GS534: Solving the 1D Poisson equation using finite differences ...

Lv = f
(19a)
followed by finding the solution u from
Uu = f
(19b)
Once the decomposition is known it is easy to find v by solving (19a) with forward substitution:
v 1 = f 1
L 11
v i = 1 ⎡
⎢⎣ f i − i−1 ⎤
L ii
Σ L ij v j ⎥⎦
j=1
and solving (19b) for u by backsubstitution:
u N =
y N
U NN
u i = 1
U ii
⎡
⎢⎣ y i −
N
j=i+1
i = 2, 3, ..., N
Σ U ij u j
⎤
⎥⎦ i = N − 1, N − 2, . . . . , 1
(20a)
(20b)
(19a)
(19b)
Example:
Consider the matrix-vector equation
⎡ 6 3 ⎤ ⎡ u 1
⎤
⎢ ⎥ ⎢ ⎥⎦ = ⎡ 3 ⎤
⎢ ⎥
⎣
2 13
⎦ ⎣
u 2 ⎣
−11
⎦
We can show that the LU decomposition can be written as
⎡ 6 3 ⎤
⎢ ⎥ = ⎡ 3 0 ⎤ ⎡ 2 1 ⎤
⎢ ⎥ ⎢ ⎥ =
⎣
2 13
⎦ ⎣
1 3
⎦ ⎣
0 4
⎦
so that, using the notation in (18) and (19)
and
v = [1, −4] T
u = [1, −1] T
(verify). The decomposition into LU follows from Crout’s algorithm which can replace the
matrix A in memory with the upper and lower triangular matrices U and L (e.g, Press et al.,
1992).
4.4. Practical implementation of routines for Gaussian elimination
The algorithms for Gaussian elimination are implemented in a variety of standard libraries and
packages and it’s generally not necessary to write user programs for these algorithms. A particular
class of subroutines for the solution of problems arising from linear algebra is LINPACK
(www.netlib.org/LINPACK) which was developed in the 70s and 80s and is now largely superseded
by LAPACK (www.netlib.org/LAPACK), but the name linpack still exists as a ’generic’
term. Most algorithms make extensive use of vector algebra, including the inner (dot) product and
outer product. These are generally available in to programmers by calls to the BLAS (Basic
(20)
(21)
(22)
8