Develop Reference

How to use Newton-Raphson method to find the square root of a BigInteger in C#

So I'm attempting to use the Newton-Raphson method to find the square root of a BigInteger.
Here is my code:
private void sqrRt(BigInteger candidate)
{
BigInteger epsilon = new BigInteger(0.0001);
BigInteger guess = candidate / 2;
while (BigInteger.Abs(guess * guess - candidate) >= epsilon)
{
// guess = guess - (((guess**2) - y)/(2*guess))
guess = BigInteger.Subtract(guess, BigInteger.Divide(BigInteger.Subtract(BigInteger.Multiply(guess, guess), candidate), BigInteger.Multiply(2, guess)));
MessageBox.Show(Convert.ToString(guess));
}
}
The problem seems to be that the BigInteger is not precise enough to fall within the degree of accuracy of the epsilon in the while loop - i.e. it needs a decimal place. My question is what/how/where do I convert to a double to make the while loop eventually return false?

You are using the wrong data type. In order to have decimal points, you would need to use double, float, decimal, or Complex.
Check the links of all these types so you can see their digits of precision.

Rounding to 2 decimal places, without using banker's rounding

.NET and Compact Framework by default use banker's rounding which means that
the value: 1,165 will be rounded to: 1,16.
sql server as a opposite rounds it to 1,17 as for me it is the correct behaviour.
Has anyone come across rounding topic and have a workaround for a Compact Framework?
(In .net there is an additional parameter which has an influence on the rounding behaviour)

Math.Floor(double) seems to be supported, so;
private static double RoundToTwo(double value)
{
return Math.Floor(100*value + .5)/100;
}
Console.WriteLine(RoundToTwo(1.165));
> 1.17
Console.WriteLine(RoundToTwo(1.16499));
> 1.16

Here is a method you can use instead of decimal.round:
public static decimal RoundHalfUp(this decimal d, int decimals)
{
if (decimals < 0)
{
throw new ArgumentException("The decimals must be non-negative",
"decimals");
}
decimal multiplier = (decimal)Math.Pow(10, decimals);
decimal number = d * multiplier;
if (decimal.Truncate(number) < number)
{
number += 0.5m;
}
return decimal.Round(number) / multiplier;
}
Taken from: Why does .NET use banker's rounding as default?
This question also asks why .Net uses bankers rounding. So I believe it will be a good read for you.
To answer why
This bankers algorithm means that results collected will be evenly spread of rounding up/down when the decimal == .5, so really it is just to even out data results.
Here's another link which describes this by Mr. Skeet

Try
double number = 1.165;
string RoundedNumber = number.ToString("f3");
Where 3 is the scale

Performing Math operations on decimal datatype in C#?

I was wondering if the above was at all possible. For example:
Math.Sqrt(myVariableHere);
When looking at the overload, it requires a double parameter, so I'm not sure if there is another way to replicate this with decimal datatypes.

I don't understand why all the answers to that question are the same.
There are several ways to calculate the square root from a number. One of them was proposed by Isaac Newton. I'll only write one of the simplest implementations of this method. I use it to improve the accuracy of double's square root.
// x - a number, from which we need to calculate the square root
// epsilon - an accuracy of calculation of the root from our number.
// The result of the calculations will differ from an actual value
// of the root on less than epslion.
public static decimal Sqrt(decimal x, decimal epsilon = 0.0M)
{
if (x < 0) throw new OverflowException("Cannot calculate square root from a negative number");
decimal current = (decimal)Math.Sqrt((double)x), previous;
do
{
previous = current;
if (previous == 0.0M) return 0;
current = (previous + x / previous) / 2;
}
while (Math.Abs(previous - current) > epsilon);
return current;
}
About speed: in the worst case (epsilon = 0 and number is decimal.MaxValue) the loop repeats less than a three times.
If you want to know more, read this (Hacker's Delight by Henry S. Warren, Jr.)

I just came across this question, and I'd suggest a different algorithm than the one SLenik proposed. This is based on the Babylonian Method.
public static decimal Sqrt(decimal x, decimal? guess = null)
{
var ourGuess = guess.GetValueOrDefault(x / 2m);
var result = x / ourGuess;
var average = (ourGuess + result) / 2m;
if (average == ourGuess) // This checks for the maximum precision possible with a decimal.
return average;
else
return Sqrt(x, average);
}
It doesn't require using the existing Sqrt function, and thus avoids converting to double and back, with the accompanying loss of precision.

In most cases involving a decimal (currency etc), it isn't useful to take a root; and the root won't have anything like the expected precision that you might expect a decimal to have. You can of course force it by casting (assuming we aren't dealing with extreme ends of the decimal range):
decimal root = (decimal)Math.Sqrt((double)myVariableHere);
which forces you to at least acknowledge the inherent rounding issues.

