It is well known that the conjugate gradient method and a quasi-Newton method, using any well-defined update matrix from the one-parameter Broyden family of updates, produce identical iterates on a quadratic problem with positive-definite Hessian. This equivalence does not hold for any...

This work deals with the use of the conjugate gradient method in conjunction with an adjoint problem formulation for the simultaneous estimation of the spatially varying diffusion coefficient and of the source term distribution in a one-dimensional nonlinear diffusion problem. In the present...

When solving linear systems and, in particular when solving large scale ill-conditioned problems it is important to understand the behaviour of the conjugate gradient method. The conjugate gradient method converges typically in three phases, an initial phase of rapid convergence but short...

Based on the NEWUOA algorithm, a new derivative-free algorithm is developed, named LCOBYQA. The main aim of the algorithm is to find a minimizer <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$x^{*} \in\mathbb{R}^{n}$</EquationSource> </InlineEquation> of a non-linear function, whose derivatives are unavailable, subject to linear inequality constraints. The algorithm is...</equationsource></inlineequation>

A family of new conjugate gradient methods is proposed based on Perry’s idea, which satisfies the descent property or the sufficient descent property for any line search. In addition, based on the scaling technology and the restarting strategy, a family of scaling symmetric Perry conjugate...

We consider a 3-term recurrence, namely CG_2step, for the iterative solution of symmetric linear systems. The new algorithm generates conjugate directions and extends some standard theoretical properties of the Conjugate Gradient (CG) method [10]. We prove the finite convergence of CG_2step, and...