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"integrand" Definitions
  1. a mathematical expression to be integrated
"integrand" Antonyms

59 Sentences With "integrand"

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The efficiency of VEGAS depends on the validity of this assumption. It is most efficient when the peaks of the integrand are well-localized. If an integrand can be rewritten in a form which is approximately separable this will increase the efficiency of integration with VEGAS.
In the alternative notation, writing \int dy \, \int dx \, f(x, y), the nestedmost integrand is computed first.
Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral. The integrand is evaluated at a finite set of points called integration points and a weighted sum of these values is used to approximate the integral. The integration points and weights depend on the specific method used and the accuracy required from the approximation. An important part of the analysis of any numerical integration method is to study the behavior of the approximation error as a function of the number of integrand evaluations.
Newton–Cotes formula for n = 2 In numerical analysis, the Newton–Cotes formulas, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulas for numerical integration (also called quadrature) based on evaluating the integrand at equally spaced points. They are named after Isaac Newton and Roger Cotes. Newton–Cotes formulas can be useful if the value of the integrand at equally spaced points is given. If it is possible to change the points at which the integrand is evaluated, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are probably more suitable.
The efficiency of VEGAS depends on the validity of this assumption. It is most efficient when the peaks of the integrand are well-localized. If an integrand can be rewritten in a form which is approximately separable this will increase the efficiency of integration with VEGAS. VEGAS incorporates a number of additional features, and combines both stratified sampling and importance sampling.
Note that Gaussian quadrature can also be adapted for various weight functions, but the technique is somewhat different. In Clenshaw–Curtis quadrature, the integrand is always evaluated at the same set of points regardless of w(x), corresponding to the extrema or roots of a Chebyshev polynomial. In Gaussian quadrature, different weight functions lead to different orthogonal polynomials, and thus different roots where the integrand is evaluated.
Making this procedure rigorous requires a limiting procedure, where the domain of integration is divided into smaller and smaller regions. For each small region, the value of the integrand cannot vary much, so it may be replaced by a single value. In a functional integral the domain of integration is a space of functions. For each function, the integrand returns a value to add up.
In numerical analysis, Romberg's method is used to estimate the definite integral : \int_a^b f(x) \, dx by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist.
Global adaptive quadrature can be more efficient (using fewer evaluations of the integrand) but is generally more complex to program and may require more working space to record information on the current set of intervals.
Here, the rapid-oscillation part of the integrand is taken into account via specialized methods for W_k, whereas the unknown function f(x) is usually better behaved. Another case where weight functions are especially useful is if the integrand is unknown but has a known singularity of some form, e.g. a known discontinuity or integrable divergence (such as 1/) at some point. In this case the singularity can be pulled into the weight function w(x) and its analytical properties can be used to compute W_k accurately beforehand.
In many applications, one needs to calculate the rate of change of a volume or surface integral whose domain of integration, as well as the integrand, are functions of a particular parameter. In physical applications, that parameter is frequently time t.
If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate. The method is named after Werner Romberg (1909–2003), who published the method in 1955.
The above integral may be expressed as an infinite truncated series by expanding the integrand in a Taylor series, performing the resulting integrals term by term, and expressing the result as a trigonometric series. In 1755, Euler derived an expansion in the third eccentricity squared.
Under this substitution, the integral becomes -\int du. The integrand involving transcendental functions has been reduced to one involving a rational function (a constant). The result is -u+c=-\cos t+c, which is of course elementary and could have been done without Bioche's rules.
36, pp. A1008–A1026 (2014) As a practical matter, high-order numeric integration is rarely performed by simply evaluating a quadrature formula for very large N. Instead, one usually employs an adaptive quadrature scheme that first evaluates the integral to low order, and then successively refines the accuracy by increasing the number of sample points, possibly only in regions where the integral is inaccurate. To evaluate the accuracy of the quadrature, one compares the answer with that of a quadrature rule of even lower order. Ideally, this lower-order quadrature rule evaluates the integrand at a subset of the original N points, to minimize the integrand evaluations.
