Lesson 16 Linear Systems - Bruce E. Shapiro

108 LESSON 16. LINEAR SYSTEMS
x = Prepend[x, x1];
Return[x];
]
For example, to solve the system
⎛
⎞ ⎛ ⎞ ⎛ ⎞
0.116093 0.230616 0.34202 x 1 3
⎝0.461232 0.897598 1.28558⎠
⎝x 2
⎠ = ⎝17⎠ (16.14)
1.02606 1.92836 2.59808 x 3 5
One could use this function by typing
In:=
Out:=
A={{0.116093, 0.230616, 0.34202},
{0.461232, 0.897598, 1.28558},
{1.02606, 1.92836, 2.59808}};
b={3, 17, 5};
gauss[A, b]
{-33612.9, 27351.9, -7024.58}
In Mathematicawe can also solve the system directly by using the built in function
LinearSolve[A,b].
Gaussian elimination can fail if we divide by zero, and is susceptible to large errors or
possible overflow if we divide by a very small number (relative to the other numbers in
the matrix). Division occurs in two places in the algorithm: during the row reduction
phase where we define m = a k1 /a 11 and during the back-substitution step at the end
of the algorithm, where we solve for x 1 (here we also divide by a 11 , but its usually
a different a 11 ). These numbers are called pivots. The solution is to rearrange the
matrix (and the corresponding elements of b): if at any step along the way the pivot
is zero, then the entire row is exchanged with a row that does not have zero in that
column. If all of the remaining elements in that column are zero then the matrix is
singular and there is no unique solution (or no solution at all).
Math 481A
California State University Northridge
2008, B.E.Shapiro
Last revised: November 16, 2011