Concept

Equation solving

In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality. A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set. An equation may be solved either numerically or symbolically. Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions. For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1. Or x and y can both be treated as unknowns, and then there are many solutions to the equation; a symbolic solution is (x, y) = (a + 1, a), where the variable a may take any value. Instantiating a symbolic solution with specific numbers gives a numerical solution; for example, a = 0 gives (x, y) = (1, 0) (that is, x = 1, y = 0), and a = 1 gives (x, y) = (2, 1). The distinction between known variables and unknown variables is generally made in the statement of the problem, by phrases such as "an equation in x and y", or "solve for x and y", which indicate the unknowns, here x and y. However, it is common to reserve x, y, z, ... to denote the unknowns, and to use a, b, c, ... to denote the known variables, which are often called parameters.

Source officielle

https://en.wikipedia.org/wiki/Equation_solving

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Cours associés (32)

MATH-250: Advanced numerical analysis I

Construction and analysis of numerical methods for the solution of problems from linear algebra, integration, approximation, and differentiation. ...

MATH-111(e): Algèbre linéaire

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications. ...

EE-548: Audio

This lecture is oriented towards the study of audio engineering, room acoustics, sound propagation, and sound radiation from sources and acoustic antennas. The learning outcomes will be the techniques for microphones and loudspeaker design, as well as room ...

Afficher plus

Unités associées (2)

Laboratoire transport et mobilité

Algorithmes numériques et calcul haute performance - Chaire CADMOS

Graph Chat

Chattez avec Graph Search

Posez n’importe quelle question sur les cours, conférences, exercices, recherches, actualités, etc. de l’EPFL ou essayez les exemples de questions ci-dessous.

AVERTISSEMENT : Le chatbot Graph n'est pas programmé pour fournir des réponses explicites ou catégoriques à vos questions. Il transforme plutôt vos questions en demandes API qui sont distribuées aux différents services informatiques officiellement administrés par l'EPFL. Son but est uniquement de collecter et de recommander des références pertinentes à des contenus que vous pouvez explorer pour vous aider à répondre à vos questions.

Source officielle

https://en.wikipedia.org/wiki/Equation_solving

À propos de ce résultat

Cours associés (32)

MATH-250: Advanced numerical analysis I

Construction and analysis of numerical methods for the solution of problems from linear algebra, integration, approximation, and differentiation. ...

MATH-111(e): Algèbre linéaire

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications. ...

EE-548: Audio

Afficher plus

Séances de cours associées (32)

Valeurs absolues : Équations et inégalités

Explore les valeurs absolues dans les équations et les inégalités, en se concentrant sur la résolution de l'équation f(x)=g(x).

Block Tiré par un printemps : Dynamique

Couvre la dynamique d'un bloc connecté à un ressort, dérivant des équations de mouvement et de résolution pour les points de temps clés.

Valeur absolue

Couvre le concept de valeur absolue et ses propriétés, y compris l'interprétation graphique et la résolution d'équations.

Afficher plus

Publications associées (20)

Inverse Kinematic Solution for Generic 3R Positional Robots Using Conformal Geometric Algebra

The inverse kinematics of generic 3R robots has been investigated through multiple approaches, mainly algebraic methods involving the solution of certain equation sets. Previous geometric interpretations of the solution, characterized as the intersection of a pair of conics have been confined to the joint-space domain. In this article, we study the Inverse Kinematic Model (IKM) of 3R robots, using the advantages of Conformal Geometric Algebra (CGA) to provide further insights on its kinematic properties. Ou

Springer Nature2026

High Order Smoothness for Stochastic Navier-stokes Equations with Transport and Stretching Noise on Bounded Domains

We obtain energy estimates for a transport and stretching noise under Leray Projection on a 2D bounded convex domain, in Sobolev Spaces of arbitrarily high order. The estimates are taken in equivalent inner products, defined through powers of the Stokes Operator with a specific choice of Navier boundary conditions. We exploit fine properties of the noise in relation to the Stokes Operator to achieve cancellation of derivatives in the presence of the Leray Projector. As a result, we achieve an additional deg

