Découvrez notre Chaîne YouTube "Ingénierie et Projets"
Découvrez notre Chaîne Secondaire "Information Neuronale et l'Ingénierie du Cerveau"

Titre: Cours MAP : Mathématiques Appliquées

Auteurs: Jocelyne Erhel

Ecole: INRIA, UR Rennes

Résumé: L’objectif de ce cours est la résolution de systèmes linéaires avec soit autant de données que d’inconnues, soit plus de données que d’inconnues (système surdéterminé), soit plus d’inconnues que de données (système sous-déterminé). Un système linéaire est un ensemble de m équations linéaires `a n inconnues.

Extrait du sommaire:

1 Introduction 7
1.1 Notations 7
1.2 Objectif du cours 7
1.3 Application au calibrage d’un modèle 8
1.3.1 Exemple : interpolation et approximation polynomiale 8
1.4 Application `a un moteur de recherche 9
2 Bases de l’algèbre linéaire 11
2.1 Matrices et vecteurs 11
2.1.1 Matrices particulières 11
2.1.2 Opérations sur les matrices 11
2.1.3 Matrices carrées symétriques 11
2.1.4 Partitions par blocs 12
2.2 Normes matricielles 12
2.3 Orthogonalité dans Rn 12
2.4 Image, noyau, rang d’une matrice 13
2.5 Valeurs propres et vecteurs propres 15
2.6 Notions de complexité et de performances 15
2.7 Bibliothèques BLAS et LAPACK 16
2.7.1 Opérations BLAS1 16
2.7.2 Opérations BLAS2 16
2.7.3 Opérations BLAS3 16
2.7.4 Bibliothèque LAPACK 17
3 Arithmétique flottante 19
3.1 Erreurs de calcul 19
3.1.1 Sources d’erreur 19
3.1.2 Mesures de l’erreur 19
3.2 Arithmétique flottante 20
3.2.1 Format flottant 20
3.2.2 Arrondis 21
3.2.3 Opérations arithmétiques 22
3.2.4 Exceptions 22
3.2.5 Norme IEEE-754 23
3.2.6 Phénomène d’absorption 23
3.2.7 Phénomène de cancellation 24
3.2.8 Extensions de la norme IEEE 25
3.3 Stabilité des problèmes 26
3.3.1 Lien avec le résidu 26
3.3.2 Exemple : effet papillon 26
3.4 Stabilité et ordre des schémas de discrétisation 27
3.5 Convergence des algorithmes itératifs 27
3.6 Stabilité des algorithmes directs 29
3.6.1 Exemple : produit scalaire 29
3.6.2 Exemple : produit extérieur 30
4 Résolution de systèmes linéaires carrés 31
4.1 Inverse d’une matrice 31
4.1.1 Matrices particulières 32
4.2 Résolution d’un système linéaire 32
4.2.1 Cas des matrices symétriques définies positives 33
4.3 Analyse des erreurs pour un système linéaire 34
4.3.1 Stabilité numérique de Gauss et de Cholesky 34
4.3.2 Analyse de perturbation d’un système linéaire 35
4.3.3 Précision de la résolution d’un système linéaire 37
5 Problèmes aux moindres carrés – cas du rang plein 39
5.1 Existence et unicité d’une solution 39
5.1.1 Existence d’une solution dans le cas général 39
5.1.2 Condition d’unicité de la solution 40
5.2 Equations normales 40
5.3 Factorisation QR 41
5.4 Analyse d’erreur pour les problèmes aux moindres carrés de rang plein 42
5.4.1 Stabilité numérique de la factorisation QR 42
5.4.2 Analyse de perturbation et conditionnement 42
5.4.3 Précision d’une résolution de problème aux moindres carrés de rang plein 43
6 Décomposition en Valeurs Singulières 45
6.1 Diagonalisation d’une matrice carrée 45
6.1.1 Diagonalisation des matrices symétriques 46
6.2 Décomposition SVD (Singular Value Decomposition) d’une matrice rectangulaire 46
6.2.1 SVD réduite. Rang, image et noyau 48
6.3 Approximation de rang k et rang numérique 48
6.4 Calcul de la SVD 49
6.4.1 Bidiagonalisation 50
6.5 Analyse d’erreur du calcul de la SVD 51
6.5.1 Stabilité numérique du calcul d’une SVD 51
6.5.2 Analyse de perturbation des valeurs singulières 51
6.5.3 Précision du calcul des valeurs singulières 51
7 Problèmes aux moindres carrés – cas général 53
7.1 Problèmes aux moindres carrés – cas du rang plein 53
7.1.1 SVD, équations normales et factorisation QR 53
7.1.2 SVD, inverse et pseudo-inverse 54
7.2 Problèmes aux moindres carrés – cas général 54
7.2.1 Résolution d’un problème aux moindres carrés avec la SVD 56
7.3 SVD et conditionnement des matrices 56
8 Conclusion 57

Mathématique appliquée cours 17

Télécharger le fichier PDF: Cours MAP : Mathématiques Appliquées

Le blog contient des publicités, elles permettent de financer l'hébergement et maintenir le blog en fonctionnement. Vous pouvez utiliser adblock pour une lecture sans publicités.
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.
Cookies settings
Privacy & Cookie policy
Privacy & Cookies policy
Cookie name Active

Privacy Policy

What information do we collect?

