FWIW, I set up the exact same problem in a 14-year-old inexpensive commercial program using their "NEWTONS" function and it worked fine starting from [1,1,1,1]
See attached PDF
--- On Tue, 8/17/10, Barton Willis <willisb at unk.edu> wrote:
> From: Barton Willis <willisb at unk.edu>
> Subject: Re: [Maxima] mnewton error
> To: "Sheldon Newhouse" <sen1 at math.msu.edu>
> Cc: maxima at math.utexas.edu, maxima-bounces at math.utexas.edu
> Date: Tuesday, August 17, 2010, 4:55 PM
> > The Jacobian determinant?
> of your function f(a,b,c,d) is 0 at [1,1,1,1].
>
>
> Nevertheless, I suspect that there is an error somewhere:
>
> Unable to compute the LU factorization
> #0: solve_by_lu(
> Maxima encountered a Lisp error:
> Error in MERROR [or a callee]: NIL is not of type
> CHARACTER.
>
> The user should get a clear message, not a Lisp error.
>
> I'd guess this is a bug in the LU code, but the mnewton
> code could use
> improvement.
> Maybe the bug is related to the (ill-advised) way that the
> mnewton code
> uses an
> undeclared array h.
>
> --Barton
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