PERGAMON-ELSEVIER SCIENCE LTD2026

Compatibilities and supercompatibility conditions in shape memory alloys determined from correspondence, metrics and symmetries

Cyril Cayron

The phenomenological theory of martensite crystallography (PTMC) explains the habit planes of bi-variant laminate martensite product and the transformation twins between the variants. It also permits to determine whether or not the austenite and martensite lattice parameters satisfy supercompatibility by three mathematical equations called “cofactor conditions”. The PTMC calculations are in great part based on mathematical tools of continuum mechanics (polar decompositions and stretch tensors). Here, we pro

Elsevier BV2026

Afficher plus

Personnes associées (27)

Marco Picasso

Expertise Computational Science and Engineering Mission I am teaching analysis and numerical analysis for engineers and mathematicians at bachelor and master levels.My research pertains to the numerical simulation of complex physical systems with partial differential equations.Here is some bibliometry: Google Scholar, Research Gate, Orcid, here is my most recent CV. Ongoing research projects: - Numerical simulation of aluminium electrolysis, collaboration with RioTinto LRF lab, Ycoorsystems SA and Jacques

Sacha Friedli

Diploma of Physics, Université de Lausanne, 1999 Phd, Institut de Physique Théorique, EPFL, 1999-2004 Post-doc, IMPA (Instituto Nacional de Matemática Pura e Aplicada), Rio de Janeiro, Brasil, 2004-2006 Adjunct professor, Departamento de Matemática (Universidade Federal de Minas Gerais), 2006-2016 Collaborateur Scientifique, Cours de Mathématiques Spéciales, EPFL, Lausanne, 2016-

Baruch Rofman

Enseignement et PhD Cours Physique numérique (pour SPH) PHYS-210 Aborder, formuler et résoudre des problèmes de physique en utilisant des méthodes numériques élémentaires. Comprendre les avantages et les limites de ces méthodes (stabilité, convergence). Illustrer différents sujets de physique traités dans d'autres cours.

Joan Decker

Enseignement et PhD Doctorant·es actuel·les Ewout Koen L. Devlaminck, Isabel Muzio, Salomon Guinchard Cours Fusion and industrial plasma technologies PHYS-632 The course provides an overview of the technologies that are essential for fusion developments and for industrial plasma applications, highlighting the synergies between the two fields. The aim is to provide a combined theoretical and experimental background to the various topics. Physique numérique (pour SPH) PHYS-210

Guillaume Alexandre C Olikier

Expertise Optimisation continue, analyse numérique Guillaume Olikier est chercheur postdoctoral (Instructeur Bernoulli) au sein de l'Institut de Mathématiques à l'EPFL. Il a obtenu un master d'ingénieur en mathématiques appliquées (2017) et un doctorat en sciences de l'ingénieur et technologie (2023) de l'UCLouvain, Louvain-la-Neuve, Belgique. De 2023 à 2025, il était chercheur postdoctoral (Starting Research Position) au centre Inria d'Université Côte d’Azur, Sophia Antipolis, France . Son principal domain

