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numerical linear algebra

Edit detail for numerical linear algebra revision 3 of 5

	1 2 3 4 5
Editor: test1 Time: 2013/05/23 17:40:25 GMT+0
Note:

removed:
-From unknown Fri Jun 24 04:47:02 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 04:47:02 -0500
-Subject: test
-Message-ID: <20050624044702-0500@page.axiom-developer.org>
-
-\begin{axiom}
-A:=[[cos(x),-sin(x)],[sin(x),cos(x)]]
-\end{axiom}
-  
-

removed:
-
-From unknown Fri Jun 24 04:54:14 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 04:54:14 -0500
-Subject: test
-Message-ID: <20050624045414-0500@page.axiom-developer.org>
-
-\begin{axiom}
-A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]]
-\end{axiom}

removed:
-\begin{axiom}
-A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]]
-eigen:=eigenvalues(A)
-\end{axiom}
-
-From unknown Fri Jun 24 07:59:17 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 07:59:17 -0500
-Subject: test
-Message-ID: <20050624075917-0500@page.axiom-developer.org>
-
-\begin{axiom}
-A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]]
-\end{axiom}
-
-From unknown Fri Jun 24 07:59:52 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 07:59:52 -0500
-Subject: test
-Message-ID: <20050624075952-0500@page.axiom-developer.org>

added:
eigen:=eigenvalues(A)

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

removed:
-\begin{axiom}
-A:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]]
-A(1,1)*A(2,2)-A(2,1)*A(1,2)
-\end{axiom}
-
-From unknown Fri Jun 24 08:06:31 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 08:06:31 -0500
-Subject: test
-Message-ID: <20050624080631-0500@page.axiom-developer.org>
-

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

removed:
-
-
-From unknown Fri Jun 24 08:07:40 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 08:07:40 -0500
-Subject: test
-Message-ID: <20050624080740-0500@page.axiom-developer.org>
-
-\begin{axiom}
-A:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]]
-solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L)
-\end{axiom}
-
-
-From unknown Fri Jun 24 08:10:32 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 08:10:32 -0500
-Subject: test
-Message-ID: <20050624081032-0500@page.axiom-developer.org>
-
-\begin{axiom}
-A:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]]
-solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L)
-L
-\end{axiom}
-
-From unknown Fri Jun 24 08:11:21 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 08:11:21 -0500
-Subject: test
-Message-ID: <20050624081121-0500@page.axiom-developer.org>
-
-\begin{axiom}
-A:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]]
-B=solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L)
-\end{axiom}
-
-From unknown Fri Jun 24 08:13:01 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 08:13:01 -0500
-Subject: test
-Message-ID: <20050624081301-0500@page.axiom-developer.org>
-
-\begin{axiom}
-A:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]]
-solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L)
-L(1)
-\end{axiom}
-
-From unknown Fri Jun 24 08:13:40 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 08:13:40 -0500
-Subject: test
-Message-ID: <20050624081340-0500@page.axiom-developer.org>
-
-\begin{axiom}
-A:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]]
-solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L)
-L.1
-\end{axiom}

changed:
-solve(x^2,x)
-\end{axiom}
-
-From unknown Fri Jun 24 08:17:43 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 08:17:43 -0500
-Subject: test
-Message-ID: <20050624081743-0500@page.axiom-developer.org>
-
-\begin{axiom}
solve(x^2 - 2,x)

removed:
-\end{axiom}
-
-From unknown Fri Jun 24 08:18:43 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 08:18:43 -0500
-Subject: 
-Message-ID: <20050624081843-0500@page.axiom-developer.org>
-
-\begin{axiom}

removed:
-
-From unknown Fri Jun 24 08:20:13 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 08:20:13 -0500
-Subject: test
-Message-ID: <20050624082013-0500@page.axiom-developer.org>
-
-\begin{axiom}
-solve(x^2=4,x)
-x
-\end{axiom}
-
-From unknown Fri Jun 24 08:21:50 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 08:21:50 -0500
-Subject: 
-Message-ID: <20050624082150-0500@page.axiom-developer.org>
-
-\begin{axiom}
-e=vector[1,2]
-e=solve(x^2=4,x)
-\end{axiom}
-
-From unknown Fri Jun 24 08:22:22 -0500 2005
-From: unknown
-Date: Fri, 24 Jun 2005 08:22:22 -0500
-Subject: 
-Message-ID: <20050624082222-0500@page.axiom-developer.org>
-
-\begin{axiom}
-e=solve(x^2=4,x)
-\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:

fricas

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

$\label{eq1}\left[ \begin{array}{cccccccccc} {19101368.0}&{61278041.0}&{70995855.0}&{34982229.0}&{73454712.0}&{3 3749409.0}&{43151276.0}&{56873386.0}&{36952224.0}&{71646262.0} \ {61278041.0}&{2968650.0}&{85736917.0}&{24332967.0}&{4553241.0}&{6 3473373.0}&{20558912.0}&{92297086.0}&{54375004.0}&{83468790.0} \ {70995855.0}&{85736917.0}&{15829350.0}&{1_19704936.0}&{365228 09.0}&{39703630.0}&{6801050.0}&{36762682.0}&{42794759.0}&{353 93317.0} \ {34982229.0}&{24332967.0}&{1_19704936.0}&{26538740.0}&{489820 62.0}&{89215221.0}&{64457881.0}&{15519476.0}&{57773751.0}&{29 448891.0} \ {73454712.0}&{4553241.0}&{36522809.0}&{48982062.0}&{46207882.0}&{4 4673466.0}&{1_00571933.0}&{60490393.0}&{96108269.0}&{59277125.0} \ {33749409.0}&{63473373.0}&{39703630.0}&{89215221.0}&{44673466.0}&{1 826916.0}&{1_19858422.0}&{67603577.0}&{82798875.0}&{76291926.0} \ {43151276.0}&{20558912.0}&{6801050.0}&{64457881.0}&{1_0057193 3.0}&{1_19858422.0}&{1_28173768.0}&{91100645.0}&{74438999.0}&{6 8433103.0} \ {56873386.0}&{92297086.0}&{36762682.0}&{15519476.0}&{60490393.0}&{6 7603577.0}&{91100645.0}&{53378192.0}&{36023315.0}&{94404719.0} \ {36952224.0}&{54375004.0}&{42794759.0}&{57773751.0}&{96108269.0}&{8 2798875.0}&{74438999.0}&{36023315.0}&{1_28828432.0}&{30785633.0} \ {71646262.0}&{83468790.0}&{35393317.0}&{29448891.0}&{59277125.0}&{7 6291926.0}&{68433103.0}&{94404719.0}&{30785633.0}&{91534560.0}$

(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):

fricas

)set messages time on

Try this::

fricas

eigen:=eigenvalues(sm)

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

(2)

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

fricas

Time: 0.01 (EV) + 0.09 (OT) = 0.10 sec
solve(rhs(eigen.1),15)

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[{\%A = -{14233140}}, \:{\%A = -{54892644}}, \:{\%A = -{9 3438316}}, \: \right. \ \ \displaystyle \left.{\%A = -{104131052}}, \:{\%A = -{139552428}}, \:{\%A ={5 90433868}}, \: \right. \ \ \displaystyle \left.{\%A ={136341412}}, \:{\%A ={118357956}}, \:{\%A ={5185 2748}}, \: \right. \ \ \displaystyle \left.{\%A ={23649452}}\right]$

(3)

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

fricas

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

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

fricas

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

$\label{eq4}\begin{array}{@{}l} \displaystyle {{\%A}^{10}}-{{514387856}\ {{\%A}^{9}}}-{{81063768099338128}\ {{\%A}^{8}}}+ \ \ \displaystyle {{18617484846001637830908928}\ {{\%A}^{7}}}+ \ \ \displaystyle {{1995236975166117972408075311659520}\ {{\%A}^{6}}}- \ \ \displaystyle {{196489116089160028107106986738280672256000}\ {{\%A}^{5}}}- \ \ \displaystyle {{ \begin{array}{@{}l}\displaystyle 17713683140651728063324774167477194777834644447 \ 232$

(4)

