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numerical linear algebra

Edit detail for numerical linear algebra revision 4 of 5

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Editor: test1 Time: 2014/10/07 13:56:49 GMT+0
Note:

changed:
-solve(rhs(eigen.1),15)
solve(rhs(eigen.1),10.0^(-15))

changed:
-Why would you expect to be able to recover the characteristic polynomial?  There is always round off error for finite precision arithmetic. For integer approximations, it would be worse.
-The command {\tt solve: (Polynomial Fraction Integer, PositiveInteger)->List Equation Polynomial Integer}
-solves the equation over the integers, so it is {\it not} accurate. For example:
To recover characteristic polynomial one needs good precision, which means very
small second argument to solve.  Giving large second arguments means very poor
accuracy:

removed:
-ev:= solve(rhs(eigen.1),1.0*10^(-50))
-cp:= reduce(*, [rhs(x)-lhs(x) for x in ev])

added:

To have use of precise solution we also need to increase precision of other
floating point computations using digits.  Unfortunately, this leads to
ugly display, so we only show final result:
\begin{axiom}
digits(120)
ev:= solve(rhs(eigen.1),1.0*10^(-100));
cp:= reduce(*, [rhs(x)-lhs(x) for x in ev]);
rhs(eigen.1) - cp
\end{axiom}

I'm new to Axiom, so maybe I'm doing things in a stupid way.

I want to get (estimates of) the eigenvalues of a 10x10 matrix of floats:

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m := matrix([[random()$Integer for i in 1..10] for j in 1..10]); sm := m + transpose(m); smf:Matrix Float := sm

$\label{eq1}\left[ \begin{array}{cccccccccc} {19101368.0}&{61278041.0}&{70995855.0}&{34982229.0}&{73454712.0}&{3 3749409.0}&{43151276.0}&{56873386.0}&{36952224.0}&{71646262.0} \ {61278041.0}&{2968650.0}&{85736917.0}&{24332967.0}&{4553241.0}&{6 3473373.0}&{20558912.0}&{92297086.0}&{54375004.0}&{83468790.0} \ {70995855.0}&{85736917.0}&{15829350.0}&{1 \ 19704936.0}&{365 22809.0}&{39703630.0}&{6801050.0}&{36762682.0}&{42794759.0}&{3 5393317.0} \ {34982229.0}&{24332967.0}&{1 \ 19704936.0}&{26538740.0}&{489 82062.0}&{89215221.0}&{64457881.0}&{15519476.0}&{57773751.0}&{2 9448891.0} \ {73454712.0}&{4553241.0}&{36522809.0}&{48982062.0}&{46207882.0}&{4 4673466.0}&{1 \ 00571933.0}&{60490393.0}&{96108269.0}&{59277 125.0} \ {33749409.0}&{63473373.0}&{39703630.0}&{89215221.0}&{44673466.0}&{1 826916.0}&{1 \ 19858422.0}&{67603577.0}&{82798875.0}&{762919 26.0} \ {43151276.0}&{20558912.0}&{6801050.0}&{64457881.0}&{1 \ 0057 1933.0}&{1 \ 19858422.0}&{1 \ 28173768.0}&{91100645.0}&{744 38999.0}&{68433103.0} \ {56873386.0}&{92297086.0}&{36762682.0}&{15519476.0}&{60490393.0}&{6 7603577.0}&{91100645.0}&{53378192.0}&{36023315.0}&{94404719.0} \ {36952224.0}&{54375004.0}&{42794759.0}&{57773751.0}&{96108269.0}&{8 2798875.0}&{74438999.0}&{36023315.0}&{1 \ 28828432.0}&{30785 633.0} \ {71646262.0}&{83468790.0}&{35393317.0}&{29448891.0}&{59277125.0}&{7 6291926.0}&{68433103.0}&{94404719.0}&{30785633.0}&{91534560.0}$

(1)

Type: Matrix(Float)

The problem is: If I now call eigenvalues(smf) on the symmetric float matrix smf Axiom 3.0 Beta (February 2005) runs for a very long time (uncomment code if you want to try it):

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)set messages time on

Try this::

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eigen:=eigenvalues(sm)

$\label{eq2}\left[ \left(\%B \mid{{{\%B}^{10}}-{{514387858}\ {{\%B}^{9}}}-{{81063769717750363}\ {{\%B}^{8}}}+{{186174851844360357608590 98}\ {{\%B}^{7}}}+{{1995237067328519443299986185143419}\ {{\%B}^{6}}}-{{196489116329546029723739477630207952322692}\ {{\%B}^{5}}}-{{17713684304188911913775711272171401718653930456014}\ {{\%B}^{4}}}+{{56122609706355950813933443516197394131149130564764128 1838}\ {{\%B}^{3}}}+{{399896551509446465317296211063275390420 68724401313082575314065916}\ {{\%B}^{2}}}-{{45738107297399937 0254434039690959701937607418001547553700564621454196664}\ \%B}-{123950107976044776612898516638225104195271993434900035759953 61443410961398217808}}\right) \right]$

