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Random() times 2^32 - 1 always returns even last digit

I have encountered this weird behaviour, which is probably best described by a small example:
Random R = new Random();
for (int i = 0; i < 10_000; i++)
{
double d = R.NextDouble() * uint.MaxValue;
}
Now, the last digit of d before the decimal mark is always even, i.e. int r = (int) (d % 10) is always 0, 2, 4, 6, or 8. There are odd digits on either side, though.
I suspected that multiplying with uint.MaxValue (2^32 - 1) could force some rounding error in the last digits, but since double has more than 50 bits of precision, this should beat uint with about 20 bits to spare after the separator. This behaviour also occurs if I explicitly store uint.MaxValue as a double before the multiplication and use that instead.
Can someone shed any light on this?

This is a deficiency in the .Net Random class.
If you inspect the source code you will see the following comment in the implementation of the private method GetSampleForLargeRange():
// The distribution of double value returned by Sample
// is not distributed well enough for a large range.
// If we use Sample for a range [Int32.MinValue..Int32.MaxValue)
// We will end up getting even numbers only.
This is used in the implementation of Next():
public virtual int Next(int minValue, int maxValue) {
if (minValue>maxValue) {
throw new ArgumentOutOfRangeException("minValue",Environment.GetResourceString("Argument_MinMaxValue", "minValue", "maxValue"));
}
Contract.EndContractBlock();
long range = (long)maxValue-minValue;
if( range <= (long)Int32.MaxValue) {
return ((int)(Sample() * range) + minValue);
}
else {
return (int)((long)(GetSampleForLargeRange() * range) + minValue);
}
}
But it is NOT used for the values returned from NextDouble() (which just returns the value returned from Sample().
So the answer is that NextDouble() is not well-distributed.
You can use RNGCryptoServiceProvider to generate better random numbers, but it's a bit of a fiddle to create the double. From this answer:
static void Main()
{
var R = new RNGCryptoServiceProvider();
var bytes = new Byte[8];
for (int i = 0; i < 10_000; i++)
{
R.GetBytes(bytes);
var ul = BitConverter.ToUInt64(bytes, 0) / (1 << 11);
var d = ul / (double)(1UL << 53);
d *= uint.MaxValue;
Console.WriteLine(d);
}
}

Get n distinct random numbers between two values whose sum is equal to a given number

I would like to find distinct random numbers within a range that sums up to given number.
Note: I found similar questions in stackoverflow, however they do not address exactly this problem (ie they do not consider a negative lowerLimit for the range).
If I wanted that the sum of my random number was equal to 1 I just generate the required random numbers, compute the sum and divided each of them by the sum; however here I need something a bit different; I will need my random numbers to add up to something different than 1 and still my random numbers must be within a given range.
Example: I need 30 distinct random numbers (non integers) between -50 and 50 where the sum of the 30 generated numbers must be equal to 300; I wrote the code below, however it will not work when n is much larger than the range (upperLimit - lowerLimit), the function could return numbers outside the range [lowerLimit - upperLimit]. Any help to improve the current solution?
static void Main(string[] args)
{
var listWeights = GetRandomNumbersWithConstraints(30, 50, -50, 300);
}
private static List<double> GetRandomNumbersWithConstraints(int n, int upperLimit, int lowerLimit, int sum)
{
if (upperLimit <= lowerLimit || n < 1)
throw new ArgumentOutOfRangeException();
Random rand = new Random(Guid.NewGuid().GetHashCode());
List<double> weight = new List<double>();
for (int k = 0; k < n; k++)
{
//multiply by rand.NextDouble() to avoid duplicates
double temp = (double)rand.Next(lowerLimit, upperLimit) * rand.NextDouble();
if (weight.Contains(temp))
k--;
else
weight.Add(temp);
}
//divide each element by the sum
weight = weight.ConvertAll<double>(x => x / weight.Sum()); //here the sum of my weight will be 1
return weight.ConvertAll<double>(x => x * sum);
}
EDIT - to clarify
Running the current code will generate the following 30 numbers that add up to 300. However those numbers are not within -50 and 50
-4.425315699
67.70219958
82.08592061
46.54014109
71.20352208
-9.554070146
37.65032717
-75.77280868
24.68786878
30.89874589
142.0796933
-1.964407284
9.831226893
-15.21652248
6.479463312
49.61283063
118.1853036
-28.35462683
49.82661159
-65.82706541
-29.6865969
-54.5134262
-56.04708803
-84.63783048
-3.18402453
-13.97935982
-44.54265204
112.774348
-2.911427266
-58.94098071

