Add new methods for abs2 (#2098)

* Add new methods for abs2; Nemo issue 2093

* Simplified implementation of abs2(ComplexFieldElem), thanks to Tommy H

Author: JohnAAbbott

src/arb/Complex.jl

@@ -531,6 +531,12 @@ function abs(x::ComplexFieldElem, prec::Int = precision(Balls))
   return z
 end
 
+function abs2(x::ComplexFieldElem, prec::Int = precision(Balls))
+  set_precision!(Balls, prec) do
+    return real(x)^2 + imag(x)^2
+  end
+end
+
 ################################################################################
 #
 #  Inversion

src/arb/Real.jl

@@ -842,6 +842,10 @@ function abs(x::RealFieldElem)
   return z
 end
 
+function abs2(x::RealFieldElem)
+  return x^2
+end
+
 ################################################################################
 #
 #  Inverse

src/arb/acb.jl

@@ -543,6 +543,10 @@ function abs(x::AcbFieldElem)
   return z
 end
 
+function abs2(x::AcbFieldElem)
+  return real(x)^2 + imag(x)^2
+end
+
 ################################################################################
 #
 #  Inversion

src/arb/arb.jl

@@ -836,6 +836,10 @@ function abs(x::ArbFieldElem)
   return z
 end
 
+function abs2(x::ArbFieldElem)
+  return x^2
+end
+
 ################################################################################
 #
 #  Inverse

src/calcium/ca.jl

@@ -818,6 +818,17 @@ function abs(a::CalciumFieldElem)
   return r
 end
 
+@doc raw"""
+    abs2(a::CalciumFieldElem)
+
+Return the square of the absolute value of `a`.
+"""
+function abs2(a::CalciumFieldElem)
+  a2 = real(a)^2 + imag(a)^2  # ***OR***  a2 = a * conj(a)
+  check_special(a2)
+  return a2
+end
+
 @doc raw"""
     conj(a::CalciumFieldElem; form::Symbol=:default)
 

src/flint/fmpq.jl

@@ -100,6 +100,10 @@ function abs(a::QQFieldElem)
   return z
 end
 
+function abs2(a::QQFieldElem)
+  return a^2
+end
+
 zero(_::QQField) = QQFieldElem(0)
 
 one(_::QQField) = QQFieldElem(1)

src/flint/fmpz.jl

@@ -357,6 +357,10 @@ function abs(x::ZZRingElem)
   return z
 end
 
+function abs2(x::ZZRingElem)
+  return x^2
+end
+
 floor(x::ZZRingElem) = x
 ceil(x::ZZRingElem) = x
 trunc(x::ZZRingElem) = x

test/arb/Complex-test.jl

@@ -138,6 +138,9 @@ end
   @test -CC(3) == CC(-3)
   @test abs(-CC(3)) == 3
   @test abs(CC(3)) == 3
+  @test abs2(-CC(3)) == 9
+  @test abs2(CC(3)) == 9
+  @test abs2(CC(0)) == 0
   @test inv(CC(2)) == CC(QQ(1,2))
 
   @test angle(CC(3)) == 0

test/arb/Real-test.jl

@@ -87,19 +87,31 @@ end
   @test_throws ErrorException Int(RR(c))
 
   @test abs(Float64(RR("2.3")) - 2.3) < 1e-10
+  @test abs(abs2(Float64(RR("2.3"))) - 5.29) < 1e-10
   @test setprecision(BigFloat, 1000) do
     set_precision!(Balls, 1000) do
       abs(BigFloat(RR("2.3")) - BigFloat("2.3")) < 1e-299
     end
   end
+  @test setprecision(BigFloat, 1000) do
+    set_precision!(Balls, 1000) do
+      abs(abs2(BigFloat(RR("2.3"))) - BigFloat("5.29")) < 1e-299
+    end
+  end
 
   @test setprecision(BigFloat, 1000) do
     abs(BigFloat(RR("2.3", 1000)) - BigFloat("2.3")) < 1e-299
   end
+  @test setprecision(BigFloat, 1000) do
+    abs(abs2(BigFloat(RR("2.3", 1000))) - BigFloat("5.29")) < 1e-299
+  end
 
   @test setprecision(BigFloat, 1000) do
     abs(BigFloat(RR("2.3"; precision=1000)) - BigFloat("2.3")) < 1e-299
   end
+  @test setprecision(BigFloat, 1000) do
+    abs(abs2(BigFloat(RR("2.3"; precision=1000))) - BigFloat("5.29")) < 1e-299
+  end
 
   for T in [Float64, BigFloat]
     x = RR(-1)/3
@@ -246,6 +258,9 @@ end
   @test -RR(3) == RR(-3)
   @test abs(-RR(3)) == 3
   @test abs(RR(3)) == 3
+  @test abs2(-RR(3)) == 9
+  @test abs2(RR(3)) == 9
+  @test abs2(RR(0)) == 0
   @test inv(RR(2)) == RR(0.5)
 end
 
@@ -561,7 +576,7 @@ end
   CC = ComplexField()
 
   tau = (1 + sqrt(CC(-23)))/2
-  a = abs(modular_weber_f2(tau))^2
+  a = abs2(modular_weber_f2(tau))
   C = lindep([RR(1), a, a^2, a^3, a^4, a^5], 20)
 
   @test C == ZZRingElem[-1, 1, 1, 0, 1, 0]

test/arb/acb-test.jl

@@ -138,6 +138,8 @@ end
   @test -CC(3) == CC(-3)
   @test abs(-CC(3)) == 3
   @test abs(CC(3)) == 3
+  @test abs2(CC(3)) == 9
+  @test abs2(CC(0)) == 0
   @test inv(CC(2)) == CC(QQ(1,2))
 end