Biomedical Engineering Reference
In-Depth Information
Direct methods compute the solution to a problem in a finite number of steps.
Examples are the Gaussian elimination, and the TDMA (TriDiagonal Matrix Algo-
rithm) which applies matrix algebra such as row operations to transform the matrix
into a form that is easier to solve. Typically direct methods are the preferred solution
method for low to medium sized systems (e.g. < 1000 equations) as it is computa-
tionally inexpensive and requires a minimum amount of storage in the core memory.
In contrast, iterative methods are based on repeated applications of an algorithm
leading to its eventual convergence after a number of repetitions. A convergence
criterion is specified to determine when a suitable solution has been reached. Itera-
tive methods are typically used for very large systems. For non-linear problems,
they are used out of necessity but are just as valuable for sparse linear systems.
Well-known point-by-point methods such as Jacobi and Gauss-Siedel are presented
in this chapter to provide the reader with some basic understanding of iterative
methods. Other variants from these two iterative methods will also be described
particularly those algorithms that are used in solving CHD problems.
5.5.1
Direct Solution Methods
One of the most basic methods for solving linear systems of algebraic equations
is the
Gaussian elimination
. The algorithm derives from the basis of systematic
reduction of large systems of equations to smaller ones. Let us suppose that the
systems of equations are written in the form:
(5.75)
AB
φ=
where
ϕ
is the unknown nodal variables. Matrix
A
contains non-zero coefficients of
the algebraic equations as:
AAA
A
11
12
13
1
n
AAA
A
21
22
23
2
n
A
=
AAA
A
31
32
33
3
n
n
AAA
A
n
1
n
2
n
3
n
while
B
comprises of known values of
ϕ
, that are given by the boundary conditions
or source/sink terms. It can be observed that the diagonal coefficients of the matrix
A
are represented by the entries of
A
11
,
A
22
,….,
A
nn
. The first step is to apply
For-
ward Elimination
which eliminates the entries below the diagonal to yield a lower
triangle of zeroes. This means eliminating the elements of
A
21
,
A
31
,
A
32
,….,
A
nn-
1 by
replacing them with zeroes. We begin the elimination process by considering the
first column elements of
A
21
,
A
31
, ….,
A
n
1 in the matrix
A
.
Multiplying the first row of the matrix by
A
21
/
A
11
and subtracting these values
from the second row, all the elements in the second row are subsequently modified,
which includes the terms in
B
on the right hand side of the equations. The other
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