Ok, here how it could be done
We will use Dirichlet Distribution, which is distribution for random numbers xi in the range [0...1] such that
Sumi xi = 1
So, after linear rescaling condition for sum would be satisfied automatically. Dirichlet distribution is parametrized by αi, but we assume all RN to be from the same marginal distribution, so there is only one parameter α for each and every index.
For reasonable large value of α, mean value of sampled random numbers would be =1/n, and variance ~1/(n * α), so larger α lead to random value more close to the mean.
Ok, now back to rescaling,
vi = A + B*xi
And we have to get A and B. As #HansKesting rightfully noted, with only two free parameters we could satisfy only two constraints, but you have three. So we would strictly satisfy low bound constraint, sum value constraint, but occasionally violate upper bound constraint. In such case we just throw whole sample away and do another one.
Again, we have a knob to turn, α getting larger means we are close to mean values and less likely to hit upper bound. With α = 1 I'm rarely getting any good sample, but with α = 10 I'm getting close to 40% of good samples. With α = 16 I'm getting close to 80% of good samples.
Dirichlet sampling is done via Gamma distribution, using code from MathDotNet.
Code, tested with .NET Core 2.1
using System;
using MathNet.Numerics.Distributions;
using MathNet.Numerics.Random;
class Program
{
static void SampleDirichlet(double alpha, double[] rn)
{
if (rn == null)
throw new ArgumentException("SampleDirichlet:: Results placeholder is null");
if (alpha <= 0.0)
throw new ArgumentException($"SampleDirichlet:: alpha {alpha} is non-positive");
int n = rn.Length;
if (n == 0)
throw new ArgumentException("SampleDirichlet:: Results placeholder is of zero size");
var gamma = new Gamma(alpha, 1.0);
double sum = 0.0;
for(int k = 0; k != n; ++k) {
double v = gamma.Sample();
sum += v;
rn[k] = v;
}
if (sum <= 0.0)
throw new ApplicationException($"SampleDirichlet:: sum {sum} is non-positive");
// normalize
sum = 1.0 / sum;
for(int k = 0; k != n; ++k) {
rn[k] *= sum;
}
}
static bool SampleBoundedDirichlet(double alpha, double sum, double lo, double hi, double[] rn)
{
if (rn == null)
throw new ArgumentException("SampleDirichlet:: Results placeholder is null");
if (alpha <= 0.0)
throw new ArgumentException($"SampleDirichlet:: alpha {alpha} is non-positive");
if (lo >= hi)
throw new ArgumentException($"SampleDirichlet:: low {lo} is larger than high {hi}");
int n = rn.Length;
if (n == 0)
throw new ArgumentException("SampleDirichlet:: Results placeholder is of zero size");
double mean = sum / (double)n;
if (mean < lo || mean > hi)
throw new ArgumentException($"SampleDirichlet:: mean value {mean} is not within [{lo}...{hi}] range");
SampleDirichlet(alpha, rn);
bool rc = true;
for(int k = 0; k != n; ++k) {
double v = lo + (mean - lo)*(double)n * rn[k];
if (v > hi)
rc = false;
rn[k] = v;
}
return rc;
}
static void Main(string[] args)
{
double[] rn = new double [30];
double lo = -50.0;
double hi = 50.0;
double alpha = 10.0;
double sum = 300.0;
for(int k = 0; k != 1_000; ++k) {
var q = SampleBoundedDirichlet(alpha, sum, lo, hi, rn);
Console.WriteLine($"Rng(BD), v = {q}");
double s = 0.0;
foreach(var r in rn) {
Console.WriteLine($"Rng(BD), r = {r}");
s += r;
}
Console.WriteLine($"Rng(BD), summa = {s}");
}
}
}
UPDATE
Usually, when people ask such question, there is an implicit assumption/requirement - all random numbers shall be distribution in the same way. It means that if I draw marginal probability density function (PDF) for item indexed 0 from the sampled array, I shall get the same distribution as I draw marginal probability density function for the last item in the array. People usually sample random arrays to pass it down to other routines to do some interesting stuff. If marginal PDF for item 0 is different from marginal PDF for last indexed item, then just reverting array will produce wildly different result with the code which uses such random values.
Here I plotted distributions of random numbers for item 0 and last item (#29) for original conditions([-50...50] sum=300), using my sampling routine. Look similar, isn't it?
Ok, here is a picture from your sampling routine, same original conditions([-50...50] sum=300), same number of samples
UPDATE II
User supposed to check return value of the sampling routine and accept and use sampled array if (and only if) return value is true. This is acceptance/rejection method. As an illustration, below is code used to histogram samples:
int[] hh = new int[100]; // histogram allocated
var s = 1.0; // step size
int k = 0; // good samples counter
for( ;; ) {
var q = SampleBoundedDirichlet(alpha, sum, lo, hi, rn);
if (q) // good sample, accept it
{
var v = rn[0]; // any index, 0 or 29 or ....
var i = (int)((v - lo) / s);
i = System.Math.Max(i, 0);
i = System.Math.Min(i, hh.Length-1);
hh[i] += 1;
++k;
if (k == 100000) // required number of good samples reached
break;
}
}
for(k = 0; k != hh.Length; ++k)
{
var x = lo + (double)k * s + 0.5*s;
var v = hh[k];
Console.WriteLine($"{x} {v}");
}

