413 - Householder Symbolic Explicit example -- Sage

A = matrix(QQ,[[2,0,4,7],[-1,3,5,2],[3,4,-1,-3],[-4,2,0,1]]) A

[ 2  0  4  7]
[-1  3  5  2]
[ 3  4 -1 -3]
[-4  2  0  1]

[ 2  0  4  7]
[-1  3  5  2]
[ 3  4 -1 -3]
[-4  2  0  1]

x = A.transpose()[0].column() print(x) e1 = identity_matrix(4)[0].column() v = x + sqrt(x[0][0]^2+x[1][0]^2+x[2][0]^2+x[3][0]^2)*e1 print print(v) normsq = (v.transpose()*v)[0][0] print(normsq) H1 = identity_matrix(4)-(2/normsq)*v*v.transpose() show(H1) A1 = H1*A show(A1.simplify_full())

[ 2]
[-1]
[ 3]
[-4]

[sqrt(30) + 2]
[          -1]
[           3]
[          -4]
(sqrt(30) + 2)^2 + 26

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -\frac{2 \, {\left(\sqrt{30} + 2\right)}^{2}}{{\left(\sqrt{30} + 2\right)}^{2} + 26} + 1 & \frac{2 \, {\left(\sqrt{30} + 2\right)}}{{\left(\sqrt{30} + 2\right)}^{2} + 26} & -\frac{6 \, {\left(\sqrt{30} + 2\right)}}{{\left(\sqrt{30} + 2\right)}^{2} + 26} & \frac{8 \, {\left(\sqrt{30} + 2\right)}}{{\left(\sqrt{30} + 2\right)}^{2} + 26} \\ \frac{2 \, {\left(\sqrt{30} + 2\right)}}{{\left(\sqrt{30} + 2\right)}^{2} + 26} & -\frac{2}{{\left(\sqrt{30} + 2\right)}^{2} + 26} + 1 & \frac{6}{{\left(\sqrt{30} + 2\right)}^{2} + 26} & -\frac{8}{{\left(\sqrt{30} + 2\right)}^{2} + 26} \\ -\frac{6 \, {\left(\sqrt{30} + 2\right)}}{{\left(\sqrt{30} + 2\right)}^{2} + 26} & \frac{6}{{\left(\sqrt{30} + 2\right)}^{2} + 26} & -\frac{18}{{\left(\sqrt{30} + 2\right)}^{2} + 26} + 1 & \frac{24}{{\left(\sqrt{30} + 2\right)}^{2} + 26} \\ \frac{8 \, {\left(\sqrt{30} + 2\right)}}{{\left(\sqrt{30} + 2\right)}^{2} + 26} & -\frac{8}{{\left(\sqrt{30} + 2\right)}^{2} + 26} & \frac{24}{{\left(\sqrt{30} + 2\right)}^{2} + 26} & -\frac{32}{{\left(\sqrt{30} + 2\right)}^{2} + 26} + 1 \end{array}\right)$

[ 2]
[-1]
[ 3]
[-4]

[sqrt(30) + 2]
[          -1]
[           3]
[          -4]
(sqrt(30) + 2)^2 + 26

x = A1.transpose()[1].column() # print(x) e2 = identity_matrix(4)[1].column() x[0] = 0 # get rid of the stuff above the diagonal on the resulting matrix v = (x + sqrt(x[0][0]^2+x[1][0]^2+x[2][0]^2+x[3][0]^2)*e2).simplify_full() # print(v) normsq = (v.transpose()*v)[0][0] print(normsq) H2 = (identity_matrix(4)-(2/normsq)*v*v.transpose()).simplify_full() # print(H2) A2 = (H2*A1).simplify_full() show(A2)

1/4*(8*sqrt(30) + 117)^2/(sqrt(30) + 15)^2 + 4*(sqrt(30) +
16)^2/(sqrt(30) + 15)^2 + 1/900*(2*sqrt(869)*(sqrt(30) + 15) +
91*sqrt(30) + 180)^2/(sqrt(30) + 2)^2

