why 10 times faster after 'vpa' the sym matrix

Hi lovely Matlab experts,

Could you please let me know the rationale behind the speed-up stated below? Is it a random case or not? Hope there's something I can keep in mind to have faster run.

symMat=subs(sym,{},{});
%symMat is a complex matrix whose elements are too long to show in command window

symMat 
%*elapsed time ~20s*, digits=32 as default

vpa(symMat,32) 
% *elapsed time ~5s*

More surprisingly is the following 'for' or 'parfor' loop will be 10 times faster after being 'vpa'ed as shown below, and I thought there's 'vpa' in the loop so escaping another vpa outside loop is good:

tic
for i=1:1000
  digits(4)
  vpa(subs(symMat(:,1),,)));
  digits(32)
  vpa(subs(symMat(:,2),,)));
end 
toc %%*the above elapse ~1200s*

VpasymMat=vpa(symMat,32)
tic
  for i=1:1000
    digits(4)
    vpa(subs(VpasymMat(:,1),,)));
    digits(32)
    vpa(subs(VpasymMat(:,2),,)));
  end 
toc %%*the above elapse 60-100s*

thanks,

best

J

Answer 1

[Walter Roberson]

Why shouldn't it be faster? You are doing different operations. When you vpa() symMat ahead of time, then the vpa() pre-calculates most things that can be calculated as numeric, using fixed-point calculations. This can permit it to consolidate a number of values.

For example, you might have

m = x * (sin(sym(pi)/17) + int(tan(y),y,0,sym(pi)/4))^(1453/611)

This involves the indefinitely-precise value of sin(pi/17) and an integral that will probably produce a transcendental number, and involves taking the sum of those two to the 1453 power and taking the 611'th root of that -- more indefinitely precise numbers, all of which have to be held as indefinitely precise because the code does not know what you are going to do with m later. Repeat this a number of times and it is going to add up in time.

If, on the other hand, you vpa() this ahead of time, then it is going to reduce to

m = 0.22127813173338713927096201193632*x

which is comparatively fast to execute many times.

[Jiyuan Liu]

Hah, Thanks a lot, Walter.

Do you think it’s better to use

SymMat=Vpa(Subs(sym,{},{}))

Rather than

      SymMat=Subs(sym,{},{}))
      SymMat=vpa(symmat)

[Jiyuan Liu]

the former superior.

[Walter Roberson]

SymMat = vpa(subs(SymMat));

would be my preference.