I am trying to find the first Eigenfunction of the Laplacian (in 1D), i.e. a solution of $$ u''(x)=k u(x)\\ u(0)=u(1)=0 $$ with minimal $k>0$ (in this trivial example, I actually know the analytic solution but this is not the point; neither is using any built-in Eigensolver. I simply want to get acquainted with Mathematica).

To do so, I want to use the Power iteration method:

Repeatedly applying the inverse Laplacian to an arbitrary initial function will produce a sequence converging to the solution.

What I currently have is

f[x_] = 0.5 - Abs[x - 0.5];
steps = 1;
 s = NDSolve[{h''[x] == f[x], h[0] == 0, h[1] == 0}, h, {x, 0, 1}];
 f[x_] = h[x]/h[0.5] /. s;
 , steps]

However, when I take steps=2 or more, I get

   Dot::rect: Nonrectangular tensor encountered.

Note that this is my very first use of Mathematica, and you might have to explain obvious things to me.

Side question: How do I properly output intermediate results in the loop? If I use

f[x_] = h[x]/h[0.5] /. s; ?f

I do get info on f but if I put

f[x_] = h[x]/h[0.5] /. s; ?f ; ?s

I get

Information::nomatch: "No symbol matching ?s found."

(Adding newlines or using commas doesn't help either. For example,

f[x_] = h[x]/h[0.5] /. s; 
?f ,
?s ,

gives no output at all, whereas

f[x_] = h[x]/h[0.5] /. s; 
 ?f ;
  ?s ;


Information::ssym: "\!\(f; Information[\"s\", LongForm -> False]\) is not a symbol or a valid string pattern"


  • $\begingroup$ So at each step the definition of f and s will be interpolating functions. What information do you want to have about them? How do you want to display it? $\endgroup$
    – Jason B.
    Commented May 11, 2016 at 10:43

2 Answers 2


To solve the first issue, write

 s = NDSolve[{h''[x] == f[x], h[0] == 0, h[1] == 0}, h, {x, 0, 1}][[1]];

instead (notice the final brackets).


This allows you to visualize the (small) changes to the function over the course of the iterations,

f[x_] = 0.5 - Abs[x - 0.5];
steps = 4;
list = Join[
   {{#, f[#]} & /@ Subdivide[1, 100]},
    s = NDSolveValue[{h''[x] == f[x], h[0] == 0, h[1] == 0}, 
      h, {x, 0, 1}];
    f[x_] = s[x]/s[0.5];
    {#, f[#]} & /@ Subdivide[1, 100]
    , steps]];
ListLinePlot[list, PlotLegends -> Range[0, 4]]

Mathematica graphics

I've also used NDSolveValue to get around your first problem.


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