O16-C054
> restart;
> with(linalg):
Warning, new definition for norm
Warning, new definition for trace
> a:=matrix(4,4,[-12,2,3,4,2,13,4,5,3,4,15,5,4,5,6,-17]);det(a);

                         [-12     2     3      4]
                         [                      ]
                         [  2    13     4      5]
                    a := [                      ]
                         [  3     4    15      5]
                         [                      ]
                         [  4     5     6    -17]


                                44484

> d:=diag(seq(a[i,i],i=1..4));

                         [-12     0     0      0]
                         [                      ]
                         [  0    13     0      0]
                    d := [                      ]
                         [  0     0    15      0]
                         [                      ]
                         [  0     0     0    -17]

> b:=matrix(4,1,[2,1,2,1]):u:=matrix(4,1,[1,0,0,0]):
> evalf(evalm(a^(-1)&*b));

                           [-.1259778797 ]
                           [             ]
                           [.05754878158 ]
                           [             ]
                           [ .1494469922 ]
                           [             ]
                           [-.01879327399]

> y:=array(0..20):y[0]:=u:for k from 1 to 20 do y[k]:=evalm(d^(-1)&*(b-(a-d)&*y[k-1])) od:
> seq(evalf(evalm(y[k])),k=0..20);

  [1.]  [-.1666666667 ]  [-.1373303167]  [-.1734414279 ]
  [  ]  [             ]  [            ]  [             ]
  [0 ]  [-.07692307692]  [.05520361991]  [ .1140155470 ]
  [  ], [             ], [            ], [             ],
  [0 ]  [-.06666666667]  [.1283559578 ]  [ .1941427853 ]
  [  ]  [             ]  [            ]  [             ]
  [0 ]  [ .1764705882 ]  [-.1441930618]  [-.02959808358]

        [-.1089944070 ]  [-.1197795596 ]  [-.1292606372 ]
        [             ]  [             ]  [             ]
        [.05525401018 ]  [.04738048914 ]  [.05861709646 ]
        [             ], [             ], [             ],
        [ .1474835009 ]  [ .1395905819 ]  [ .1500428123 ]
        [             ]  [             ]  [             ]
        [.002421690338]  [-.01616509246]  [-.02380425302]

        [-.1273211985 ]  [-.1251751842 ]  [-.1257390025 ]
        [             ]  [             ]  [             ]
        [.05979779155 ]  [.05722249168 ]  [.05709169197 ]
        [             ], [             ], [             ],
        [ .1514889860 ]  [ .1491985579 ]  [ .1490180963 ]
        [             ]  [             ]  [             ]
        [-.01904118779]  [-.01772717177]  [-.01878805476]

        [-.1261595455 ]  [-.1260207864 ]  [-.1259368069 ]
        [             ]  [             ]  [             ]
        [.05764199181 ]  [.05764277111 ]  [.05752437328 ]
        [             ], [             ], [             ],
        [ .1495193676 ]  [ .1495350049 ]  [ .1494271049 ]
        [             ]  [             ]  [             ]
        [-.01902288073]  [-.01878305986]  [-.01874466240]

        [-.1259707157 ]  [-.1259870177 ]  [-.1259789727 ]
        [             ]  [             ]  [             ]
        [.05752988508 ]  [.05755493381 ]  [.05755253326 ]
        [             ], [             ], [             ],
        [ .1494290826 ]  [ .1494521097 ]  [ .1494505769 ]
        [             ]  [             ]  [             ]
        [-.01879780774]  [-.01880346715]  [-.01879180846]

        [-.1259758697 ]  [-.1259777432 ]  [-.1259783167 ]
        [             ]  [             ]  [             ]
        [.05754728307 ]  [.05754805114 ]  [.05754913795 ]
        [             ], [             ], [             ],
        [ .1494457218 ]  [ .1494462860 ]  [ .1494472984 ]
        [             ]  [             ]  [             ]
        [-.01879116252]  [-.01879369015]  [-.01879370596]

        [-.1259778877 ]  [-.1259777857 ]
        [             ]  [             ]
        [.05754892074 ]  [.05754869853 ]
        [             ], [             ]
        [ .1494471285 ]  [ .1494469200 ]
        [             ]  [             ]
        [-.01879316393]  [-.01879318682]

> # Condition nécessaire de convergence : a est inversible
> # Condition nécessaire et suffisante : soit a inversible de x0=a^(-1)b. Alors y[k]-x0=(-d^(-1)a+I_4)(y[k-1]-x0)=m^k(u-x0) avec m=-d^(-1)a+I_4 donc (y[k]) a une limite pour tout u ssi sp(m) est dans la boule ouverte B(0,1)