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Approximation of a function by its tangent line at a point
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are
Linear_approximation
Finding linear approximation of function at given point
mathematics, linearization (British English: linearisation) is finding the linear approximation to a function at a given point. The linear approximation of a
Linearization
Least squares approximation of linear functions to data
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems
Linear_least_squares
Arithmetic operation
− 1 1 {\displaystyle z={^{-1}1}} . A linear approximation (solution to the continuity requirement, approximation to the differentiability requirement)
Tetration
Form of cryptanalysis
In cryptography, linear cryptanalysis is a general form of cryptanalysis based on finding affine approximations to the action of a cipher. Attacks have
Linear_cryptanalysis
Something roughly the same as something else
An approximation is anything that is intentionally similar but not exactly equal to something else. The word approximation is derived from Latin approximatus
Approximation
Approximation of powers of some binomials
f(x)=(1+x)^{\alpha }} is a smooth function for x near 0. Thus, standard linear approximation tools from calculus apply: one has f ′ ( x ) = α ( 1 + x ) α − 1
Binomial_approximation
Approximation of a function by a polynomial
polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are
Taylor's_theorem
Type of mathematical function
function Triangular function An approximation to a known curve can be found by sampling the curve and interpolating linearly between the points. An algorithm
Piecewise_linear_function
Function returning one of only two values
of all linear functions of output bits, known as its components. The set of Walsh transforms of the components is known as a linear approximation table
Boolean_function
Approximation method in statistics
is known as the shift vector. At each iteration the model is linearized by approximation to a first-order Taylor polynomial expansion about β k {\displaystyle
Non-linear_least_squares
Analysis and synthesis of electronic circuits
piecewise approximation, harmonic balance or other methods. Software such as the PLECS interface to Simulink uses piecewise linear approximation of the equations
Electronic_circuit_design
Matrix of partial derivatives of a vector-valued function
y to f(x) + Jf(x) ⋅ (y – x) is the best linear approximation of f(y) for all points y close to x. The linear map h → Jf(x) ⋅ h is known as the derivative
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Assemblage of connected electrical elements
linear circuit matrix can be solved with Gaussian elimination. Software such as the PLECS interface to Simulink uses piecewise-linear approximation of
Electrical_network
Projection of data onto lower-dimensional manifolds
quality of data approximation and some penalty terms for the bending of the manifold. The popular initial approximations are generated by linear PCA and Kohonen's
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
Optimization algorithm
Frank–Wolfe algorithm considers a linear approximation of the objective function, and moves towards a minimizer of this linear function (taken over the same
Frank–Wolfe_algorithm
Study of rates of change
derivative of a function at a point generally determines the best linear approximation to the function at that point. Differential calculus is one of the
Differential_calculus
Linear map or polynomial function of degree one
Piecewise linear function Linear approximation Linear interpolation Discontinuous linear map Linear least squares "The term linear function means a linear form
Linear_function
Algorithm for finding zeros of functions
f(x0)): that is, the improved guess, x1, is the unique root of the linear approximation of f at the initial guess, x0. The process is repeated as x n + 1
Newton's_method
Measure of a substance's ability to resist or conduct electric current
{\displaystyle \kappa } [clarification needed]. Because the linear approximation is only an approximation, α {\displaystyle \alpha } is different for different
Electrical resistivity and conductivity
Electrical_resistivity_and_conductivity
Force needed to pull a spring grows linearly with distance
equation can be assumed is said to be linear-elastic or Hookean. Hooke's law is a first-order linear approximation to the real response of springs and other
Hooke's_law
Algorithms for calculating square roots
the approximation, though not all approximations are polynomial. Common methods of estimating include scalar, linear, hyperbolic and logarithmic. A decimal