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differential equations, and in the path integral approach to the quantum mechanics of particles and fields. In an ordinary integral (in the sense of Lebesgue integration) there is a function to be integrated (the integrand) and a region of space over which to integrate the function (the domain of integration). The process of integration consists of adding up the values of the integrand for each point of the domain of integration.
Concretely, the integral from 0 to any particular t is a random variable, defined as a limit of a certain sequence of random variables. The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. So with the integrand a stochastic process, the Itô stochastic integral amounts to an integral with respect to a function which is not differentiable at any point and has infinite variation over every time interval. The main insight is that the integral can be defined as long as the integrand H is adapted, which loosely speaking means that its value at time t can only depend on information available up until this time.
Itô integral Yt(B) () of a Brownian motion B () with respect to itself, i.e., both the integrand and the integrator are Brownian. It turns out Yt(B) = (B2 \- t)/2. Itô calculus, named after Kiyoshi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process).
The symbol is separated from the integrand by a space (as shown). A function is said to be integrable if the integral of the function over its domain is finite. The points and are called the limits of the integral. An integral where the limits are specified is called a definite integral.
The classic method of Gaussian quadrature evaluates the integrand at N+1 points and is constructed to exactly integrate polynomials up to degree 2N+1. In contrast, Clenshaw–Curtis quadrature, above, evaluates the integrand at N+1 points and exactly integrates polynomials only up to degree N. It may seem, therefore, that Clenshaw–Curtis is intrinsically worse than Gaussian quadrature, but in reality this does not seem to be the case. In practice, several authors have observed that Clenshaw–Curtis can have accuracy comparable to that of Gaussian quadrature for the same number of points. This is possible because most numeric integrands are not polynomials (especially since polynomials can be integrated analytically), and approximation of many functions in terms of Chebyshev polynomials converges rapidly (see Chebyshev approximation).
The integral with respect to of a real-valued function of a real variable on the interval is written as : \int_a^b f(x)\,dx. The integral sign represents integration. The symbol , called the differential of the variable , indicates that the variable of integration is . The function to be integrated is called the integrand.
In complex analysis, the integrand is a complex-valued function of a complex variable instead of a real function of a real variable . When a complex function is integrated along a curve \gamma in the complex plane, the integral is denoted as follows :\int_\gamma f(z)\,dz. This is known as a contour integral.
The Gauss–Bonnet theorem is a special case when M is a 2-dimensional manifold. It arises as the special case where the topological index is defined in terms of Betti numbers and the analytical index is defined in terms of the Gauss–Bonnet integrand. As with the two-dimensional Gauss–Bonnet theorem, there are generalizations when M is a manifold with boundary.
If the loop does not lie in a single plane, for example, there is no one obvious choice. The answer is that it does not matter; by Stokes' theorem, the integral is the same for any surface with boundary , since the integrand is the curl of a smooth field (i.e. exact). In practice, one usually chooses the most convenient surface (with the given boundary) to integrate over.
A Gaussian quadrature rule is typically more accurate than a Newton–Cotes rule, which requires the same number of function evaluations, if the integrand is smooth (i.e., if it is sufficiently differentiable). Other quadrature methods with varying intervals include Clenshaw–Curtis quadrature (also called Fejér quadrature) methods, which do nest. Gaussian quadrature rules do not nest, but the related Gauss–Kronrod quadrature formulas do.
Distributed ray tracing samples the integrand at many randomly chosen points and averages the results to obtain a better approximation. It is essentially an application of the Monte Carlo method to 3D computer graphics, and for this reason is also called "stochastic ray tracing". Path tracing is a rendering technique that combines all of these integration domains into a single, high-dimensional domain and samples it in a unified way.
A differential -form can be integrated over an oriented -dimensional manifold. When the -form is defined on an -dimensional manifold with , then the -form can be integrated over oriented -dimensional submanifolds. If , integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, with according to the orientation of those points. Other values of correspond to line integrals, surface integrals, volume integrals, and so on.
