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Lecture

Direct Methods for Linear Systems of Equations

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 direct methods for solving linear systems of equations, focusing on the Gauss elimination method and LU decomposition. It explains the process of forward and back substitution, as well as the importance of pivoting in reducing errors. The Cholesky decomposition method is also introduced, providing insights into efficient solutions for symmetric positive definite matrices.

Instructors (2)

Kevin Sivula

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.

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

LU decomposition

In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.

Cholesky decomposition

In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ʃəˈlɛski ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices, and posthumously published in 1924. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.

Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations , the matrix A can be decomposed via the LU decomposition.

QR decomposition

In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Any real square matrix A may be decomposed as where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning ) and R is an upper triangular matrix (also called right triangular matrix).

Gaussian elimination

In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855).

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