(2)

Type: List(Union(Fraction(Polynomial(Integer)),SuchThat?(Symbol,Polynomial(Integer))))

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Time: 0.01 (IN) + 0.01 (EV) + 0.07 (OT) = 0.09 sec
solve(rhs(eigen.1),10.0^(-15))

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[{\%B = -{14233142.7839271088 \_ 23}}, \: \right. \ \ \displaystyle \left.{\%B = -{54892641.0998019923 \_ 75}}, \: \right. \ \ \displaystyle \left.{\%B = -{93438314.6687504400 \_ 27}}, \: \right. \ \ \displaystyle \left.{\%B = -{104131052.5421902803 \_ 1}}, \: \right. \ \ \displaystyle \left.{\%B = -{139552430.7909528369 \_ 9}}, \: \right. \ \ \displaystyle \left.{\%B ={590433871.2921979769 \_ 2}}, \: \right. \ \ \displaystyle \left.{\%B ={136341415.0492967303 \_ 4}}, \: \right. \ \ \displaystyle \left.{\%B ={118357957.8969694609 \_ 4}}, \: \right. \ \ \displaystyle \left.{\%B ={51852746.7256929047 \_ 26}}, \: \right. \ \ \displaystyle \left.{\%B ={23649448.9214655855 \_ 95}}\right]$

(3)

Type: List(Equation(Polynomial(Float)))

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Time: 0.03 (EV) + 0.01 (OT) = 0.04 sec

Thank you! This helps, but doesn't answer everything. Since interestingly:

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charpol := reduce(*, [ rhs(x) - lhs(x) for x in % ])

$\label{eq4}\begin{array}{@{}l} \displaystyle {{\%B}^{10}}-{{514387858.0}\ {{\%B}^{9}}}-{{8106376 \ 971775 0363.001}\ {{\%B}^{8}}}+ \ \ \displaystyle {{0.1861748518 \ 4436035761 E 26}\ {{\%B}^{7}}}+ \ \ \displaystyle {{0.1995237067 \ 3285194434 E 34}\ {{\%B}^{6}}}- \ \ \displaystyle {{0.1964891163 \ 2954602973 E 42}\ {{\%B}^{5}}}- \ \ \displaystyle {{0.1771368430 \ 4188911913 E 50}\ {{\%B}^{4}}}+ \ \ \displaystyle {{0.5612260970 \ 6355950813 E 57}\ {{\%B}^{3}}}+ \ \ \displaystyle {{0.3998965515 \ 0944646532 E 65}\ {{\%B}^{2}}}- \ \ \displaystyle {{0.4573810729 \ 7399937026 E 72}\ \%B}- \ \ \displaystyle {0.1239501079 \ 7604477661 E 80}$

(4)

Type: Polynomial(Float)

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Time: 0 sec

we cannot recover the characteristic polynomial from this solution: Even if a large number is passed to solve, accuracy does not increase.

To recover characteristic polynomial one needs good precision, which means very small second argument to solve. Giving large second arguments means very poor accuracy:

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solve(x+11/10,3)

$\label{eq5}\left[{x = - 1}\right]$

(5)

Type: List(Equation(Polynomial(Fraction(Integer))))

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Time: 0.01 (IN) = 0.01 sec

To have use of precise solution we also need to increase precision of other floating point computations using digits. Unfortunately, this leads to ugly display, so we only show final result:

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digits(120)

$\label{eq6}20$

(6)

Type: PositiveInteger?

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Time: 0 sec
ev:= solve(rhs(eigen.1),1.0*10^(-100));

Type: List(Equation(Polynomial(Float)))

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Time: 0.09 (EV) = 0.09 sec
cp:= reduce(*, [rhs(x)-lhs(x) for x in ev]);

Type: Polynomial(Float)

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Time: 0 sec
rhs(eigen.1) - cp