We collect information from you when you register on our site or place an order. When ordering or registering on our site, as appropriate, you may be asked to enter your: name, e-mail address or mailing address.

What do we use your information for?

Any of the information we collect from you may be used in one of the following ways: To personalize your experience (your information helps us to better respond to your individual needs) To improve our website (we continually strive to improve our website offerings based on the information and feedback we receive from you) To improve customer service (your information helps us to more effectively respond to your customer service requests and support needs) To process transactions Your information, whether public or private, will not be sold, exchanged, transferred, or given to any other company for any reason whatsoever, without your consent, other than for the express purpose of delivering the purchased product or service requested. To administer a contest, promotion, survey or other site feature To send periodic emails The email address you provide for order processing, will only be used to send you information and updates pertaining to your order.

How do we protect your information?

We implement a variety of security measures to maintain the safety of your personal information when you place an order or enter, submit, or access your personal information. We offer the use of a secure server. All supplied sensitive/credit information is transmitted via Secure Socket Layer (SSL) technology and then encrypted into our Payment gateway providers database only to be accessible by those authorized with special access rights to such systems, and are required to?keep the information confidential. After a transaction, your private information (credit cards, social security numbers, financials, etc.) will not be kept on file for more than 60 days.

Do we use cookies?

Yes (Cookies are small files that a site or its service provider transfers to your computers hard drive through your Web browser (if you allow) that enables the sites or service providers systems to recognize your browser and capture and remember certain information We use cookies to help us remember and process the items in your shopping cart, understand and save your preferences for future visits, keep track of advertisements and compile aggregate data about site traffic and site interaction so that we can offer better site experiences and tools in the future. We may contract with third-party service providers to assist us in better understanding our site visitors. These service providers are not permitted to use the information collected on our behalf except to help us conduct and improve our business. If you prefer, you can choose to have your computer warn you each time a cookie is being sent, or you can choose to turn off all cookies via your browser settings. Like most websites, if you turn your cookies off, some of our services may not function properly. However, you can still place orders by contacting customer service. Google Analytics We use Google Analytics on our sites for anonymous reporting of site usage and for advertising on the site. If you would like to opt-out of Google Analytics monitoring your behaviour on our sites please use this link (https://tools.google.com/dlpage/gaoptout/)

Do we disclose any information to outside parties?

We do not sell, trade, or otherwise transfer to outside parties your personally identifiable information. This does not include trusted third parties who assist us in operating our website, conducting our business, or servicing you, so long as those parties agree to keep this information confidential. We may also release your information when we believe release is appropriate to comply with the law, enforce our site policies, or protect ours or others rights, property, or safety. However, non-personally identifiable visitor information may be provided to other parties for marketing, advertising, or other uses.


The minimum information we need to register you is your name, email address and a password. We will ask you more questions for different services, including sales promotions. Unless we say otherwise, you have to answer all the registration questions. We may also ask some other, voluntary questions during registration for certain services (for example, professional networks) so we can gain a clearer understanding of who you are. This also allows us to personalise services for you. To assist us in our marketing, in addition to the data that you provide to us if you register, we may also obtain data from trusted third parties to help us understand what you might be interested in. This ‘profiling’ information is produced from a variety of sources, including publicly available data (such as the electoral roll) or from sources such as surveys and polls where you have given your permission for your data to be shared. You can choose not to have such data shared with the Guardian from these sources by logging into your account and changing the settings in the privacy section. After you have registered, and with your permission, we may send you emails we think may interest you. Newsletters may be personalised based on what you have been reading on theguardian.com. At any time you can decide not to receive these emails and will be able to ‘unsubscribe’. Logging in using social networking credentials If you log-in to our sites using a Facebook log-in, you are granting permission to Facebook to share your user details with us. This will include your name, email address, date of birth and location which will then be used to form a Guardian identity. You can also use your picture from Facebook as part of your profile. This will also allow us and Facebook to share your, networks, user ID and any other information you choose to share according to your Facebook account settings. If you remove the Guardian app from your Facebook settings, we will no longer have access to this information. If you log-in to our sites using a Google log-in, you grant permission to Google to share your user details with us. This will include your name, email address, date of birth, sex and location which we will then use to form a Guardian identity. You may use your picture from Google as part of your profile. This also allows us to share your networks, user ID and any other information you choose to share according to your Google account settings. If you remove the Guardian from your Google settings, we will no longer have access to this information. If you log-in to our sites using a twitter log-in, we receive your avatar (the small picture that appears next to your tweets) and twitter username.

Children’s Online Privacy Protection Act Compliance

We are in compliance with the requirements of COPPA (Childrens Online Privacy Protection Act), we do not collect any information from anyone under 13 years of age. Our website, products and services are all directed to people who are at least 13 years old or older.

Updating your personal information

We offer a ‘My details’ page (also known as Dashboard), where you can update your personal information at any time, and change your marketing preferences. You can get to this page from most pages on the site – simply click on the ‘My details’ link at the top of the screen when you are signed in.

Online Privacy Policy Only

This online privacy policy applies only to information collected through our website and not to information collected offline.

Your Consent

By using our site, you consent to our privacy policy.

Changes to our Privacy Policy

If we decide to change our privacy policy, we will post those changes on this page.
Save settings
Cookies settings

You have successfully subscribed to the newsletter

There was an error while trying to send your request. Please try again.

FPGA | Arduino | Matlab | Cours will use the information you provide on this form to be in touch with you and to provide updates and marketing.