Dimitrios Lignos

Prof. Lignos joined the École Polytechnique Féderale de Lausanne (EPFL) in 2016 from McGill University in Canada where he was a tenured Associate Professor and a William Dawson Scholar for Infrastructure Resilience. He holds a diploma (National Technical University of Athens, NTUA, 2003), M.S. (Stanford University, 2004) and Ph.D. (Stanford University, 2008). In addition, he was a post-doctoral scientist at Stanford University (2009) and in Kyoto University (2010). Prof. Lignos teaches graduate and undergraduate courses in seismic design, nonlinear behaviour of steel and composite structures as well as supplemental damping systems, Structural Stability, Nonlinear Analysis and Performance-based Earthquake Engineering. His awards for teaching, research and service in Civil Engineering include the 2011 Outstanding Teaching Award (Faculty of Engineering, McGill University), as well as the Outstanding reviewer (2012, 2013) award from ASCE, the 2013 State-of-the-Art in Civil Engineering Award by ASCE and the 2014 Christophe Pierre Award for Research Excellence - Early Career. Just recently, he received the 2019 Walter L. Huber Civil Engineering Research Prize from ASCE for significant contributions in developing state of the art methods to simulate extreme limit states in steel structures.Prof. Lignos is a member of ASCE and the Earthquake Engineering Research Institute. He acts as an Associate Editor for Metal Structures and Seismic Effects of the ASCE Journal of Structural Engineering. He joined the Editorial Board of Earthquake Spectra and Earthquake Engineering and Structural Dynamics International journals. He serves as an acting member of the CEN/TC 250/SC 8/WG 2 and has been selected as a member of the Project Team (PT2) for the Eurocode 8-Part 1 Current Revisions for Steel and Composite Structures. He is also a member of the Canadian Standards Association (CSA) S16 technical committee for Steel Structures. Prof. Lignos is involved as a NEHRP consultant in numerous research-to-practice projects related to the behaviour and nonlinear modelling and analysis of structures applicable to the engineering practice through the Applied Technology Council (ATC). Detailed Curriculum Vitae (last update September 2018)

Nikolaos Skretas

Simone Deparis

Expertise Simone Deparis est professeur à l’EPFL, spécialisé en mathématiques appliquées, analyse numérique et didactique des mathématiques à l’université. Ses recherches portent à la fois sur la modélisation computationnelle pour les applications biomédicales et sur l’innovation pédagogique, avec un accent sur l’équité, l’auto-efficacité et la réforme de l’évaluation dans les grands cursus scientifiques. Mission Mes recherches portent sur les mathématiques appliquées et les sciences de l’apprentissage. En

Ronan Boulic

I come from Brittany, France, where I have completed my PhD degree in Computer Science in 1986 from the University of Rennes, France, at the INRIA-IRISA research institute. I also received the Habilitation degree from the University of Grenoble, France, in march 1995. I was hired in 1989 as First Assistant in the VRLAB, I became scientific collaborator, and senior researcher. I'm presently Senior Scientist (MER) and leader of the Immersive Interaction research Group (IIG). I'm co-author of around 150 research papers among which 43 appeared in international peer-reviewed journals. I have contributed to multiple SNF projects and EU projects. Please check iig.epfl.ch for more details.

Karim Achouri

Enseignement et PhD Doctorant·es actuel·les Mahmoud Abouelatta, Hossein Allahverdizadeh, Mustafa Yücel Cours Applied Electromagnetics for Metamaterial Design MICRO-449 This course provides the students with the appropriate electromagnetics foundation to understand, model and design metamaterials, especially two-dimensional electromagnetic and optical structures such as metasurfaces. This course covers concepts that apply to both the microwave and optical regimes. Ingénierie optique (pour

Afficher plus

Unités associées (2)

Laboratoire transport et mobilité

Algorithmes numériques et calcul haute performance - Chaire CADMOS

Concepts associés (10)

Algèbre

L'algèbre (de l’arabe الجبر, al-jabr) est une branche des mathématiques qui permet d'exprimer les propriétés des opérations et le traitement des équations et aboutit à l'étude des structures algébriques. Selon l’époque et le niveau d’études considérés, elle peut être décrite comme : une arithmétique généralisée, étendant à différents objets ou grandeurs les opérations usuelles sur les nombres ; la théorie des équations et des polynômes ; depuis le début du , l’étude des structures algébriques (on parle d'algèbre générale ou abstraite).

Factorisation

En mathématiques, la factorisation consiste à écrire une expression algébrique (notamment une somme), un nombre, une matrice sous la forme d'un produit. Cette transformation peut se faire suivant différentes techniques détaillées ci-dessous. Les enjeux de la factorisation sont très divers : à un niveau élémentaire, le but peut être de ramener la résolution d'une équation à celle d'une équation produit-nul, ou la simplification d'une écriture fractionnaire ; à un niveau intermédiaire, la difficulté algorithmique présumée de la factorisation des nombres entiers en produit de facteurs premiers est à la base de la fiabilité du cryptosystème RSA.