$\label{eq7}\begin{array}{@{}l} \displaystyle -{{0.5714936956 \ 4 E - 100}\ {{\%B}^{9}}}+ \ \ \displaystyle {{0.2473701638 \ 8 E - 91}\ {{\%B}^{8}}}+{{0.1465027643 \ 5 E - 83}\ {{\%B}^{7}}}- \ \ \displaystyle {{0.7490726307 \ 8 E - 75}\ {{\%B}^{6}}}-{{0.3974731329 E - 67}\ {{\%B}^{5}}}+ \ \ \displaystyle {{0.4873818394 \ 7 E - 59}\ {{\%B}^{4}}}+{{0.3998713024 \ 3 6 E - 51}\ {{\%B}^{3}}}+ \ \ \displaystyle {{0.145275416 E - 44}\ {{\%B}^{2}}}-{{0.1071484665 \ 3 E - 3 5}\ \%B}- \ \ \displaystyle {0.1609012932 \ 1 E - 28}$

(7)

Type: Polynomial(Float)

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Time: 0 sec

test --unknown, Fri, 24 Jun 2005 04:48:11 -0500 reply

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A:=[[a,b],[c,d]]

$\label{eq8}\left[{\left[ a , \: b \right]}, \:{\left[ c , \: d \right]}\right]$

(8)

Type: List(List(Symbol))

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Time: 0.01 (OT) = 0.01 sec

test --unknown, Fri, 24 Jun 2005 04:55:05 -0500 reply

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A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]]

$\label{eq9}\left[ \begin{array}{cc} {\cos \left({x}\right)}& -{\sin \left({x}\right)} \ {\sin \left({x}\right)}&{\cos \left({x}\right)}$

(9)

Type: Matrix(Expression(Integer))

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Time: 0.04 (OT) = 0.04 sec
A(1,1)

$\label{eq10}\cos \left({x}\right)$

(10)

Type: Expression(Integer)

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Time: 0 sec
eigen:=eigenvalues(A)

   There are 1 exposed and 1 unexposed library operations named 
      eigenvalues having 1 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                           )display op eigenvalues
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      eigenvalues with argument type(s) 
                         Matrix(Expression(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Unfortunately, currently eigenvalues does not work for general expressions, which causes the failure above.

test --unknown, Fri, 24 Jun 2005 08:05:16 -0500 reply

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A:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]]

$\label{eq11}\left[ \begin{array}{cc} {{\cos \left({x}\right)}- L}& -{\sin \left({x}\right)} \ {\sin \left({x}\right)}&{{\cos \left({x}\right)}- L}$

(11)

Type: Matrix(Expression(Integer))

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Time: 0 sec
A(1,1)*A(2,2)

$\label{eq12}{{\cos \left({x}\right)}^{2}}-{2 \ L \ {\cos \left({x}\right)}}+{{L}^{2}}$

(12)

Type: Expression(Integer)

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Time: 0 sec
A(2,1)*A(1,2)

$\label{eq13}-{{\sin \left({x}\right)}^{2}}$

(13)

Type: Expression(Integer)

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Time: 0 sec
A(1,1)*A(2,2)-A(2,1)*A(1,2)

$\label{eq14}{{\sin \left({x}\right)}^{2}}+{{\cos \left({x}\right)}^{2}}-{2 \ L \ {\cos \left({x}\right)}}+{{L}^{2}}$

(14)

Type: Expression(Integer)

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Time: 0 sec
B := solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L)

$\label{eq15}\left[{L ={{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}}, \:{L ={-{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}}\right]$

(15)

Type: List(Equation(Expression(Integer)))

fricas

Time: 0.01 (IN) + 0.01 (EV) + 0.01 (OT) = 0.03 sec
B(1)

$\label{eq16}L ={{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}$

(16)

Type: Equation(Expression(Integer))

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Time: 0 sec

test --unknown, Fri, 24 Jun 2005 08:16:56 -0500 reply

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solve(x^2 - 2,x)

$\label{eq17}\left[{{{{x}^{2}}- 2}= 0}\right]$

(17)

Type: List(Equation(Fraction(Polynomial(Integer))))

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Time: 0.01 (EV) = 0.01 sec
sqrt(2)

$\label{eq18}\sqrt{2}$

(18)

Type: AlgebraicNumber?

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Time: 0 sec
solve(x^2=4,x)

$\label{eq19}\left[{x = 2}, \:{x = - 2}\right]$

(19)

Type: List(Equation(Fraction(Polynomial(Integer))))

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Time: 0.01 (IN) = 0.01 sec

... --unknown, Fri, 07 Oct 2005 16:14:35 -0500 reply

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P:=matrix[[a, b], [1.0 - a, 1.0 - b]]

$\label{eq20}\left[ \begin{array}{cc} a & b \ {-{{1.0}\ a}+{1.0}}&{-{{1.0}\ b}+{1.0}}$

(20)

Type: Matrix(Polynomial(Float))

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Time: 0 sec
eigenvectors(P)

   Internal Error
   The function eigenvectors with signature hashcode is missing from 
      domain EigenPackage(Float)