> with(LinearAlgebra): > 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/bp.ncsu.edu/dist/maple70/lib" > with(LsqPkg); [Gram_Schmidt] > v1:=<1,1,1,1>; [1] [ ] [1] v1 := [ ] [1] [ ] [1] > v2:=<1,-2,1,0>; [ 1] [ ] [-2] v2 := [ ] [ 1] [ ] [ 0] > v3:=<0,0,0,1>; [0] [ ] [0] v3 := [ ] [0] [ ] [1] > GramSchmidt([v1,v2,v3]); [1] [ 1] [-1/4] [ ] [ ] [ ] [1] [-2] [-1/4] [[ ], [ ], [ ]] [1] [ 1] [-1/4] [ ] [ ] [ ] [1] [ 0] [3/4 ] > GramSchmidt([v1,v2,v3],normalized); [1/2] [ 1/6 sqrt(6) ] [- 1/6 sqrt(3)] [ ] [ ] [ ] [1/2] [- 1/3 sqrt(6)] [- 1/6 sqrt(3)] [[ ], [ ], [ ]] [1/2] [ 1/6 sqrt(6) ] [- 1/6 sqrt(3)] [ ] [ ] [ ] [1/2] [ 0 ] [ 1/2 sqrt(3) ] > printlevel:=3: > A := ; [1 1 0] [ ] [1 -2 0] A := [ ] [1 1 0] [ ] [1 0 1] > QRD:=QRDecomposition(A); [1/2 1/6 sqrt(6) - 1/6 sqrt(3)] [ ] [1/2 - 1/3 sqrt(6) - 1/6 sqrt(3)] QRD := [ ], [1/2 1/6 sqrt(6) - 1/6 sqrt(3)] [ ] [1/2 0 1/2 sqrt(3) ] [2 0 1/2 ] [ ] [0 sqrt(6) 0 ] [ ] [0 0 1/2 sqrt(3)] > QRD[1]; [1/2 1/6 sqrt(6) - 1/6 sqrt(3)] [ ] [1/2 - 1/3 sqrt(6) - 1/6 sqrt(3)] [ ] [1/2 1/6 sqrt(6) - 1/6 sqrt(3)] [ ] [1/2 0 1/2 sqrt(3) ] > QRD[2]; [2 0 1/2 ] [ ] [0 sqrt(6) 0 ] [ ] [0 0 1/2 sqrt(3)] > QRD[1] . QRD[2]; [1 1 0] [ ] [1 -2 0] [ ] [1 1 0] [ ] [1 0 1] > Q := Gram_Schmidt(A, 'R'); [1] [ ] [1] v[1]=, [ ] [1] [ ] [1] [1] [ ] [1] u[1]=, [ ] [1] [ ] [1] ||u[1]||^2 = , 4 [ 1] [ ] [-2] v[2]=, [ ] [ 1] [ ] [ 0] [ 1] [ ] [-2] u[2]=, [ ] [ 1] [ ] [ 0] =, 0 ||u[2]||^2 = , 6 [0] [ ] [0] v[3]=, [ ] [0] [ ] [1] [-1/4] [ ] [-1/4] u[3]=, [ ] [-1/4] [ ] [3/4 ] =, 0 =, 0 ||u[3]||^2 = , 3/4 [1 1 -1/4] [ ] [1 -2 -1/4] Q := [ ] [1 1 -1/4] [ ] [1 0 3/4 ] > R; [1 0 1/4] [ ] [0 1 0 ] [ ] [0 0 1 ] > Q . R; [1 1 0] [ ] [1 -2 0] [ ] [1 1 0] [ ] [1 0 1] > rnd:=rand(-10..10); rnd := proc() local t; global _seed; _seed := irem(a*_seed, p); t := _seed; to concats do _seed := irem(a*_seed, p); t := s*t + _seed end do; irem(t, divisor) + offset end proc > rnd(); -4 > B := Matrix(6,6,(i,j)->rnd()); [ 7 8 10 -6 -8 -5] [ ] [ 7 6 -6 0 5 -10] [ ] [ 1 1 -3 -10 5 -4] B := [ ] [-8 -1 -10 2 3 -1] [ ] [-9 8 4 -5 10 -2] [ ] [ 5 7 7 -7 4 -8] > QB:=Gram_Schmidt(B, 'RB'); [ 7] [ ] [ 7] [ ] [ 1] v[1]=, [ ] [-8] [ ] [-9] [ ] [ 5] [ 7] [ ] [ 7] [ ] [ 1] u[1]=, [ ] [-8] [ ] [-9] [ ] [ 5] ||u[1]||^2 = , 269 [ 8] [ ] [ 6] [ ] [ 1] v[2]=, [ ] [-1] [ ] [ 8] [ ] [ 7] [1662] [----] [269 ] [ ] [1124] [----] [269 ] [ ] [199 ] [--- ] [269 ] u[2]=, [ ] [291 ] [--- ] [269 ] [ ] [2782] [----] [269 ] [ ] [1533] [----] [269 ] =, 0 52935 ||u[2]||^2 = , ----- 269 [ 10] [ ] [ -6] [ ] [ -3] v[3]=, [ ] [-10] [ ] [ 4] [ ] [ 7] [ 70562 ] [ ----- ] [ 17645 ] [ ] [-578818] [-------] [ 52935 ] [ ] [-200153] [-------] [ 52935 ] u[3]=, [ ] [-132054] [-------] [ 17645 ] [ ] [103996 ] [------ ] [52935 ] [ ] [ 35783 ] [ ----- ] [ 17645 ] =, 0 =, 0 11319274 ||u[3]||^2 = , -------- 52935 [ -6] [ ] [ 0] [ ] [-10] v[4]=, [ ] [ 2] [ ] [ -5] [ ] [ -7] [ 1042915 ] [ ------- ] [ 5659637 ] [ ] [ 17350055 ] [ -------- ] [ 5659637 ] [ ] [-110091721] [----------] [ 11319274 ] u[4]=, [ ] [ 737600 ] [ ------- ] [ 5659637 ] [ ] [ 1857553 ] [ ------- ] [ 5659637 ] [ ] [-20434461 ] [--------- ] [11319274 ] =, 0 =, 0 =, 0 1215818327 ||u[4]||^2 = , ---------- 11319274 [-8] [ ] [ 5] [ ] [ 5] v[5]=, [ ] [ 3] [ ] [10] [ ] [ 4] [-7880472154] [-----------] [1215818327 ] [ ] [1954601593 ] [---------- ] [1215818327 ] [ ] [-276695367 ] [---------- ] [1215818327 ] u[5]=, [ ] [-5239150072] [-----------] [1215818327 ] [ ] [2260742173 ] [---------- ] [1215818327 ] [ ] [4038253655 ] [---------- ] [1215818327 ] =, 0 =, 0 =, 0 =, 0 94476294496 ||u[5]||^2 = , ----------- 1215818327 [ -5] [ ] [-10] [ ] [ -4] v[6]=, [ ] [ -1] [ ] [ -2] [ ] [ -8] [4923063767 ] [----------- ] [23619073624 ] [ ] [24256074965 ] [----------- ] [47238147248 ] [ ] [23265842357 ] [----------- ] [47238147248 ] u[6]=, [ ] [-6902106235 ] [----------- ] [5904768406 ] [ ] [33126908745 ] [----------- ] [47238147248 ] [ ] [-81114776037] [------------] [47238147248 ] =, 0 =, 0 =, 0 =, 0 =, 0 126513241969 ||u[6]||^2 = , ------------ 23619073624 QB := [ 1662 70562 1042915 -7880472154 4923063767 ] [7 , ---- , ----- , ------- , ----------- , -----------] [ 269 17645 5659637 1215818327 23619073624] [ 1124 -578818 17350055 1954601593 24256074965] [7 , ---- , ------- , -------- , ---------- , -----------] [ 269 52935 5659637 1215818327 47238147248] [ 199 -200153 -110091721 -276695367 23265842357] [1 , --- , ------- , ---------- , ---------- , -----------] [ 269 52935 11319274 1215818327 47238147248] [ 291 -132054 737600 -5239150072 -6902106235] [-8 , --- , ------- , ------- , ----------- , -----------] [ 269 17645 5659637 1215818327 5904768406 ] [ 2782 103996 1857553 2260742173 33126908745] [-9 , ---- , ------ , ------- , ---------- , -----------] [ 269 52935 5659637 1215818327 47238147248] [ 1533 35783 -20434461 4038253655 -81114776037] [5 , ---- , ----- , --------- , ---------- , ------------] [ 269 17645 11319274 1215818327 47238147248 ] > RB; [ 70 104 -58 -110 -123] [1 , --- , --- , --- , ---- , ----] [ 269 269 269 269 269 ] [ ] [ 28228 -12007 28144 -7693] [0 , 1 , ----- , ------ , ----- , -----] [ 52935 17645 52935 10587] [ ] [ -1332333 -5307473 2429870] [0 , 0 , 1 , -------- , -------- , -------] [ 11319274 11319274 5659637] [ ] [ -433805879 237506910 ] [0 , 0 , 0 , 1 , ---------- , ----------] [ 1215818327 1215818327] [ ] [ -5312618603] [0 , 0 , 0 , 0 , 1 , -----------] [ 47238147248] [ ] [0 , 0 , 0 , 0 , 0 , 1] > QB . RB; [ 7 8 10 -6 -8 -5] [ ] [ 7 6 -6 0 5 -10] [ ] [ 1 1 -3 -10 5 -4] [ ] [-8 -1 -10 2 3 -1] [ ] [-9 8 4 -5 10 -2] [ ] [ 5 7 7 -7 4 -8] >