Square_root_algorithms
Type of activation function
sum of an exponential number of linear models that share parameters. They then proposed ReLU as a good approximation to it. Specifically, they began by
Rectified_linear_unit
Mathematical analysis technique
is linear with L, which can be deduced from the fact that the partial with respect to (w.r.t.) L does not depend on L. Thus the linear "approximation" turns
Experimental uncertainty analysis
Experimental_uncertainty_analysis
Musical interval unit
the function 2x increases almost linearly from 1.00000 to 1.05946, allowing for a piecewise linear approximation. Thus, although cents represent a logarithmic
Cent_(music)
System used in chess to estimate the strength of a player
new rating is 2635 + 10 × (10.5 − 9.66) = 2643.4. Elo devised a linear approximation to his full system, negating the need for look-up tables of expected
Chess_rating_system
Linear approximation of smooth maps on tangent spaces
In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that φ : M → N {\displaystyle
Pushforward_(differential)
Type of derivative in mathematics
the linear part of the best affine approximation to the function near the point. In one-variable calculus, this is the tangent line approximation. In
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Mathematical concept
The approximation error in a given data value represents the significant discrepancy that arises when an exact, true value is compared against some approximation
Approximation_error
Theorem in mathematics
have an inverse function. The essential idea is that if the best linear approximation to the function at a point is invertible, then with sufficient regularity
Inverse_function_theorem
Physical quantity that expresses internal forces in a continuous material
strain, and the strain rate can be quite complicated, although a linear approximation may be adequate in practice if the quantities are sufficiently small
Stress_(mechanics)
Class of algorithms that find approximate solutions to optimization problems
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems
Approximation_algorithm
Theory of stochastic processes
error is necessarily smaller than the error of a linear approximation which selects the M approximation vectors independently of f. Let us sort { | ⟨ f
Kosambi–Karhunen–Loève theorem
Kosambi–Karhunen–Loève_theorem
Analytical expression in statistics
Laplace's approximation or the quadratic approximation (QUAP) provides an analytical expression for a posterior probability distribution by fitting a Gaussian
Laplace's_approximation
Multivariate derivative (mathematics)
^{n}} characterizes the best linear approximation to f {\displaystyle f} at x 0 {\displaystyle x_{0}} . The approximation is as follows: f ( x ) ≈ f (
Gradient
Method of curve fitting
"curvier" the function is, the worse the approximations made with simple linear interpolation become. Linear interpolation has been used since antiquity
Linear_interpolation
Law of electrical current and voltage
is linear in current. The voltage drop across the circuit includes additionally the Seebeck thermoelectromotive force which again is again linear in current
Ohm's_law
Instantaneous rate of change (mathematics)
graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. The derivative is often described
Derivative
Exponent of a power of two
approximated as log 2 x ≈ E + m {\displaystyle \log _{2}x\approx E+m} . This approximation is exact at both ends of the range of mantissas but underestimates the
Binary_logarithm
Partial differential equation with nonlinear terms
equation has to be studied as a separate problem. The distinction between a linear and a nonlinear partial differential equation is usually made in terms of
Nonlinear partial differential equation
Nonlinear_partial_differential_equation
Thought experiments
usually derived by using the implicit function theorem to calculate a linear approximation to the system of equations that defines the equilibrium, under the
Comparative_statics
Concept in integral mathematics
implies that the approximation ratio in the linear programming relaxation might be bad, and it may be better to look for other approximation schemes for that
Linear_programming_relaxation
Root-finding algorithm
that I x {\displaystyle I_{x}} is a scaled and shifted piecewise-linear approximation of log 2 ( x ) {\displaystyle \log _{2}(x)} , as illustrated in
Fast_inverse_square_root
Calculus of vector-valued functions
theorems reduce to the Green's theorem: Linear approximations are used to replace complicated functions with linear functions that are almost the same. Given
