n = 2 인 경우에는 Newton-Raphson 의 방법이 Heron 의 방법과 동일하다.
Julia 언어에는 지수 연산자 ^ 를 (밑수)^(지수) 의 형식으로 언어 자체에서 지원하고 있다. 하지만 차후 필요한 데가 있을 것 같아서 이와 유사한 n 제곱 함수와 n 제곱근 함수를 구현해 보았다. (참고로 Julia 언어에는 pow() 함수가 정의되어 있지 않다.)
지수가 정수인 거듭제곱을 계산하는 함수도 nPow(), gPow, mPow() 세 개 구현해 놓았는데, 이들 세 함수는 절차적 언어의 성능상 재귀호출이 아니고 단순 반복 기법을 사용하는 함수이다. 이 세 함수 중 mPow() 의 성능이 가장 우수하다. 큰 지수의 경우 for 반복문의 반복회수를 따져 보면 성능 비교를 할 수 있을 것이다. (성능 비교를 위해 세 가지를 모두 소스에 남겨 두었다.) mPow() 함수는 n 제곱근을 구하는 재귀함수 newtonNthRoot(int, float) 의 구현에 사용되기도 한다. if ... else ... 구문이 많아 소스가 복잡하게 보일지 모르겠으나 이는 밑수나 지수가 음수이거나 0인 경우의 처리를 위함이다. 구현된 모든 함수의 구현에는 예외상황(예를 들어, 음수의 짝수 제곱근 같은 예외상황) 처리 과정이 있다.
#!/usr/bin/env julia
# Filename: testNthRoot.jl
#
# Approximate square roots, cubic roots and n-th roots of a given number.
#
# Execute: julia testNthRoot.jl
#
# Compile: ipy %IRONPYTHON_HOME%\tools\scripts\pyc.py /main:testNthRoot.py /target:exe
# Execute: testNthRoot
#
# Date: 2013. 3. 7.
# Copyright (c) 2013 PH Kim (pkim __AT__ scripts.pe.kr)
MAX_ITER = 20000
M_EPSILON = 1.0e-15
#
# Compute the n-th root of x to a given scale, x > 0.
#
function nPow(a, n)
if n > 0
if n == 1
return a
else
if a == 0.0 || a == 1.0
return a
elseif a == -1.0
if n % 2 == 1
return -1.0
else
return 1.0
end
elseif a < 0.0
if n % 2 == 1
return -nPow(-a, n)
else
return nPow(-a, n)
end
else
y = 1.0
for i = 1:n
y *= a
end
return y
end
end
elseif n == 0
return 1.0
else # when n < 0
if a == 0.0
println("Error") # throws "Negative powering exception of zero."
else
if n == -1
return 1.0/a
else
return 1.0/nPow(a, -n)
end
end
end
end
#
# Compute the n-th root of x to a given scale, x > 0.
#
function gPow(a, n)
if n > 0
if n == 1
return a
else
if a == 0.0 || a == 1.0
return a
elseif a == -1.0
if n % 2 == 1
return -1.0
else
return 1.0
end
elseif a < 0.0
if n % 2 == 1
return -gPow(-a, n)
else
return gPow(-a, n)
end
else
y = 1.0
r = a
m = 8*4 - 1 # 8*sizeof(int) - 1;
one = 1
for i = 0:m
if (n & one) == 0
y *= 1.0
else
y *= r
end
r = r*r
one <<= 1
if one > n
break
end
end
return y
end
end
elseif n == 0
return 1.0
else # when n < 0
if a == 0.0
throw("Negative powering exception of zero.")
else
if n == -1
return 1.0/a
else
return 1.0/gPow(a, -n)
end
end
end
end
#
# Compute the n-th root of x to a given scale, x > 0.
#
function mPow(a, n)
if n > 0
if n == 1
return a
else
if a == 0.0 || a == 1.0
return a
elseif a == -1.0
if n % 2 == 1
return -1.0
else
return 1.0
end
elseif a < 0.0
if n % 2 == 1
return -mPow(-a, n)
else
return mPow(-a, n)
end
else
y = 1.0
r = a
m = n
while m > 0
if (m & 0x1) == 1
y *= r
end
r = r*r
m >>= 1
end
return y
end
end
elseif n == 0
return 1.0
else # when n < 0
if a == 0.0
throw("Negative powering exception of zero.")