Algorithme de recherche d'un zéro d'une fonction

Un algorithme de recherche d'un zéro d’une fonction est une méthode numérique ou un algorithme de recherche d’une valeur approchée d’un x vérifiant , pour une fonction donnée f. Ici, x est un nombre réel appelé zéro de f ou lorsque f est polynomiale, racine de f. Lorsque x est un vecteur, les algorithmes pour trouver x tel que sont généralement appelés « algorithmes de résolution numérique d'un système d'équations ». Ces algorithmes sont une généralisation des algorithmes de recherche d’un zéro d’une fonction et peuvent s’appliquer à des équations linéaires ou non linéaires.

Afficher plus

MOOCs associés (3)

Physique des plasmas : Introduction

Le plasma, quatrième état de la matière, est de loin la forme la plus abondante de matière connue dans l'univers. Son comportement est très différent des autres états de la matière que nous connaissons habituellement. Pour le comprendre, un formalisme rigoureux est nécessaire. Ce formalisme est essentiel non seulement pour expliquer d'importants phénomènes astrophysiques, mais aussi pour optimiser de nombreuses applications industrielles et médicales et pour obtenir de l'énergie de fusion sur Terre. Ce co

Fonctions Trigonométriques, Logarithmiques et Exponentielles

Ce cours dispense les connaissances fondamentales liées à ces fonctions. Il est articulé autour de 9 chapitres s'étendant sur 12 semaines: L'angle et sa mesure Les fonctions trigonométriques sinus et cosinus et leurs inverses Les fonctions trigonométriques tangente et cotangente et leurs inverses Les formules trigonométriques (addition, bissection, etc) Les oscillations harmoniques Les équations trigonométriques Les relations dans le triangle (théorèmes du sinus et du cosinus) Continuité et dérivabilité de

Physique des plasmas : Applications

Ce cours de physique, enseigné par des experts de renommée mondiale dans ce domaine, vous donnera un aperçu des applications de la physique des plasmas. De l'étude d'objets astrophysiques très éloignés, en passant par diverses applications dans l'industrie et la médecine, jusqu'à l'objectif ultime de la production durable d'électricité à partir de la fusion nucléaire. Dans la première partie de ce cours, vous apprendrez comment la fusion nucléaire alimente notre Soleil et les étoiles de l'Univers. Vous ex

Exercices associés (11)

Forces de frottement

A sports car is driving around a circular track of radius R with a horizontal velocity of constant norm v. The track is on an incline with angle α from the the horizontal, such that the car is tilted inwards. The static friction coefficient between the tires and the track is μ.

Draw a diagram, considering the vertical plane perpendicular to the car’s velocity and represent all forces acting on the car.
What’s the condition on v for the car not to drift outwards ?

Numerical application: $g=10\

Le liévre et la tortue

A hare and a turtle race over a distance L. The turtle starts running at constant speed

v_{\rm t}

and the hare at constant speed

v_{\rm h}

. The hare is not taking this seriously, and so

v_{\rm h} < v_{\rm t}

. When the turtle reaches a bridge that stands at a distance L′u2004

Plot sur plan incliné

A block of mass M on an inclined plane of angle α with respect to the horizontal plane is attached with a massless cable to a mass m, hanging in the air. The friction is dry.

Determine the mass m needed to overcome the dry friction characterised by the static friction coefficient μ_(s).
In motion, the block is subjected to a dry friction characterised by the kinetic friction coefficient μ_(c). Establish the equations of motion.
Overcoming the static friction amounts to finding the

Afficher plus

Notebooks associés (5)

Designing FIR Filters Using the Parks-McClellan Algorithm

The Parks-McClellan Filter Design Method[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

The Parks-McClellan (PM) algorithm is a clever application of advanced polynomial fitting techniques to the problem of FIR filter design. In this notebook we will explore the key ideas behind the method by considering the design of a simple Type-I lowpass filter. NOTE: This notebook uses interactive widgets. If the interactive controls do not work or show up, please fo

