Numerical recipes

74 Chapter 2. Solution of Linear Algebraic Equations
The whole procedure requires only 3N 2 multiplies and a like number of adds (an
even smaller number if u or v is a unit vector).
The Sherman-Morrison formula can be directly applied to a class of sparse
problems. If you already have a fast way of calculating the inverse of A (e.g., a
tridiagonal matrix, or some other standard sparse form), then (2.7.4)–(2.7.5) allow
you to build up to your related but more complicated form, adding for example a
row or column at a time. Notice that you can apply the Sherman-Morrison formula
more than once successively, using at each stage the most recent update of A −1
(equation 2.7.5). Of course, if you have to modify every row, then you are back to
an N 3 method. The constant in front of the N 3 is only a few times worse than the
better direct methods, but you have deprived yourself of the stabilizing advantages
of pivoting — so be careful.
For some other sparse problems, the Sherman-Morrison formula cannot be
directly applied for the simple reason that storage of the whole inverse matrix A −1
is not feasible. If you want to add only a single correction of the form u ⊗ v,
and solve the linear system
(A + u ⊗ v) · x = b (2.7.6)
then you proceed as follows. Using the fast method that is presumed available for
the matrix A, solve the two auxiliary problems
A · y = b A· z = u (2.7.7)
for the vectors y and z. In terms of these,
� �
v · y
x = y −
z
1+(v · z)
(2.7.8)
as we see by multiplying (2.7.2) on the right by b.
Cyclic Tridiagonal Systems
So-called cyclic tridiagonal systems occur quite frequently, and are a good
example of how to use the Sherman-Morrison formula in the manner just described.
The equations have the form
⎡
⎤ ⎡ ⎤ ⎡ ⎤
b1 c1 0 ··· β
⎢ a2
⎢
⎣
b2 c2 ···
···
··· aN−1 bN−1 cN−1
α ··· 0 aN bN
⎥
⎦ ·
⎢
⎣
x1
x2
···
xN−1
xN
⎥
⎦ =
⎢
⎣
r1
r2
···
rN−1
rN
⎥
⎦ (2.7.9)
This is a tridiagonal system, except for the matrix elements α and β in the corners.
Forms like this are typically generated by finite-differencing differential equations
with periodic boundary conditions (§19.4).
We use the Sherman-Morrison formula, treating the system as tridiagonal plus
a correction. In the notation of equation (2.7.6), define vectors u and v to be
⎡ ⎤ ⎡ ⎤
γ
⎢ 0.
⎥
⎢ ⎥
u = ⎢ .
⎥
⎣
0
⎦
α
⎢
v = ⎢
⎣
1
0.
.
0
β/γ
⎥
⎦
(2.7.10)
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