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MathGroup Archive 2004

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Re: Slow LinearSolve.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49684] Re: Slow LinearSolve.
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 27 Jul 2004 07:01:37 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <ce2gcm$8tu$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <ce2gcm$8tu$1 at smc.vnet.net>,
 Aaron Fude <aaronfude at yahoo.com> wrote:

>  Within 24 hours and counting, Mathematica was not able to solve Ax = b for
>  the following A and the following B. Another system does this in a matter of 
>  seconds.

The other system does it numerically, I assume? Mathematica is trying to 
compute the answer exactly. If you enter

   LinearSolve[N[A], b]

and you will get an answer immediately. 

I note that your matrix A is a Vandermonde matrix

   Table[Exp[I n m Pi/12], {m, -6, 6}, {n, 0, 12}] == A

or

   x[m_] = Exp[I m Pi/12]

   Table[x[m]^n, {m, -6, 6}, {n, 0, 12}] == A

(see http://mathworld.wolfram.com/VandermondeMatrix.html) and the system you
want to solve is related to computing the (inverse) discrete Fourier 
transform of b.

Cheers,
Paul

>  How do make Mathematica do the same? I use "LinearSolve[A, b]"
>  
>  A = {{1, -I, -1, I, 1, -I, -1, I, 1, -I, -1, I, 1}, {1, E^(((-5*I)/12)*Pi),
>  E^(((-5*I)/6)*Pi),
>  E^(((3*I)/4)*Pi), E^((I/3)*Pi), E^((-I/12)*Pi), -I, E^(((-11*I)/12)*Pi),
>  E^(((2*I)/3)*Pi),
>  E^((I/4)*Pi), E^((-I/6)*Pi), E^(((-7*I)/12)*Pi), -1},
>  {1, E^((-I/3)*Pi), E^(((-2*I)/3)*Pi), -1, E^(((2*I)/3)*Pi), E^((I/3)*Pi), 1,
>  E^((-I/3)*Pi),
>  E^(((-2*I)/3)*Pi), -1, E^(((2*I)/3)*Pi), E^((I/3)*Pi), 1},
>  {1, E^((-I/4)*Pi), -I, E^(((-3*I)/4)*Pi), -1, E^(((3*I)/4)*Pi), I,
>  E^((I/4)*Pi), 1,
>  E^((-I/4)*Pi), -I, E^(((-3*I)/4)*Pi), -1}, {1, E^((-I/6)*Pi),
>  E^((-I/3)*Pi), -I,
>  E^(((-2*I)/3)*Pi), E^(((-5*I)/6)*Pi), -1, E^(((5*I)/6)*Pi),
>  E^(((2*I)/3)*Pi), I,
>  E^((I/3)*Pi), E^((I/6)*Pi), 1}, {1, E^((-I/12)*Pi), E^((-I/6)*Pi),
>  E^((-I/4)*Pi),
>  E^((-I/3)*Pi), E^(((-5*I)/12)*Pi), -I, E^(((-7*I)/12)*Pi),
>  E^(((-2*I)/3)*Pi),
>  E^(((-3*I)/4)*Pi), E^(((-5*I)/6)*Pi), E^(((-11*I)/12)*Pi), -1},
>  {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, E^((I/12)*Pi), E^((I/6)*Pi),
>  E^((I/4)*Pi),
>  E^((I/3)*Pi), E^(((5*I)/12)*Pi), I, E^(((7*I)/12)*Pi), E^(((2*I)/3)*Pi),
>  E^(((3*I)/4)*Pi),
>  E^(((5*I)/6)*Pi), E^(((11*I)/12)*Pi), -1}, {1, E^((I/6)*Pi), E^((I/3)*Pi),
>  I,
>  E^(((2*I)/3)*Pi), E^(((5*I)/6)*Pi), -1, E^(((-5*I)/6)*Pi),
>  E^(((-2*I)/3)*Pi), -I,
>  E^((-I/3)*Pi), E^((-I/6)*Pi), 1}, {1, E^((I/4)*Pi), I, E^(((3*I)/4)*Pi), -1,
>  E^(((-3*I)/4)*Pi), -I, E^((-I/4)*Pi), 1, E^((I/4)*Pi), I,
>  E^(((3*I)/4)*Pi), -1},
>  {1, E^((I/3)*Pi), E^(((2*I)/3)*Pi), -1, E^(((-2*I)/3)*Pi), E^((-I/3)*Pi), 1,
>  E^((I/3)*Pi),
>  E^(((2*I)/3)*Pi), -1, E^(((-2*I)/3)*Pi), E^((-I/3)*Pi), 1},
>  {1, E^(((5*I)/12)*Pi), E^(((5*I)/6)*Pi), E^(((-3*I)/4)*Pi), E^((-I/3)*Pi),
>  E^((I/12)*Pi), I,
>  E^(((11*I)/12)*Pi), E^(((-2*I)/3)*Pi), E^((-I/4)*Pi), E^((I/6)*Pi),
>  E^(((7*I)/12)*Pi), -1},
>  {1, I, -1, -I, 1, I, -1, -I, 1, I, -1, -I, 1}}
>  
>  b = {0, (-2*I)/5, 0, (-2*I)/3, 0, -2*I, Pi, 2*I, 0, (2*I)/3, 0, (2*I)/5, 0}
>

-- 
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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