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interpolant
interpolant Math. (ɪnˈtɜːpələnt) [f. L. interpolant-em, pres. pple. of interpolāre (see interpolate v.), or interpolate v. + -ant1.] A value or expression (given or calculated) used in finding some other value by interpolation.1920 Tracts for Computers ii. 17 Forward difference formulae, central dif... Oxford English Dictionary
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Craig interpolation
This suffices to show that φ is a suitable interpolant in this case. know that Hence, ρ is a suitable interpolant for φ and ψ. wikipedia.org
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Craig interpolants equivalent to given intermediate predicates Let us consider a first-order predicate (or a propositional formula) K to be "intermediate" between A and B iff it is weaker than A and stronger than B (t...
Therefore, no interpolant exists which is logically equivalent to K.
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Radial basis function interpolation
The interpolant takes the form of a weighted sum of radial basis functions, like for example Gaussian distributions. points where it will sharply peak the so-called "bed-of-nails interpolant" (as seen in the plot to the right). wikipedia.org
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How to show the interpolant polynomial? Consider a uniform grid $\\{x_1, \ldots, x_N\\}$, with $x_{j+1}-x_j=h$ for each $j=1,2,...,N$, and a set of corresponding data values $\\{u_1,\ldots,u_N \\}$. Let $p_j$ be the ...
Let $p_j(x) = a+b(x_j-x)+c(x_j-x)^2$, where $a,b,c \in R$ are unknowns. Then we have $a = p_j(x_j)=u_j$. So, $p_j(x) = u_j+b(x_j-x)+c(x_j-x)^2$. Now, using $x_j-x_{j-1}=x_{j+1}-x_j=h$, we get $p_j(x_{j-1})=u_j+bh+ch^2=u_{j-1}$ and $p_j(x_{j+1})=u_j-bh+ch^2=u_{j+1}$. Solving for $b$ and $c$ gives $b=...
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Discrete spline interpolation
In the mathematical field of numerical analysis, discrete spline interpolation is a form of interpolation where the interpolant is a special type of piecewise This interpolant agrees with the values of f(x) at x0, x1, . . ., xn. wikipedia.org
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Fitting a quadratic interpolant to $y = cos x$ at three nodes Lets say we want quadratic interpolation of $y = cos x$ at the nodes $\\{0, \pi/2, \pi\\}$. And we would like to match the derivative of $y(x)$ at the cent...
.$$ This is the third equation for the first interpolant; the second interpolant will be similar.
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Piecewise linear continuation
Over the whole mesh of triangles, this piecewise linear interpolant is continuous. The contour of the linear interpolant over a triangle The contour of the piecewise linear interpolant is a set of curves made up of these line segments wikipedia.org
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Error term for a cubic interpolation I have a question on one interpolation problem. The problem is below. For the given points, $x_0 = -1, x_1 = 0, x_2 = 3$ and $x_3 = 4,$ find the error term $e_3(\bar{x}) = f(\bar{...
The error term is related to the article in wiki: < at the section "Interpolation error" Since you are given the original function and the closed interval containing the points that you interpolate the function with, then you can apply the formula in wiki directly to calculate an upperbound for the ...
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What does it mean, that the interpolants are choosen from BL-space of distributions? I would like to understand this paper: Reconstruction and Representation of 3D Objects with Radial Basis Functions It is clear, unt...
This allows to define different norms and choose for each the smoothest interpolant -- different RBFs.
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Marching cubes
Later, Nielson and Hamann in 1991 observed the existence of ambiguities in the interpolant behavior on the face of the cube. They proposed a test called Asymptotic Decider to correctly track the interpolant on the faces of the cube. wikipedia.org
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Cubic Spline: Prove S(3/2) = (3/2)^3 Let S(x) be the not-a-knot cubic spline interpolant of the points (0, 0), (1, 1), (2, 8), and (3, 27). Explain why $S(3/2) = (3/2)^3$ .
The not-a-knot condition requires the third derivative of the interpolant is continuous at point (1, 1) and at point (2, 8).
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Quadrature rule for spline interpolation Consider an integrable function $f$ on $[-1,1]$. We denote $\left(x_j\right)_{-N}^{N}$ the equally spaced grid on $[-1,1]$, and wish to compute the integral $I = \int\limits_{-...
The resulting integration rule depends on the choice of end point conditions. If clamped conditions are used you get what is variably called the 'corrected trapezoidal rule' or the 'composite Hermite rule', $$ \int_a^b f(x) dx = h \left( \frac{1}{2}f_0 + f_2 + \dots + f_{n-1} + \frac{1}{2}f_{n} \rig...
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Lagrange Polynomial Interpolation, centered coefficients My textbook states that the Lagrange Interpolant on the interval $[a,b]$, with the data points $(x_0,y_0),...,(x_n,y_n)$, written as: $\prod_nf(x)=\sum_{i=0}^...
If I understood your question correctly, two answers linked below should help solve your problem 1. if $p(x)=a_n x^n + a_{n-1}x^{n-1}+\cdots+a_0$ then $\displaystyle a_n = \sum_{i=0}^n y_i \prod_{j=0,j\neq i}^n \frac{1}{x_i-x_j}$ 2. recentering a polynomial: $ \sum_{k=0}^p a_k \left(z+z'\right)^k = ...
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Is the order of the interpolation points in Newton's interpolation polynomial important? I have been given an assignment to compute the polynomial interpolant of a function, but in the assignment question it says $x_0...
For example, if you have a Newton interpolant $a+b(x-x_0)$ for two points and you add a third point $(x_2,y_2)$ (irrespective of how $x_2$ compares to
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