else
if n == -1
return 1.0/a
else
return 1.0/mPow(a, -n)
end
end
end
end
#
# Compute the square root of x to a given scale, x > 0.
#
function heronSqrt(a)
if a < 0.0
throw("Cannot find the sqrt of a negative number.")
elseif a == 0.0 || a == 1.0
return a
else
x1 = a
x2 = (x1 + a/x1)/2.0
er = x1 - x2
counter = 0
while x1 + er != x1
x1 = x2
x2 = (x1 + a/x1)/2.0
er = x1 - x2
if abs(er) < abs(M_EPSILON*x1)
break
end
counter += 1
if counter > MAX_ITER
break
end
end
if counter >= MAX_ITER
throw("Inaccurate sqrt exception by too many iterations.")
end
return x2
end
end
#
# Compute the cubic root of x to a given scale, x > 0.
#
function newtonCbrt(a)
if a == 0.0 || a == 1.0 || a == -1.0
return a
elseif a < 0.0
return -newtonCbrt(-a)
else
x1 = a
x2 = (2.0*x1 + a/(x1*x1))/3.0
# println(" x2 = $x2")
er = x1 - x2
counter = 0
while x1 + er != x1
x1 = x2
x2 = (2.0*x1 + a/(x1*x1))/3.0
# println(" x2 = $x2")
er = x1 - x2
# println(" abs(er) = $(abs(er))")
# println(" abs(M_EPSILON*x1) = $(abs(M_EPSILON*x1))")
if abs(er) < abs(M_EPSILON*x1)
break
end
counter += 1
if counter > MAX_ITER
break
end
end
if counter >= MAX_ITER
throw("Inaccurate cbrt exception by too many iterations.")
end
return x2
end
end
#
# Compute the n-th root of x to a given scale, x > 0.
#
function newtonNthRoot(n, a)
if n == 0
return 1.0
elseif n == 1
return a
elseif n > 0
if a == 0.0 || a == 1.0
return a
elseif a == -1.0
if n % 2 == 1
return a
else
throw("Cannot find the even n-th root of a negative number.")
end
elseif a < 0.0
if n % 2 == 1
return -newtonNthRoot(n, -a)
else
throw("Cannot find the even n-th root of a negative number.")
end
elseif a < 1.0
return 1.0/newtonNthRoot(n, 1.0/a)
else
x1 = a
xn = mPow(x1, n - 1)
x2 = ((n - 1)*x1 + a/xn)/n
er = x1 - x2
counter = 0
while x1 + er != x1
x1 = x2
xn = mPow(x1, n - 1)
x2 = ((n - 1)*x1 + a/xn)/n
er = x1 - x2
if abs(er) < abs(M_EPSILON*x1)
break
end
counter += 1
if counter > MAX_ITER
break
end
end
if counter >= MAX_ITER
throw("Inaccurate n-th root exception by too many iterations.")
end
return x2
end
else
if a == 0.0
throw("Cannot find the negative n-th root of zero.")
