Symbolic expansions, is it possible to express a solution as a series of exponentials

Lets say I have something like

f_t = (A*cos(a*t) + B*sin(b*t))*(C*cos(c*t) + D*sin(c*t)),

what I would like is something of the form

f_t = c_1 exp(k1*t) + c_2 exp(k2*t) + ... + c_N exp(kN*t) .

I can't find anything within the symbolic toolbox that allows expansion in this form. I don't really want to write one either until I know for sure there isn't one included / on the FEX. Anyone that can help?

Answer 1

[Star Strider]

There is one in the Symbolic Math Toolbox: rewrite!

To wit:

syms A a t
f_t = (A*cos(a*t) + B*sin(b*t))*(C*cos(c*t) + D*sin(c*t));
f_t = rewrite(f_t, 'exp')

produces:

f_t =

(A*(exp(-a*t*i)/2 + exp(a*t*i)/2) + B*((exp(-b*t*i)*i)/2 - (exp(b*t*i)*i)/2))*(C*(exp(-c*t*i)/2 + exp(c*t*i)/2) + D*((exp(-c*t*i)*i)/2 - (exp(c*t*i)*i)/2))

You’ll have to play with it it get the result in your example. There may not be any direct way to do that.

[David H]

Not quite but that is none the less a step in the right direction!

[Star Strider]

You may have to do the coefficient definitions yourself. Your second f_t expression looks like a nonlinear curve-fit of some sort, with exponential terms. It’s going to be difficult to do that in the Symbolic Toolbox, since you’ll have to generate symbolic matrices and the do the equivalent of solving A*x=b with them.

Other possibilities are the taylor function, or writing your own series expansion.

The ‘c_1...c_n’ might not be difficult to estimate, but the exponent constants ‘k1...kn’ might be next to impossible to express analytically in terms of the arguments of your trigonometric functions.

[David H]

Currently the only way I can really see to do this after the rewrite to then apply "expand". This is products of exponentials

(A*C*exp(-a*t*i)*exp(-c*t*i))/4 + (A*C*exp(-a*t*i)*exp(c*t*i))/4 + (A*C*exp(a*t*i)*exp(-c*t*i))/4 + (A*C*exp(a*t*i)*exp(c*t*i))/4 + (A*D*exp(-a*t*i)*exp(-c*t*i)*i)/4 - (A*D*exp(-a*t*i)*exp(c*t*i)*i)/4 +...

I suppose now I would need a string search to identify exponential products and combine them into a single exponential, running this recursively until no more exponential products can be found. After this I would have to collate the coefficients on exponentials with identical exponents.

Edit: I wrote something to do this final step, regretably it is a bit of a hash but it seems to work for the examples I have tried! On the FEX (46905-express-as-exponential-series)

[Star Strider]

It would seem so.

I also experimented with subexpr, but wasn’t sure that’s what you wanted to do, so didn’t suggest it earlier. If the constants are related in some way, subs may also help. To break them out into fractional numerators and denominators, numden could help.

Explore the options in the Formula Rearrangement and Rewriting section of the documentation. I’m not certain you can do exactly what you want, but you can probably get close. Delving into MuPad directly may also provide options the Symbolic Math Toolbox does not, at least easily.

EDIT — I’ll take a look at it!