Cool examples given in Gilbert Strang's Introduction to Linear Algebra are
>From Section 6.2 -- The solution to u(k+1) = A uk starting from u0 is uk = A^k u0 = S Lambda^k S^-1 u0 , so that uk = c1 lambda1^k x1 + ... + cn lambdan^k xn provided u0 = c1 x1 + ... + cn xn (xk is eigenvector corrresponding to eigenvalue lamdak) Lambda is a diagonal matrix with the eigenvalues of A on the diagonal, and S is a square matrix whose columns are the eigenvectors. Strang illustrates with the Fibonnaci sequence F0 F1 F2 ... , setting uk = ( F(k+1) , Fk ) and u0 = (1 , 0) . >From Section 6.3 -- The solution to u' = A u starting from u(0) is u(t) = c1 e^(lambda1 t) x1 + ... + cn e^(lambdan t) xn provided u(0) = c1 x1 + ... + cn xn . The solution can be expressed as u(t) = e^(A t) u(0) with the matrix exponential e^(A t) . Equations involving y'' reduce to u' = A u by combining y' and y into u = (y' , y) --Kip Murray Sent from my iPad > On Oct 3, 2013, at 6:10 AM, Raul Miller <[email protected]> wrote: > > I was looking at those the other day and ran into a variety of difficulties. > > I'll not bore you with the details, but I'll admit that I would love > to see some pages devoted to example usages. > > I'm not really looking for comprehensive, in-depth documentation - > that's already available through web searching ... if I can understand > what terms I need to use to search on. What I'm looking for are cool > examples - things that probe the possibilities, bits of prose perhaps > which hint at relevant search terms. Failed attempts might also be > useful, as stepping stones for finding or writing more better or > slightly beautiful expositions. > > Thanks, > > -- > Raul > >> On Wed, Oct 2, 2013 at 12:27 PM, km <[email protected]> wrote: >> Here is a resource that should be better known. To use it you load >> ~addons/math/mt/mt.ijs --Kip Murray >> >> >>>>> On Mon, Apr 8, 2013 at 3:17 AM, Kip Murray <[email protected]> wrote: >>>> >>>> Igor Zhuravlof provides j routines that model LAPACK routines for >>>> eigenvalues and eigenvectors. See "matrix toolbox" >>>> >>>> ~addons/math/mt >>>> >>>> with contents summarized in >>>> >>>> ~addons/math/mt/mt.ijs >>>> >>>> I have not tried them but would expect them to run in j701JHS. >> >> Sent from my iPad >> >>> On Oct 2, 2013, at 9:21 AM, Raul Miller <[email protected]> wrote: >>> >>> J's support for mechanisms to compute eigenvalues has been rather >>> messy. And, by messy I mean that it looks like we rarely exercise >>> these mechanisms - we don't have unit tests on the entry points to be >>> run before uploading library updates, we don't have particularly good >>> documentation on the code we have and there are other problems. >>> >>> Here's an example I stumbled over today: >>> >>> docs_jlapack_'' >>> |value error: dirs >>> | dirs jpathsep path,'doc/*.lap' >>> require'dirs' >>> not found: /Users/rdmiller/Applications/j64-801/bin/dirs >>> |file name error: script >>> | 0!:0 y[4!:55<'y' >>> require 'dir' >>> docs_jlapack_'' >>> |value error: dirs >>> | dirs jpathsep path,'doc/*.lap' >>> getscripts_j_ 'dir' >>> >>> >>> So here's a question: does anyone have the time and energy to put into >>> this mess? >>> >>> Thanks, >>> >>> -- >>> Raul >>> ---------------------------------------------------------------------- >>> For information about J forums see http://www.jsoftware.com/forums.htm >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
