Matrix representations of multivalued projections and least squares problems

Least Squares Solutions of Inconsistent Fuzzy Linear Matrix Equations

Linear Least Squares Problems

A fundamental task in scientific computing is to estimate parameters in a mathematical model from collected data which are subject to errors. The influence of the errors can be reduced by using a greater number of data than the number of unknowns. If the model is linear, the resulting problem is then to “solve” an in general inconsistent linear system Ax = b, where A ∈ Rm×n and m ≥ n. In other ...

Least Squares Problems

Here we solve linear systems Ax = b that do not have a solution. If b is not in the column space of A, there is no x such that Ax = b. The best we can do is to find a vector y that brings left and right hand side of the linear system as close as possible, in other words y is chosen to make the distance between Ay and b as small as possible. That is, we want to minimize the distance ‖Ax − b‖2 ov...

Dykstra's algorithm for constrained least-squares doubly symmetric matrix problems

In this work we apply Dykstra’s alternating projection algorithm for minimizing ‖AX − B‖ where ‖ · ‖ is the Frobenius norm and A ∈ Rm×n, B ∈ Rm×n and X ∈ Rn×n are doubly symmetric positive definite matrices with entries within prescribed intervals. We first solve the constrained least-squares matrix problem by using the special structure properties of doubly symmetric matrices, and then use the...

Least-Squares Covariance Matrix Adjustment

We consider the problem of finding the smallest adjustment to a given symmetric n × n matrix, as measured by the Euclidean or Frobenius norm, so that it satisfies some given linear equalities and inequalities, and in addition is positive semidefinite. This least-squares covariance adjustment problem is a convex optimization problem, and can be efficiently solved using standard methods when the ...