Re: linear equations with indexed variables?

• To: mathgroup at smc.vnet.net
• Subject: [mg6934] Re: linear equations with indexed variables?
• From: sherod at boussinesq.Colorado.EDU (Scott Herod)
• Date: Tue, 29 Apr 1997 20:48:14 -0400 (EDT)
• Organization: /usr/local/lib/rn/organization
• Sender: owner-wri-mathgroup at wolfram.com

```Just off the bat, you could easily convert your equations to a full
system using something like

====================================================================
In[1]:=
eqns = Table[3 x[i] == 2 x[i+1] + x[i-1], {i,1,3}]

Out[1]=
{3 x[1]==x[0]+2 x[2],3 x[2]==x[1]+2 x[3],3 x[3]==x[2]+2 x[4]}

In[2]:=
moreEqns = Flatten[Union[eqns, {x[0] == 0, x[4] == 1}]]

Out[2]=
{x[0]==0,3 x[1]==x[0]+2 x[2],3 x[2]==x[1]+2 x[3],3 x[3]==x[2]+2 x[4],x[4]==1}

In[3]:=
Solve[%]
======================================================================

But I notice that you are solving a difference equation with a pair
of boundary conditions.  It might be easier to do something like the
following if this is always the case.

====================================================================
In[1]:= x[i_] := r^i

In[3]:= Factor[3 x[i] - 2 x[i+1] - x[i-1]]

-1 + i
Out[3]= -((-1 + r) r       (-1 + 2 r))

In[4]:= Solve[% == 0, r]

Solve::ifun: Inverse functions are being used by Solve, so some solutions may
not be found.

1                    1/(-1 + i)
Out[4]= {{r -> -}, {r -> 1}, {r -> 0          }}
2

(* I need to figure out a slick way to get rid of the r^(i-1) term. *)

In[9]:= y[i_] := (C[1] x[i] /. Out[4][[1]]) + (C[2] x[i] /. Out[4][[2]])

In[10]:= y[i]

C[1]
Out[10]= ---- + C[2]
i
2

In[11]:= Solve[{y[0] == 0, y[4] == 1}]

16           16
Out[11]= {{C[1] -> -(--), C[2] -> --}}
15           15

=======================================================================

Scott Herod
Applied Math
Univ. of Colorado, Boulder

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In article <5jqr2i\$hb8 at smc.vnet.net> hxie at gradin.cis.upenn.edu (Hong-liang Xie) writes:
>I am new to Mathematica and I need to solve a system of
>linear equations whose variables are indexed (i.e., x[1],
>x[2], x[3]), for example,
>
>       3x[i] = 2x[i+1] + x[i-1] ( 1 <= i <= 3)
>       x[4 ]  = 1
>       x[0]   = 0
>
>The above is a system of 5 linear equations with
>5 variables x[0], x[1], x[2], x[3], x[4].  I can certainly
>rewrite it into 5 equations using 5 variables x0, x1, x2,
>x3, x4.  However, if the range of i gets bigger, or, worse,
>if each x has two indexes as in x[i,j], this rewriting could
>get of hand quickly.  I tried different equation solving
>functions in Mathematica but with no luck.  I would therefore
>appreciate help from experts here on how to solve this kind of
>equations directly.  Thanks a lot!
>
>Hong
>CIS Dept
>Univ of Pennsylvania
>
>
>
>
>
>

```

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