singular matrix- integral
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From: Rob Johnson on 17 May 2010 05:40 In article <918aaf91-14cb-45e0-a384-6d0036fbaf32(a)h39g2000yqn.googlegroups.com>, Magdalena Moczydlowska <magdamoczydlowska(a)gmail.com> wrote: >On 14 Maj, 13:11, r...(a)trash.whim.org (Rob Johnson) wrote: >> In article <cce824e6-0270-404f-b8fa-ba6abc5b2...(a)k17g2000yqf.googlegroups.com>, >> >> Magdalena Moczydlowska <magdamoczydlow...(a)gmail.com> wrote: >> >On 14 Maj, 09:57, Magdalena Moczydlowska <magdamoczydlow...(a)gmail.com> >> >wrote: >> >> You have right and this is simple. Thank You! >> >> >> But the second problem is how to calculate Int_{0}^{h} ( exp(h- >> >> s)A ) ds . The same integral but without A. >> >> >> Magdalena >> >> >Of course the assumption about A are the same as in the first message. >> >> Consider the function E(x) = (exp(x)-1)/x. What is the derivative of >> tE(tA) with respect to t? > >It is -1+exp(tA)+A exp(At)/ tA but A is not inverible Consider E(x) as a power series that converges for all x: E(x) = 1 + x/2 + x^2/6 + x^3/24 + ... Apply this power series to tA: E(tA) = I + t/2 A + t^2/6 A^2 + t^3/24 A^3 + ... multiply by t: t E(tA) = t I + t^2/2 A + t^3/6 A^2 + t^4/24 A^3 + ... Differentiate with respect to t: (d/dt) t E(tA) = I + t A + t^2/2 A^2 + t^3/6 A^3 + ... The fact that A is not invertible does not impede the computation of of either tE(tA) or its derivative. So, again, the question is: what is the derivative of tE(tA) with respect to t? Rob Johnson <rob(a)trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font |