subsection 'Difference equation' in Strang's linear algebra section 6.3

The author is using finite differences to explain its usefulness and limitations by leveraging on the concepts described previously. The example is still $y''+y=0$ which was re-written as $y''=-y$ in the first paragraph of the section titled "Difference Equations".

The value of $y''$ is approximated by numerically by the Eq. 11 (FCB). Then there is a renaming of variables $Z$ stands for the discretized first derivative (not the second) of the solution, while $Y$ is the discretized solution (this is no different from the vector $textbf{u}$ in Eq. 10 -and I mention the vector, not the equation as a whole).

The author abandon the 2nd derivative in favor of the phase portrait diagram that contains $u=(y,y')$, see Fig. 6.3. as a qualitative representation of the solution space of the ODE. Then, he uses the forward scheme to determine $Z$ and $Y$ at the time step $n+1$ knowing the solution at the step $n$ in an incremental fashion in Eq. 11 F (which can be obtain from Taylor expansion by getting rid of higher order terms). Eq. 12 contains the same information as Eq. 11 in matrix form where $U_n=(Y_n : Z_n)^T$, the vector on the LHS stands for the homologous vector $U_{n+1}$ and the matrix contains the coefficients multiplying $Y_n$ and $Z_n$ on the RHS of Eq. 11F.

The construction of the diagrams is essentially analogous to an integration where the pair $(Y_n,Z_n)$ denotes the point $n$ in the phase portrait (See figures 6.3 and 6.4) The author also claims that there are numerical errors that produce inaccurate results (ideally all points should lie in a circle) for a fixed steplength equivalent to 32 pts along the perimeter of the circle. The errors are reduced if the number of points are increased.