THE LINEAR ALGEBRA OF BLOCK QUASI-NEWTON ALGORITHMS

The quasi-Newton family of algorithms for minimizing functions and sol ving systems of nonlinear equations has achieved a great deal of compu tational success and forms the core of many software libraries for sol ving these problems. In this work we extend the theory of the quasi-Ne wton algorithms to the block case, in which we minimize a collection o f functions having a common Hessian matrix, or we solve a collection o f nonlinear equations having a common Jacobian matrix. This paper focu ses on the linear algebra: update formulas, positive definiteness, lea st-change secant properties, relation to block conjugate gradient algo rithms, finite termination for quadratic function minimization or solv ing linear systems, and the use of the quasi-Newton matrices as precon ditioners.