In an animated scene, motion blur can be simulated by distributing rays in time. Distributing rays in the spectrum allows for the rendering of dispersion effects, such as rainbows and prisms. Mathematically, in order to evaluate the rendering equation, one must evaluate several integrals. Conventional ray tracing estimates these integrals by sampling the value of the integrand at a single point in the domain, which is a very bad approximation, except for narrow domains.
Simpson's rule, which is based on a polynomial of order 2, is also a Newton–Cotes formula. Quadrature rules with equally spaced points have the very convenient property of nesting. The corresponding rule with each interval subdivided includes all the current points, so those integrand values can be re-used. If we allow the intervals between interpolation points to vary, we find another group of quadrature formulas, such as the Gaussian quadrature formulas.
In mathematical analysis, Darboux's formula is a formula introduced by for summing infinite series by using integrals or evaluating integrals using infinite series. It is a generalization to the complex plane of the Euler–Maclaurin summation formula, which is used for similar purposes and derived in a similar manner (by repeated integration by parts of a particular choice of integrand). Darboux's formula can also be used to derive the Taylor series from calculus.
When the limits are omitted, as in :\int f(x) \,dx, the integral is called an indefinite integral, which represents a class of functions (the antiderivative) whose derivative is the integrand. The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. Occasionally, limits of integration are omitted for definite integrals when the same limits occur repeatedly in a particular context. Usually, the author will make this convention clear at the beginning of the relevant text.
However, Morokoff and Caflisch gave examples where the advantage of the quasi-Monte Carlo is less than expected theoretically. Still, in the examples studied by Morokoff and Caflisch, the quasi-Monte Carlo method did yield a more accurate result than the Monte Carlo method with the same number of points. Morokoff and Caflisch remark that the advantage of the quasi-Monte Carlo method is greater if the integrand is smooth, and the number of dimensions s of the integral is small.
Since a function of a real or complex variable cannot be entered into a digital computer, the solution of continuous problems involves partial information. To give a simple illustration, in the numerical approximation of an integral, only samples of the integrand at a finite number of points are available. In the numerical solution of partial differential equations the functions specifying the boundary conditions and the coefficients of the differential operator can only be sampled. Furthermore, this partial information can be expensive to obtain.
In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that X is adapted if and only if, for every realisation and every n, Xn is known at time n. The concept of an adapted process is essential, for instance, in the definition of the Itō integral, which only makes sense if the integrand is an adapted process.
A sample path of an Itō process together with its surface of local times. In the mathematical theory of stochastic processes, local time is a stochastic process associated with semimartingale processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level. Local time appears in various stochastic integration formulas, such as Tanaka's formula, if the integrand is not sufficiently smooth. It is also studied in statistical mechanics in the context of random fields.
The Tanh-Sinh method is quite insensitive to endpoint behavior. Should singularities or infinite derivatives exist at one or both endpoints of the (−1, +1) interval, these are mapped to the (−∞,+∞) endpoints of the transformed interval, forcing the endpoint singularities and infinite derivatives to vanish. This results in a great enhancement of the accuracy of the numerical integration procedure, which is typically performed by the Trapezoidal Rule. In most cases, the transformed integrand displays a rapid roll-off (decay), enabling the numerical integrator to quickly achieve convergence.
The multiple integral expands the concept of the integral to functions of any number of variables. Double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space. Fubini's theorem guarantees that a multiple integral may be evaluated as a repeated integral or iterated integral as long as the integrand is continuous throughout the domain of integration. The surface integral and the line integral are used to integrate over curved manifolds such as surfaces and curves.
In pseudospectral methods, integration is approximated by quadrature rules, which provide the best numerical integration result. For example, with just N nodes, a Legendre-Gauss quadrature integration achieves zero error for any polynomial integrand of degree less than or equal to 2N-1. In the PS discretization of the ODE involved in optimal control problems, a simple but highly accurate differentiation matrix is used for the derivatives. Because a PS method enforces the system at the selected nodes, the state-control constraints can be discretized straightforwardly.