Vector_calculus
Property of a mass in motion
ρvj by a velocity vi is equal to ρvjvj.[dubious – discuss] In the linear approximation that leads to the above acoustic equation, the time average of this
Momentum
Equations describing classical electromagnetism
however, the linear approximation can break down in common materials like iron leading to phenomena like hysteresis. Even the linear case can have various
Maxwell's_equations
Differential equation parameter in thermal physics
temperature and α Δ T ≪ 1 {\displaystyle \alpha \Delta T\ll 1} , a linear approximation will be useful in estimating the value R of a property at a temperature
Temperature_coefficient
Numerical approximation algorithm
improving approximate solutions for a class of problems, in which the i-th approximation (called an "iterate") is derived from the previous ones. A specific
Iterative_method
Free swinging suspended body
K\left(\sin {\frac {10^{\circ }}{2}}\right)\approx 2.0102{\text{ s}}.} The linear approximation gives 2 π 1 m g ≈ 2.0064 s . {\displaystyle 2\pi {\sqrt {\frac
Pendulum_(mechanics)
Form of cryptanalysis
characteristic have a linear approximation defined, and we expect that for each chosen plaintext pair, the probability of the linear approximation holding for one
Differential-linear_attack
Algorithm for approximating nonlinear relationships
tabulation (ISAT) is an algorithm for the approximation of nonlinear relationships. ISAT is based on multiple linear regressions that are dynamically added
In_situ_adaptive_tabulation
Approximation valid for weakly non-linear and fairly long waves
the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph
Boussinesq approximation (water waves)
Boussinesq_approximation_(water_waves)
Approach to feature extraction
two dimensions for the 2-D linear approximation occurs differently. This can be corrected by applying the linear approximation in the form: F ( x + h )
Kanade–Lucas–Tomasi feature tracker
Kanade–Lucas–Tomasi_feature_tracker
Ratio of competing statistical models
not necessary; for instance, it could also be a non-linear model compared to its linear approximation. The Bayes factor can be thought of as a Bayesian
Bayes_factor
Formula for the derivative of a product
} are differentiable at x {\displaystyle x} , then we can write linear approximations: f ( x + h ) = f ( x ) + f ′ ( x ) h + ε 1 ( h ) {\displaystyle
Product_rule
Relations between flows and forces, or gradients, in thermodynamic systems
to the principle of detailed balance and follow from them in the linear approximation near equilibrium. Experimental verifications of the Onsager reciprocal
Onsager_reciprocal_relations
Numerical method for solving physical or engineering problems
equations are often partial differential equations (PDEs). To explain the approximation of this process, FEM is commonly introduced as a special case of the
Finite_element_method
Polynomial function of degree at most one
f ′ ( c ) {\displaystyle f\,'(c)} is the slope of this linear function, and the approximation is: f ( x ) ≈ f ′ ( c ) ( x − c ) + f ( c ) {\displaystyle
Linear_function_(calculus)
Branch of mathematics
disciplines. For example, it can be used with linear algebra to find the "best fit" linear approximation for a set of points in a domain. Or, it can be
Calculus
Technique in numerical linear algebra
In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization
Low-rank_approximation
Branch of mathematics
studies functions, spaces, and operators through quantitative methods of approximation and convergence. It grew out of calculus, especially the use of derivatives
Mathematical_analysis
theorem Journals: Constructive Approximation Journal of Approximation Theory Extrapolation Linear predictive analysis — linear extrapolation Unisolvent functions
List of numerical analysis topics
List_of_numerical_analysis_topics
Branch of mathematics
systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential
Linear_algebra
Concept in mathematics
linear approximation, the parameters α {\displaystyle \displaystyle \alpha } and β {\displaystyle \displaystyle \beta } are the same as in the linear
Nonlinear conjugate gradient method
Nonlinear_conjugate_gradient_method
Type of mathematical system
Linear dynamical systems are dynamical systems whose evolution functions are linear. While dynamical systems, in general, do not have closed-form solutions
Linear_dynamical_system
Opposition to the passage of an electric current
parameter fitted from measurement data. Because the linear approximation is only an approximation, α {\displaystyle \alpha } is different for different
