Page 28 - Matrix Analysis & Applied Linear Algebra

20 Chapter 1 Linear Equations

Notice the pattern of the entries in the coefficient matrix in the above ex-
ample. The nonzero elements occur only on the subdiagonal, main-diagonal, and
superdiagonal lines—such a system (or matrix) is said to be tridiagonal. This
is characteristic in the sense that when finite difference approximations are ap-
plied to the general two-point boundary value problem, a tridiagonal system is
the result.
Tridiagonal systems are particularly nice in that they are inexpensive to
solve. When Gaussian elimination is applied, only two multiplications/divisions
are needed at each step of the triangularization process because there is at most
only one nonzero entry below and to the right of each pivot. Furthermore, Gaus-
sian elimination preserves all of the zero entries that were present in the original
tridiagonal system. This makes the back substitution process cheap to execute
because there are at most only two multiplications/divisions required at each
substitution step. Exercise 3.10.6 contains more details.

Exercises for section 1.4

1.4.1. Divide the interval [0, 1] into five equal subintervals, and apply the finite
difference method in order to approximate the solution of the two-point
boundary value problem

y (t) = 125t, y(0) = y(1)=0
at the four interior grid points. Compare your approximate values at
the grid points with the exact solution at the grid points. Note: You
should not expect very accurate approximations with only four interior
grid points.

1.4.2. Divide [0, 1] into n+1 equal subintervals, and apply the finite difference
approximation method to derive the linear system associated with the
two-point boundary value problem

y (t) − y (t)= f(t), y(0) = y(1) = 0.

1.4.3. Divide [0, 1] into five equal subintervals, and approximate the solution
to

y (t) − y (t) = 125t, y(0) = y(1) = 0

at the four interior grid points. Compare the approximations with the
exact values at the grid points.