Re: linear equations with indexed variables?

• To: mathgroup at smc.vnet.net
• Subject: [mg6940] Re: [mg6896] linear equations with indexed variables?
• From: "w.meeussen" <w.meeussen.vdmcc at vandemoortele.be>
• Date: Tue, 29 Apr 1997 20:48:19 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```At 14:00 25-04-97 -0400, Hong-liang Xie wrote:
>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
>
>
For larger (or unbounded) ranges of i,

In[1]:=
<<DiscreteMath`RSolve`
In[2]:=
RSolve[  {3x[i] == 2x[i+1] + x[i-1] ,x[4 ]  == 1,
x[0]   == 0},x[i],i]

Solve::"svars":
"Equations may not give solutions for all \"solve\" variables."

Out[2]=
4 - i
{{x[i] -> 2 + 2      (-1 + C[1]) - C[1] +

14 If[i == 0, 1, 0] - 15 C[1] If[i == 0, 1, 0] -

(-6 + 7 C[1] - C[3]) If[i == 1, 1, 0] -

(-2 + 3 C[1] - C[2]) If[i == 2, 1, 0]}}
*****
the occurence of undetermined constants C[..] shows that the eqn is
underdetermined,
the profusion of "If[i=..,..,..]" shows it is not "simple" or "clean".
*****
In[6]:=Table[{i,x[i]/.%2},{i,0,10}]//Simplify
Out[6]=
{{0, {0}}, {1, {C[3]}}, {2, {C[2]}}, {3, {C[1]}}, {4, {1}},

1                    1
{5, {- (3 - C[1])}}, {6, {- (7 - 3 C[1])}},
2                    4

1                       1
{7, {- (15 - 7 C[1])}}, {8, {-- (31 - 15 C[1])}},
8                       16

1                          1
{9, {-- (63 - 31 C[1])}}, {10, {-- (127 - 63 C[1])}}}
32                         64

why not play 'round with the package, and see what it can do for you.

Dr. Wouter L. J. MEEUSSEN
eu000949 at pophost.eunet.be
w.meeussen.vdmcc at vandemoortele.be

```

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