else
return 1.0/newtonNthRoot(-n, a)
end
end
end
x = 16.0
u = sqrt(x)
println("[ Testing heronSqrt(double) ]--------------------")
@printf("x = %f\n", x)
@printf("u = sqrt(%f) = %f\n", x, u)
y = heronSqrt(x)
@printf("y = heronSqrt(%f) = %f\n", x, y)
@printf("y*y = %f\n", y*y)
println()
println("[ Testing newtonCbrt(double) ]--------------------")
x = -216.0
@printf("x = %f\n", x)
@printf("-exp(log(-x)/3.0) = %f\n", -exp(log(-x)/3.0))
w = newtonCbrt(x)
@printf("w = newtonCbrt(%f) = %f\n", x, w)
@printf("w*w*w = %f\n", w*w*w)
println()
x = 729000000000.0
@printf("x = %f\n", x)
@printf("exp(log(x)/3.0) = %f\n", exp(log(x)/3.0))
w = newtonCbrt(x)
@printf("w = newtonCbrt(%f) = %f\n", x, w)
@printf("w*w*w = %f\n", w*w*w)
println()
println("[ Testing newtonNthRoot(int, double) ]--------------------")
z = newtonNthRoot(3, x)
@printf("x = %f\n", x)
@printf("z = newtonNthRoot(3, %f) = %f\n", x, z)
@printf("z*z*z = %f\n", z*z*z)
println()
x = 12960000000000000000.0
z = newtonNthRoot(4, x)
println("x = $x")
println("z = newtonNthRoot(4, x) = newtonNthRoot(4, $x) = $z")
@printf("z*z*z*z = %f\n", z*z*z*z)
println()
x = 1.0/12960000000000000000.0
z = newtonNthRoot(4, x)
println("x = $x")
@printf("exp(log(x)/4.0) = %f\n", exp(log(x)/4.0) )
println("z = newtonNthRoot(4, x) = newtonNthRoot(4, $x) = $z")
println("z*z*z*z = $(z*z*z*z)" )
println()
try
x = -4.0
println("[ Test Exception heronSqrt(double) ]--------------------")
println("x = $x")
println("Calculating heronSqrt($x)")
y = heronSqrt(x)
println("y = heronSqrt($x) = $y")
println("y*y = $(y*y)")
println()
catch ex
println("$ex\nCaught some exception in calculating heronSqrt($x)")
println()
end
try
x = -4.0
println("[ Test Exception in newtonCbrt(double) ]--------------------")
println("x = $x")
println("Calculating newtonCbrt($x)")
y = newtonCbrt(x)
println("y = newtonCbrt($x) = $y")
println("y*y*y = $(y*y*y)")
println()
catch ex
println("$ex\nCaught some exception in calculating newtonCbrt$x)")
println()
end
println("[ Test calculations by powering ]-----------------------------")
x = 200.0
z = newtonNthRoot(10, x)
println("x = $x")
println("exp(log(x)/10.0) = $(exp(log(x)/10.0))")
println("z = newtonNthRoot(10, x) = newtonNthRoot(10, $x) = $z")
println("z^10 = $(z^10)")
println()
x = 3001.0
z = newtonNthRoot(99, x)
println("x = $x")
println("exp(log(x)/99.0) = $(exp(log(x)/99.0))")
println("z = newtonNthRoot(99, x) = newtonNthRoot(99, $x) = $z")
println("z^99 = $(z^99)")
println()
x = 3001.0
z = newtonNthRoot(-99, x)
println("x = $x")
println("exp(log(x)/-99.0) = $(exp(log(x)/-99.0))")
println("z = newtonNthRoot(-99, x) = newtonNthRoot(-99, $x) = $z")
println("1.0/z^99 = $(1.0/(z^99))")
println("z^(-99) = $(z^(-99))")
println()
println("2.1^2.1 = $(2.1^ 2.1)")
println("2.1^2.1 * 2.1^(-2.1) = $(2.1^2.1 * 2.1^(-2.1))")
println("2.1^2.1 = exp(2.1*log(2.1)) = $(exp(2.1*log(2.1)))")
println("2.1^(-2.1) = exp(-2.1*log(2.1)) = $(exp(-2.1*log(2.1)))")
println("2.1^2.1 * 2.1^(-2.1) = exp(2.1*log(2.1)) * exp(-2.1*log(2.1)) = $(exp(2.1*log(2.1))) * $(exp(-2.1*log(2.1))) = $( exp(2.1*log(2.1)) * exp(-2.1*log(2.1)) )")
println()
k = 301
x = -1.029
t1 = nPow(x, k)
t2 = gPow(x, k)
t3 = mPow(x, k)
println("$x^$k = $(x^k)")
println("t1 = nPow($x, $k) = $t1")
println("t2 = gPow($x, $k) = $t2")
println("t3 = mPow($x, $k) = $t3")
println("t1 / t2 = $(t1 / t2)")
println("t1 - t2 = $(t1 - t2)")
print("t1 == t2 ? ")
if t1 == t2
print("yes")
else
print("no")
end
println()
println("t1 / t3 = $(t1 / t3)")
println("t1 - t3 = $(t1 - t3)")
print("t1 == t3 ? ")
if t1 == t3
print("yes")
else
print("no")
end
println()
println("t2 / t3 = $(t2 / t3)")
println("t2 - t3 = $(t2 - t3)")
print("t2 == t3 ? ")
if t2 == t3
print("yes")
else
print("no")
end
println()
println()
println("Done.")
