<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>This article describes an approach known as the conjugate Gram-Schmidt method for estimating gradients and Hessian using function evaluations and difference quotients, and uses the Gram-Schmidt conjugate direction algorithm to minimize functions and compares it to other techniques for solving ∇<italic>f</italic> = 0. Comparable minimization algorithms are also used to demonstrate convergence rates using quotient and root convergence factors, as described by Ortega and Rheinbolt to determine the optimal minimization technique to obtain similar results to the Newton method, between the Gram-Schmidt approach and other minimizing approaches. A survey of the existing literature in order to compare Hestenes’ Gram-Schmidt conjugate direction approach without derivative to other minimization methods is conducted and the further analytical and computational details are provided.</p>
</abstract>

This article describes an approach known as the conjugate Gram-Schmidt method for estimating gradients and Hessian using function evaluations and difference quotients, and uses the Gram-Schmidt conjugate direction algorithm to minimize functions and compares it to other techniques for solving ∇f = 0. Comparable minimization algorithms are also used to demonstrate convergence rates using quotient and root convergence factors, as described by Ortega and Rheinbolt to determine the optimal minimization technique to obtain similar results to the Newton method, between the Gram-Schmidt approach and other minimizing approaches. A survey of the existing literature in order to compare Hestenes’ Gram-Schmidt conjugate direction approach without derivative to other minimization methods is conducted and the further analytical and computational details are provided.

A comparison of the convergence rates of Hestenes’ conjugate Gram-Schmidt method without derivatives with other numerical optimization methods

Department of Mathematics and Computer Science

International Journal of Mathematics and Computer in Engineering

This work is licensed under the Creative Commons Attribution 4.0 International License.

This article describes an approach known as the conjugate Gram-Schmidt method for estimating gradients and Hessian using function evaluations and difference...

A comparison of the convergence rates of Hestenes’ conjugate Gram-Schmidt method without derivatives with other numerical optimization methods

Article Category: Original Study

Published Online: Jun 03, 2024

Page range: -

Received: Sep 26, 2023

Accepted: Feb 12, 2024

DOI: https://doi.org/10.2478/ijmce-2025-0010

Keywords
Conjugate direction methods, numerical optimization, convergence rates, conjugate Gram-Schmidt without derivative

© 2025 Md Nurul Raihen, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

A comparison of the convergence rates of Hestenes’ conjugate Gram-Schmidt method without derivatives with other numerical optimization methods

Article Category: Original Study

Published Online: Jun 03, 2024

Page range: -

Received: Sep 26, 2023

Accepted: Feb 12, 2024

DOI: https://doi.org/10.2478/ijmce-2025-0010

KeywordsConjugate direction methods, numerical optimization, convergence rates, conjugate Gram-Schmidt without derivative

© 2025 Md Nurul Raihen, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Conjugate direction methods, numerical optimization, convergence rates, conjugate Gram-Schmidt without derivative