Finally define b(a) to be the birth rate per capita for mothers of age a. All of these quantities can be viewed in the continuous limit, producing the following integral expression for B: : B(t) = \int_0^t B(t - a )\ell(a)b(a) \, da. The integrand gives the number of births a years in the past multiplied by fraction of those individuals still alive at time t multiplied by the reproduction rate per individual of age a. We integrate over all possible ages to find the total rate of births at time t.
In statistics and physics, multicanonical ensemble (also called multicanonical sampling or flat histogram) is a Markov chain Monte Carlo sampling technique that uses the Metropolis–Hastings algorithm to compute integrals where the integrand has a rough landscape with multiple local minima. It samples states according to the inverse of the density of states, which has to be known a priori or be computed using other techniques like the Wang and Landau algorithm. Multicanonical sampling is an important technique for spin systems like the Ising model or spin glasses.
They are encoded in the positive geometry of the amplituhedron, via the singularity structure of the integrand for scattering amplitudes. Arkani-Hamed suggests this is why amplituhedron theory simplifies scattering-amplitude calculations: in the Feynman-diagrams approach, locality is manifest, whereas in the amplituhedron approach, it is implicit. Since the planar limit of the N = 4 supersymmetric Yang–Mills theory is a toy theory that does not describe the real world, the relevance of this technique for more realistic quantum field theories is currently unknown, but it provides promising directions for research into theories about the real world.
A Newton–Cotes formula of any degree n can be constructed. However, for large n a Newton–Cotes rule can sometimes suffer from catastrophic Runge's phenomenon where the error grows exponentially for large n. Methods such as Gaussian quadrature and Clenshaw–Curtis quadrature with unequally spaced points (clustered at the endpoints of the integration interval) are stable and much more accurate, and are normally preferred to Newton–Cotes. If these methods cannot be used, because the integrand is only given at the fixed equidistributed grid, then Runge's phenomenon can be avoided by using a composite rule, as explained below.
Practically, an ensemble of chains is generally developed, starting from a set of points arbitrarily chosen and sufficiently distant from each other. These chains are stochastic processes of "walkers" which move around randomly according to an algorithm that looks for places with a reasonably high contribution to the integral to move into next, assigning them higher probabilities. Random walk Monte Carlo methods are a kind of random simulation or Monte Carlo method. However, whereas the random samples of the integrand used in a conventional Monte Carlo integration are statistically independent, those used in MCMC are autocorrelated.
If the interval is subdivided, the Gauss evaluation points of the new subintervals never coincide with the previous evaluation points (except at zero for odd numbers), and thus the integrand must be evaluated at every point. Gauss–Kronrod rules are extensions of Gauss quadrature rules generated by adding points to an -point rule in such a way that the resulting rule is of order . This allows for computing higher-order estimates while re-using the function values of a lower-order estimate. The difference between a Gauss quadrature rule and its Kronrod extension is often used as an estimate of the approximation error.
Although the integrand of the line integral is time-independent, because the Faraday disc that forms part of the boundary of line integral is moving, the full-time derivative is non-zero and returns the correct value for calculating the electromotive force. Alternatively, the disc can be reduced to a conductive ring along the disc's circumference with a single metal spoke connecting the ring to the axle. The Lorentz force law is more easily used to explain the machine's behaviour. This law, formulated thirty years after Faraday's death, states that the force on an electron is proportional to the cross product of its velocity and the magnetic flux vector.
The two general-purpose routines most suitable for use without further analysis of the integrand are QAGS for integration over a finite interval and QAGI for integration over an infinite interval. These two routines are used in GNU Octave (the `quad` command) and R (the `integrate` function). ;QAGS : uses global adaptive quadrature based on 21-point Gauss–Kronrod quadrature within each subinterval, with acceleration by Peter Wynn's epsilon algorithm. ;QAGI : is the only general-purpose routine for infinite intervals, and maps the infinite interval onto the semi-open interval (0,1] using a transformation then uses the same approach as QAGS, except with 15-point rather than 21-point Gauss–Kronrod quadrature.
