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Hello,

I'm solving a math problem in Ruby.
I need to determine if a number is a perfect square.
If the number is small, you may do like the following.

def perfect_square? n
sqrt = n ** 0.5
sqrt - sqrt.to_i == 0
end

But Float number has limitation on precision.
Thus the function won't work correctly for big numbers like
(123456789123456789).

How would you solve such a case?
It should be fast as well as correct because I will use it repeatedly.

Thanks in advance.

Sam

Sam Kong wrote:
> Hello,
>
> I'm solving a math problem in Ruby.
> I need to determine if a number is a perfect square.
> If the number is small, you may do like the following.
>
> def perfect_square? n
> sqrt = n ** 0.5
> sqrt - sqrt.to_i == 0
> end
>
> But Float number has limitation on precision.
> Thus the function won't work correctly for big numbers like
> (123456789123456789).
>
> How would you solve such a case?
> It should be fast as well as correct because I will use it repeatedly.

Easy: compare integers rather than floats.

x = 123456789123456789

sqrt = Math::sqrt(x)
p(x == sqrt.floor**2)

--
vjoel : Joel VanderWerf : path berkeley edu : 510 665 3407

On Feb 11, 2007, at 8:30 PM, Sam Kong wrote:

> Hello,
>
> I'm solving a math problem in Ruby.
> I need to determine if a number is a perfect square.
> If the number is small, you may do like the following.
>
> def perfect_square? n
> sqrt = n ** 0.5
> sqrt - sqrt.to_i == 0
> end
>
> But Float number has limitation on precision.
> Thus the function won't work correctly for big numbers like
> (123456789123456789).
>
> How would you solve such a case?
> It should be fast as well as correct because I will use it repeatedly.

There are faster algorithms based on the fact that most non-squares
aren't quadratic residues modulo some integers. I remember some is
explained in Henri Cohen's "A Course in Computational Algebraic
Number Theory", but do not have the book at hand.

Perhaps you can take a look at the function that implements this test
in GNU MP, explained here,

http://www.swox.com/gmp/manual/Perfe...Algorithm.html

or the one in Pari:

http://www.ufr-mi.u-bordeaux.fr/~belabas/pari/

-- fxn

On Feb 11, 2007, at 10:48 PM, Xavier Noria wrote:

> There are faster algorithms based on the fact that most non-squares
> aren't quadratic residues modulo some integers. I remember some is
> explained in Henri Cohen's "A Course in Computational Algebraic
> Number Theory", but do not have the book at hand.

I've been able to consult the book today. There is a specialised
algorithm to compute integer roots using integer arithmetic, and the
integer square test itself. Thus, they guarantee a correct result. I
attach them below translated to Ruby.

Some operations would be written using bitwise stuff, but anyway the
code performs very poorly (compared to Math::sqrt) according to the
benchmarks. If performance is important a solution in C would be
worth exploring.

-- fxn


# From Henri Cohen's "A Course in Computational Algebraic Number
# Theory".

# Algorithm 1.7.1 (Integer Square Root) Given a positive integer n,
# this algorithm computes the integer part of the square root of n,
# i.e. the number m such that m^2 <= n < (m + 1)^2.
def isqrt(n)
x = n
loop do
y = ((x + n/x)/2)
if y < x
x = y
else
return x
end
end
end

# Cache the squares modulus 11, 63, 64, and 65. This is used to check
# for non-squares, since a square is a square mod k for all k. The
# choice of these numbers is based on the probability that a non-square
# is a square mod any of them, which is 6/715, less than a 1%.
$squares = {}
[11, 63, 64, 65].each do |m|
$squares[m] = [false] * m
(0...(m/2)).each {|i| $squares[m][i**2 % m] = true}
end


# Algorithm 1.7.3 (Square Test). Given a positive integer n,
# this algorithm determines whether n is a square or not,
# and if it is, outputs the square root of n. We assume the
# precomputations made above.
def issquare(n)
return false unless $squares[64][n % 64]

r = n % 45045 # 45045 = 63*65*11
return false unless $squares[63][r % 63]
return false unless $squares[65][r % 65]
return false unless $squares[11][r % 11]

q = isqrt(n)
return q**2 == n ? q : false
end

require 'benchmark'

