Linear Least‐Squares Estimation: Solution Techniques
Bruce P. Gibbs · 2011
This chapter discusses numerical methods for solving least - squares problems, sensitivity to numerical errors, and practical implementation issues. These topics necessarily involve discussions of matrix inversion methods (Gauss - Jordan elimination, Cholesky factorization), orthogonal transformations, and iterative refinement of solutions. The orthogonalization methods (Givens rotations, Householder transformations, and Gram - Schmidt orthogonalization) are used in the QR and Singular Value Decomposition (SVD) methods for computing least - squares solutions. The concepts of observability, numerical conditioning, and pseudo - inverses are also discussed. Examples demonstrate numerical accuracy and computational speed issues. The SVD can also be used to solve the least - squares problem. It has the advantages of automatically computing the solution condition number and indicating which linear combinations of states are unobservable. Controlled Vocabulary Terms least squares approximations; singular value decomposition