Understanding the Conjugate Gradient Method
The conjugate gradient method is an iterative algorithm used for solving systems of linear equations, particularly when the matrix is symmetric and positive definite. A key aspect of this method is its relationship with the residual vector and how it converges to the solution.
Key Concepts
- Residual Vector (v(k)): The residual vector at iteration k is defined as v(k) = b - Ax^(k). It measures how far the current estimate x^(k) is from the exact solution.
- Condition v(k) = 0: If at some iteration k, v(k) = 0, this implies that the current estimate x^(k) satisfies the equation Ax^(k) = b.
Proof Explanation
1. Definition of Residual:
- The residual vector indicates the difference between the actual right-hand side (b) and the product of the matrix A and the current estimate x^(k).
2. Setting Residual to Zero:
- If v(k) = 0, then by definition:
v(k) = b - Ax^(k) = 0.
3. Rearranging the Equation:
- Rearranging gives us:
Ax^(k) = b.
4. Conclusion:
- Therefore, when the residual vector becomes zero, it confirms that the current solution x^(k) is indeed the exact solution to the system of equations.
Implications
- Convergence: This condition signifies that the conjugate gradient method has converged to the solution in k iterations.
- Efficiency: It highlights the power of the method in minimizing the error iteratively to reach the exact solution efficiently.
In summary, if v(k) = 0 during the iterative process, we conclude that the current approximation x^(k) is the exact solution to the linear system Ax = b.