Type: Polynomial(Fraction(Integer))

fricas

Time: 0.01 (OT) = 0.01 sec

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

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 solve: (Polynomial Fraction Integer, PositiveInteger)->List Equation Polynomial Integer solves the equation over the integers, so it is {\it not} accurate. For example:

fricas

solve(x+11/10,3)

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

(5)

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

fricas

Time: 0.01 (IN) = 0.01 sec
ev:= solve(rhs(eigen.1),1.0*10^(-50))

$\label{eq6}\begin{array}{@{}l} \displaystyle \left[{\%A = -{14233142.7839271088_23}}, \: \right. \ \ \displaystyle \left.{\%A = -{54892641.0998019923_75}}, \: \right. \ \ \displaystyle \left.{\%A = -{93438314.6687504400_27}}, \: \right. \ \ \displaystyle \left.{\%A = -{104131052.5421902803_1}}, \: \right. \ \ \displaystyle \left.{\%A = -{139552430.7909528369_9}}, \: \right. \ \ \displaystyle \left.{\%A ={590433871.2921979769_2}}, \: \right. \ \ \displaystyle \left.{\%A ={136341415.0492967303_4}}, \: \right. \ \ \displaystyle \left.{\%A ={118357957.8969694609_4}}, \: \right. \ \ \displaystyle \left.{\%A ={51852746.7256929047_26}}, \: \right. \ \ \displaystyle \left.{\%A ={23649448.9214655855_95}}\right]$

(6)

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

fricas

Time: 0.07 (EV) = 0.07 sec
cp:= reduce(*, [rhs(x)-lhs(x) for x in ev])

$\label{eq7}\begin{array}{@{}l} \displaystyle {{\%A}^{10}}-{{514387858.0}\ {{\%A}^{9}}}-{{8106376_971775036 3.001}\ {{\%A}^{8}}}+ \ \ \displaystyle {{0.1861748518_4436035761 E 26}\ {{\%A}^{7}}}+ \ \ \displaystyle {{0.1995237067_3285194434 E 34}\ {{\%A}^{6}}}- \ \ \displaystyle {{0.1964891163_2954602973 E 42}\ {{\%A}^{5}}}- \ \ \displaystyle {{0.1771368430_4188911913 E 50}\ {{\%A}^{4}}}+ \ \ \displaystyle {{0.5612260970_6355950813 E 57}\ {{\%A}^{3}}}+ \ \ \displaystyle {{0.3998965515_0944646532 E 65}\ {{\%A}^{2}}}- \ \ \displaystyle {{0.4573810729_7399937026 E 72}\ \%A}- \ \ \displaystyle {0.1239501079_7604477661 E 80}$

(7)

Type: Polynomial(Float)

fricas

Time: 0 sec

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

fricas

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

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

(8)

Type: List(List(Symbol))

fricas

Time: 0.01 (OT) = 0.01 sec

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

fricas

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))

fricas

Time: 0.01 (EV) + 0.03 (OT) = 0.04 sec
A(1,1)

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

(10)

Type: Expression(Integer)

fricas

Time: 0 sec
eigen:=eigenvalues(A)

   There are 1 exposed and 0 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

fricas

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))

fricas

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)

fricas

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

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

(13)

Type: Expression(Integer)

fricas

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)

fricas

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))

fricas

Time: 0 sec

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

fricas

solve(x^2 - 2,x)

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

(17)

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

fricas

Time: 0 sec
sqrt(2)

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

(18)

Type: AlgebraicNumber?

fricas

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

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

(19)

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

fricas

Time: 0 sec

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

fricas

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))

fricas

Time: 0.01 (IN) = 0.01 sec
eigenvectors(P)

$\label{eq21}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{eigval ={-{{1.0}\ b}+ a}}, \:{eigmult = 1}, \: \right. \ \ \displaystyle \left.{eigvec ={\left[{\left[ \begin{array}{c} -{1.0} \ {1.0}$

(21)

Type: List(Record(eigval: Union(Fraction(Polynomial(Float)),SuchThat?(Symbol,Polynomial(Float))),eigmult: NonNegativeInteger?,eigvec: List(Matrix(Fraction(Polynomial(Float))))))

fricas

Time: 0 sec