octave:2> A=rand(4)
A =

   0.529737   0.371992   0.692179   0.496653
   0.947285   0.352989   0.097161   0.334264
   0.992025   0.467582   0.225537   0.693762
   0.955836   0.366773   0.839654   0.742790

octave:3> expm(A)
ans =

   3.7212   1.4105   2.0245   2.0335
   2.5994   2.1661   1.2389   1.5110
   3.3867   1.6173   2.8411   2.3935
   3.9804   1.8165   2.7725   3.9032

octave:4> phi1m(A)
ans =

   1.84326   0.45917   0.69143   0.65270
   0.90419   1.39003   0.35301   0.47443
   1.12225   0.53462   1.54149   0.79158
   1.27175   0.56394   0.92598   1.93959

octave:5> a=1e-15
a =  1.0000e-15
octave:6> (exp(a)-1)/a
ans =  1.1102
octave:7> a=1e-10
a =  1.0000e-10
octave:8> (exp(a)-1)/a
ans =  1.0000
octave:9> a=1e-20
a =  1.0000e-20
octave:10> (exp(a)-1)/a
ans = 0
octave:11> A
A =

   0.529737   0.371992   0.692179   0.496653
   0.947285   0.352989   0.097161   0.334264
   0.992025   0.467582   0.225537   0.693762
   0.955836   0.366773   0.839654   0.742790

octave:12> det(A)
ans =  0.034726
octave:13> phi1m(A)
ans =

   1.84326   0.45917   0.69143   0.65270
   0.90419   1.39003   0.35301   0.47443
   1.12225   0.53462   1.54149   0.79158
   1.27175   0.56394   0.92598   1.93959

octave:14> A\(expm(A)-eye(size(A)))
ans =

   1.84326   0.45917   0.69143   0.65270
   0.90419   1.39003   0.35301   0.47443
   1.12225   0.53462   1.54149   0.79158
   1.27175   0.56394   0.92598   1.93959

octave:15> format long e
octave:16> A\(expm(A)-eye(size(A)))
ans =

 Columns 1 through 3:

   1.84326477172993e+00   4.59173340146895e-01   6.91434570688754e-01
   9.04189474860063e-01   1.39003102710163e+00   3.53007334553534e-01
   1.12224728297697e+00   5.34621759045839e-01   1.54148765081066e+00
   1.27175280487114e+00   5.63942063526924e-01   9.25984438589806e-01

 Column 4:

   6.52703747009379e-01
   4.74428009713175e-01
   7.91579947128987e-01
   1.93958782961182e+00

octave:17> phi1m(A)
ans =

 Columns 1 through 3:

   1.84326477172993e+00   4.59173340146895e-01   6.91434570688753e-01
   9.04189474860066e-01   1.39003102710163e+00   3.53007334553533e-01
   1.12224728297697e+00   5.34621759045839e-01   1.54148765081065e+00
   1.27175280487114e+00   5.63942063526926e-01   9.25984438589810e-01

 Column 4:

   6.52703747009379e-01
   4.74428009713175e-01
   7.91579947128987e-01
   1.93958782961182e+00

octave:18> A=100*toeplitz([-2,1,zeros(1,8)]);
octave:19> y=ones(10,1);
octave:20> expm((1-0)*A)*y
ans =

   1.07990465570348e-04
   2.07232185864215e-04
   2.89685186916627e-04
   3.48669616949197e-04
   3.79406908234599e-04
   3.79406908234599e-04
   3.48669616949197e-04
   2.89685186916627e-04
   2.07232185864215e-04
   1.07990465570348e-04

octave:21> qui
error: `qui' undefined near line 21 column 1
octave:21> quit