Linear approximation, the Glossary
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).[1]
Table of Contents
35 relations: Affine transformation, Amplitude, Angle, Banach space, Binomial approximation, Concave function, Convex function, Euclidean vector, Euler method, Finite difference, Finite difference method, Fréchet derivative, Frequency, Function (mathematics), Geometrical optics, Gravitational acceleration, Inflection point, Isochronous timing, Jacobian matrix and determinant, Length, Linear function, Mass, Mathematics, Newton's method, Optical axis, Paraxial approximation, Pendulum (mechanics), Power series, Real number, Second derivative, Series (mathematics), Sphere, Tangent, Taylor series, Taylor's theorem.
- First order methods
Affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
See Linear approximation and Affine transformation
Amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period).
See Linear approximation and Amplitude
Angle
In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
See Linear approximation and Angle
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
See Linear approximation and Banach space
Binomial approximation
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that It is valid when |x| and |\alpha x| \ll 1 where x and \alpha may be real or complex numbers.
See Linear approximation and Binomial approximation
Concave function
In mathematics, a concave function is one for which the value at any convex combination of elements in the domain is greater than or equal to the convex combination of the values at the endpoints.
See Linear approximation and Concave function
Convex function
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points.
See Linear approximation and Convex function
Euclidean vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction.
See Linear approximation and Euclidean vector
Euler method
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. Linear approximation and Euler method are first order methods.
See Linear approximation and Euler method
Finite difference
A finite difference is a mathematical expression of the form. Linear approximation and finite difference are numerical analysis.
See Linear approximation and Finite difference
Finite difference method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences.
See Linear approximation and Finite difference method
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on normed spaces.
See Linear approximation and Fréchet derivative
Frequency
Frequency (symbol f), most often measured in hertz (symbol: Hz), is the number of occurrences of a repeating event per unit of time.
See Linear approximation and Frequency
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of.
See Linear approximation and Function (mathematics)
Geometrical optics
Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of rays.
See Linear approximation and Geometrical optics
Gravitational acceleration
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag).
See Linear approximation and Gravitational acceleration
Inflection point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. Linear approximation and inflection point are differential calculus.
See Linear approximation and Inflection point
Isochronous timing
A sequence of events is isochronous if the events occur regularly, or at equal time intervals.
See Linear approximation and Isochronous timing
Jacobian matrix and determinant
In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. Linear approximation and Jacobian matrix and determinant are differential calculus.
See Linear approximation and Jacobian matrix and determinant
Length
Length is a measure of distance.
See Linear approximation and Length
Linear function
In mathematics, the term linear function refers to two distinct but related notions.
See Linear approximation and Linear function
Mass
Mass is an intrinsic property of a body.
See Linear approximation and Mass
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Linear approximation and Mathematics
Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.
See Linear approximation and Newton's method
Optical axis
An optical axis is an imaginary line that passes through the geometrical center of an optical system such as a camera lens, microscope or telescopic sight.
See Linear approximation and Optical axis
Paraxial approximation
In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens).
See Linear approximation and Paraxial approximation
Pendulum (mechanics)
A pendulum is a body suspended from a fixed support such that it freely swings back and forth under the influence of gravity.
See Linear approximation and Pendulum (mechanics)
Power series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n.
See Linear approximation and Power series
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.
See Linear approximation and Real number
Second derivative
In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of. Linear approximation and second derivative are differential calculus.
See Linear approximation and Second derivative
Series (mathematics)
In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
See Linear approximation and Series (mathematics)
Sphere
A sphere (from Greek) is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.
See Linear approximation and Sphere
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point.
See Linear approximation and Tangent
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point.
See Linear approximation and Taylor series
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial.
See Linear approximation and Taylor's theorem
See also
First order methods
- Barzilai-Borwein method
- Euler method
- Finite differences
- Frank–Wolfe algorithm
- Gradient descent
- Gradient method
- Linear approximation
- Proximal gradient methods for learning
- Schild's ladder
- Structured sparsity regularization
References
[1] https://en.wikipedia.org/wiki/Linear_approximation
Also known as Affine approximation, Approximation of functions, Approximation of functions, linear methods, Tangent Line Approximation, Tangent Plane Approximation.