Simple: Cast your decimal to a double and call the function, get the result and cast that back to a decimal. That will probably be faster than any sqrt function you could make on your own, and save a lot of effort.

Math.Sqrt((double)myVariableHere);
Will give you back a double that's the square root of your decimal myVariableHere.

Why does this floating-point calculation give different results on different machines?

I have a simple routine which calculates the aspect ratio from a floating point value. So for the value 1.77777779, the routine returns the string "16:9". I have tested this on my machine and it works fine.
The routine is given as :
public string AspectRatioAsString(float f)
{
bool carryon = true;
int index = 0;
double roundedUpValue = 0;
while (carryon)
{
index++;
float upper = index * f;
roundedUpValue = Math.Ceiling(upper);
if (roundedUpValue - upper <= (double)0.1 || index > 20)
{
carryon = false;
}
}
return roundedUpValue + ":" + index;
}
Now on another machine, I get completely different results. So on my machine, 1.77777779 gives "16:9" but on another machine I get "38:21".

Here's an interesting bit of the C# specifiction, from section 4.1.6:
Floating-point operations may be
performed with higher precision than
the result type of the operation. For
example, some hardware architectures
support an “extended” or “long double”
floating-point type with greater range
and precision than the double type,
and implicitly perform all
floating-point operations using this
higher precision type. Only at
excessive cost in performance can such
hardware architectures be made to
perform floating-point operations with
less precision, and rather than
require an implementation to forfeit
both performance and precision, C#
allows a higher precision type to be
used for all floating-point
operations. Other than delivering more
precise results, this rarely has any
measurable effects.
It is possible that this is one of the "measurable effects" thanks to that call to Ceiling. Taking the ceiling of a floating point number, as others have noted, magnifies a difference of 0.000000002 by nine orders of magnitude because it turns 15.99999999 into 16 and 16.00000001 into 17. Two numbers that differ slightly before the operation differ massively afterwards; the tiny difference might be accounted for by the fact that different machines can have more or less "extra precision" in their floating point operations.
Some related issues:
C# XNA Visual Studio: Difference between "release" and "debug" modes?
CLR JIT optimizations violates causality?
To address your specific problem of how to compute an aspect ratio from a float: I'd possibly solve this a completely different way. I'd make a table like this:
struct Ratio
{
public int X { get; private set; }
public int Y { get; private set; }
public Ratio (int x, int y) : this()
{
this.X = x;
this.Y = y;
}
public double AsDouble() { return (double)X / (double)Y; }
}
Ratio[] commonRatios = {
new Ratio(16, 9),
new Ratio(4, 3),
// ... and so on, maybe the few hundred most common ratios here.
// since you are pinning results to be less than 20, there cannot possibly
// be more than a few hundred.
};
and now your implementation is
public string AspectRatioAsString(double ratio)
{
var results = from commonRatio in commonRatios
select new {
Ratio = commonRatio,
Diff = Math.Abs(ratio - commonRatio.AsDouble())};
var smallestResult = results.Min(x=>x.Diff);
return String.Format("{0}:{1}", smallestResult.Ratio.X, smallestResult.Ratio.Y);
}
Notice how the code now reads very much like the operation you are trying to perform: from this list of common ratios, choose the one where the difference between the given ratio and the common ratio is minimized.

I wouldn't use floating point numbers unless I really had to. They're too prone to this sort of thing due to rounding errors.
Can you change the code to work in double precision? (decimal would be overkill). If you do this, does it give more consistent results?
As to why it's different on different machines, what are the differences between the two machines?
32 bit vs 64 bit?
Windows 7 vs Vista vs XP?
Intel vs AMD processor? (thanks Oded)
Something like this might be the cause.

Try Math.Round instead of Math.Ceiling. If you end up with 16.0000001 and round up you'll incorrectly discard that answer.
Miscellaneous other suggestions:
Doubles are better than floats.
(double) 0.1 cast is unnecessary.
Might want to throw an exception if you can't figure out what the aspect ratio is.
If you return immediately upon finding the answer you can ditch the carryon variable.
A perhaps more accurate check would be to calculate the aspect ratio for each guess and compare it to the input.
Revised (untested):
public string AspectRatioAsString(double ratio)
{
for (int height = 1; height <= 20; ++height)
{
int width = (int) Math.Round(height * ratio);
double guess = (double) width / height;
if (Math.Abs(guess - ratio) <= 0.01)
{
return width + ":" + height;
}
}
throw ArgumentException("Invalid aspect ratio", "ratio");
}

