> restart;
> f := proc(x) 2*x + 1; end:
> rando := rand(-10^5 .. 10^5):
> for i from 1 to 10 do
>     x[i] := i;
>     y[i] := f(i) + i*rando()/(10.0^5)/10;
> od;
> 

                              x[1] := 1


                         y[1] := 3.031994000


                              x[2] := 2


                         y[2] := 4.975450000


                              x[3] := 3


                         y[3] := 7.166623000


                              x[4] := 4


                         y[4] := 9.289180000


                              x[5] := 5


                         y[5] := 11.13348500


                              x[6] := 6


                         y[6] := 12.98072800


                              x[7] := 7


                         y[7] := 14.73242500


                              x[8] := 8


                         y[8] := 17.05532800


                              x[9] := 9


                         y[9] := 19.83168100


                             x[10] := 10


                         y[10] := 21.09528000

> points := []:for i from 1 to 10 do
>     points := [op(points), [x[i], y[i]]];
> od:
> 
> points;

  [[1, 3.031994000], [2, 4.975450000], [3, 7.166623000],

        [4, 9.289180000], [5, 11.13348500], [6, 12.98072800],

        [7, 14.73242500], [8, 17.05532800], [9, 19.83168100],

        [10, 21.09528000]]

> A := Matrix(10,2): b := Vector(10):
> x_coords := []: y_coords:=[]: yw_coords:=[]:
> for i from 1 to 10 do
>     x_coords := [op(x_coords), x[i]]; A[i,1] := 1; A[i,2] := x[i];
>     y_coords := [op(y_coords), y[i]]; b[i] := y[i];
>     yw_coords := [op(yw_coords), Weight(y[i],1/i)]; b[i] := y[i];
> od:
> 
> A;

                              [1     1]
                              [       ]
                              [1     2]
                              [       ]
                              [1     3]
                              [       ]
                              [1     4]
                              [       ]
                              [1     5]
                              [       ]
                              [1     6]
                              [       ]
                              [1     7]
                              [       ]
                              [1     8]
                              [       ]
                              [1     9]
                              [       ]
                              [1    10]

> b;

                            [3.031994000]
                            [           ]
                            [4.975450000]
                            [           ]
                            [7.166623000]
                            [           ]
                            [9.289180000]
                            [           ]
                            [11.13348500]
                            [           ]
                            [12.98072800]
                            [           ]
                            [14.73242500]
                            [           ]
                            [17.05532800]
                            [           ]
                            [19.83168100]
                            [           ]
                            [21.09528000]

> with(LinearAlgebra):
> xhat := LinearSolve(Transpose(A) . A, Transpose(A) . b);

                            [.989760933333321824]
                    xhat := [                   ]
                            [2.02535572121212315]

> x_coords;

                   [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

> y_coords;

  [3.031994000, 4.975450000, 7.166623000, 9.289180000, 11.13348500,

        12.98072800, 14.73242500, 17.05532800, 19.83168100,

        21.09528000]

> model := stats[fit,leastsquare[[x,y],y='c'[1]*x + 'c'[2]]]([x_coords,
> y_coords]);

               model := y = 2.025355721 x + .9897609333

# 
> read("/afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlgebra/Maple/
> LeastSqus/incsolve.mpl"):
> xhat_inf := incsolve(convert(A,'matrix'), convert(b,vector));

                           w = .4496671250


                xhat_inf := [.4823660000, 2.099960875]


> Norm(A . xhat - b, infinity);

                         .613718575757570050

> Norm(A . Transpose(convert(xhat_inf,'Vector')) - b, infinity);

                         .604116669999996247

> yw_coords;

  [Weight(3.031994000, 1), Weight(4.975450000, 1/2),

        Weight(7.166623000, 1/3), Weight(9.289180000, 1/4),

        Weight(11.13348500, 1/5), Weight(12.98072800, 1/6),

        Weight(14.73242500, 1/7), Weight(17.05532800, 1/8),

        Weight(19.83168100, 1/9), Weight(21.09528000, 1/10)]

> modelw := stats[fit,leastsquare[[x,y],y='c'[1]*x + 'c'[2]]]([x_coords,
> yw_coords]);

              modelw := y = 2.022392924 x + 1.006056316

> xhatw := Vector([op([2,2],modelw),op([2,1,1],modelw)]);

                                 [1.006056316]
                        xhatw := [           ]
                                 [2.022392924]

> Norm(A . xhatw - b, 2);

                         .829063295213279038

> model;

                   y = 2.025355721 x + .9897609333

>