Editing Conjugate gradient method (section)

=== Application to Sparse Systems===

The finite termination property also has practical implications in solving large sparse systems, which frequently arise in scientific and engineering applications. For instance, discretizing the two-dimensional Laplace equation <math>\nabla^2 u = 0</math> using finite differences on a uniform grid leads to a sparse linear system <math>A \mathbf{x} = \mathbf{b}</math>, where <math>A</math> is symmetric and positive definite.

Using a <math>5 \times 5</math> interior grid yields a <math>25 \times 25</math> system, and the coefficient matrix <math>A</math> has a five-point stencil pattern. Each row of <math>A</math> contains at most five nonzero entries corresponding to the central point and its immediate neighbors. For example, the matrix generated from such a grid may look like:

<math>
A = \begin{bmatrix}
4 & -1 & 0 & \cdots & -1 & 0 & \cdots \\
-1 & 4 & -1 & \cdots & 0 & 0 & \cdots \\
0 & -1 & 4 & -1 & 0 & 0 & \cdots \\
\vdots & \vdots & \ddots & \ddots & \ddots & \vdots \\
-1 & 0 & \cdots & -1 & 4 & -1 & \cdots \\
0 & 0 & \cdots & 0 & -1 & 4 & \cdots \\
\vdots & \vdots & \cdots & \cdots & \cdots & \ddots
\end{bmatrix}
</math>

Although the system dimension is 25, the conjugate gradient method is theoretically guaranteed to terminate in at most 25 iterations under exact arithmetic. In practice, convergence often occurs in far fewer steps due to the matrix's spectral properties. This efficiency makes CGM particularly attractive for solving large-scale systems arising from partial differential equations, such as those found in heat conduction, fluid dynamics, and electrostatics.