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2 results

  • 12 - Optimization and Root Finding of Algebraic Systems

  • from Part III - Least Squares and Optimization
  • Kevin W. Cassel, Illinois Institute of Technology
  • Book:
    Matrix, Numerical, and Optimization Methods in Science and Engineering
    Published online:
    18 February 2021
    Print publication:
    04 March 2021, pp 594-641

  • Summary

    Optimization and root finding are closely aligned techniques for determining the extremums and zeros, respectively, of a function.Newton's method is the workhorse of both types of algorithms for nonlinear functions, and the conjugate-gradient and GMRES methods are also covered.Optimization of linear, quadratic, and nonlinear functions are addressed with and without constraints, which may be equality or inequality.In the linear programming case, emphasis is placed on the simplex method.

  • 10 - Least-Squares Methods

  • from Part III - Least Squares and Optimization
  • Kevin W. Cassel, Illinois Institute of Technology
  • Book:
    Matrix, Numerical, and Optimization Methods in Science and Engineering
    Published online:
    18 February 2021
    Print publication:
    04 March 2021, pp 529-559

  • Summary

    Least-squares methods provide the mathematical foundation for optimization of algebraic systems.They can be applied to overdetermined systems, having more equations than unknowns, or undertermined systems, having fewer equations than unknowns.The optimization may involve constraints or be subject to a penalty function.Numerical methods, namely the conjugate-gradient and GMRES methods, that are based on least-squares optimization method are discussed in detail and put into the context of other Krylov-based methods.