Electrical resistance and conductance
Electrical_resistance_and_conductance
Mathematical approximation of a function
axis is studied by replacing the sine function with its linear approximation. Such approximations are used throughout mathematics, physics, and engineering
Taylor_series
Method to solve optimization problems
LPs commonly arise as a linear programming relaxation of a combinatorial problem and are important in the study of approximation algorithms. For example
Linear_programming
Method in cosmology and astrophysics
1970. The approximation provides a relatively simple way to model the growth of structure in the early universe, bridging the gap between linear theory and
Zeldovich_approximation
approximations: ultra_fast_sigmoid, which is a multi-part piecewise approximation and hard_sigmoid, which is a 3-part piecewise linear approximation (output
Hard_sigmoid
Theory of getting acceptably close inexact mathematical calculations
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing
Approximation_theory
Unstable behavior of two contacting fluids of different densities
exponentially with time. In the non-linear phase, perturbation amplitude is too large for a linear approximation, and non-linear equations are required to describe
Rayleigh–Taylor_instability
Property of artificial neural networks
In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate
Universal approximation theorem
Universal_approximation_theorem
Solution process for some optimization problems
into subclasses to be solved with convex (minimization problem) or linear approximations that form a lower bound on the overall cost within the subdivision
Nonlinear_programming
Statistics model
logarithm of a Weibull distribution, the complementary log-log model. Linear approximation Cox, D. R. (1970). "Simple Regression". Analysis of Binary Data.
Linear_probability_model
Rate of separation of infinitesimally close trajectories
diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by | δ ( t ) | ≈ e λ t | δ 0 | {\displaystyle
Lyapunov_exponent
Generalized function whose value is zero everywhere except at zero
is 1). In some situations such as numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking η1 to
Dirac_delta_function
structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved are also topological
Discontinuous_linear_map
Mathematics of real numbers and real functions
approximation is. One example of this is the Taylor remainder, which gives an effective and computable constant that determines how well the linear approximation
Real_analysis
Apparent temperature estimating how humans are affected
equation from a four-degree polynomial to a linear relationship (the authors noted that the linear approximation was not tested for air temperatures under
Wet-bulb_globe_temperature
Solution method for linear differential equations
In mathematical physics, the WKB approximation or WKB method is a technique for finding approximate solutions to linear differential equations with spatially
WKB_approximation
Method for estimating new data outside known data points
that eventually diverge away from the x-axis even faster than the linear approximation. This divergence is a specific property of extrapolation methods
Extrapolation
Function in statistics
359-366. H. Guo, B. Makki, M. -S. Alouini and T. Svensson, "A Semi-Linear Approximation of the First-Order Marcum Q-Function With Application to Predictor
Marcum_Q-function
Statistical method for resampling
(n-1)} obtained by omitting one observation. The jackknife is a linear approximation of the bootstrap. The jackknife technique was developed by Maurice
Jackknife_resampling
Number whose square is a given number
is close to the square root, for which a polynomial or piecewise-linear approximation can be used. The time complexity for computing a square root with
Square_root
American mathematician
the content-rich („inhaltsreiche“) Linear Approximation is an important contribution to the theory of approximation of integrals, derivatives, function
Arthur_Sard
Approximation method in statistics
theorem supports the idea that this is a good approximation in many cases. The Gauss–Markov theorem. In a linear model in which the errors have expectation
Least_squares
Method for solving continuous operator problems (such as differential equations)
method, one also gives the name along with typical assumptions and approximation methods used: Ritz–Galerkin method (after Walther Ritz) typically assumes
Galerkin_method
Image luminance mapping function
because the floating-point format already provides a piecewise linear approximation of a logarithmic curve. Although gamma encoding was developed originally