Here you go. It'll probably run for centuries before actually returning the list, but it'll comply :)
public List<double> TheThing(int qty, double lowest, double highest, double sumto)
{
if (highest * qty < sumto)
{
throw new Exception("Impossibru!");
// heresy
highest = sumto / 1 + (qty * 2);
lowest = -highest;
}
double rangesize = (highest - lowest);
Random r = new Random();
List<double> ret = new List<double>();
while (ret.Sum() != sumto)
{
if (ret.Count > 0)
ret.RemoveAt(0);
while (ret.Count < qty)
ret.Add((r.NextDouble() * rangesize) + lowest);
}
return ret;
}

I come up with this solution which is fast. I am sure it couldbe improved, but for the moment it does the job.
n = the number of random numbers that I will need to find
Constraints
the n random numbers must add up to finalSum the n random numbers
the n random numbers must be within lowerLimit and upperLimit
The idea is to remove from the initial list (that sums up to finalSum) of random numbers the numbers outside the range [lowerLimit, upperLimit].
Then count the number left of the list (called nValid) and their sum (called sumOfValid).
Now, iteratively search for (n-nValid) random numbers within the range [lowerLimit, upperLimit] whose sum is (finalSum-sumOfValid)
I tested it with several combinations for the inputs variables (including negative sum) and the results looks good.
static void Main(string[] args)
{
int n = 100;
int max = 5000;
int min = -500000;
double finalSum = -1000;
for (int i = 0; i < 5000; i++)
{
var listWeights = GetRandomNumbersWithConstraints(n, max, min, finalSum);
Console.WriteLine("=============");
Console.WriteLine("sum = " + listWeights.Sum());
Console.WriteLine("max = " + listWeights.Max());
Console.WriteLine("min = " + listWeights.Min());
Console.WriteLine("count = " + listWeights.Count());
}
}
private static List<double> GetRandomNumbersWithConstraints(int n, int upperLimit, int lowerLimit, double finalSum, int precision = 6)
{
if (upperLimit <= lowerLimit || n < 1) //todo improve here
throw new ArgumentOutOfRangeException();
Random rand = new Random(Guid.NewGuid().GetHashCode());
List<double> randomNumbers = new List<double>();
int adj = (int)Math.Pow(10, precision);
bool flag = true;
List<double> weights = new List<double>();
while (flag)
{
foreach (var d in randomNumbers.Where(x => x <= upperLimit && x >= lowerLimit).ToList())
{
if (!weights.Contains(d)) //only distinct
weights.Add(d);
}
if (weights.Count() == n && weights.Max() <= upperLimit && weights.Min() >= lowerLimit && Math.Round(weights.Sum(), precision) == finalSum)
return weights;
/* worst case - if the largest sum of the missing elements (ie we still need to find 3 elements,
* then the largest sum is 3*upperlimit) is smaller than (finalSum - sumOfValid)
*/
if (((n - weights.Count()) * upperLimit < (finalSum - weights.Sum())) ||
((n - weights.Count()) * lowerLimit > (finalSum - weights.Sum())))
{
weights = weights.Where(x => x != weights.Max()).ToList();
weights = weights.Where(x => x != weights.Min()).ToList();
}
int nValid = weights.Count();
double sumOfValid = weights.Sum();
int numberToSearch = n - nValid;
double sum = finalSum - sumOfValid;
double j = finalSum - weights.Sum();
if (numberToSearch == 1 && (j <= upperLimit || j >= lowerLimit))
{
weights.Add(finalSum - weights.Sum());
}
else
{
randomNumbers.Clear();
int min = lowerLimit;
int max = upperLimit;
for (int k = 0; k < numberToSearch; k++)
{
randomNumbers.Add((double)rand.Next(min * adj, max * adj) / adj);
}
if (sum != 0 && randomNumbers.Sum() != 0)
randomNumbers = randomNumbers.ConvertAll<double>(x => x * sum / randomNumbers.Sum());
}
}
return randomNumbers;
}

Why do different algorithms of summing not match?

