Let Hn=Qn Rn be the Q R factorization of the n ×n Hilbert matrix (1.72). (a) Find Qn and Rn for

step 1 For this exercise A, we have that Q sub 1 equals Q sub 2 S. That S equals Q sub 2 transpose time

Let Hn=Qn Rn be the Q R factorization of the n ×n Hilbert...

Let $H_n=Q_n R_n$ be the $Q R$ factorization of the $n \times n$ Hilbert matrix (1.72). (a) Find $Q_n$ and $R_n$ for $n=2,3,4$. (b) Use a computer to find $Q_n$ and $R_n$ for $n=10$ and 20 . (c) Let $\mathbf{x}^{\star} \in \mathbb{R}^n$ denote the vector whose $i^{\text {th }}$ entry is $x_i^{\star}=(-1)^i i /(i+1)$. For the values of $n$ in parts (a) and (b), compute $\mathbf{y}^{\star}=H_n \mathbf{x}^{\star}$. Then solve the system $H_n \mathbf{x}=\mathbf{y}^{\star}(i)$ directly using Gaussian Elimination; (ii) using the $Q R$ factorization based on(4.34); (iii) using Householder's Method. Compare the results to the correct solution $\mathbf{x}^{\star}$ and discuss the pros and cons of each method.

Key Concepts

Hilbert Matrix

The Hilbert matrix is a classic example in numerical linear algebra known for its severe ill-conditioning. It is a square matrix with entries defined by a formula that makes it symmetric and positive definite, but as its dimension increases, it becomes increasingly sensitive to numerical errors. This property makes it an ideal test case for analyzing the stability and accuracy of various numerical algorithms and linear system solvers.

QR Factorization

QR factorization is a method of decomposing a matrix into the product of an orthogonal matrix Q and an upper triangular matrix R. This factorization is useful in solving linear systems, performing eigenvalue computations, and least-squares data fitting. It provides an effective way to address numerical stability issues by leveraging the orthogonality of Q, which preserves the 2-norm of vectors, and simplifying matrix inversions through the triangular structure of R.

Gaussian Elimination

Gaussian elimination is a standard algorithm for solving systems of linear equations. It involves using elementary row operations to reduce a matrix to an upper triangular form, from which the solution can be obtained by back-substitution. While it is conceptually simple and widely used, Gaussian elimination can be numerically unstable, especially for matrices with a high condition number, unless techniques such as partial pivoting are employed.

Householder Transformation

Householder transformation is a technique used to perform QR factorization more stably by reflecting a vector about a plane or hyperplane. This method systematically zeroes out components below the diagonal of a matrix without explicitly forming the orthogonal matrix Q. Its use improves numerical stability when solving linear systems or performing eigenvalue decompositions, especially for ill-conditioned problems, because it avoids some of the pitfalls associated with more elementary algorithms.

Numerical Stability and Conditioning

Numerical stability and conditioning refer to a system or algorithm's sensitivity to small perturbations in input data or rounding errors during computations. In the context of solving linear systems, an ill-conditioned matrix like the Hilbert matrix can amplify these small errors, leading to large deviations in the solution. Analyzing the performance of different methods, such as Gaussian elimination and QR factorization, helps in understanding how well these algorithms cope with such error propagation and maintain the accuracy of the solution.

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