> 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, /afs/eos.ncsu.edu/users/k/kaltofen/www/courses/\

        LinAlgebra/Maple, "/afs/bp.ncsu.edu/dist/maple551/update",

        "/afs/bp.ncsu.edu/dist/maple551/lib"

> with(lsqpkg);

                            [gram_schmidt]

> ?gram_schmidt
> with(linalg):
Warning, new definition for norm
Warning, new definition for trace
> printlevel:=3:
> A := matrix(4,3,[1,1,0, 1,-2,0,1,1,0,1,0,1]);

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

> gram_schmidt(A);

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


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


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


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


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


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


                           ||u[2]||^2 = , 6


                         v[3]=, [0, 0, 0, 1]


                    u[3]=, [-1/4, -1/4, -1/4, 3/4]


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


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


                          ||u[3]||^2 = , 3/4


                          [1     1    -1/4]
                          [               ]
                          [1    -2    -1/4]
                          [               ]
                          [1     1    -1/4]
                          [               ]
                          [1     0    3/4 ]

> A:=randmatrix(5,5);

                     [-12    -18     31    -26    -62]
                     [                               ]
                     [  1    -47    -91    -47    -61]
                     [                               ]
                A := [ 41    -58    -90     53     -1]
                     [                               ]
                     [ 94     83    -86     23    -84]
                     [                               ]
                     [ 19    -50     88    -53     85]

> gram_schmidt(A);

                     v[1]=, [-12, 1, 41, 94, 19]


                     u[1]=, [-12, 1, 41, 94, 19]


                         ||u[1]||^2 = , 11023


                   v[2]=, [-18, -47, -58, 83, -50]


                [-142698  -522724  -829697  478467  -639367]
         u[2]=, [-------, -------, -------, ------, -------]
                [ 11023    11023    11023   11023    11023 ]


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


                                      146940129
                       ||u[2]||^2 = , ---------
                                        11023


                    v[3]=, [31, -91, -90, -86, 88]


            [1076461  -215142280  -103763909  -553722  294563447]
     u[3]=, [-------, ----------, ----------, -------, ---------]
            [ 50547    2577897     2577897    286433    2577897 ]


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


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


                                     18989582765
                      ||u[3]||^2 = , -----------
                                       859299


                    v[4]=, [-26, -47, 53, 23, -53]


         [-5349990232108  -12182875979650  49498194263231
  u[4]=, [--------------, ---------------, --------------,
         [ 360802072535    216481243521    1082406217605

        -6021676109166  -24368345813963]
        --------------, ---------------]
         360802072535    1082406217605 ]


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


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


                           <u[3], u[4]>=, 0


                                  2259902021960334
                   ||u[4]||^2 = , ----------------
                                    360802072535


                    v[5]=, [-62, -61, -1, -84, 85]


         [-261055063889116460  83338813844221664  -101086293856771886
  u[5]=, [-------------------, -----------------, -------------------,
         [ 3389853032940501    10169559098821503   10169559098821503

        -10119079955104960  169682030331264698]
        ------------------, ------------------]
         1129951010980167   10169559098821503 ]


                           <u[1], u[5]>=, 0


                           <u[2], u[5]>=, 0


                           <u[3], u[5]>=, 0


                           <u[4], u[5]>=, 0


                                65646657556951761032
                 ||u[5]||^2 = , --------------------
                                 10169559098821503


        [      -142698   1076461   -5349990232108
        [-12 , ------- , ------- , -------------- ,
        [       11023     50547     360802072535

        -261055063889116460]
        -------------------]
         3389853032940501  ]

        [    -522724   -215142280   -12182875979650
        [1 , ------- , ---------- , --------------- ,
        [     11023     2577897      216481243521

        83338813844221664]
        -----------------]
        10169559098821503]

        [     -829697   -103763909   49498194263231
        [41 , ------- , ---------- , -------------- ,
        [      11023     2577897     1082406217605

        -101086293856771886]
        -------------------]
         10169559098821503 ]

        [     478467   -553722   -6021676109166   -10119079955104960]
        [94 , ------ , ------- , -------------- , ------------------]
        [     11023    286433     360802072535     1129951010980167 ]

        [     -639367   294563447   -24368345813963
        [19 , ------- , --------- , --------------- ,
        [      11023     2577897     1082406217605

        169682030331264698]
        ------------------]
        10169559098821503 ]

> evalf(%);

        [-12. , -12.94547764 , 21.29623914 , -14.82804740 ,

        -77.01073213]

        [1. , -47.42121020 , -83.45650738 , -56.27682002 ,

        8.194928908]

        [41. , -75.26961807 , -40.25137893 , 45.72977636 ,

        -9.940086180]

        [94. , 43.40624150 , -1.933164126 , -16.68969379 ,

        -8.955326255]

        [19. , -58.00299374 , 114.2650180 , -22.51312439 ,

        16.68528878]

>