Re: Piecewise function involving matrices

*To*: mathgroup at smc.vnet.net*Subject*: [mg91230] Re: Piecewise function involving matrices*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>*Date*: Sun, 10 Aug 2008 01:53:41 -0400 (EDT)*Organization*: The Open University, Milton Keynes, UK*References*: <g7k0gn$ieh$1@smc.vnet.net>

kaveh wrote: > I am trying to define a piecewise matrix function of time with > Mathematica that I later want to use to transform another matrix as a > function of time and integrate it. To make it simple I have > > In[163]:= A={{1,.1},{.1,-1}}; > In[171]:= H0={{0,b},{b,0}}; > In[172]:= F[t_]:=Piecewise[{{exp[-I t] A,0<t<1},{exp[+I t] A,1<t<2}}] Syntax error: Except if you have already defined the function exp somewhere else, exp should be written Exp, which is the standard exponential function in Mathematica. > In[173]:= F[0.1] > Out[173]= {{exp[-0.1 \[ImaginaryI]],0.1 exp[-0.1 \[ImaginaryI]]},{0.1 > exp[-0.1 \[ImaginaryI]],-exp[-0.1 \[ImaginaryI]]}} > > In[181]:= H[t_]:= Inverse[F[t]].H0.F[t]+H0.H0 > > In[182]:= H[0.4] > Out[182]= {{0.19802 b+b^2,-0.980198 b},{-0.980198 b,-0.19802 b+b^2}} > > The problem is that when I evaluate H[t] for a given number, it > returns the right value but if t is a symbol Mathematica naturally > leaves it as unevaluated. This by itself is nice, however when I want > to integrate H[t], I run into problems. > Say > In[183]:= Integrate[H[x],{x,0,1}] <snip> > Notice that the output is not a 2x2 matrix but has dimensions > {2,2,2,2}! > Same with: > In[186]:= Integrate[H[x][[1,1]],{x,0,1}] > Out[186]= {{0.00990099 (20. b+101. b^2),0.00990099 (-99. b+101. b^2)}, > {0.00990099 (-99. b+101. b^2),0.00990099 (-20. b+101. b^2)}} > which should be just a number but is a 2x2 matrix! > > Obviously there is something wrong with the treatment of piecewise as > an array instead of a flat thing. Since Mathematica 6.0, Integrate works well with *continuous* piecewise functions. In your case, you should use *Boole* to write the piecewise function. For instance, In[1]:= A = {{1, .1}, {.1, -1}}; H0 = {{0, b}, {b, 0}}; F[t_] := Boole[0 < t < 1] Exp[-I t] A + Boole[1 < t < 2] Exp[+I t] A F[0.1] H[t_] := Inverse[F[t]].H0.F[t] + H0.H0 H[0.4] Out[4]= {{0.995004- 0.0998334 I, 0.0995004- 0.00998334 I}, {0.0995004- 0.00998334 I, -0.995004 + 0.0998334 I}} Out[6]= {{(0.19802- 6.93889*10^-18 I) b + b^2, (-0.980198 - 4.33681*10^-19 I) b}, {(-0.980198 - 4.33681*10^-19 I) b, (-0.19802 + 6.93889*10^-18 I) b + b^2}} In[7]:= Integrate[H[x], {x, 0, 1}] Out[7]= {{0.00990099 (20. b + 101. b^2), -0.980198 b}, {-0.980198 b, 0.00990099 (-20. b + 101. b^2)}} In[8]:= % // Dimensions Out[8]= {2, 2} In[9]:= Integrate[H[x][[1, 1]], {x, 0, 1}] Out[9]= 0.00990099 (20. b + 101. b^2) In[10]:= % // Dimensions Out[10]= {2} Regards, -- Jean-Marc