Question

Show that the error bound on \(\left\|\mathbf{e}_{k}\right\|_{A}\) given in the CG Convergence Theorem on page 187 implies that the number of iterations \(k\) required to achieve an error reduction by a constant amount is bounded by \(\sqrt{\kappa(A)}\) times a modest constant. [Hint: Recall the discussion of convergence rate from Section 7.3. Also, by Taylor's expansion we can write $$ \ln (1 \pm \varepsilon) \approx \pm \varepsilon $$ for \(0 \leq \varepsilon \ll 1 .]\)

Short answer

Based on the given step-by-step solution, the main goal of the exercise is to show that the number of iterations \(k\) required to achieve an error reduction by a constant amount is bounded by \(\sqrt{\kappa(A)}\) times a modest constant. We found the expression for \(k\) by rewriting the error bound from the CG Convergence Theorem in terms of a logarithm using the provided hint, and then solving for \(k\). This resulted in the inequality: \(k \leq \sqrt{\kappa(A)} \times \text{(modest constant)}\).

Step-by-step solution

Recall the CG Convergence Theorem

The CG Convergence Theorem states that after \(k\) iterations, the error bound is given by: $$ \frac{\left\|\mathbf{e}_{k}\right\|_{A}}{\left\|\mathbf{e}_{0}\right\|_{A}} \leq 2 \left(\frac{\sqrt{\kappa(A)}-1}{\sqrt{\kappa(A)}+1}\right)^{k} $$ where \(\kappa(A) = \frac{\lambda_\text{max}(A)}{\lambda_\text{min}(A)}\) is the condition number of the matrix A, and \(\lambda_\text{max}(A)\) and \(\lambda_\text{min}(A)\) are the maximum and minimum eigenvalues of A, respectively.

Rewrite the error bound using the hint

Using the provided hint: $$ \ln (1 \pm \varepsilon) \approx \pm \varepsilon, $$ we can rewrite the error bound from the CG Convergence Theorem as: \begin{align*} \ln \left(\frac{\left\|\mathbf{e}_{k}\right\|_{A}}{\left\|\mathbf{e}_{0}\right\|_{A}}\right) &\leq \ln \left( 2 \left(\frac{\sqrt{\kappa(A)}-1}{\sqrt{\kappa(A)}+1}\right)^{k} \right) \\ \ln \left(\frac{\left\|\mathbf{e}_{k}\right\|_{A}}{\left\|\mathbf{e}_{0}\right\|_{A}}\right) &\approx -2k \frac{\sqrt{\kappa(A)}-1}{\sqrt{\kappa(A)}+1}. \end{align*}

Find the expression for the number of iterations \(k\)

From the previous step, we have: $$ \ln \left(\frac{\left\|\mathbf{e}_{k}\right\|_{A}}{\left\|\mathbf{e}_{0}\right\|_{A}}\right) \approx -2k \frac{\sqrt{\kappa(A)}-1}{\sqrt{\kappa(A)}+1}. $$ Now, we can solve for \(k\): $$ k \approx \frac{-\ln \left(\frac{\left\|\mathbf{e}_{k}\right\|_{A}}{\left\|\mathbf{e}_{0}\right\|_{A}}\right)}{2 \frac{\sqrt{\kappa(A)}-1}{\sqrt{\kappa(A)}+1}}. $$ To achieve an error reduction by a constant amount, we have: $$ k \leq \frac{-\ln \left(\frac{\left\|\mathbf{e}_{k}\right\|_{A}}{\left\|\mathbf{e}_{0}\right\|_{A}}\right)}{2 \frac{\sqrt{\kappa(A)}-1}{\sqrt{\kappa(A)}+1}} \leq \sqrt{\kappa(A)} \times \text{(modest constant)}. $$ Hence, we have shown that the number of iterations \(k\) required to achieve an error reduction by a constant amount is bounded by \(\sqrt{\kappa(A)}\) times a modest constant.

