The method of least squares is used to solve overdetermined systems. Least squares is often applied in statistical contexts, particularly regression analysis.Least squares can be interpreted as a method of fitting data. The best fit in the least-squares sense is that instance of the model for which the sum of squared residuals has its least value, a residual being the difference between an observed value and the value given by the model. The method was first described by Carl Friedrich Gauss around 1794.[1] Least squares corresponds to the maximum likelihood criterion if the experimental errors have a normal distribution and can also be derived as a method of moments estimator. Regression analysis is available in most statistical software packages.
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and both of unit length. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by linear ones.
A linear function is a real or complex function f with the functional equation y=f(x)=m⋅x+b, where m and b are real (or complex) numbers. The equation y=f(x)=m⋅x+b is called (general) linear equation. The graph of a real linear function is a straight line.
Introduction to applied linear algebra and linear dynamical systems, with applications to circuits, signal processing, communications, and control systems. Topics include: Least-squares aproximations of over-determined equations and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical interpretation. Matrix exponential, stability, and asymptotic behavior. Multi-input multi-output systems, impulse and step matrices; convolution and transfer matrix descriptions.
Convex optimization is a subfield of mathematical optimization. Given a real vector space together with a convex, real-valued function
defined on a convex subset of , the problem is to find the point in for which the number is smallest, i.e., the point such that for all .
The convexity of and make the powerful tools of convex analysis applicable: the Hahn–Banach theorem and the theory of subgradients lead to a particularly satisfying and complete theory of necessary and sufficient conditions for optimality, a duality theory comparable in perfection to that for linear programming, and effective computational methods. Convex optimization has applications in a wide range of disciplines, such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, statistics, and finance. Modern computing power has improved the tractability of convex optimization problems to a level roughly equal to that of linear programming.
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Lecture by Professor Stephen Boyd for Convex Optimization II (EE 364B) in the Stanford Electrical Engineering department. Professor Boyd lectures on the localization and cutting-plane methods and then moves into the Analytic center cutting-plane methods.
The computing environment of tomorrow will not be anything like the stand-alone environment of yesterday. Computing infrastructure is changing drastically, presenting unique opportunities as well as challenges for applications and application scientists. Despite the rapid onset of distributed infrastructure, the majority of scientists still use the same programming methods that they are accustomed to using on isolated computers. On the other hand, the underlying infrastructure does not support seamless extensibility or scaling of applications from desktops to distributed heterogenous resources. An important feature of this theme will be its focus on applications, and thus a consequent "top-down" approach to distributed programming abstractions. In particular, this Theme will examine how scientific applications can be programmed such that they can easily utilise distributed infrastructure, and how computer scientists can help design and implement applications that are shielded from future trends and changes in computing environments. The Theme aims to address these issues by focusing, amongst many others, on the following questions: What are the main barriers to wide-spread distributed High Performance Computing applications? Is there a level at which simple yet powerful programming abstractions can be effective in distributed environments?
Over the course of this theme, there will be a series of correlated workshops and meetings to explore many of these issues. In addition to surveying the status of the field and addressing the high level questions, this theme aims to spawn focussed research groups that will examine in greater detail some of the specific ideas.
Special relativity (SR) (also known as the special theory of relativity or STR) is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein (after considerable contributions of Hendrik Lorentz and Henri Poincaré) in the paper "On the Electrodynamics of Moving Bodies". It generalizes Galileo's principle of relativity – that all uniform motion is relative, and that there is no absolute and well-defined state of rest (no privileged reference frames) – from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be. In addition, special relativity incorporates the principle that the speed of light is the same for all inertial observers regardless of the state of motion of the source.
This theory has a wide range of consequences which have been experimentally verified. Special relativity overthrows Newtonian notions of absolute space and time by stating that time and space are perceived differently by observers in different states of motion. It yields the equivalence of matter and energy, as expressed in the mass-energy equivalence formula E = mc2, where c is the speed of light in a vacuum. The predictions of special relativity agree well with Newtonian mechanics in their common realm of applicability, specifically in experiments in which all velocities are small compared to the speed of light.
The theory is termed "special" because it applies the principle of relativity only to inertial frames. Einstein developed general relativity to apply the principle generally, that is, to any frame, and that theory includes the effects of gravity. Strictly, special relativity cannot be applied in accelerating frames or in gravitational fields.
Special relativity reveals that c is not just the velocity of a certain phenomenon, namely the propagation of electromagnetic radiation (light)—but rather a fundamental feature of the way space and time are unified as spacetime. A consequence of this is that it is impossible for any particle that has mass to be accelerated to the speed of light.