Assume that I want to get sum of all squares from M to N. I googled a bit and found this formula:
(1^2 + 2^2 + 3^2 + ... + N^2) = (N * (N + 1) * (2N + 1)) / 6
so I write this code:
static void Main(string[] args)
{
const int from = 10;
const int to = 50000;
Console.WriteLine(SumSquares(from, to));
Console.WriteLine(SumSquares2(from, to));
}
static long SumSquares(int m, int n)
{
checked
{
long x = m - 1;
long y = n;
return (((y*(y + 1)*(2*y + 1)) - (x*(x + 1)*(2*x + 1)))/6);
}
}
static long SumSquares2(int m, int n)
{
long sum = 0;
for (int i = m; i <= n; ++i)
{
sum += i * i;
}
return sum;
}
it works fine until 40k, but when N becomes 50k it fails. Output for 50k:
41667916674715
25948336371355
Press any key to continue . . .
I think it's an overflow or something, so I added checked keyword and tried to change long to double, but I got the same result. How can it be explained? How to get correct result without loops?

Your second method is overflowing because you are using an int in the loop. Change it to a long as follows (and also add checked):
static long SumSquares2(int m, int n)
{
checked
{
long sum = 0;
for (long i = m; i <= n; ++i)
{
sum += i*i;
}
return sum;
}
}
What was going wrong is that i*i was being calculated internally as an int data type even though the result was being cast to a long data type (i.e. the variable sum), and so it overflowed.

While you are using long for the result, you are still using int for the operators. I would define M and N as long or even BigInteger, and the same for the result. If you do not, you are probably doing int arithmetic still, even though your result is of type long.
I tried your code, and got the results you got. But then I changed every int to long and got the two numbers to match, up to an N of 1600000.
Using BigInteger, I am up to 160000000 and still working ok (result for m=10 and n=160000000 is 13653333461333333359999715, both ways).
To use BigInteger, you will need to add a reference to the System.Numerics dll to your project, and you will need to have a statement at the top of your code including that library.
using System.Numerics;
namespace ConsoleFiddle
{
class Program
{
static void Main(string[] args)
{
BigInteger from = 10;
BigInteger to = 160000000;
Console.WriteLine(SumSquares(from, to));
Console.WriteLine(SumSquares2(from, to));
Console.ReadKey();
}
static BigInteger SumSquares(BigInteger m, BigInteger n)
{
checked
{
BigInteger x = m - 1;
BigInteger y = n;
return (((y * (y + 1) * (2 * y + 1)) - (x * (x + 1) * (2 * x + 1))) / 6);
}
}
static BigInteger SumSquares2(BigInteger m, BigInteger n)
{
checked
{
BigInteger sum = 0;
for (BigInteger i = m; i <= n; ++i)
{
sum += i * i;
}
return sum;
}
}
For an M of 4000000000000000000 (4 x 10^18), and an N of 4000000000100000000. This code still works and gives an immediate result with the first method (1600000016040000000400333333338333333350000000). With the second method it takes it a little while (100 million loop iterations) but gives the same result.

Most probably you are experiencing integer overflow, as the range of long is limited. Probably you have disabled exceptions for integer overflow, so no exception is thrown. The exceptions for integer overflow can be disabled and enabled in the project properties in Visual Studio, if I'm not mistaken.

how to set the values of sill,nugget,range in ordinary kriging?

im work on temperature data ,and i want to interpolate the data to any location that i specify . my programming language is asp.net4 c#. i dont know how can i determine the sill,range,nugget for experimental variogram or theoratical variogram?
i make the following code to calculate these variables in experimental variogram :
public void Calculate_EmpericalVari_nugget_sill_range()
{
double H=0,VaiogramValue;
int N = KnownPoints.Count;
for (int j = 0; j < N; j++)//calculate the emperical variogram
{
for (int i = 0; i < N; i++)
{
H = AucledianDistance(KnownPoints[i].X, KnownPoints[i].Y,
KnownPoints[j].X, KnownPoints[j].Y);
VaiogramValue = 0.5 * Math.Pow((KnownPoints[i].Z - KnownPoints[j].Z), 2);
Emper_Variogram[i][j] = VaiogramValue;
if (H==0) nugget = VaiogramValue;
if (sill < VaiogramValue)
{
sill = VaiogramValue;
range = H;
}
}
}
}
I work according to the following that sill is the max value and range is the distance when i get the sill and nugget is the value at distance equal zero?
does this right? and do they remain constants when i calculate the theoretical semivariogram

Is "(float)integer == integer" guaranteed to be equal in C#?

