New Mexico Tech Hyd 510
Hydrology Program Lab Notes. 5 Linear Algebraic
Quantitative Methods in Hydrology Equations and Solvers
-5.10-
Linear Equations: Exercises in building solvers
1. Write a code to perform Gaussian Elimination
Given a square (nxn) non-symmetric coefficient matrix A, and (nx1) load vector b, write a Gaussian Elimination
code to solve the system of linear algebraic equations1 Ax=b for unknown vector x. A suggested pseudo-code
(generic algorithm outline) is:
Input n, an nxn matrix A, and an nx1 vector b
Forward Elimination
for j= 1 to n-1
for i = j+1 to n
m
ij = aij/ajj
a
ij = 0; bi = bi – mij*bj
for k = j +1 to n
a
ik = aik – mij * ajk
Backsubstitution
for i= n downto 1
x
i = bi
for j = i+1 to n
x
i = xi – aij*xj
x
i = xi/aii
Output Solution vector x
2. Write a code to perform Jacobi iterations
Given a square (nxn) non-symmetric coefficient matrix A, and (nx1) load vector b, write an code to solve the
system of linear algebraic equations Ax=b for unknown (nx1) vector x using Jacobi iteration.
,...1,0 &,...2,1for ,
1
1
1==
−= ∑
≠
=
+knixab
a
x
n
ij
j
jiji
ii
k
i
To start this process you need an initial guess, 0
i
x, for each value of x. If the problem is sufficiently well behaved,
and for decent initial guesses, the method converges. We need a stopping criteria, to stop the process when the
solution is “sufficiently close”. Let’s use a comparison of old and new values of x. If, for any i, the absolute value
1 Note the shift in the notation for the vector of unknowns, from z to x. It is traditional to use x for this purpose, as
in the text, Chapter 20, so I’ve made the shift in this assignment.
