Householder QR Decomposition for Stable Least Squares

Summary

This article explains QR decomposition, which factors a matrix into an orthogonal matrix and an upper triangular matrix. It connects the method to least-squares problems used in regression and quantitative analysis, emphasizing that QR is more numerically stable than some alternatives, though it can take longer to compute.

The main algorithm is Householder reflection: successive transformations eliminate entries below the diagonal, producing the triangular factor while accumulating the orthogonal factor. The article contrasts this approach with Gram–Schmidt and demonstrates a worked matrix decomposition, then compares a library implementation with a pure Python version. Both produce an upper triangular result and closely matching factors, with small floating-point residuals in the hand-written computation. The example is illustrative rather than a performance benchmark or a regression study. For practical work, the article recommends using an optimized numerical library; the derivation and implementation serve primarily to explain the algorithm and its numerical considerations.

Key ideas

• QR decomposition represents a matrix as an orthogonal factor times an upper triangular factor.
• Householder reflections eliminate entries below the diagonal through successive transformations.
• QR supports least-squares solutions used in regression analysis.
• The article presents QR as more numerically stable than Gram–Schmidt, with a speed tradeoff.
• Floating-point arithmetic can leave small residual values even when the resulting factor is effectively triangular.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.