Exploring Chemical Potential for Fermions in 3D at Finite Temperature

This notebook gives a simple illustration to the variation of chemical potential for Fermions in 3D, following the Fermi-Dirac statistics in finite temperature. " In 3D, we have the density of states

\\rho \\propto \\sqrt{E}

. At zero temperature, the distribution of electrons just follows

g(E)=a \\sqrt{E}

and the Fermi energy is determined by the total occupied number

N_{occ}

E_f

is the energy such that

N_{occ}=\\int_{0}^{E_f} g(E) dE

\n", " \n", "In finite temperature, however, o

Band Structure Investigation of Monolayer and Bilayer Graphene Using the Tight Binding Model

In [1]: import matplotlib.pyplot as plt import numpy as np from mpl_toolkits import mplot3d import itertools %matplotlib widget Band Structure of Mono and Bilayer Graphene[1]¶ If you have questions about this notebook, e-mail Oleg Malanyuk (oleg.malanyuk@epfl.ch) The purpose of this notebook is to investigate the band model of Mono and Bilayer graphene modeled with tight binding model. Tight binding model for monolayer graphene[2]¶ Monolayer graphene is a sheet of carbons aranged in a he

Voice Transformation Techniques Using Digital Signal Processing

A Gallery of Voice Transformers[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In this notebook we will explore various ways to implement a system that changes the voice of a speaker. Typical applications for these techniques, strictly sorted by plausibility, include international spy missions and prank telephone calls, and devices such as the one displayed here can be easily bought on the web... or in a novelty shop. No description has been provided for t

Investigating the Effect of Phase on Sound Perception

Hearing the phase of a sound[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In this notebook we will investigate the effect of phase on the perceptual quality of a sound. It is often said that the human ear is commercial-grade audio equipment takes place in the magnitude domain only. In [1]: %matplotlib inline import matplotlib import matplotlib.pyplot as plt import numpy as np import IPython from scipy.io import wavfile In [2]: plt.rcParams["figure.fig

Personnes associées (27)

Guillaume Alexandre C Olikier

Exercices associés (11)

Forces de frottement

Draw a diagram, considering the vertical plane perpendicular to the car’s velocity and represent all forces acting on the car.
What’s the condition on v for the car not to drift outwards ?

Numerical application: $g=10\

Le liévre et la tortue

A hare and a turtle race over a distance L. The turtle starts running at constant speed

v_{\rm t}

and the hare at constant speed

v_{\rm h}

. The hare is not taking this seriously, and so

v_{\rm h} < v_{\rm t}

. When the turtle reaches a bridge that stands at a distance L′u2004

Plot sur plan incliné

A block of mass M on an inclined plane of angle α with respect to the horizontal plane is attached with a massless cable to a mass m, hanging in the air. The friction is dry.

Determine the mass m needed to overcome the dry friction characterised by the static friction coefficient μ_(s).
In motion, the block is subjected to a dry friction characterised by the kinetic friction coefficient μ_(c). Establish the equations of motion.
Overcoming the static friction amounts to finding the

Afficher plus

Notebooks associés (5)

Designing FIR Filters Using the Parks-McClellan Algorithm

The Parks-McClellan Filter Design Method[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

Exploring Chemical Potential for Fermions in 3D at Finite Temperature

This notebook gives a simple illustration to the variation of chemical potential for Fermions in 3D, following the Fermi-Dirac statistics in finite temperature. " In 3D, we have the density of states

\\rho \\propto \\sqrt{E}

. At zero temperature, the distribution of electrons just follows

g(E)=a \\sqrt{E}

and the Fermi energy is determined by the total occupied number

N_{occ}

E_f

is the energy such that

N_{occ}=\\int_{0}^{E_f} g(E) dE

\n", " \n", "In finite temperature, however, o

Band Structure Investigation of Monolayer and Bilayer Graphene Using the Tight Binding Model

Voice Transformation Techniques Using Digital Signal Processing

A Gallery of Voice Transformers[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

Investigating the Effect of Phase on Sound Perception

Hearing the phase of a sound[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL