate * x
imag x < 0 =>
x := complex(- imag x, real x)
[- c.unit * imaginary(), x, c.associate * imaginary()]
[c.unit ::%, x, c.associate ::%]
else
unitNormal x ==
zero? x => [1,x,1]
(u := recip x) case % => [x, 1, u]
zero? real x =>
c := unitNormal imag x
[complex(0, c.unit), (c.associate * imag x)::%,
complex(0, - c.associate)]
c := unitNormal real x
x := c.associate * x
[c.unit ::%, x, c.associate ::%]
if R has EuclideanDomain then
if R has additiveValuation then
euclideanSize x == max(euclideanSize real x,
euclideanSize imag x)
else
euclideanSize x == euclideanSize(real(x)**2 + imag(x)**2)$R
if R has IntegerNumberSystem then
x rem y ==
zero? imag y =>
yr:=real y
complex(symmetricRemainder(real(x), yr),
symmetricRemainder(imag(x), yr))
divide(x, y).remainder
x quo y ==
zero? imag y =>
yr:= real y
xr:= real x
xi:= imag x
complex((xr-symmetricRemainder(xr,yr)) quo yr,
(xi-symmetricRemainder(xi,yr)) quo yr)
divide(x, y).quotient
else
x rem y ==
zero? imag y =>
yr:=real y
complex(real(x) rem yr,imag(x) rem yr)
divide(x, y).remainder
x quo y ==
zero? imag y => complex(real x quo real y,imag x quo real y)
divide(x, y).quotient
divide(x, y) ==
r := norm y
y1 := conjugate y
xx := x * y1
x1 := real(xx) rem r
a := x1
if x1^=0 and sizeLess?(r, 2 * x1) then
a := x1 - r
if sizeLess?(x1, a) then a := x1 + r
x2 := imag(xx) rem r
b := x2
if x2^=0 and sizeLess?(r, 2 * x2) then
b := x2 - r
if sizeLess?(x2, b) then b := x2 + r
y1 := (complex(a, b) exquo y1)::%
[((x - y1) exquo y)::%, y1]
if R has TranscendentalFunctionCategory then
half := recip(2::R)::R
if R has RealNumberSystem then
atan2loc(y: R, x: R): R ==
pi1 := pi()$R
pi2 := pi1 * half
x = 0 => if y >= 0 then pi2 else -pi2
-- Atan in (-pi/2,pi/2]
theta := atan(y * recip(x)::R)
while theta <= -pi2 repeat theta := theta + pi1
while theta > pi2 repeat theta := theta - pi1
x >= 0 => theta -- I or IV
if y >= 0 then
theta + pi1 -- II
else
theta - pi1 -- III
argument x == atan2loc(imag x, real x)
else
-- Not ordered so dictate two quadrants
argument x ==
zero? real x => pi()$R * half
atan(imag(x) * recip(real x)::R)
pi() == pi()$R :: %
if R is DoubleFloat then
stoc ==> S_-TO_-C$Lisp
ctos ==> C_-TO_-S$Lisp
exp x == ctos EXP(stoc x)$Lisp
log x == ctos LOG(stoc x)$Lisp
sin x == ctos SIN(stoc x)$Lisp
cos x == ctos COS(stoc x)$Lisp
tan x == ctos TAN(stoc x)$Lisp
asin x == ctos ASIN(stoc x)$Lisp
acos x == ctos ACOS(stoc x)$Lisp
atan x == ctos ATAN(stoc x)$Lisp
sinh x == ctos SINH(stoc x)$Lisp
cosh x == ctos COSH(stoc x)$Lisp
tanh x == ctos TANH(stoc x)$Lisp
asinh x == ctos ASINH(stoc x)$Lisp
acosh x == ctos ACOSH(stoc x)$Lisp
atanh x == ctos ATANH(stoc x)$Lisp
else
atan x ==
ix := imaginary()*x
- imaginary() * half * (log(1 + ix) - log(1 - ix))
log x ==
complex(log(norm x) * half, argument x)
exp x ==
e := exp real x
complex(e * cos imag x, e * sin imag x)
cos x ==
e := exp(imaginary() * x)
half * (e + recip(e)::%)
sin x ==
e := exp(imaginary() * x)
- imaginary() * half * (e - recip(e)::%)
if R has RealNumberSystem then
polarCoordinates x ==
[sqrt norm x, (negative?(t := argument x) => t + 2 * pi(); t)]
x:% ** q:Fraction(Integer) ==
zero? q =>
zero? x => error "0 ** 0 is undefined"
1
zero? x => 0
rx := real x
zero? imag x and positive? rx => (rx ** q)::%
zero? imag x and denom q = 2 => complex(0, (-rx)**q)
ax := sqrt(norm x) ** q
tx := q::R * argument x
complex(ax * cos tx, ax * sin tx)
else if R has RadicalCategory then
x:% ** q:Fraction(Integer) ==
zero? q =>
zero? x => error "0 ** 0 is undefined"
1
r := real x
zero?(i := imag x) => (r ** q)::%
t := numer(q) * recip(denom(q)::R)::R * argument x
e:R :=
zero? r => i ** q
norm(x) ** (q / (2::Fraction(Integer)))
complex(e * cos t, e * sin t)
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