[R] Solving a tridiagonal system

[Gabor Grothendieck]

The following will create a matrix x with given sub diagonal,
diagonal and super diagonal.

# define test vectors for a, b and c
va <- -(1:4); vb <- 11:15; vc <- 1:4

# diag.num is a matrix whose ith super diagonal equals i and 
#  sub diagonal equals -i
diag.num <- -outer(seq(vb),seq(vb),"-")

x <- diag(vb)
x[diag.num == 1] <- va
x[diag.num == -1] <- vc

---

Date: Wed, 1 Oct 2003 05:56:34 -0700 (PDT) 
From: Will Harvey <will_harvey03 at yahoo.com>
To: <r-help at stat.math.ethz.ch> 
Subject: [R] Solving a tridiagonal system 
 
I need to find solutions to a tridiagonal system. By
this I mean a set of linear equations Ax = d where A
is a square matrix containing elements A[i,i-1],
A[i,i] and A[i,i+1] for i in 1:nrow, and zero
elsewhere. R is probably not the ideal way to do this,
but this is part of a larger problem that requires R.

In my application it is much easier (and much faster)
to generate the diagonal and off-diagonal elements of
A as vectors, i.e. a = A[i,i-1], b = A[i,i] and c =
A[i,i+1]. So I have three vectors that define A, along
with a solution vector d. The conventional method of
solving such systems is to use the so-called "Thomas
algorithm", see e.g. 
<http://www.enseeiht.fr/hmf/travaux/CD0001/travaux/optmfn/hi/01pa/hyb74/node24.html>;.
This is very easy to code, but much more difficult to
"vectorize". Is anyone aware of a library that
contains a fast implementation of this algorithm?

Another alternative is to use backsolve. I can easily
eliminate the lower diagonal a, but I'm still left
with b and c, whereas backsolve requires a matrix.
Again, I can write a function to read b and c into a
matrix, but this requires loops, and is too slow. Is
there a vectorized way of doing it? Of course, the
diag command works for b, but what about c? In Octave,
diag allows for an offset, but R apparently does not.

I would appreciate any and all assistance you experts
can offer. Thanks in advance.

Will Harvey

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