Skip to main content

Minimization and Least Squares

  • 191k Accesses

Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

Because Nature seems to strive for efficiency, many systems arising in physical applications are founded on a minimization principle. In a mechanical system, the stable equilibrium configurations minimize the potential energy. In an electrical circuit, the current adjusts itself to minimize the power. In optics and relativity, light rays follow the paths of minimal distance — the geodesics on the curved space-time manifold. Solutions to most of the boundary value problems arising in applications to continuum mechanics are also characterized by a minimization principle, which is then employed to design finite element numerical approximations to their solutions, [61, 81]. Optimization — finding minima or maxima — is ubiquitous throughout mathematical modeling, physics, engineering, economics, and data science, including the calculus of variations, differential geometry, control theory, design and manufacturing, linear programming, machine learning, and beyond.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-91041-3_5
  • Chapter length: 66 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-91041-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   74.99
Price excludes VAT (USA)
Hardcover Book
USD   74.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Olver, P.J., Shakiban, C. (2018). Minimization and Least Squares. In: Applied Linear Algebra. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-91041-3_5

Download citation