Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points.

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1answer
65 views

Cubic Spline Interpolation

My problem is to find a interpolating cubic spline to the points $$\left\{(0,0), \left(\frac{\pi}{2}, 1\right), \left(\pi,0\right), \left(\frac{3\pi}{2}, -1\right),(2\pi,0)\right\}$$ I did as ...
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0answers
34 views

Least square polynomial interpolation

Given an arbitrary continuous function f(x), let Pn(x) be the polynomial of degree at most n that approximates f(x) in the least squares sense. Is it true that Pn(x) interpolates f(x) at n + 1 points? ...
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30 views

Algorithm Identification

Background I'm currently working with a system that has a 4-dimensional function. Currently, an algorithm is used to speed up calculation of the final value via interpolation, and two of the ...
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1answer
18 views

build function passing for specific points

I have to solve a problem very similar to this how-to-create-a-function-passing-through-given-points I need a function that draw a curve like the blue one in the picture here thus passing as ...
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1answer
57 views

Multistep Method: Gear's Formula Interpolation

Please explain how to do this. How can we use Lagrange Interpolation to derive this formula? Thanks in advance.
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20 views

Interpolate with smoothing parameter

I need to implement in C++ interpolation with smoothing parameter. To the non-familiar with this function: The smoothing parameter gets a value from 0 to 1. 0 brings absoulte linear interpolation (...
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0answers
28 views

Given two entire functions $f_1,f_2$ without common zeros, prove that one can find some entire functions $g_1,g_2$ such that $f_1g_1+f_2g_2=1$ [duplicate]

The question Let $f_1,f_2$ be some entire functions without zeros in common, so for every $z∈ℂ$ we have $|f_1(z)|^2+|f_2(z)|^2≠0$. Prove that there exist two entire functions $g_1,g_2$ such that: $$ ...
2
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1answer
35 views

Approximating Fresnel integrals with standard functions

I would like to approximate the Fresnel S and Fresnel C with standard functions. I've started with the $ S(x) $ function: $$ approxS(x) = sgn(x) * \left ( sgn(x)* \left ( \frac{ \sin( \frac{x^2}{2} ...
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0answers
21 views

Why settle for Lagrange Interpolation when doing linear multistep ODE integration?

Say that we have some initial value problem: $y'(t) = f(t,y(t)) ; y(0) = y_0$ with $y_0$ and $f(t,y(t))$ known. If we use Euler's method to numerically approximate the first k points, then we have ...
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0answers
13 views

Comparing smoothness among approximations

We are interpolating a missing fragment of a 2D curve given a set of sample points. Our method generates several candidates of curve pieces to fill the missing part, but we want to select the solution ...
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0answers
7 views

Roll forward a payment

If I earned $100 per month from Jan 1, 2016 to April 30, 2016 how do I determine my projected 2016 salary if I am assuming an annual trend rate of 7.8 % starting May 1st? I would think it would be ...
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11 views

interpolation preserving boundedness property

I'm trying to construct interpolation for a function $m$ such that \begin{equation*} 0\leq m(x)\leq 1,\quad\forall x\in\Omega\subset \mathbb{R}^1. \end{equation*} I tried to use Lagrange-...
2
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1answer
30 views

Fourier series on incomplete data [closed]

Given a periodic function that's only partly specified, e.g.: $$f(\theta)=\begin{cases}1 & \text{if } \cos(\theta)>a\\ -1 & \text{if } \cos(\theta)<-a\end{cases}$$ Obviously the ...
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1answer
29 views

Second Order Accurate Interpolation

On a grid I am having the values of a physical quantity say for example Temperature, at the E,W,N,S and P node all of them being calculated using a second order discretization scheme. I want a second ...
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1answer
18 views

Some doubts on Simpsons Rule by the Method of Undetermined Coefficients

There is this note about Quadratic Interpolation by Simpsons Rule that I don't quite understand how to get the LHS. Simpsons Rule by the Method of Undetermined Coefficients We seek an approximation $...
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1answer
30 views

Number of continuous derivatives of a piecewise quadratic polynomial

I've been trying to reason through the following problem: Suppose that we interpolate $n+1$ data points with a piecewise quadratic polynomial. How many continuous derivatives can this interpolating ...
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3answers
61 views

SmoothStep: Looking for a continuous family of interpolation functions

Background: SmoothStep is a simple sigmoid-like function defined as S(x) = 3x^2 - 2x^3. It is monotonically increasing from (0, 0) to (1, 1), is rotationally symmetric over that interval, and has ...
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19 views

Is there a standard name for interpolation parameters?

