Simplifying output in Matlab

If you get an expression like this as output

syms x 
fx = -(1.0*(1.82e+39*x^2 + 1.81e+32*x - 3.04e+39))/(6.35e+41*x + 4.27e+37)

How do you simplify the output and use smaller numbers, for example

fx = -(1.0*(1.82e+7*x^2 + 1.81*x - 3.04e+7))/(6.35e+9*x + 4.27e+5)

[Star Strider]

Likely the best to be hoped for is —

syms x 
fx = -(1.0*(1.82e+39*x^2 + 1.81e+32*x - 3.04e+39))/(6.35e+41*x + 4.27e+37)

fx = 

fx = vpa(fx, 3)

fx = 

fx = vpa(simplifyFraction(fx), 3)

fx = 

.

[Aleem Andrew]

Thank you for you help

[Walter Roberson]

Is the rule that you want the numerator coefficient of x^1 to be in the range [1, 10) ?

[Aleem Andrew]

No, I wanted the output simplified because I was using ode45 to plot the function and initially when using the original expression the computation time was very large. For example, in the code below the plot takes a long time to generate.

tspan = [0 10];
x0 = 0;
[t,x] = ode45(@(t,x) -(1.0*(6.84e+45*x^2 + 5.24e+32*x - 2.49e+42))/(2.47e+39*x + 7.12e+37), tspan, x0);
plot(t,x,'b')

I was looking for a way to decrease the time, like simplifying it using smaller numbers although that doesn't always work.

[Star Strider]

The integration proceeded very quickly, running it in Input argument for ode45 function type error (running R2021b), however the coefficient magnitudes are large, although not so different between them that this could be regarded as a ‘stiff’ system. It could be worth experimenting with ode15s or other stiff solvers to see if the speed increases significantly.

[Jan]

The computation of smaller numbers is easier for human, but does not change the speed when performing the calculations on a computer.

Answer 1

[Jan]

The magnitude of the variables does not matter. For a computer, the addition of pi + 1.23456789, pi + 1.0 and pi+ 1.23456789e37 takes exactly the same time. Therefore the simplification does not accelerate the evaluation. And by the way, this is very fast in Matlab at all:

x = rand;
tic;
for k = 1:1e9
  fx = -(1.0*(1.82e+39*x^2 + 1.81e+32*x - 3.04e+39))/(6.35e+41*x + 4.27e+37);
end
toc

Elapsed time is 0.777027 seconds.

tic;
for k = 1:1e9
  fx = -(1.0*(1.82e+7*x^2 + 1.81*x - 3.04e+7))/(6.35e+9*x + 4.27e+5);
end
toc

Elapsed time is 0.706243 seconds.

Both take about 0.4 seconds on my i5-mobile Matlab R2018b. A tiny acceleration is measurable, if you omit the meaningless multiplication by 1.0.

The slow computation has another cause: ODE45 is a solver for non-stiff equations. Use a stiff solver for this equation instead:

tic
tspan = [0 10];
x0 = 0;
[t,x] = ode23s(@(t,x) -(1.0*(6.84e+45*x^2 + 5.24e+32*x - 2.49e+42))/(2.47e+39*x + 7.12e+37), tspan, x0);
plot(t,x,'b');

toc

Elapsed time is 0.207868 seconds.

There is a very sharp knee at the start. ODE45 struggles massively with it, because the stepsize controller drives crazy.