When index is 9, you would expect to get something like upper = 16.0000001 or upper = 15.9999999. Which one you get will depend on rounding error, which may differ on different machines. When it's 15.999999, roundedUpValue - upper <= 0.1 is true, and the loop ends. When it's 16.0000001, roundedUpValue - upper <= 0.1 is false and the loop keeps going until you get to index > 20.
Instead maybe you should try rounding upper to the nearest integer and checking if the absolute value of its difference from that integer is small. In otherwords, use something like if (Math.Abs(Math.Round(upper) - upper) <= (double)0.0001 || index > 20)

We had printf()-statements with floating point values that gave different roundings on computer 1 versus computer 2, even though both computers contained the same Visual Studio 2019 version and build.
The difference was found however in a slightly older Windows 10 SDK versus the newest version. How strange it may seem... After fixing that the differences were gone.

Comparing double values in C#

I've a double variable called x.
In the code, x gets assigned a value of 0.1 and I check it in an 'if' statement comparing x and 0.1
if (x==0.1)
{
----
}
Unfortunately it does not enter the if statement
Should I use Double or double?
What's the reason behind this? Can you suggest a solution for this?

It's a standard problem due to how the computer stores floating point values. Search here for "floating point problem" and you'll find tons of information.
In short – a float/double can't store 0.1 precisely. It will always be a little off.
You can try using the decimal type which stores numbers in decimal notation. Thus 0.1 will be representable precisely.
You wanted to know the reason:
Float/double are stored as binary fractions, not decimal fractions. To illustrate:
12.34 in decimal notation (what we use) means
1 * 101 + 2 * 100 + 3 * 10-1 + 4 * 10-2
The computer stores floating point numbers in the same way, except it uses base 2: 10.01 means
1 * 21 + 0 * 20 + 0 * 2-1 + 1 * 2-2
Now, you probably know that there are some numbers that cannot be represented fully with our decimal notation. For example, 1/3 in decimal notation is 0.3333333…. The same thing happens in binary notation, except that the numbers that cannot be represented precisely are different. Among them is the number 1/10. In binary notation that is 0.000110011001100….
Since the binary notation cannot store it precisely, it is stored in a rounded-off way. Hence your problem.

double and Double are the same (double is an alias for Double) and can be used interchangeably.
The problem with comparing a double with another value is that doubles are approximate values, not exact values. So when you set x to 0.1 it may in reality be stored as 0.100000001 or something like that.
Instead of checking for equality, you should check that the difference is less than a defined minimum difference (tolerance). Something like:
if (Math.Abs(x - 0.1) < 0.0000001)
{
...
}

You need a combination of Math.Abs on X-Y and a value to compare with.
You can use following Extension method approach
public static class DoubleExtensions
{
const double _3 = 0.001;
const double _4 = 0.0001;
const double _5 = 0.00001;
const double _6 = 0.000001;
const double _7 = 0.0000001;
public static bool Equals3DigitPrecision(this double left, double right)
{
return Math.Abs(left - right) < _3;
}
public static bool Equals4DigitPrecision(this double left, double right)
{
return Math.Abs(left - right) < _4;
}
...
Since you rarely call methods on double except ToString I believe its pretty safe extension.
Then you can compare x and y like
if(x.Equals4DigitPrecision(y))

Comparing floating point number can't always be done precisely because of rounding. To compare
(x == .1)
the computer really compares
(x - .1) vs 0
Result of sybtraction can not always be represeted precisely because of how floating point number are represented on the machine. Therefore you get some nonzero value and the condition evaluates to false.
To overcome this compare
Math.Abs(x- .1) vs some very small threshold ( like 1E-9)

From the documentation:
Precision in Comparisons
The Equals method should be used with caution, because two apparently equivalent values can be unequal due to the differing precision of the two values. The following example reports that the Double value .3333 and the Double returned by dividing 1 by 3 are unequal.
...
Rather than comparing for equality, one recommended technique involves defining an acceptable margin of difference between two values (such as .01% of one of the values). If the absolute value of the difference between the two values is less than or equal to that margin, the difference is likely to be due to differences in precision and, therefore, the values are likely to be equal. The following example uses this technique to compare .33333 and 1/3, the two Double values that the previous code example found to be unequal.
So if you really need a double, you should use the techique described on the documentation.
If you can, change it to a decimal. It' will be slower, but you won't have this type of problem.

Use decimal. It doesn't have this "problem".

Exact comparison of floating point values is know to not always work due to the rounding and internal representation issue.
Try imprecise comparison:
if (x >= 0.099 && x <= 0.101)
{
}
The other alternative is to use the decimal data type.

double (lowercase) is just an alias for System.Double, so they are identical.
For the reason, see Binary floating point and .NET.
In short: a double is not an exact type and a minute difference between "x" and "0.1" will throw it off.

