> read("/afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlgebra/Maple/
> LsqPkg/InitPkg.mpl");

  libname := /afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlg\

        ebra/Maple, "/usr/local/maple/lib"

> with(LsqPkg);

                            [Gram_Schmidt]

> ?Gram_Schmidt
# Examples:
> printlevel:=3:
> with(LsqPkg, Gram_Schmidt);

                            [Gram_Schmidt]

> A := Matrix(3, 4, [[1,1,2,1],[1,1,2,0],[1,0,1,0]]);

                            [1    1    2    1]
                            [                ]
                       A := [1    1    2    0]
                            [                ]
                            [1    0    1    0]

> Q := Gram_Schmidt(A, 'R');

                                     [1]
                                     [ ]
                              v[1]=, [1]
                                     [ ]
                                     [1]


                                     [1]
                                     [ ]
                              u[1]=, [1]
                                     [ ]
                                     [1]


                           ||u[1]||^2 = , 3


                                     [1]
                                     [ ]
                              v[2]=, [1]
                                     [ ]
                                     [0]


                                   [1/3 ]
                                   [    ]
                            u[2]=, [1/3 ]
                                   [    ]
                                   [-2/3]


                           <u[1], u[2]>=, 0


                          ||u[2]||^2 = , 2/3


                                     [2]
                                     [ ]
                              v[3]=, [2]
                                     [ ]
                                     [1]


                                     [0]
                                     [ ]
                              u[3]=, [0]
                                     [ ]
                                     [0]


                           <u[1], u[3]>=, 0


                           <u[2], u[3]>=, 0


                           ||u[3]||^2 = , 0


                                     [1]
                                     [ ]
                              v[4]=, [0]
                                     [ ]
                                     [0]


                                   [1/2 ]
                                   [    ]
                            u[4]=, [-1/2]
                                   [    ]
                                   [ 0  ]


                           <u[1], u[4]>=, 0


                           <u[2], u[4]>=, 0


                          ||u[4]||^2 = , 1/2


                            [1    1/3     1/2 ]
                            [                 ]
                       Q := [1    1/3     -1/2]
                            [                 ]
                            [1    -2/3     0  ]

> print(R);

                       [1    2/3    5/3    1/3]
                       [                      ]
                       [0     1      1     1/2]
                       [                      ]
                       [0     0      0      1 ]

> Q . R;

                          [1    1    2    1]
                          [                ]
                          [1    1    2    0]
                          [                ]
                          [1    0    1    0]

>