Error while using Polyval

While solving a problem using polyval, I encountered an error. Here is the screenshot of the code attached (just the part which focuses on the use of polyval)

It seems that polyval isn't accurate when number of digits of the array exceed 15. I am unable to understand what is the cause of this discrepancy.

Answer 1

[Steven Lord]

Not all numbers in the vicinity of large values can be exactly represented in double precision. In particular 12345678901234567 is not exactly representable in double.

v = [1:9 0:7];
format longg
p = polyval(v, 10)

p = 
      1.23456789012346e+16

s = sym(p)

s = 

12345678901234568

What's the distance between p and the next largest representable number?

eps(p)

ans = 
     2

If we add something smaller than eps(p) to p it's not enough to get us to the next largest double precision number.

p2 = p + 0.5;
p == p2 % true

ans = logical
   1

[Dyuman Joshi]

Thank you for your answer, @Steven!

I still have another query - Is there a way to represent such large number in string format rather than just typing them directly?