Not exactly the exact answer using Symbolic Math Toolbox

I'm finding the RMS value of sine across one period, and I expect the result to be A/sqrt(2).

The result I get is very close, but not exactly the same.

Can anyone please help?

syms t

f = sin(t);

frms = sqrt(1/(2*pi)*int(f^2,t,[0 2*pi])) % rms value from definition

frms = 

frms_simplified = simplify(frms) % here I expected 1/sqrt(2)

frms_simplified = 

var = vpa(frms_simplified) % decimal value

var = 

0.70710678118654754625835857353067

var2 = vpa(1/sqrt(2)) % for comparison

var2 = 

0.70710678118654752440084436210485

Thanks in advance :-)

Answer 1

[Torsten]

syms t
f = sin(t);
frms = sqrt(1/(2*sym(pi))*int(f^2,t,[0 2*sym(pi)])) % rms value from definition

frms = 

[Kristian Lomholdt]

Thanks - I wasn't aware of the difference :-)

Answer 2

[John D'Errico]

Your problem stems from a common mistake.

Whaen you write this:

syms t
f = sin(t);
frms = sqrt(1/(2*pi)*int(f^2,t,[0 2*pi])) % rms value from definition

frms = 

And there, MATLAB has chosen a way to express the result using large integers.

In that third line, do you think MATLAB "knows" that something like sqrt(1/(2*pi)) really is EXACTLY the number you wrote? It is not. In fact, pi in MATLAB is the number

format long g
pi

ans = 
          3.14159265358979

which is darn close to the true value of pi, but it is a DOUBLE PRECISION NUMBER. It is not exactly the number pi, but ony a 52 binary bit approximation to pi. They are entirely different things.

So let me change what you wrote by telling MATLAB to use pie as a symbolic constant instead.

pie = sym('pi');
frms = sqrt(1/(2*pie)*int(f^2,t,[0 2*pie])) % rms value from definition

frms = 

Which is indeed sqrt(1/2), even though simplify did not notice that sin(4*pie) would be zero. Stupid computers. Of course, when I say that, it is most of the time my own foolishness I am referring to, since the computer just does what I tell it to do. ;-) I could probably convince MATLAB to fix that of course, but if I did, then MATLAB would just ignore everything I type. Oh well.

[Torsten]

Defining

pie = sym('pi')

pi is used as a symbolic variable with name "pi", not as the numerical constant pi.

[John D'Errico]

yes

[Walter Roberson]

syms t
f = sin(t);
pie = sym(pi);
frms = sqrt(1/(2*pie)*int(f^2,t,[0 2*pie])) % rms value from definition

frms = 

[Kristian Lomholdt]

Thanks for clarifiyng :-)