$r = 1000

# square of 32248581868698698768678697647823648238462348
$s =
10399710325421624583100997305362730327341924241351 3150637645310539211244
0875299413673104

# non-square, the previous number minus 1
$ns =
10399710325421624583100997305362730327341924241351 3150637645310539211244
0875299413673103

# Just for the sake of curiosity, since the code based on Math::sqrt
is not correct.
Benchmark.benchmark do |x|
x.report("builtin is square (true)") do
1.upto($r) do
sqrt = Math::sqrt($s)
$s == sqrt.floor**2
end
end
x.report("modular is square (true)") do
1.upto($r) do
issquare($s)
end
end
x.report("builtin is square (false)") do
1.upto($r) do
sqrt = Math::sqrt($ns)
$ns == sqrt.floor**2
end
end
x.report("modular is square (false)") do
1.upto($r) do
issquare($ns)
end
end
end

Hi Joel,

On Feb 11, 11:37 am, Joel VanderWerf <v...@path.berkeley.edu> wrote:
> Sam Kong wrote:
> > Hello,
>
> > I'm solving a math problem in Ruby.
> > I need to determine if a number is a perfect square.
> > If the number is small, you may do like the following.
>
> > def perfect_square? n
> > sqrt = n ** 0.5
> > sqrt - sqrt.to_i == 0
> > end
>
> > But Float number has limitation on precision.
> > Thus the function won't work correctly for big numbers like
> > (123456789123456789).
>
> > How would you solve such a case?
> > It should be fast as well as correct because I will use it repeatedly.
>
> Easy: compare integers rather than floats.
>
> x = 123456789123456789
>
> sqrt = Math::sqrt(x)
> p(x == sqrt.floor**2)

Yes. Your approach is better than mine.
But it gives a wrong answer for big numbers like 55833579873437812.


>
> --
> vjoel : Joel VanderWerf : path berkeley edu : 510 665 3407

Thanks anyway.

Sam

Hi Xavier,

I really appreciate your kind help.
I put my comment inline.

On Feb 12, 10:32 am, Xavier Noria <f...@hashref.com> wrote:
> On Feb 11, 2007, at 10:48 PM, Xavier Noria wrote:
>
> > There are faster algorithms based on the fact that most non-squares
> > aren't quadratic residues modulo some integers. I remember some is
> > explained in Henri Cohen's "A Course in Computational Algebraic
> > Number Theory", but do not have the book at hand.
>
> I've been able to consult the book today. There is a specialised
> algorithm to compute integer roots using integer arithmetic, and the
> integer square test itself. Thus, they guarantee a correct result. I
> attach them below translated to Ruby.
>
> Some operations would be written using bitwise stuff, but anyway the
> code performs very poorly (compared to Math::sqrt) according to the
> benchmarks. If performance is important a solution in C would be
> worth exploring.
>
> -- fxn
>
> # From Henri Cohen's "A Course in Computational Algebraic Number
> # Theory".
>
> # Algorithm 1.7.1 (Integer Square Root) Given a positive integer n,
> # this algorithm computes the integer part of the square root of n,
> # i.e. the number m such that m^2 <= n < (m + 1)^2.
> def isqrt(n)
> x = n
> loop do
> y = ((x + n/x)/2)
> if y < x
> x = y
> else
> return x
> end
> end
> end

Actually you posted this message, I implemented an algorithm myself.
It's just the way we calculate sqrt with pen and paper.
It might be slow but correct even for big numbers.

def max_digit n, m
result = 0
(0..9).each do |i|
break if ((n * 10) + i) * i > m
result = i
end
result
end

def perfect_square? n
arr = []
a = n
while true
a, b = a / 100, a % 100
arr.unshift b
break if a == 0
end
remain = 0
carried = 0
arr.each do |i|
num = remain * 100 + i
digit = max_digit(carried, num)
remain = num - (carried * 10 + digit) * digit
carried = (carried * 10 + digit) + digit
end
remain == 0
end