I often use interpolations between two values in game / UI programming, for animations etc. - e.g. a linear interpolation: $$x = x_1 + a(x_2 - x_1)$$ Or a 'cubic' sigmoid type interpolation like: $$...
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0answers
12 views

Interpolationspace for H^{-1}\cap H^{1}

For which $\theta\in[1,\infty)$ does hold $(H^{-1}(\Omega),H^1(\Omega))_{1-\frac{1}{\theta},\theta}=L^2(\Omega)$ if $\Omega$ is a bounded domain with smooth boundary and is three dimensional. I don't ...
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14 views

Fitting with initial conditions

i try to do some fits but i have ten initials conditions and i think it will be difficult to evaluate the sensitivity of my conditions. Do you know some methods which allow to know the sensitivity of ...
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0answers
41 views

The butcher array of explicit Runge- Kutta method

Just a quick question, for a family of explicit Runge-Kutta methods parametrized by order q, by applying up to $p-1$ passes of deferred correction to p steps of Euler's method. When $p=2$, should its ...
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0answers
39 views

Lagrange interpolation for ellipse

Consider the ellipse $$\frac{x^2}{4} + \frac{y^2}{2} =1$$ The line integral $I$ of the ellipse in the first quadrant is $$I=\int^2_0 \Big[ 1+(y'(x))^2 \Big]^{1/2} dx$$ Find the cubic polynomial $...
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38 views

Interpolate solution derived with Matlab PDE tool

I have tried to solve a bvp problem using Matlab. Matlab succesfully returned the result of the numerical procedure based on an internal finite element method. However I don't know how to interpolate ...
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1answer
35 views

Cubic Spline for a function

I have the function $f(x)=x^3$ and I need to find the cubic spline. The given points are: $\{-1, 0, 1\}$. What is the cubic spline for this function and what would a demonstration to this be? I would ...
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0answers
75 views

Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
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22 views

Effect of location of nodes for interpolation

I've been doing some numerical experiments to see how the location of the interpolating nodes affects the performance of the interpolator. I am just curious about this because it seems like the ...
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1answer
10 views

Adding two functions represented by a table of values with a different step size?

Let $f(t)$ be some numerically obtained $T$-periodic function represented by a table of values over one period or a set of points $(t, y)$ with a time step $\Delta t.$ Now let's change the frequency/...
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1answer
28 views

Calculate what function is approximated with the Lagrange Polynomial

I would like to find out what the sum estimates and prove that it estimates that function. $$\sum_{j=0}^ml_j(x)*x_j^k=?$$ From the Lagrange interpolation polynomials we know that $$l_k(x) = \prod_{i=0}...
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1answer
22 views

n-order B-splines interpolation

I am wondering if the following statements are correct: (1) zero-order B-splines interpolation is equivalent to nearest-neighbor interpolation. $C^0$ continuity thus is not differentiable. (2) first-...
2
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2answers
32 views

Numerically stable interpolation of circular curve

Suppose I start with two points $p_1, p_2 \in \mathbb{R}^3$. I want to interpolate along a circular arc between these two points, given normal vectors $n_1, n_2$ at each point. It's fairly ...
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24 views

Quadratic polynomial interpolation from a transformation

Some modeling considerations have mandated a search for a function $$ u(x) = \gamma_{0}\exp(\gamma_{1}x + \gamma_{2}x^{2}) $$ where the unknown coefficients $\gamma_{1}$ and $\gamma_{2}$ are ...
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1answer
34 views

Lagrange Polynomial Interpolation - Polynomyal Differences Depending Upon the Degree?

My question is simple: Give the table: | x |0|2|4|6| |f(x)|1|3|5|7| Why when calculating Lagrange Polynomial Interpolation for: | x |0|2| |f(x)|1|3| P1(x) = x+1 And when calculating Lagrange ...
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2answers
43 views

Proof of Lagrange Polynomial

I am trying to prove the following concepts of the Lagrange Polynomial: $\sum_{j=0}^n L_j(x)=1$ $\sum_{j=0}^n x_j^m(x)L_j(x)=x^m, m \le n $ This is my work so far, but I am a little stuck on ...
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1answer
20 views

How to interpolate elliptically

Given two orthogonal axes with different weightings along each axis, how do I interpolate elliptically between the two weightings? This is in 2d cartesian space. For example, axis1 might be Vector(2,...
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2answers
111 views

NURBS Curves to Interpolate Points and Derivatives on a Surface of Revolution

Problem in Prose My starting point is a set of conic segments on a plane. Each of these conic segments interpolates between three points and known slopes on the two outer points. I want to find a ...
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1answer
19 views

Alternatives to Shephard interpolation?

