Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Multi-input and Multi-variable systems01:22

Multi-input and Multi-variable systems

508
Cruise control systems in cars are designed as multi-input systems to maintain a driver's desired speed while compensating for external disturbances such as changes in terrain. The block diagram for a cruise control system typically includes two main inputs: the desired speed set by the driver and any external disturbances, such as the incline of the road. By adjusting the engine throttle, the system maintains the vehicle's speed as close to the desired value as possible.
In the absence of...
508
Synthetic Disvision of Polynomials01:28

Synthetic Disvision of Polynomials

348
Synthetic division is an efficient algorithmic approach for dividing a polynomial by a linear binomial of the form x - c, where c is a real number. This method is helpful due to its streamlined process, which avoids the more cumbersome steps involved in the traditional long division of polynomials. It simplifies computation and serves as a practical tool for evaluating polynomials and identifying their factors.To perform synthetic division, one begins by listing the coefficients of the...
348
Vector Algebra: Method of Components01:08

Vector Algebra: Method of Components

20.8K
It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
In many applications, the magnitudes and directions of...
20.8K
Fundamental Theorem of Algebra01:30

Fundamental Theorem of Algebra

459
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as:  with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the...
459
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

457
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
457
Linearization and Approximation01:26

Linearization and Approximation

213
Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
213

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Engineering multispecific antibodies with complete killing selectivity through the closed-loop integration of machine learning and high-throughput experimentation.

mAbs·2025
Same author

Elliptic PDE learning is provably data-efficient.

Proceedings of the National Academy of Sciences of the United States of America·2023
Same author

A global synchronization theorem for oscillators on a random graph.

Chaos (Woodbury, N.Y.)·2022
Same author

Data-driven discovery of Green's functions with human-understandable deep learning.

Scientific reports·2022
Same author

Sufficiently dense Kuramoto networks are globally synchronizing.

Chaos (Woodbury, N.Y.)·2021
Same author

Exponential node clustering at singularities for rational approximation, quadrature, and PDEs.

Numerische mathematik·2021
Same journal

Computational modelling distinguishes diverse contributors to aneurysmal progression in the Marfan aorta.

Proceedings. Mathematical, physical, and engineering sciences·2025
Same journal

Inferring the shape of data: a probabilistic framework for analysing experiments in the natural sciences.

Proceedings. Mathematical, physical, and engineering sciences·2023
Same journal

The Elbert range of magnetostrophic convection. I. Linear theory.

Proceedings. Mathematical, physical, and engineering sciences·2022
Same journal

Soft wetting with (a)symmetric Shuttleworth effect.

Proceedings. Mathematical, physical, and engineering sciences·2022
Same journal

The quantum theory of time: a calculus for q-numbers.

Proceedings. Mathematical, physical, and engineering sciences·2022
Same journal

Integrable nonlinear evolution equations in three spatial dimensions.

Proceedings. Mathematical, physical, and engineering sciences·2022
See all related articles

Related Experiment Video

Updated: Apr 18, 2026

Author Spotlight: Integrated Multi-Omics Analysis for Unveiling Multicellular Immune Signatures in Clinical Heart Attack Cohorts
08:51

Author Spotlight: Integrated Multi-Omics Analysis for Unveiling Multicellular Immune Signatures in Clinical Heart Attack Cohorts

Published on: September 20, 2024

2.4K

Continuous analogues of matrix factorizations.

Alex Townsend1, Lloyd N Trefethen2

  • 1Department of Mathematics , MIT , Cambridge, MA 02139, USA.

Proceedings. Mathematical, Physical, and Engineering Sciences
|January 9, 2015
PubMed
Summary
This summary is machine-generated.

This study introduces new matrix factorization methods for quasimatrices and cmatrices, extending singular value decomposition (SVD) and other techniques. The research addresses challenges in generalizing matrix operations to continuous variables and proves convergence for cmatrix factorizations under specific conditions.

Keywords:
ChebfunCholeskyLUQRsingular value decomposition

More Related Videos

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

9.8K
Author Spotlight: A Novel Approach for Embedding Cell-Free Protein Synthesis Reactions in Hydrogels
06:38

Author Spotlight: A Novel Approach for Embedding Cell-Free Protein Synthesis Reactions in Hydrogels

Published on: June 23, 2023

2.0K

Related Experiment Videos

Last Updated: Apr 18, 2026

Author Spotlight: Integrated Multi-Omics Analysis for Unveiling Multicellular Immune Signatures in Clinical Heart Attack Cohorts
08:51

Author Spotlight: Integrated Multi-Omics Analysis for Unveiling Multicellular Immune Signatures in Clinical Heart Attack Cohorts

Published on: September 20, 2024

2.4K
Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

9.8K
Author Spotlight: A Novel Approach for Embedding Cell-Free Protein Synthesis Reactions in Hydrogels
06:38

Author Spotlight: A Novel Approach for Embedding Cell-Free Protein Synthesis Reactions in Hydrogels

Published on: June 23, 2023

2.0K

Area of Science:

  • Numerical analysis
  • Linear algebra
  • Scientific computing

Background:

  • Traditional matrix factorizations like SVD, QR, LU, and Cholesky are fundamental in numerical analysis.
  • These methods are typically applied to discrete matrices.
  • Extending these powerful tools to continuous domains remains a significant challenge.

Purpose of the Study:

  • To develop analogues of standard matrix factorizations for quasimatrices (continuous in one dimension) and cmatrices (continuous in two dimensions).
  • To generalize concepts of triangular structure and pivoting to continuous variables.
  • To investigate the convergence properties of these new factorization methods.

Main Methods:

  • Generalization of triangular structure and pivoting for quasimatrices and cmatrices.
  • Development of new algorithms based on these generalized concepts.
  • Mathematical proofs to establish convergence criteria for cmatrix factorizations.

Main Results:

  • Analogues of SVD, QR, LU, and Cholesky factorizations are presented for quasimatrices and cmatrices.
  • A novel definition of a 'triangular quasimatrix' is introduced, enabling generalized pivoting.
  • Theorems are proven demonstrating the convergence of cmatrix factorizations for sufficiently smooth functions.

Conclusions:

  • The study successfully extends discrete matrix factorization techniques to continuous domains.
  • The introduced generalizations of triangularity and pivoting are crucial for applying these methods.
  • The convergence proofs provide a theoretical foundation for the practical application of cmatrix factorizations.