I felt ashamed to see the code you posted.OTL

>
> # Cache the squares modulus 11, 63, 64, and 65. This is used to check
> # for non-squares, since a square is a square mod k for all k. The
> # choice of these numbers is based on the probability that a non-square
> # is a square mod any of them, which is 6/715, less than a 1%.
> $squares = {}
> [11, 63, 64, 65].each do |m|
> $squares[m] = [false] * m
> (0...(m/2)).each {|i| $squares[m][i**2 % m] = true}
> end
>
> # Algorithm 1.7.3 (Square Test). Given a positive integer n,
> # this algorithm determines whether n is a square or not,
> # and if it is, outputs the square root of n. We assume the
> # precomputations made above.
> def issquare(n)
> return false unless $squares[64][n % 64]
>
> r = n % 45045 # 45045 = 63*65*11
> return false unless $squares[63][r % 63]
> return false unless $squares[65][r % 65]
> return false unless $squares[11][r % 11]
>
> q = isqrt(n)
> return q**2 == n ? q : false
> end
>
> require 'benchmark'
>
> $r = 1000
>
> # square of 32248581868698698768678697647823648238462348
> $s =
> 10399710325421624583100997305362730327341924241351 3150637645310539211244
> 0875299413673104
>
> # non-square, the previous number minus 1
> $ns =
> 10399710325421624583100997305362730327341924241351 3150637645310539211244
> 0875299413673103
>
> # Just for the sake of curiosity, since the code based on Math::sqrt
> is not correct.
> Benchmark.benchmark do |x|
> x.report("builtin is square (true)") do
> 1.upto($r) do
> sqrt = Math::sqrt($s)
> $s == sqrt.floor**2
> end
> end
> x.report("modular is square (true)") do
> 1.upto($r) do
> issquare($s)
> end
> end
> x.report("builtin is square (false)") do
> 1.upto($r) do
> sqrt = Math::sqrt($ns)
> $ns == sqrt.floor**2
> end
> end
> x.report("modular is square (false)") do
> 1.upto($r) do
> issquare($ns)
> end
> end
> end

Thank you for your support.

Sam

Sam Kong wrote:
> Hi Joel,
>
> On Feb 11, 11:37 am, Joel VanderWerf <v...@path.berkeley.edu> wrote:
>> Sam Kong wrote:
>>> Hello,
>>> I'm solving a math problem in Ruby.
>>> I need to determine if a number is a perfect square.
>>> If the number is small, you may do like the following.
>>> def perfect_square? n
>>> sqrt = n ** 0.5
>>> sqrt - sqrt.to_i == 0
>>> end
>>> But Float number has limitation on precision.
>>> Thus the function won't work correctly for big numbers like
>>> (123456789123456789).
>>> How would you solve such a case?
>>> It should be fast as well as correct because I will use it repeatedly.
>> Easy: compare integers rather than floats.
>>
>> x = 123456789123456789
>>
>> sqrt = Math::sqrt(x)
>> p(x == sqrt.floor**2)
>
> Yes. Your approach is better than mine.
> But it gives a wrong answer for big numbers like 55833579873437812.

Wrong how?

irb(main):001:0> x = 55833579873437812
=> 55833579873437812
irb(main):002:0> sqrt = Math::sqrt(x)
=> 236291303.0
irb(main):003:0> sqrt.floor**2 - x
=> -3

Ok, I can see that one problem with my approach is that I should have
used #round instead of #floor.

