Developments in the theory of linear least-squares estimation in the last thirty years or so are outlined. Particular attention is paid to early mathematica[ work in the field and to more modern developments showing some of the many connections between least-squares filtering and other fields.

A view of three decades of linear filtering theory

Matrix quadratic equation solution derivation applied in finding steady state solutions of Riccati differential equations with constant coefficients

Matrix Quadratic Solutions

Oscillation theory of ordinary linear differential equations

Publisher Summary This chapter highlights generalized inverses, normal solvability, and iteration for singular operator equations. It explores the generalized inverse for a bounded linear operator, whose range is not necessarily closed from a point of view that depends heavily on projection operators and that preserves the best approximation solution property in the case of Hilbert spaces. The chapter presents several results on the convergence of the method of successive approximations (with a fixed averaging parameter) and the steepest descent and conjugate gradient methods to least squares solutions for the linear operator equation Tx = y. The chapter also presents some results on normal solvability of nonlinear operator equations obtained by Pohozaev and highlights some directions for extending generalized inverses to nonlinear operators under certain conditions.

Generalized Inverses, Normal Solvability, and Iteration for Singular Operator Equations

Algebraic Properties of Generalized Inverses

Generalized Green's Matrices for Compatible Systems of Differential Equations

A Boundary Value Problem Associated with the Calculus of Variations

Discontinuous Solutions in the Non-Parametric Problem of Mayer in the Calculus of Variations

An Integro-Differential Boundary Value Problem

The Theory of the Second Variation for the Non-Parametric Problem of Bolza

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