I am a chemist, so I have little experience in the field of math. My program is that I have a set of points (approx. 20000) in some larger dimensional space (like 10-20 dimensions), and I want to be ...
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1answer
29 views

Deriving a tridiagonal system for cubic spline interpolation

Can anyone explain how $B_{i-1} = 1/4$ and $B_{i+1} = 1/4$ were chosen in line 6 of the picture, just above the matrix? I'm trying to understand cubic splines but this result seems like it came out ...
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0answers
17 views

Vector/Multidimensional version of Newton Divided Difference

newton divided difference polynomial (NDDP) finds an y=f(x) relation by interpolating a polynomial, is there a y=f(x,z) version for n dimensions? Any help appreciated.
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1answer
22 views

Derivation of linear interpolation?

Anyone know a good derivation of the linear interpolation: $$\frac{y-y_0}{x-x_0}=\frac{y_1-y_0}{x_1-x_0}$$ Wikipedia gives one, which I don't understand.
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0answers
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Find 2 datapoints too interpolate sin(x)

I have the function $f:\left [ 0,\pi \right ]\rightarrow \mathbb{R}, x \mapsto \sin(x)$. How can I choose two points $x_0, x_1 \in \left [ 0,\pi \right ]$ for my polynomial interpolation such that I ...
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2answers
43 views

Interpolation by splines: how to set up the equation system for finding the coefficients of the spline (in a B-spline basis)

Problem. I want to interpolate a function $f$ in some equidistant points $x_0<x_1<x_2<x_3<x_4$ using a quadratic spline. My attempt. I assume that we can use the interpolation points as ...
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1answer
52 views

comparison of piecewise linear interpolation, cubic interpolation, cubic spline interpolation?

What are the advantages and disadvantages of piecewise linear interpolation, cubic interpolation, and cubic spline interpolation? I know that piecewise linear interpolation is not smooth and may not ...
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1answer
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Why do I have the wrong ratio for this linear interpolation question

$$f(x)=5x^3-8x^2+1$$ There is a root between x=1 and x=2, use linear interpolation (using similar triangles) to find the root correct to 1 dp So I tried doing this: $$f(1)=-2$$ $$f(2)=9$$ so the ...
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31 views

Method of Undetermined Coefficients (Vandermonde) on data points?

I know how to interpolate using Vandermonde and obtain p(x) if the data points are given as something like p(-1)=1 p(0)=0 p(1)=1 But what if derivatives, p'(x) ...
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2answers
39 views

The error formula for the Romberg integration [closed]

I am just wondering if there exists a error formula for the Romberg integration. Since it just applies Richardson extrapolation to Trapezoidal rule, is its error formula the same as that of the ...
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1answer
22 views

An example of hermite interpolation [closed]

I found this example on wikipedia. What I don't understand is that on the right hand side, the column starts with $-10$. Why isn't the column $-10, -4, 4, 10$? Why do the numbers in the middle place ...
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1answer
50 views

Radial Basis Function RBF Gaussian based Interpolation

Based on short description below (an image), how do I find the highlighted f function value? I understand that it is a value associated with the vertex, sorry I am not a good math student to ...
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0answers
27 views

Distance between a map and a point using bicubic interpolation

I have an image (i.e., a two dimensional regular grid) with pixel values that represent elevations. To interpolate points of the surface described by the grid, I use bicubic interpolation. The image, ...
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1answer
24 views

Approximation of continuous functions

Let $f$ $\in$ C([0,1]), $f(0)=0$ and $\epsilon > 0$. Prove there exists a polynomial $p$ such that $p(0)=f(0)=0$, $p´(0)=0$ and $||p-f|| < \epsilon$ . The norm is sup-norm I Know that by ...
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1answer
29 views

Find root with chord method

With chord method find real root of the equation $x^3 - 2x+1-{e^x\over2} = 0$ accurate to $0.001$ I can not perform first condition in the method of chords