--
vjoel : Joel VanderWerf : path berkeley edu : 510 665 3407

On Feb 12, 12:12 pm, Joel VanderWerf <v...@path.berkeley.edu> wrote:
> Sam Kong wrote:
> > Hi Joel,
>
> > On Feb 11, 11:37 am, Joel VanderWerf <v...@path.berkeley.edu> wrote:
> >> Sam Kong wrote:
> >>> Hello,
> >>> I'm solving a math problem in Ruby.
> >>> I need to determine if a number is a perfect square.
> >>> If the number is small, you may do like the following.
> >>> def perfect_square? n
> >>> sqrt = n ** 0.5
> >>> sqrt - sqrt.to_i == 0
> >>> end
> >>> But Float number has limitation on precision.
> >>> Thus the function won't work correctly for big numbers like
> >>> (123456789123456789).
> >>> How would you solve such a case?
> >>> It should be fast as well as correct because I will use it repeatedly.
> >> Easy: compare integers rather than floats.
>
> >> x = 123456789123456789
>
> >> sqrt = Math::sqrt(x)
> >> p(x == sqrt.floor**2)
>
> > Yes. Your approach is better than mine.
> > But it gives a wrong answer for big numbers like 55833579873437812.
>
> Wrong how?
>
> irb(main):001:0> x = 55833579873437812
> => 55833579873437812
> irb(main):002:0> sqrt = Math::sqrt(x)
> => 236291303.0
> irb(main):003:0> sqrt.floor**2 - x
> => -3
>
> Ok, I can see that one problem with my approach is that I should have
> used #round instead of #floor.

I think even if you use #round, the problem won't go away.
Float type cannot generate correct result due to its limited
precision.

Sam

>
> --
> vjoel : Joel VanderWerf : path berkeley edu : 510 665 3407

On Feb 12, 2:07 pm, "Sam Kong" <sam.s.k...@gmail.com> wrote:
> On Feb 12, 12:12 pm, Joel VanderWerf <v...@path.berkeley.edu> wrote:
> > Wrong how?
>
> > irb(main):001:0> x = 55833579873437812
> > => 55833579873437812
> > irb(main):002:0> sqrt = Math::sqrt(x)
> > => 236291303.0
> > irb(main):003:0> sqrt.floor**2 - x
> > => -3
>
> I think even if you use #round, the problem won't go away.
> Float type cannot generate correct result due to its limited
> precision.

For example:
irb(main):004:0>
x=981723462487562983749812734972342334123435635465 656432452
=> 98172346248756298374981273497234233412343563546565 6432452

irb(main):005:0> y=x**2
=>
96378095679856948493754946318049293808086637413854 24520224844154985436178651763483837924660159303671 23924038732304

irb(main):006:0> z=Math.sqrt(y)
=> 9.81723462487563e+056

irb(main):007:0> z==x
=> true

irb(main):010:0> z==(x+1000)
=> true

irb(main):011:0> z==(x+10000)
=> true

Sam Kong wrote:
>
> On Feb 12, 12:12 pm, Joel VanderWerf <v...@path.berkeley.edu> wrote:
>> Sam Kong wrote:
>>> Hi Joel,
>>> On Feb 11, 11:37 am, Joel VanderWerf <v...@path.berkeley.edu> wrote:
>>>> Sam Kong wrote:
>>>>> Hello,
>>>>> I'm solving a math problem in Ruby.
>>>>> I need to determine if a number is a perfect square.
>>>>> If the number is small, you may do like the following.
>>>>> def perfect_square? n
>>>>> sqrt = n ** 0.5
>>>>> sqrt - sqrt.to_i == 0
>>>>> end
>>>>> But Float number has limitation on precision.
>>>>> Thus the function won't work correctly for big numbers like
>>>>> (123456789123456789).
>>>>> How would you solve such a case?
>>>>> It should be fast as well as correct because I will use it repeatedly.
>>>> Easy: compare integers rather than floats.
>>>> x = 123456789123456789
>>>> sqrt = Math::sqrt(x)
>>>> p(x == sqrt.floor**2)
>>> Yes. Your approach is better than mine.
>>> But it gives a wrong answer for big numbers like 55833579873437812.
>> Wrong how?
>>
>> irb(main):001:0> x = 55833579873437812
>> => 55833579873437812
>> irb(main):002:0> sqrt = Math::sqrt(x)
>> => 236291303.0
>> irb(main):003:0> sqrt.floor**2 - x
>> => -3
>>
>> Ok, I can see that one problem with my approach is that I should have
>> used #round instead of #floor.
>
> I think even if you use #round, the problem won't go away.
> Float type cannot generate correct result due to its limited
> precision.

I agree, but I don't think the problem shows up until you have much
larger numbers. Is that the case for your program?

In this case, 55833579873437812 cannot be exactly represented by a
float, but it doesn't matter for purposes of this calculation.

--
vjoel : Joel VanderWerf : path berkeley edu : 510 665 3407