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# Edit detail for numerical linear algebra revision 1 of 5

 1 2 3 4 5 Editor: 127.0.0.1 Time: 2007/11/15 20:17:51 GMT-8 Note: transferred from axiom-developer

changed:
-
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:

\begin{axiom}
m := matrix([[random()$Integer for i in 1..10] for j in 1..10]); sm := m + transpose(m); smf:Matrix Float := sm \end{axiom} 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): \begin{axiom} )set messages time on -- eigenvalues(smf) \end{axiom} Try this:: \begin{axiom} eigen:=eigenvalues(sm) solve(rhs(eigen.1),15) \end{axiom} Thank you! This helps, but doesn't answer everything. Since interestingly: \begin{axiom} charpol := reduce(*, [ rhs(x) - lhs(x) for x in % ]) \end{axiom} 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 {\tt solve: (Polynomial Fraction Integer, PositiveInteger)->List Equation Polynomial Integer} solves the equation over the integers, so it is {\it not} accurate. For example: \begin{axiom} solve(x+11/10,3) ev:= solve(rhs(eigen.1),1.0*10^(-50)) cp:= reduce(*, [rhs(x)-lhs(x) for x in ev]) \end{axiom} 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} From unknown Fri Jun 24 04:48:11 -0500 2005 From: unknown Date: Fri, 24 Jun 2005 04:48:11 -0500 Subject: test Message-ID: <20050624044811-0500@page.axiom-developer.org> \begin{axiom} A:=[[a,b],[c,d]] \end{axiom} 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} From unknown Fri Jun 24 04:55:05 -0500 2005 From: unknown Date: Fri, 24 Jun 2005 04:55:05 -0500 Subject: test Message-ID: <20050624045505-0500@page.axiom-developer.org> \begin{axiom} A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]] eigen:=eigenvalues(A) \end{axiom} From unknown Fri Jun 24 07:57:44 -0500 2005 From: unknown Date: Fri, 24 Jun 2005 07:57:44 -0500 Subject: test Message-ID: <20050624075744-0500@page.axiom-developer.org> \begin{axiom} A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]] \end{axiom From unknown Fri Jun 24 07:59:17 -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> \begin{axiom} A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]] A(1,1) \end{axiom} From unknown Fri Jun 24 08:05:16 -0500 2005 From: unknown Date: Fri, 24 Jun 2005 08:05:16 -0500 Subject: test Message-ID: <20050624080516-0500@page.axiom-developer.org> \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> \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 08:07:40 -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} From unknown Fri Jun 24 08:16:56 -0500 2005 From: unknown Date: Fri, 24 Jun 2005 08:16:56 -0500 Subject: test Message-ID: <20050624081656-0500@page.axiom-developer.org> \begin{axiom} 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} sqrt(2) \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} solve(x^2=4,x) \end{axiom} 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} From unknown Fri Oct 7 16:14:35 -0500 2005 From: unknown Date: Fri, 07 Oct 2005 16:14:35 -0500 Subject: Message-ID: <20051007161435-0500@www.axiom-developer.org> \begin{axiom} P:=matrix[[a, b], [1.0 - a, 1.0 - b]] eigenvectors(P) \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: axiomm := matrix([[random()$Integer for i in 1..10] for j in 1..10]); sm := m + transpose(m); smf:Matrix Float := sm (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):

axiom)set messages time on

Try this::

axiomeigen:=eigenvalues(sm) (2)
Type: List Union(Fraction Polynomial Integer,SuchThat?(Symbol,Polynomial Integer))
axiomTime: 0.55 (EV) + 0.15 (OT) + 0.30 (GC) = 1.00 sec
solve(rhs(eigen.1),15) (3)
Type: List Equation Polynomial Fraction Integer
axiomTime: 0.02 (IN) + 0.48 (EV) + 0.02 (OT) + 0.35 (GC) = 0.87 sec

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

axiomcharpol := reduce(*, [ rhs(x) - lhs(x) for x in % ]) (4)
Type: Polynomial Fraction Integer
axiomTime: 0 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:

axiomsolve(x+11/10,3) (5)
Type: List Equation Polynomial Fraction Integer
axiomTime: 0.02 (IN) + 0.01 (EV) + 0.01 (OT) = 0.04 sec
ev:= solve(rhs(eigen.1),1.0*10^(-50)) (6)
Type: List Equation Polynomial Float
axiomTime: 0.01 (IN) + 0.49 (EV) + 0.07 (OT) + 0.28 (GC) = 0.85 sec
cp:= reduce(*, [rhs(x)-lhs(x) for x in ev]) (7)
Type: Polynomial Float
axiomTime: 0 sec

axiomA:=[[cos(x),-sin(x)],[sin(x),cos(x)]] (8)
Type: List List Expression Integer
axiomTime: 0.02 (IN) + 0.02 (EV) + 0.08 (OT) = 0.12 sec

axiomA:=[[a,b],[c,d]] (9)
Type: List List Symbol
axiomTime: 0.01 (OT) = 0.01 sec

axiomA:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]] (10)
Type: Matrix Expression Integer
axiomTime: 0.01 (EV) = 0.01 sec