Gamma_correction
Part of mathematics that addresses the stability of solutions
greater than 1. This is because near the point a, the function f has a linear approximation with slope f'(a): f ( x ) ≈ f ( a ) + f ′ ( a ) ( x − a ) . {\displaystyle
Stability_theory
Predicted physical phenomenon
has been suggested as a way of validating general relativity. A linear approximation of Einstein's equations was first proposed in 1974 by the Soviet
Gravitational_memory_effect
Several equations of degree 1 to be solved simultaneously
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. For example
System_of_linear_equations
Linear perturbations to solutions of nonlinear Einstein field equations
In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime
Linearized_gravity
Statistical modeling method
In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory
Linear_regression
Methods for numerical approximations
systems of linear equations, and the simplex method of linear programming. In practice, finite precision is used and the result is an approximation of the
Numerical_analysis
Machine learning technique
Hence, moving a small amount γ {\displaystyle \gamma } such that the linear approximation remains valid: F m ( x ) = F m − 1 ( x ) − γ ∑ i = 1 n ∇ F m − 1
Gradient_boosting
Branch of actuarial mathematics
best linear approximation to the mean of the Bayesian predictive density, which is why credibility theory has many results in common with linear filtering
Credibility_theory
Concept in mathematics
Sparse approximation (also known as sparse representation) theory deals with sparse solutions for systems of linear equations. Techniques for finding these
Sparse_approximation
Regression analysis
}}^{-1}(\mathbf {d} -\mathbf {Y{\bar {m}})} } (see also linear least squares). The linear approximation introduces bias into the statistics. Therefore, more
Nonlinear_regression
Technique pioneered by Nico van Kampen
an approximation to the solution of a master equation with nonlinear transition rates. The leading order term of the expansion is given by the linear noise
System_size_expansion
LINEAR APPROXIMATION
LINEAR APPROXIMATION
Surname or Lastname
English
English : variant of Lanier 1.Dutch : variant of Leonard.Jewish (western Ashkenazic) : name taken by someone who was good at chanting the Pentateuch at public worship in the synagogue or who regularly did so, from West Yiddish layner ‘reader’ (a derivative of West Yiddish laynen ‘to read’, which comes ultimately from Latin legere ‘to read’).Jewish (Ashkenazic) : occupational name for a flax grower or merchant, from German Lein ‘flax’ + agent suffix -er.
Surname or Lastname
English (Cornish)
English (Cornish) : habitational name from a place named with Cornish lan ‘church’. In England this surname is now found chiefly in the southern counties of Wiltshire and Hampshire, and Berkshire; it has no doubt moved there from Cornwall.
Female
English
English name probably derived from Germanic lindi, LINDA means "serpent."Â In some cases, it may have been derived from the Spanish word for "pretty."
Surname or Lastname
English
English : habitational name from Lingart, Lancashire, or Lingards Wood in Marsden, West Yorkshire, both named from Old English līn ‘flax’ + garðr ‘enclosure’.
Female
Scottish
Variant spelling of Scottish Lilias, LILEAS means "lily."
Surname or Lastname
English
English : occupational name for a whitewasher, Middle English limer, lymer, an agent derivative of Old English līm ‘lime’.
Surname or Lastname
English (Devon; of Cornish origin)
English (Devon; of Cornish origin) : topographic name for someone who lived by a menhir, i.e. a tall standing stone erected in prehistoric times (Cornish men ‘stone’ + hir ‘long’).
Male
English
Irish Anglicized form of Gaelic Fionnbarr, FINBAR means "fair-headed."
Boy/Male
Sikh
Love unending
Girl/Female
Irish
Eimear possessed the “Six Gifts of Womanhood†– “beauty, a gentle voice, sweet words, wisdom, needlework and chastity!†She was bethrothed to the warrior Cuchulainn (read the legend) when they were children and they loved each other very deeply. But Cuchulainn had “a wandering eye†and Eimear endured this, realizing “everything new is fair,†but when he made love to Fand, wife of the sea god Manannan, Eimear confronted the lovers. After seeing the strength of Fand’s love she offered to withdraw. Touched by this display of unselfishness, Fand left Cuchulainn and returned to the sea. When Cuchulainn died Eimear spoke movingly and lovingly at his graveside.
Female
English
Variant spelling of English Linsey, LINSAY means "Lincoln's wetlands."