The conjecture of Kontsevich and Zagier would imply that equality of periods is also decidable: inequality of computable reals is known recursively enumerable; and conversely if two integrals agree, then an algorithm could confirm so by trying all possible ways to transform one of them into the other one. It is not expected that Euler's number e and Euler-Mascheroni constant γ are periods. The periods can be extended to exponential periods by permitting the product of an algebraic function and the exponential function of an algebraic function as an integrand. This extension includes all algebraic powers of e, the gamma function of rational arguments, and values of Bessel functions.
The VEGAS algorithm approximates the exact distribution by making a number of passes over the integration region which creates the histogram of the function f. Each histogram is used to define a sampling distribution for the next pass. Asymptotically this procedure converges to the desired distribution.Lepage, 1978 In order to avoid the number of histogram bins growing like Kd, the probability distribution is approximated by a separable function: :g(x_1, x_2, \ldots) = g_1(x_1) g_2(x_2) \ldots so that the number of bins required is only Kd. This is equivalent to locating the peaks of the function from the projections of the integrand onto the coordinate axes.
The VEGAS algorithm approximates the exact distribution by making a number of passes over the integration region while histogramming the function f. Each histogram is used to define a sampling distribution for the next pass. Asymptotically this procedure converges to the desired distribution. In order to avoid the number of histogram bins growing like K^d with dimension d the probability distribution is approximated by a separable function: g(x_1, x_2, \ldots) = g_1(x_1) g_2(x_2) \cdots so that the number of bins required is only Kd. This is equivalent to locating the peaks of the function from the projections of the integrand onto the coordinate axes.
Using standard methods of numerical evaluation for Fourier integrals, such as Gaussian or tanh-sinh quadrature, is likely to lead to completely incorrect results, as the quadrature sum is (for most integrands of interest) highly ill- conditioned. Special numerical methods which exploit the structure of the oscillation are required, an example of which is Ooura's method for Fourier integralsTakuya Ooura, Masatake Mori, A robust double exponential formula for Fourier-type integrals, Journal of computational and applied mathematics 112.1-2 (1999): 229-241. This method attempts to evaluate the integrand at locations which asymptotically approach the zeros of the oscillation (either the sine or cosine), quickly reducing the magnitude of positive and negative terms which are summed.
Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the integrand in terms of Chebyshev polynomials. Equivalently, they employ a change of variables x = \cos \theta and use a discrete cosine transform (DCT) approximation for the cosine series. Besides having fast-converging accuracy comparable to Gaussian quadrature rules, Clenshaw–Curtis quadrature naturally leads to nested quadrature rules (where different accuracy orders share points), which is important for both adaptive quadrature and multidimensional quadrature (cubature). Briefly, the function f(x) to be integrated is evaluated at the N extrema or roots of a Chebyshev polynomial and these values are used to construct a polynomial approximation for the function.
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional to a change in a function on which the functional depends. In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integral of a functional, if a function is varied by adding to it another function that is arbitrarily small, and the resulting integrand is expanded in powers of , the coefficient of in the first order term is called the functional derivative. For example, consider the functional : J[f] = \int_a^b L( \, x, f(x), f \, '(x) \, ) \, dx \ , where .
Visualisation of the Cavaliere integral for the function f(x)=(2x+8)^3 Cavalieri's principle can be used to calculate areas bounded by curves using Riemann–Stieltjes integrals.T. L. Grobler, E. R. Ackermann, A. J. van Zyl & J. C. Olivier Cavaliere integration from Council for Scientific and Industrial Research The integration strips of Riemann integration are replaced with strips that are non-rectangular in shape. The method is to transform a "Cavaliere region" with a transformation h, or to use g = h^{-1} as integrand. For a given function f(x) on an interval [a,b], a "translational function" a(y) must intersect (x,f(x )) exactly once for any shift in the interval.