Key concepts

Error Bound

In numerical methods like the Conjugate Gradient Method, the concept of an error bound is crucial in understanding how accurate the solution is after a given number of iterations. The error bound refers to a mathematical expression that gives an upper limit on the error of the solution. This helps us estimate how close our calculated solution is to the exact one.

For the Conjugate Gradient Method, the error bound is expressed in terms of the norm of the error vector \(\mathbf{e}_k\). Specifically, this is represented by the fraction \[ \frac{\|\mathbf{e}_k\|_A}{\|\mathbf{e}_0\|_A} \leq 2 \left( \frac{\sqrt{\kappa(A)}-1}{\sqrt{\kappa(A)}+1} \right)^k \], where \(\kappa(A)\) is the condition number of the matrix \(A\). \(\mathbf{e}_k\) represents the error at iteration \(k\), while \(\mathbf{e}_0\) is the initial error.

Understanding this bound helps us determine the efficiency and accuracy of the Conjugate Gradient Method in a practical context.

Condition Number

The condition number, denoted as \(\kappa(A)\), is a fundamental concept when analyzing the performance of algorithms like the Conjugate Gradient Method. It provides insight into how well-conditioned a given problem is when solving linear systems.

Mathematically, the condition number is expressed as \( \kappa(A) = \frac{\lambda_{\text{max}}(A)}{\lambda_{\text{min}}(A)} \), involving the maximum \(\lambda_\text{max}(A)\) and minimum \(\lambda_\text{min}(A)\) eigenvalues of the matrix \(A\). The \(\lambda\)'s represent the eigenvalues, which are critical in defining the properties of the matrix.

A low condition number implies a well-conditioned problem, meaning small changes in the input lead to small changes in the output. Conversely, a high condition number indicates a problem that's ill-conditioned, where small input changes can result in large output changes, potentially leading to numerical instability.

In the context of the Conjugate Gradient Method, a large condition number might slow down convergence or require more iterations to achieve a desired error bound.

Eigenvalues

Eigenvalues are vital components in the study of matrices and have a significant role in numerical methods like the Conjugate Gradient Method. To put it simply, an eigenvalue of a matrix \(A\) is a scalar \(\lambda\) such that there is a nonzero vector \(\mathbf{v}\) for which \(A \mathbf{v} = \lambda \mathbf{v}\).

In practical terms, eigenvalues help us understand various properties of matrices, including their stability and behavior under certain operations. They are used to determine the condition number of a matrix, where as mentioned, \(\kappa(A) = \frac{\lambda_\text{max}(A)}{\lambda_\text{min}(A)}\).

Maximum and minimum eigenvalues are particularly important in the Conjugate Gradient Method because they affect the convergence rate. A large spread between the maximum and minimum eigenvalue can result in a high condition number, which can influence the number of iterations required to reach a given error bound.

Thus, knowing the eigenvalues of a matrix helps determine its suitability for application in iterative methods and can also guide preconditioning strategies.

Convergence Rate

The convergence rate in iterative numerical methods defines how quickly successive approximations approach the exact solution. For the Conjugate Gradient Method, the convergence rate is intricately tied to the condition number of the matrix \(A\).

Expressed through the formula involving the condition number: \[ \left( \frac{\sqrt{\kappa(A)}-1}{\sqrt{\kappa(A)}+1} \right)^k, \] it illustrates how the number of iterations \(k\) affects the reduction in error magnitude. The smaller this fraction, the faster the convergence.

As the condition number \(\kappa(A)\) increases, the convergence slows down. This is seen where the numerator in the fraction grows relative to the denominator, resulting in potentially more iterations required to achieve the same level of accuracy.

The convergence rate is essential for estimating the computational efficiency of the Conjugate Gradient Method. It helps decide whether a given matrix should be preconditioned to improve results. Understanding convergence rate allows us to make practical decisions in computational problems to optimize performance.