While "we all know" that x == y can be problematic, where x and y are floating point values, this question is a bit more specific:
int x = random.Next(SOME_UPPER_LIMIT);
float r = x;
// Is the following ALWAYS true?
r == x
Now, since the range of float of is much larger than that of integers (but the precision is insufficient to uniquely present integers at the edges), it would be nice if responses to this question also addressed which values of x the above can be guaranteed for, if it can be guaranteed at all.
Currently my code is making this assumption (for relatively small values of x) - I would like to make sure that I won't get bitten :)
This will fail with "not equal: 16777217" (cast float -> int):
for (int i = 0; i < int.MaxValue; i++) {
float f = i;
if ((int)f != i) throw new Exception("not equal " + i);
}
This similar code will not fail (only int -> float); however, due to loss in the conversion, there are several floats that can "equal" the same integer, and may represent a silent bug:
for (int i = 0; i < int.MaxValue; i++) {
float f = i;
if (f != i) throw new Exception("not equal " + i);
}

Yes, the comparison will always be true, whatever value the int is.
The int will be converted to a float to do the conversion, and the first conversion to float will always give the same result as the second conversion.
Consider:
int x = [any integer value];
float y = x;
float z = x;
The values of y and z will always be the same. If the conversion loses precision, both conversions will lose the precision in exactly the same way.
If you convert the float back to int to to the comparison, that's another matter.
Also, note that even if a specific int value converted to float always results in the same float value, that doesn't mean that the float value has to be unique for that int value. There are int values where (float)x == (float)(x+1) would be true.

When comparing an int and a float, the int is implicitly cast to a float. This ensures the same loss of precision happens, and so the comparison will happen to always be true. As long as you don't disturb the implicit cast or do arithmetic, the equality should hold. For example, if you write this:
bool AlwaysTrue(int i) {
return i == (float)i;
}
there is an implicit cast, so it's equivalent to this function that should always return true:
bool AlwaysTrue(int i) {
return (float)i == (float)i;
}
but if you write this:
bool SometimesTrue(int i) {
return i == (int)(float)i;
}
then there is no more implicit cast and the loss of precision only happens on the right side. The result may be false. Similarly, if you write this:
bool SometimesTrue(int i) {
return 1 + i == 1 + (float)i;
}
then the loss of precision might not be equivalent on both sides. The result may be false.

The following experiment reveals that the answer is you do not have that edge case where equality is not true
static void Main(string[] args)
{
Parallel.For(int.MinValue, int.MaxValue, (x) =>
{
float r = x;
// Is the following ALWAYS true?
bool equal = r == x;
if (!equal) Console.WriteLine("Unequal: " + x);
});
Console.WriteLine("Done");
Console.ReadKey();
return;
}
It seems reasonable that the conversions
float f = i;
and
if ((int)f != i)
should follow the same rules. This proves that int -> float and float -> int conversions are a bijection.
NOTE: the experiment code actually doesn't test the edge case int.MaxValue because Parallel.For's to parameter is exclusive, but I tested that value separately and it also passes the test.

I ran this code without an exception being thrown:
for (int x = Int16.MinValue; x < Int16.MaxValue; x++)
{
float r = x;
if (r != x)
{
throw new Exception("Failed at: " + x);
}
}
Still waiting on (didn't complete this test because it took too long, never threw an exception though while running):
for (long x = Int64.MinValue; x < Int64.MaxValue; x++)
{
float r = x;
if (r != x)
{
throw new Exception("Failed at: " + x);
}
}
Went back and ran your example with a caveat, this was the output:
[Exception: not equal 16777217 ?= 1.677722E+07 ?= 16777216]
for (int i = 0; i < int.MaxValue; i++)
{
float f = i;
if ((int)f != i) throw new Exception("not equal " + i + " ?= " + f + " ?= " + (int)f);
}

My understanding of floating point arithmetic calculations is that they are handled by the CPU, which solely determines your precision. Therefore there is no definite value above which floats lose precision.
I had thought that the x86 architecture, for instance, guaranteed a minimum, but I've been proven wrong.