Male
Scandinavian
Scandinavian form of Old Norse Einarr, EINAR means "lone warrior."
Surname or Lastname
English
English : variant of Lingard.French : occupational name for a maker of or dealer in linen goods, from Old French linge ‘linen (goods)’ (see Linge 1).
Male
Greek
(ΑἰνÎας) Variant spelling of Greek AineÃas, AINEAS means "praiseworthy."
Boy/Male
Irish
Meaning “â€fair-haired,â€â€ the name has been popular since the sixth century when St. Finbar came to an area of Cork that was being tormented by a serpent. The people begged him to do something to help them. One night he went to where the serpent was sleeping and sprinkled it with holy water. The angry serpent tore and devoured the land until she slithered into the sea at Cork Harbor. The track she left behind filled with water and became the River Lee and that’s why St. Finbar is the patron saint of Cork. It is said that the sun didn’t set for two weeks after Finbar’s death.
Boy/Male
Hindu
Lingam
Surname or Lastname
English
English : metronymic from Line.
Male
Yiddish
 Variant spelling of Yiddish Lieber, LIBER means "beloved." Compare with another form of Liber.
Boy/Male
Hindu
The Sun
Surname or Lastname
Swedish
Swedish : ornamental name from lind ‘lime tree’ + either the German suffix -er denoting an inhabitant, or the surname suffix -ér, derived from the Latin adjectival ending -er(i)us.English (mainly southeastern) : variant of Lind 2.German : habitational name from any of numerous places called Linden or Lindern, named with German Linden ‘lime trees’.
LINEAR APPROXIMATION
LINEAR APPROXIMATION
Male
Polish
Polish form of Roman Latin Lucius, ÅUCJUSZ means "light."
Boy/Male
Hindu
Girl/Female
Australian, Hebrew
Musical Grace; Poetic
Boy/Male
Gaelic
Hero.
Surname or Lastname
English
English : habitational name from places in Berkshire and Devon. The former is probably named with Old French bray ‘marsh’, the latter from the Cornish element bre ‘hill’.
Girl/Female
Tamil
Vimple | விமà¯à®ªà¯à®²
Loved one
Girl/Female
Welsh American
meaning bitter. Favored prefix for blended names like Maribel.
Male
English
Masculine variant spelling of English unisex Aubrey, AUBRY means "elf ruler."
Boy/Male
Arabic, Indian, Muslim
Holly Water
Boy/Male
Indian
Lord Ganesh, Against desire
LINEAR APPROXIMATION
LINEAR APPROXIMATION
LINEAR APPROXIMATION
LINEAR APPROXIMATION
LINEAR APPROXIMATION
a.
Linear.
a.
Like a line; narrow; of the same breadth throughout, except at the extremities; as, a linear leaf.
n.
Made of linen; as, linen cloth; a linen stocking.
adv.
In a linear manner; with lines.
prep. & adv.
Near.
v. t.
To mark with a line or lines; to cover with lines; as, to line a copy book.
v. t.
To convert into vinegar; to make like vinegar; to render sour or sharp.
a.
In the direction of a line; of or pertaining to a line; measured on, or ascertained by, a line; linear; as, lineal magnitude.
a.
Of, pertaining to, or included by, two lines; as, bilinear coordinates.
a.
Formed by right lines; rectilineal; as, a right-lined angle.
n.
One who lines, as, a liner of shoes.
a.
Of or pertaining to a line; consisting of lines; in a straight direction; lineal.
n.
A lunar distance.
a.
Descending in a direct line from an ancestor; hereditary; derived from ancestors; -- opposed to collateral; as, a lineal descent or a lineal descendant.
n.
A dealer in linen; a linen draper.
a.
Composed of lines; delineated; as, lineal designs.
n.
A vessel belonging to a regular line of packets; also, a line-of-battle ship; a ship of the line.
n.
One who adjusts things to a line or lines or brings them into line.
a.
Of a linear shape.
n.
Alt. of Lingam