axiomA:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]] (11)
Type: Matrix Expression Integer
axiomTime: 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
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 "$" to specify which version of the function you need. axiomA:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]] (12) Type: Matrix Expression Integer axiomTime: 0 sec \end{axiom Line 2: \end{axiom ....AB Error A: Missing mate. Error B: syntax error at top level Error B: Possibly missing a } 3 error(s) parsing <div class="commentsheading"><a name="msg20050624075917-0500@page.axiom-developer.org"></a><b>test</b> --unknown, <a href="http://axiom-wiki.newsynthesis.org/NumericalLinearAlgebra#msg20050624075917-0500@page.axiom-developer.org">Fri, 24 Jun 2005 07:59:17 -0500</a> <a href="http://axiom-wiki.newsynthesis.org/NumericalLinearAlgebra?subject=test&in_reply_to=%3C20050624075917-0500%40page.axiom-developer.org%3E#bottom">reply</a></div>\begin{axiom} There are no library operations named < having 1 argument(s) though there are 4 exposed operation(s) and 1 unexposed operation(s) having a different number of arguments. Use HyperDoc Browse, or issue )what op < to learn what operations contain " < " in their names, or issue )display op < to learn more about the available operations. Cannot find a definition or applicable library operation named < with argument type(s) Variable b Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.
A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]] (13)
Type: Matrix Expression Integer
axiomTime: 0.01 (IN) = 0.01 sec

axiomA:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]] (14)
Type: Matrix Expression Integer
axiomTime: 0.01 (OT) = 0.01 sec
A(1,1) (15)
Type: Expression Integer
axiomTime: 0 sec

axiomA:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]] (16)
Type: Matrix Expression Integer
axiomTime: 0.01 (IN) = 0.01 sec
A(1,1)*A(2,2)-A(2,1)*A(1,2) (17)
Type: Expression Integer
axiomTime: 0 sec

axiomA:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]] (18)
Type: Matrix Expression Integer
axiomTime: 0 sec
A(1,1)*A(2,2) (19)
Type: Expression Integer
axiomTime: 0 sec
A(2,1)*A(1,2) (20)
Type: Expression Integer
axiomTime: 0 sec

axiomA:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]] (21)
Type: Matrix Expression Integer
axiomTime: 0.01 (OT) = 0.01 sec
solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L) (22)
Type: List Equation Expression Integer
axiomTime: 0.02 (IN) + 0.10 (EV) + 0.03 (OT) + 0.07 (GC) = 0.22 sec

axiomA:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]] (23)
Type: Matrix Expression Integer
axiomTime: 0 sec
solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L) (24)
Type: List Equation Expression Integer
axiomTime: 0.01 (IN) = 0.01 sec
L (25)
Type: Variable L
axiomTime: 0 sec

axiomA:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]] (26)
Type: Matrix Expression Integer
axiomTime: 0.01 (OT) = 0.01 sec
B=solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L)
There are 3 exposed and 0 unexposed library operations named
equation having 2 argument(s) but none was determined to be
applicable. Use HyperDoc Browse, or issue
)display op equation
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
equation with argument type(s)
Variable B
List Equation Expression Integer
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. axiomA:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]] (27) Type: Matrix Expression Integer axiomTime: 0.01 (OT) = 0.01 sec solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L) (28) Type: List Equation Expression Integer axiomTime: 0 sec L(1) There are no library operations named L Use HyperDoc Browse or issue )what op L to learn if there is any operation containing " L " in its name. Cannot find a definition or applicable library operation named L with argument type(s) PositiveInteger Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

axiomA:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]] (29)
Type: Matrix Expression Integer
axiomTime: 0 sec
solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L) (30)
Type: List Equation Expression Integer
axiomTime: 0.01 (OT) = 0.01 sec
L.1
There are no library operations named L
Use HyperDoc Browse or issue
)what op L
to learn if there is any operation containing " L " in its name.
Cannot find a definition or applicable library operation named L
with argument type(s)
PositiveInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. axiomsolve(x^2,x) (31) Type: List Equation Fraction Polynomial Integer axiomTime: 0.01 (IN) = 0.01 sec axiomsqrt(2) (32) Type: AlgebraicNumber? axiomTime: 0 sec axiomsolve(x^2=4,x) (33) Type: List Equation Fraction Polynomial Integer axiomTime: 0.02 (IN) + 0.01 (OT) = 0.03 sec axiomsolve(x^2=4,x) (34) Type: List Equation Fraction Polynomial Integer axiomTime: 0 sec x (35) Type: Variable x axiomTime: 0 sec axiome=vector[1,2] There are 3 exposed and 0 unexposed library operations named equation having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op equation to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named equation with argument type(s) Variable e Vector PositiveInteger Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.
e=solve(x^2=4,x)
There are 3 exposed and 0 unexposed library operations named
equation having 2 argument(s) but none was determined to be
applicable. Use HyperDoc Browse, or issue
)display op equation
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
equation with argument type(s)
Variable e
List Equation Fraction Polynomial Integer
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need. axiome=solve(x^2=4,x) There are 3 exposed and 0 unexposed library operations named equation having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op equation to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named equation with argument type(s) Variable e List Equation Fraction Polynomial Integer Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

axiomP:=matrix[[a, b], [1.0 - a, 1.0 - b]] (36)
Type: Matrix Polynomial Float
axiomTime: 0.01 (IN) = 0.01 sec
eigenvectors(P) (37)
Type: List Record(eigval: Union(Fraction Polynomial Float,SuchThat?(Symbol,Polynomial Float)),eigmult: NonNegativeInteger?,eigvec: List Matrix Fraction Polynomial Float)
axiomTime: 0.01 (EV) + 0.01 (OT) = 0.02 sec