In Eigen3 the norm() method does not provide the precise answer, why - eigen3

Question

I have a situation that using Eigen3 library the norm() does not provide the correct answer. The norm() should be just the square root of the coeficients of a vector to the square:
NORM= sqrt( v[1]*v[1] + v[2]*v[2] + .... + v[N]*v[N])
However the following function calculates the norm() in two ways: with the norm() method of Eigen3 and by hand. The results are slighly different:
void mytest()
{
double mvec[3];
mvec[0] = -3226.9276456286984;
mvec[1] = 6153.3425006471571;
mvec[2] = 2548.5894934614853;
Vector3d v;
v(0) = mvec[0];
v(1) = mvec[1];
v(2) = mvec[2];
double normEigen = v.norm();
double normByHand = sqrt( v(0)*v(0) + v(1)*v(1) + v(2)*v(2));
double mdiff = abs((normEigen - normByHand));
std::cout.precision(17);
std::cout << "normEigen= " << normEigen << std::endl;
std::cout << "normByHand= " << normByHand << std::endl;
std::cout << "mdiff= " << mdiff << std::endl;
}
The output of this function is:
normEigen= 7400.8103858007089
normByHand= 7400.8103858007107
mdiff= 1.8189894035e-12
From digit 15 they are different, why? where is rounding some number?
Thanks in advance
PedroC.

Answer 1

The calculation is one that uses floating point computations. As such, the order of operations, as well as things like vectorization, can result in (usually) slightly different results (due to different roundings, different orders of magnitude, etc.).
In this case, the difference is just in the 15th digit. The maximal accuracy on a 64 bit floating point number is around the 16th digit.
If we look at the distance in ULPs using boost:
#include <boost/math/special_functions/next.hpp>
#include <iostream>
int main()
{
double normEigen = 7400.8103858007089;
double normByHand = 7400.8103858007107;
std::cout << boost::math::float_distance(normEigen, normByHand);
return 0;
}
we see that the distance (at least on my system) is 2. So the binary number is e.g. 0101...011 instead of 0101...001. Such a small difference is almost always due to the reasons I listed above.

Answer 2

Going deeper I see that the sum of the square values introduces the discrepancy, when I calculate the squaredNorm in a unique vector and in
3 vectors using only one tem I see that the total is not identical.
void mytest2()
{
double mvec[3];
mvec[0] = -3226.9276456286984;
mvec[1] = 6153.3425006471571;
mvec[2] = 2548.5894934614853;
Vector3d v, v1, v2, v3;
v(0) = mvec[0];
v(1) = mvec[1];
v(2) = mvec[2];
v1(0) = mvec[0]; v1(1) = v1(2) = 0.0;
v2(0) = 0.0; v2(1) = mvec[1]; v2(2) = 0.0;
v3(0) = v3(1) = 0.0; v3(2) = mvec[2];
double squnorm = v.squaredNorm();
double squnorm1 = v1.squaredNorm();
double squnorm2 = v2.squaredNorm();
double squnorm3 = v3.squaredNorm();
double squnormbyhand = squnorm1 + squnorm2 + squnorm3;
double sqdiff = abs(squnorm - squnormbyhand);
std::cout.precision(17);
std::cout << "normEigen= " << squnorm << std::endl;
std::cout << "normByHand= " << squnormbyhand << std::endl;
std::cout << "mdiff= " << sqdiff << std::endl;
}
The ouput of this function is:
normEigen= 54771994.366575643
normByHand= 54771994.366575658
mdiff= 1.49011161193847656e-8
For some reason when adding the square values Eigen introduces a rounding difference.
Thanks for your answer anyway.
Pedro