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This a general question about Picard iterations and is as follows.

Let A be a $n\times n$ matrix. show that the Picard method for solving $X^{'}=AX$, $X(0)=X_{0}$ gives the solution $e^{tA}X_{0}$

I really don't even have a clue where to start? i have used it to do some very simple problems but im totally stumped on this one.

Sepideh Bakhoda
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Faust
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  • I'm not exactly sure what you mean. The Picard iteration looks for a fixed point of the operator $(Tx)(t) = x_0 + A \int_0^t x(\tau) d \tau$. You could define a sequence $x_0(t) = x_0$ (slight abuse of notation), and $x_{n+1} = T x_n$ and show that it converges in some appropriate sense. Then expand the solution to show that it equals $e^{At} x_0$. – copper.hat Mar 26 '13 at 19:01
  • Im not sure what i mean eitheir that how its written in my textbook =) i think what you have stated above is what they want me to do. Which makes some sense you using induction to show that eventually that will be the solution? – Faust Mar 26 '13 at 19:12
  • Do you know how the method works for a simple ode $y'(x)=ay(x)$? – Mhenni Benghorbal Mar 26 '13 at 19:14
  • Yes but that is it. – Faust Mar 26 '13 at 19:22

1 Answers1

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Here is a start,

$$ X_{n+1}(t) = X_0 + A \int_0^t X_n(\tau) d \tau$$

$$= X_0 + \int_0^t A\left(X_0 + A \int_0^t X_{n-1}(\tau) d \tau \right) d \tau $$

$$ = X_0 + AX_0 t + \int_{0}^{t}(t-\tau)A^2 X_{n-1}(\tau) d\tau $$

$$ = X_0 + AX_0 t + \int_{0}^{t}(t-\tau)A^2 \left(X_0 + A \int_0^t X_{n-2}(\tau) d \tau \right) d\tau $$

$$ = X_0 + AX_0 \,t +\int_{0}^{t}(t-\tau)A^2X_0 d\tau + \frac{1}{2!}\int_0^t (t-\tau)^2 A^3X_{n-2}(\tau) d \tau $$

$$ = X_0 + AX_0 \,t +A^2X_0 \frac{t^2}{2!}+ \frac{1}{2}\int_0^t (t-\tau)^2 A^3X_{n-2}(\tau) d \tau = \ldots. $$

I leave it here for you to finish the problem. Work out few other terms to see the pattern. See here for the iterated integration.

Community
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Mhenni Benghorbal
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    wow that looks exactly like the $e^{At}$ expansion haha!

    Tyvm i can finish it from here for sure.

     – Faust Mar 26 '13 at 20:14
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    This is a nice, intuitive explanation, but to show that this converges, you need some bounds on $X_n$. – copper.hat Mar 26 '13 at 20:15
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    @copper.hat: Thanks for the cooment In my opinion, I think students need to learn how to compute things first, then they get exposed to advanced theory. Just like in the case of calculus and real analysis. – Mhenni Benghorbal Mar 26 '13 at 20:31
  • Is there a proof that we can actually apply picard iteration to matrices ? How do we know it's not just a trick? Thankss – qubitz Jul 30 '23 at 20:50