"""
[ Testing heronSqrt(double) ]--------------------
x = 16.000000
u = sqrt(16.000000) = 4.000000
y = heronSqrt(16.000000) = 4.000000
y*y = 16.000000
[ Testing newtonCbrt(double) ]--------------------
x = -216.000000
-exp(log(-x)/3.0) = -6.000000
w = newtonCbrt(-216.000000) = -6.000000
w*w*w = -216.000000
x = 729000000000.000000
exp(log(x)/3.0) = 9000.000000
w = newtonCbrt(729000000000.000000) = 9000.000000
w*w*w = 729000000000.000000
[ Testing newtonNthRoot(int, double) ]--------------------
x = 729000000000.000000
z = newtonNthRoot(3, 729000000000.000000) = 9000.000000
z*z*z = 729000000000.000000
x = 1.296e19
z = newtonNthRoot(4, x) = newtonNthRoot(4, 1.296e19) = 60000.0
z*z*z*z = 12960000000000000000.000000
x = 7.716049382716049e-20
exp(log(x)/4.0) = 0.000017
z = newtonNthRoot(4, x) = newtonNthRoot(4, 7.716049382716049e-20) = 1.6666666666
666667e-5
z*z*z*z = 7.716049382716051e-20
[ Test Exception heronSqrt(double) ]--------------------
x = -4.0
Calculating heronSqrt(-4.0)
Cannot find the sqrt of a negative number.
Caught some exception in calculating heronSqrt(-4.0)
[ Test Exception in newtonCbrt(double) ]--------------------
x = -4.0
Calculating newtonCbrt(-4.0)
y = newtonCbrt(-4.0) = -1.5874010519681994
y*y*y = -3.999999999999999
[ Test calculations by powering ]-----------------------------
x = 200.0
exp(log(x)/10.0) = 1.6986464646342472
z = newtonNthRoot(10, x) = newtonNthRoot(10, 200.0) = 1.6986464646342472
z^10 = 199.9999999999999
x = 3001.0
exp(log(x)/99.0) = 1.0842361893258805
z = newtonNthRoot(99, x) = newtonNthRoot(99, 3001.0) = 1.0842361893258805
z^99 = 3000.999999999995
x = 3001.0
exp(log(x)/-99.0) = 0.9223082662659932
z = newtonNthRoot(-99, x) = newtonNthRoot(-99, 3001.0) = 0.9223082662659932
1.0/z^99 = 3001.000000000005
z^(-99) = 3001.0000000000045
2.1^2.1 = 4.749638091742242
2.1^2.1 * 2.1^(-2.1) = 0.9999999999999999
2.1^2.1 = exp(2.1*log(2.1)) = 4.749638091742242
2.1^(-2.1) = exp(-2.1*log(2.1)) = 0.21054235726688478
2.1^2.1 * 2.1^(-2.1) = exp(2.1*log(2.1)) * exp(-2.1*log(2.1)) = 4.74963809174224
2 * 0.21054235726688478 = 1.0
-1.029^301 = -5457.928015771622
t1 = nPow(-1.029, 301) = -5457.92801577163
t2 = gPow(-1.029, 301) = -5457.928015771692
t3 = mPow(-1.029, 301) = -5457.928015771692
t1 / t2 = 0.9999999999999887
t1 - t2 = 6.184563972055912e-11
t1 == t2 ? no
t1 / t3 = 0.9999999999999887
t1 - t3 = 6.184563972055912e-11
t1 == t3 ? no
t2 / t3 = 1.0
t2 - t3 = 0.0
t2 == t3 ? yes
Done.
"""