In mathematical finance, the described evaluation strategy of the integral is conceptualized as that we are first deciding what to do, then observing the change in the prices. The integrand is how much stock we hold, the integrator represents the movement of the prices, and the integral is how much money we have in total including what our stock is worth, at any given moment. The prices of stocks and other traded financial assets can be modeled by stochastic processes such as Brownian motion or, more often, geometric Brownian motion (see Black–Scholes). Then, the Itô stochastic integral represents the payoff of a continuous-time trading strategy consisting of holding an amount Ht of the stock at time t.
The technique approximates part of the integrand using radial basis functions (local interpolating functions) and converts the volume integral into boundary integral after collocating at selected points distributed throughout the volume domain (including the boundary). In the dual-reciprocity BEM, although there is no need to discretize the volume into meshes, unknowns at chosen points inside the solution domain are involved in the linear algebraic equations approximating the problem being considered. The Green's function elements connecting pairs of source and field patches defined by the mesh form a matrix, which is solved numerically. Unless the Green's function is well behaved, at least for pairs of patches near each other, the Green's function must be integrated over either or both the source patch and the field patch.
That is true, anyway, for orientable manifolds, an orientation being in differential form terms an n-form that is never zero (and two being equivalent if related by a positive scalar field). The duality can be reformulated, to great advantage, in terms of the Hodge dual—intuitively, 'divide into' an orientation form—as it was in the years succeeding the theorem. Separating out the homological and differential form sides allowed the coexistence of 'integrand' and 'domains of integration', as cochains and chains, with clarity. De Rham himself developed a theory of homological currents, that showed how this fitted with the generalised function concept. The influence of de Rham’s theorem was particularly great during the development of Hodge theory and sheaf theory.
On non-orientable manifolds, one cannot define a volume form globally due to the non-orientability, but one can define a volume element, which is formally a density, and may also be called a pseudo-volume form, due to the additional sign twist (tensoring with the sign bundle). The volume element is a pseudotensor density according to the first definition. A change of variables in multi-dimensional integration may be achieved through the incorporation of a factor of the absolute value of the determinant of the Jacobian matrix. The use of the absolute value introduces a sign change for improper coordinate transformations to compensate for the convention of keeping integration (volume) element positive; as such, an integrand is an example of a pseudotensor density according to the first definition.
Historically, the symbol dx was taken to represent an infinitesimally "small piece" of the independent variable x to be multiplied by the integrand and summed up in an infinite sense. While this notion is still heuristically useful, later mathematicians have deemed infinitesimal quantities to be untenable from the standpoint of the real number system.In the 20th century, nonstandard analysis was developed as a new approach to calculus that incorporates a rigorous concept of infinitesimals by using an expanded number system called the hyperreal numbers. Though placed on a sound axiomatic footing and of interest in its own right as a new area of investigation, nonstandard analysis remains somewhat controversial from a pedagogical standpoint, with proponents pointing out the intuitive nature of infinitesimals for beginning students of calculus and opponents criticizing the logical complexity of the system as a whole.
Shiing-Shen Chern published his proof of the theorem in 1944 while at the Institute for Advanced Study. This was historically the first time that the formula was proven without assuming the manifold to be embedded in a Euclidean space, which is what it means by "intrinsic". The special case for a hypersurface (an n-1-dimensional submanifolds in an n-dimensional Euclidean space) was proved by H. Hopf in which the integrand is the Gauss-Kronecker curvature (the product of all principal curvatures at a point of the hypersurface). This was generalized independently by Allendoerfer in 1939 and Fenchel in 1940 to a Riemannian submanifold of a Euclidean space of any codimension, for which they used the Lipschitz-Killing curvature (the average of the Gauss-Kronecker curvature along each unit normal vector over the unit sphere in the normal space; for an even dimensional submanifold, this is an invariant only depending on the Riemann metric of the submanifold).

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