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -\frac{17 \, \sqrt{30} + 60}{2 \, \sqrt{30} + 17} & -\frac{\sqrt{30} + 2}{2 \, {\left(\sqrt{30} + 15\right)}} & 0 & \frac{17 \, \sqrt{30} + 60}{30 \, {\left(2 \, \sqrt{30} + 17\right)}} \\ 0 & -\frac{869 \, {\left(\sqrt{869} {\left(14706325487093615786109 \, \sqrt{30} + 80549861901818071735018\right)} + 414811060035386008668402 \, \sqrt{30} + 2272013751676494559284709\right)}}{\sqrt{869} {\left(2272013751676494559284709 \, \sqrt{30} + 12444331801061580260052060\right)} + 69997829992679904337730642 \, \sqrt{30} + 383393905448530563543861630} & -\frac{22 \, {\left(\sqrt{869} {\left(8418336037819938344170181 \, \sqrt{30} + 46109125435874690854944360\right)} + 237450130239941837016927953 \, \sqrt{30} + 1300567926427990058397130155\right)}}{\sqrt{869} {\left(86704528428532670559808677 \, \sqrt{30} + 474900260479883674033855906\right)} + 2671255333585007090196443256 \, \sqrt{30} + 14631068033731052842167774578} & \frac{119 \, {\left(\sqrt{869} {\left(8418336037819938344170181 \, \sqrt{30} + 46109125435874690854944360\right)} + 237450130239941837016927953 \, \sqrt{30} + 1300567926427990058397130155\right)}}{15 \, {\left(\sqrt{869} {\left(86704528428532670559808677 \, \sqrt{30} + 474900260479883674033855906\right)} + 2671255333585007090196443256 \, \sqrt{30} + 14631068033731052842167774578\right)}} \\ 0 & 0 & -\frac{2 \, {\left(\sqrt{869} {\left(754617436114418005990290661 \, \sqrt{30} + 4133209920515324789529634200\right)} + 21738424698227650464940943153 \, \sqrt{30} + 119066255717966485864526754075\right)}}{\sqrt{869} {\left(237450130239941837016927953 \, \sqrt{30} + 1300567926427990058397130155\right)} + 7315534016865526421083887289 \, \sqrt{30} + 40068830003775106352946648840} & -\frac{\sqrt{869} {\left(1641045513308603376177771109 \, \sqrt{30} + 8988376454330203356769355940\right)} + 48268100981474732293636128527 \, \sqrt{30} + 264375277185539429289318663195}{\sqrt{869} {\left(237450130239941837016927953 \, \sqrt{30} + 1300567926427990058397130155\right)} + 7315534016865526421083887289 \, \sqrt{30} + 40068830003775106352946648840} \\ 0 & 0 & \frac{4 \, {\left(\sqrt{869} {\left(8071784101750996010827993 \, \sqrt{30} + 44210981885743553923674510\right)} + 652101505389614136285206074 \, \sqrt{30} + 3571707055859153537822790030\right)}}{\sqrt{869} {\left(237450130239941837016927953 \, \sqrt{30} + 1300567926427990058397130155\right)} + 7315534016865526421083887289 \, \sqrt{30} + 40068830003775106352946648840} & \frac{5 \, {\left(\sqrt{869} {\left(190857707957538590774795465 \, \sqrt{30} + 1045370718827551806981221193\right)} + 6124788388568370834767753651 \, \sqrt{30} + 33546847615705926649366744644\right)}}{\sqrt{869} {\left(237450130239941837016927953 \, \sqrt{30} + 1300567926427990058397130155\right)} + 7315534016865526421083887289 \, \sqrt{30} + 40068830003775106352946648840} \end{array}\right)$

1/4*(8*sqrt(30) + 117)^2/(sqrt(30) + 15)^2 + 4*(sqrt(30) + 16)^2/(sqrt(30) + 15)^2 + 1/900*(2*sqrt(869)*(sqrt(30) + 15) + 91*sqrt(30) + 180)^2/(sqrt(30) + 2)^2

x = A2.transpose()[2].column() # print(x) e3 = identity_matrix(4)[2].column() x[0] = 0 # get rid of the stuff above the diagonal on the resulting matrix x[1] = 0 print(x[1][0]) v = (x + sqrt(x[0][0]^2+x[1][0]^2+x[2][0]^2+x[3][0]^2)*e3).simplify_full() print(n(v)) normsq = (v.transpose()*v)[0][0] print(normsq) H3 = (identity_matrix(4)-(2/normsq)*v*v.transpose()).simplify_full() # print(H3) A3 = (H3*A2).simplify_full() show(n(A3))

0
[  0.000000000000000]
[  0.000000000000000]
[0.00503058044563949]
[  0.248698744891207]
4*(sqrt(869)*(sqrt(79)*(754617436114418005990290661*sqrt(30) +
4133209920515324789529634200) -
3*sqrt(83)*(237450130239941837016927953*sqrt(30) +
1300567926427990058397130155)) +
11*sqrt(79)*(1976220427111604587721903923*sqrt(30) +
10824205065269680533138795825) -
2607*sqrt(83)*(8418336037819938344170181*sqrt(30) +
46109125435874690854944360))^2/(sqrt(869)*sqrt(79)*(23745013023994183701\
6927953*sqrt(30) + 1300567926427990058397130155) +
869*sqrt(79)*(8418336037819938344170181*sqrt(30) +
46109125435874690854944360))^2 +
16*(sqrt(869)*(8071784101750996010827993*sqrt(30) +
44210981885743553923674510) + 652101505389614136285206074*sqrt(30) +
3571707055859153537822790030)^2/(sqrt(869)*(237450130239941837016927953*\
sqrt(30) + 1300567926427990058397130155) +
7315534016865526421083887289*sqrt(30) + 40068830003775106352946648840)^2

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -5.47722557505166 & -0.182574185835055 & 0.000000000000000 & 0.182574185835055 \\ 0.000000000000000 & -5.38206899497458 & -2.04382366897769 & 0.737015201843469 \\ 0.000000000000000 & 0.000000000000000 & -6.15002315543226 & -6.91157220747708 \\ 0.000000000000000 & 0.000000000000000 & 0.000000000000000 & -3.82800795379024 \end{array}\right)$

0
[  0.000000000000000]
[  0.000000000000000]
[0.00503058044563949]
[  0.248698744891207]
4*(sqrt(869)*(sqrt(79)*(754617436114418005990290661*sqrt(30) + 4133209920515324789529634200) - 3*sqrt(83)*(237450130239941837016927953*sqrt(30) + 1300567926427990058397130155)) + 11*sqrt(79)*(1976220427111604587721903923*sqrt(30) + 10824205065269680533138795825) - 2607*sqrt(83)*(8418336037819938344170181*sqrt(30) + 46109125435874690854944360))^2/(sqrt(869)*sqrt(79)*(237450130239941837016927953*sqrt(30) + 1300567926427990058397130155) + 869*sqrt(79)*(8418336037819938344170181*sqrt(30) + 46109125435874690854944360))^2 + 16*(sqrt(869)*(8071784101750996010827993*sqrt(30) + 44210981885743553923674510) + 652101505389614136285206074*sqrt(30) + 3571707055859153537822790030)^2/(sqrt(869)*(237450130239941837016927953*sqrt(30) + 1300567926427990058397130155) + 7315534016865526421083887289*sqrt(30) + 40068830003775106352946648840)^2

413 - Householder Symbolic Explicit example

1306 days ago by Professor413