A  =  {{1, 2, 3},  
       {4, 5, 
          6}, 
       {7, 8, 10}}

b  =  {{-1}, {0}, {1}}

x = LinearSolve[A, b]

MatrixForm[A . x  -  b]

Aa  =  N[A]

ba  =  N[b]

xa  =  LinearSolve[Aa, ba]

MatrixForm[Aa . xa  -  ba]

Ai  =  {{Interval[{1, 1}],  Interval[{2, 2}],  
          Interval[{3, 3}]},  
{Interval[{4, 4}],  
          Interval[{5, 5}],  
          Interval[{6, 6}]}, 
{Interval[{7, 7}],  
          Interval[{8, 8}], 
Interval[{10, 
              10}]}}

bi  =  {{Interval[{-1, -1}]},  
         {Interval[{0, 
              0}]}, 
         {Interval[{1, 
              1}]}}

xi = LinearSolve[Ai, bi]

Aia  =  N[Ai, 20]

bia  =  N[bi, 20]

xia  =  N[LinearSolve[Aia, bia], 20]

Clear[x]

IN[f_, x_, xp_]  :=  xp  -  f[xp]/f'[x]

g[x_]  :=  x^2  -  2

x  =  Interval[{1, 2}]

xp  =  1.5

xnew  =  IN[g, x, xp]

x  =  xnew

xp  =  (Min[xnew] + Max[xnew])/2

xnew  =  IN[g, x, xp]