Lecture

Solving Systems of Nonlinear Equations

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In course

ChE-312: Numerical methods

This course introduces students to modern computational and mathematical techniques for solving problems in chemistry and chemical engineering. The use of introduced numerical methods will be demonstrated using the Python programming language. ...

Description

This lecture covers the Newton-Raphson method for multidimensional root finding, the use of partial derivatives in the Jacobian matrix, and iterative schemes for solving systems of nonlinear equations with examples and numerical methods.

Instructors (2)

Ljubisa Miskovic

Ljubisa Miskovic earned his Ph.D. degree in Automatic Control from the Swiss Federal Institute of Technology (EPFL) under the co-supervision of Dominique Bonvin and Alireza Karimi, in 2006. He pursued his postdoctoral studies at the Centre for Systems Engineering and Applied Mechanics, Universite Catholique de Louvain with Michel Gevers before moving to the laboratory of Vassily Hatzimanikatis at the EPFL. In 2010, he became a research scientist. His research interests include systems biology, metabolic engineering, synthetic biology, data-driven control design, system identification, stochastic processes and estimation theory.

Kevin Sivula

See webpage for more info limno.epfl.ch Teaching & PhD PhD Students Odile Louise Andres, Arthur Eliot Bouchez, Benjamin Goldman, Nicolas Johannes Diercks, Aurélie Fanny Marie Célérien, Jiajun Li, Louise Marie Foltzer-De Cian, Melanie Victoria Johanning, Nils William Laurent Golay Past EPFL PhD Students Andrea Gasperini, Pauline Bornoz, Xavier Adrian Jeanbourquin, Xiaoyun Yu, Mathieu Steven Prévot, Muhammad Aiman Bin Rahmanudin Hamdan, Nhât-Tân Vuong, Yongpeng Liu, Arvindh Sekar, Barbara Alexandra Primera

Official source

Related concepts (22)

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Newton's method in optimization

In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function F, which are solutions to the equation F (x) = 0. As such, Newton's method can be applied to the derivative f ′ of a twice-differentiable function f to find the roots of the derivative (solutions to f ′(x) = 0), also known as the critical points of f. These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article.

Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.

Numerical methods for partial differential equations

Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist. Finite difference method In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.

Numerical methods for linear least squares

Numerical methods for linear least squares entails the numerical analysis of linear least squares problems. A general approach to the least squares problem can be described as follows. Suppose that we can find an n by m matrix S such that XS is an orthogonal projection onto the image of X. Then a solution to our minimization problem is given by simply because is exactly a sought for orthogonal projection of onto an image of X (see the picture below and note that as explained in the next section the image of X is just a subspace generated by column vectors of X).

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