QR Decomposition

Jose Divasón 🌐 and Jesús Aransay 🌐

February 12, 2015

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.

Abstract

QR decomposition is an algorithm to decompose a real matrix A into the product of two other matrices Q and R, where Q is orthogonal and R is invertible and upper triangular. The algorithm is useful for the least squares problem; i.e., the computation of the best approximation of an unsolvable system of linear equations. As a side-product, the Gram-Schmidt process has also been formalized. A refinement using immutable arrays is presented as well. The development relies, among others, on the AFP entry "Implementing field extensions of the form Q[sqrt(b)]" by René Thiemann, which allows execution of the algorithm using symbolic computations. Verified code can be generated and executed using floats as well.
BSD License

Change history

[2015-06-18] The second part of the Fundamental Theorem of Linear Algebra has been generalized to more general inner product spaces.

Topics

Theories of QR_Decomposition