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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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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10 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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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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9 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 ...
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1answer
18 views

Fourier series on incomplete data [on hold]

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
25 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
17 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
22 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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45 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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18 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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10 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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29 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
60 views

Where did divided differences come from? [closed]

I'm learning about interpolation and I'm confused about where did Newton's divided differences come from? Anyone can provide a derivation starting from the ideas behind it?
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33 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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21 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
29 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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68 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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0answers
19 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
9 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 ...
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1answer
26 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) = ...
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1answer
15 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) ...
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2answers
29 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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20 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
33 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
35 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
19 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 ...
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2answers
74 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
17 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
25 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 ...
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Complexity of computing the coefficients of the Newton interpolating polynomial

Could you help me solve this problem? Thank you very much for your help. Prove that the algorithm for computing the coefficients $c_i$ in the Newton from of the interpolating polynomial involves ...
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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
21 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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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 ...
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2answers
36 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
40 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
27 views

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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28 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
34 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
17 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
45 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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22 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
22 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
25 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
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1answer
79 views

What is this polynomial zero everywhere but $n!$ at point $n$?

For an integer $n \geq 2$, define a set of $n$ points $(p_{x,1},p_{y,1}),\ldots,(p_{x,n},p_{y,n})$ as follows. For $1 \leq i \leq n-1$, let $(p_{x,i},p_{y,i}) = (i,0)$, and $(p_{x,n},p_{y,n}) = ...
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1answer
55 views

How to choose new points after grouping/resampling?

I'm resampling a signal (which takes values [0,1]) of N samples (blu points) to one with N/5 samples, where (for each group of 5 samples) I store in two arrays the max and the min values of the ...
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22 views

Best 2D interpolation for a set of points (x/y)

Given a set of X/Y points in a bounded area ( 0<=X<=Xmax and 0<=Y<=Ymax ), what are the best 'interpolation' ...
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2answers
43 views

Improve the order of accuracy for second derivative using central difference approximation

The formula $$D^{m}(h)=(4^mD^{m-1}(h/2)-D^{m-1}(h))/3$$ for $m=2,3,\ldots$improves the order of accuracy for first derivative using central diff approx. where ...
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2answers
43 views

The quadratic spline

I'd like to fit the data in table as blow x f(x) 3.0 2.5 4.5 1.0 7.0 2.5 9.0 0.5 when $x=5$, I want to find value of $f(x)$ by using ...
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10 views

Error Function using Chebyshev Node Interpolation

Given a set of measurements (at Chebyshev nodes), does an error function exist that allows you to roughly approximate the error (like the error function for Simpson's Rule)?