Double (called float in some languages) is fraut with problems due to rounding issues, it's good only if you need approximate values.
The Decimal data type does what you want.
For reference decimal and Decimal are the same in .NET C#, as are the double and Double types, they both refer to the same type (decimal and double are very different though, as you've seen).
Beware that the Decimal data type has some costs associated with it, so use it with caution if you're looking at loops etc.

Official MS help, especially interested "Precision in Comparisons" part in context of the question.
https://learn.microsoft.com/en-us/dotnet/api/system.double.equals
// Initialize two doubles with apparently identical values
double double1 = .333333;
double double2 = (double) 1/3;
// Define the tolerance for variation in their values
double difference = Math.Abs(double1 * .00001);
// Compare the values
// The output to the console indicates that the two values are equal
if (Math.Abs(double1 - double2) <= difference)
Console.WriteLine("double1 and double2 are equal.");
else
Console.WriteLine("double1 and double2 are unequal.");

1) Should i use Double or double???
Double and double is the same thing. double is just a C# keyword working as alias for the class System.Double
The most common thing is to use the aliases! The same for string (System.String), int(System.Int32)
Also see Built-In Types Table (C# Reference)

Taking a tip from the Java code base, try using .CompareTo and test for the zero comparison. This assumes the .CompareTo function takes in to account floating point equality in an accurate manner. For instance,
System.Math.PI.CompareTo(System.Math.PI) == 0
This predicate should return true.

// number of digits to be compared
public int n = 12
// n+1 because b/a tends to 1 with n leading digits
public double MyEpsilon { get; } = Math.Pow(10, -(n+1));
public bool IsEqual(double a, double b)
{
// Avoiding division by zero
if (Math.Abs(a)<= double.Epsilon || Math.Abs(b) <= double.Epsilon)
return Math.Abs(a - b) <= double.Epsilon;
// Comparison
return Math.Abs(1.0 - a / b) <= MyEpsilon;
}
Explanation
The main comparison function done using division a/b which should go toward 1. But why division? it simply puts one number as reference defines the second one. For example
a = 0.00000012345
b = 0.00000012346
a/b = 0.999919002
b/a = 1.000081004
(a/b)-1 = 8.099789405475458e-5‬
1-(b/a) = 8.100445524503848e-5‬
or
a=12345*10^8
b=12346*10^8
a/b = 0.999919002
b/a = 1.000081004
(a/b)-1 = 8.099789405475458e-5‬
1-(b/a) = 8.100445524503848e-5‬
by division we get rid of trailing or leading zeros (or relatively small numbers) that pollute our judgement of number precision. In the example, the comparison is of order 10^-5, and we have 4 number accuracy, because of that in the beginning code I wrote comparison with 10^(n+1) where n is number accuracy.

Adding onto Valentin Kuzub's answer above:
we could use a single method that supports providing nth precision number:
public static bool EqualsNthDigitPrecision(this double value, double compareTo, int precisionPoint) =>
Math.Abs(value - compareTo) < Math.Pow(10, -Math.Abs(precisionPoint));
Note: This method is built for simplicity without added bulk and not with performance in mind.

As a general rule:
Double representation is good enough in most cases but can miserably fail in some situations. Use decimal values if you need complete precision (as in financial applications).
Most problems with doubles doesn't come from direct comparison, it use to be a result of the accumulation of several math operations which exponentially disturb the value due to rounding and fractional errors (especially with multiplications and divisions).
Check your logic, if the code is:
x = 0.1
if (x == 0.1)
it should not fail, it's to simple to fail, if X value is calculated by more complex means or operations it's quite possible the ToString method used by the debugger is using an smart rounding, maybe you can do the same (if that's too risky go back to using decimal):
if (x.ToString() == "0.1")

Floating point number representations are notoriously inaccurate because of the way floats are stored internally. E.g. x may actually be 0.0999999999 or 0.100000001 and your condition will fail. If you want to determine if floats are equal you need to specify whether they're equal to within a certain tolerance.
I.e.:
if(Math.Abs(x - 0.1) < tol) {
// Do something
}

My extensions method for double comparison:
public static bool IsEqual(this double value1, double value2, int precision = 2)
{
var dif = Math.Abs(Math.Round(value1, precision) - Math.Round(value2, precision));
while (precision > 0)
{
dif *= 10;
precision--;
}
return dif < 1;
}

To compare floating point, double or float types, use the specific method of CSharp:
if (double1.CompareTo(double2) > 0)
{
// double1 is greater than double2
}
if (double1.CompareTo(double2) < 0)
{
// double1 is less than double2
}
if (double1.CompareTo(double2) == 0)
{
// double1 equals double2
}
https://learn.microsoft.com/en-us/dotnet/api/system.double.compareto?view=netcore-3.1