Question

(a) Why can't one directly extend the LU decomposition to a long and skinny matrix in order to solve least squares problems? (b) When writing \(\mathbf{x}=A^{\dagger} \mathbf{b}=\cdots=R^{-1} Q^{T} \mathbf{b}\) we have somehow moved from the conditioning \([\kappa(A)]^{2}\) to the conditioning \(\kappa(A)\) through mathematical equalities. Where is the improvement step hidden? Explain.

Short answer

Answer: In the context of least squares problems, QR decomposition leads to a reduction in the conditioning number compared to the normal equations. Specifically, the conditioning number is reduced from \([\kappa(A)]^2\) (in the normal equations) to \(\kappa(A)\) when using QR decomposition. This reduction in the conditioning number leads to more stable and accurate solutions, as the problem becomes less sensitive to small measurement errors in the data.

Step-by-step solution

Part (a) - The LU Decomposition and the Least Squares Problem

The LU decomposition is a technique used for solving linear systems of the form \(AX = \mathbf{b}\), by decomposing the matrix \(A\) into the product of two matrices, \(L\) and \(U\), where \(L\) is a lower triangular matrix and \(U\) is an upper triangular matrix. However, in the case of a long and skinny matrix, the matrix is not square, and the LU decomposition is not directly applicable because the product of two lower and upper triangular matrices cannot cover all the possibilities for a non-square matrix. Furthermore, when trying to solve a least squares problem (minimizing the distance between a set of points and a function), the underlying problem is different. The least squares solution is defined as \(\mathbf{x}=\underset{\mathbf{x}}{\operatorname{argmin}} \| A\mathbf{x}-\mathbf{b} \|\), so the objective is to minimize the distance between the projection of \(\mathbf{b}\) onto the column space of \(A\) and the actual value of \(A\mathbf{x}\).

Part (a) - QR Decomposition for Least Squares Problems

Instead of LU decomposition, the QR decomposition is more suitable for least squares problems. The matrix \(A\) is decomposed into the product of two matrices \(Q\) and \(R\), where \(Q\) is an orthogonal matrix and \(R\) is an upper triangular matrix. This decomposition allows for a better approach to solve least squares problems. The normal equations \(A^TA\mathbf{x}=A^T\mathbf{b}\) lead to \(R^TQ^TQR\mathbf{x}=R^TQ^T\mathbf{b}\), and since \(Q^TQ=I_n\) and \(R^T\) is also upper triangular, this simplifies to \(R^TR\mathbf{x}=R^TQ^T\mathbf{b}\), which can be solved efficiently.

Part (b) - Conditioning and its Improvement

Conditioning refers to how small changes in the input of a problem can cause large changes in the output. The conditioning number \(\kappa(A)\) is a measure of the ill-conditioning of the matrix \(A\). In the context of least squares problems, a high conditioning number means that the problem is more sensitive to small measurement errors in the data. To analyze where the improvement step is hidden, let's break down the equation \(\mathbf{x}=A^{\dagger} \mathbf{b}=R^{-1} Q^{T} \mathbf{b}\). Here, \(A^{\dagger}\) represents the Moore-Penrose pseudoinverse of \(A\), which gives the best least squares solution when \(A\) is not invertible. The equation can be rewritten as \(\mathbf{x}=R^{-1} Q^{T} \mathbf{b}\). The improvement step is hidden within the QR decomposition itself. In the normal equations, we have a conditioning number of \([\kappa(A)]^2\), due to the squaring process involved in premultiplying by \(A^T\). However, in the QR decomposition, the matrix \(Q\) being orthogonal preserves the conditioning number of \(A\), meaning the conditioning number is reduced to \(\kappa(A)\) when moving from the normal equations to the equation \(\mathbf{x}=R^{-1} Q^{T} \mathbf{b}\). This means that the sensitivity to small measurement errors is reduced, leading to more stable and accurate solutions in the least squares problem.

Key concepts

Least Squares Problems

Solving least squares problems is all about minimizing the error between estimated values and actual data. Imagine you have a dataset and you want to find a line (or function) that best fits this data. The goal of a least squares problem is to find this 'best fit' line.

In mathematical terms, given a matrix equation of the form \( A\mathbf{x} = \mathbf{b} \), the least squares solution minimizes the Euclidean distance \( ||A\mathbf{x} - \mathbf{b}|| \). This equates to finding an \( \mathbf{x} \) that brings \( A\mathbf{x} \) as close as possible to \( \mathbf{b} \).

This is often useful in real-world data scenarios where noise and approximations are present in data collection. The method fits models to the data with graphical applications seen in trend-lines and regression models. This is why least squares are widely used in fields like data science and engineering.

Instead of solving it through direct inversion of \( A \), which can be unstable for non-square or ill-conditioned matrices, techniques like QR decomposition are often employed to make the solution more robust and reliable.

QR Decomposition

QR decomposition is a matrix factorization method that can help solve least squares problems more effectively than LU decomposition, especially for non-square matrices. Unlike LU, QR decomposition breaks down a matrix \( A \) into the form \( A = QR \), where:

• \( Q \) is an orthogonal matrix, meaning \( Q^TQ = I \)
• \( R \) is an upper triangular matrix

This factorization transforms a potentially difficult problem into a simpler one.

For least squares, it simplifies to solving \( R\mathbf{x} = Q^T\mathbf{b} \), which is often much simpler and more stable than taking the pseudoinverse. The orthogonality of \( Q \) helps maintain the numerical stability and reduces potential errors in the solutions.

Using the QR decomposition, \( A^TA \), which normally increases the condition number and therefore the instability, is avoided. Instead, the decomposition allows for directly calculating \( R\mathbf{x} \), making this a preferred method when dealing with complex data where precision is critical. It is an essential concept in numerical linear algebra used in various computational algorithms.

Conditioning Number

Conditioning number is a critical concept in understanding the sensitivity of a solution to small changes in input data. It's often a source of difficulty in numerical computations, especially in least squares problems.

The conditioning number for a matrix \( A \) is denoted as \( \kappa(A) \) and defined as the ratio of the largest to smallest singular value of \( A \). A high conditioning number indicates that the matrix is ill-conditioned, meaning small errors in data or computation could cause large errors in results.

In the context of least squares problems, solving \( A^TA\mathbf{x} = A^T\mathbf{b} \) translates to a conditioning factor \( [\kappa(A)]^2 \), potentially compounding errors. However, QR decomposition bypasses this escalation. The orthogonality of \( Q \) ensures that the problem's conditioning can be preserved just as \( \kappa(A) \).

Understanding and managing conditioning numbers form a major part of creating stable and accurate computational algorithms. This concept helps predict how reliable a solution might be when the underlying data has uncertainties or errors, commonly encountered in practical applications.