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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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
41 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 ...
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2answers
88 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
18 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
26 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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16 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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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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17 views

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
38 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
44 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
29 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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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
35 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
47 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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23 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
27 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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28 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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45 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
45 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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0answers
11 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)?
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24 views

Padua points or Chebyshev grid for more than two dimensions?

I'm looking for a good point grid in order to sample from a (polynomial) function $f:R^n\rightarrow R$ at discrete points lying in a rectangle. I don't need any weights or interpolating polynomial. I ...
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0answers
28 views

Proof of Runge's phenomenon for a concrete case

Let $f(x)=\frac{1}{1+25x^2}$ and range is $[-1,1]$. Given $n+1$ equidistant points $x_0 = -1,x_1,...,x_n = 1$ and their values $f(x_0),f(x_1),..,f(x_n)$, perform polynomial interpolation by the $n+1$ ...
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1answer
27 views

Find points $x$ such that $p_n(x)-f(x)$ does not converge to $0$ as $n\to\infty$

Let $f(x)=\frac{1}{1+25x^2}$ and range is $[-1,1]$. Given $n+1$ equidistant points $x_0 = -1,x_1,...,x_n = 1$ and their values $f(x_0),f(x_1),..,f(x_n)$, perform polynomial interpolation by the $n+1$ ...
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14 views

Does this operator produce an interpolating spline of it's argument?

I'm studying a bit of spline functions theory: In this book, chapter 6, a lot of error bounds are given when spline functions are used to approximate functions belonging to a specific function spaces. ...
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1answer
61 views

Proof that lagrange and newton's interpolation are the same

Its known that newton's interpolation and lagrange interpolation gives the same value All i need is to prove it
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1answer
54 views

Pyramidal Histogram Of Oriented Gradients - Trilinear interpolation

Hello im struggling with an implementation of this article: https://goo.gl/8mpIuq I performed bilinear interpolation over the histogram bins and the results are better with this interpolation, ...
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40 views

Use polynomial interpolation to find a formula

Any idea how to Use polynomial interpolation to find $P_{k+1}$ where $P_{k+1}(n)=\sum_{j=1}^n j^k$ for $2\le k \le 10$? Thanks!
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58 views

find the equation of sine (or relevant) function with trend to fit a curve or fill gap

I want to fill a gap of temperature (hourly) data. After plotting it appears to be the gap can be fit by using a sine function with a trend. Now i would like to find the equation. Any suggestions will ...
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1answer
43 views

Find an equation fitting a set of points

In a game that I play, there is a power value that is an overall evaluation on how powerful your account is. There is a function where you can level up a skill, and levelling it up gives a bonus to ...
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1answer
20 views

Interpolation using barycentric coordinates of a triangle

If I have a triangle given by the coordinates of its three vertices, $r_1, r_2, r_3$, and another point inside that triangle $r$, I can find the barycentric coordinates of $r$: $$r = \lambda_1 r_1 + ...
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1answer
42 views

Lagrange interpolation not working?

I have been trying to use Lagrange Interpolation on some data to find the equation of a curve that fits the following points (x,y): (0,5.2) (1,3.8) (2,4.5) (3,5.5) (4,6) (5,6.25) (6,6.25) (7,6) ...
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1answer
37 views

Interpolation of rounded data

I would like to interpolate (not fit) a data set whose points have been rounded. Lets say I have some observations $y_i$ of a function sampled at $x_i$. The sample locations $x_i$ may be considered ...
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1answer
24 views

Find $p(x)=a_1 x+a_2 x^2$ with $p(x_i)=f_i, i=0,1$

Given $x_0,x_1,x_0\neq x_1, x_i\neq 0, i=0,1$, and $f_0,f_1$, find $p(x)=a_1 x+a_2 x^2$ with $p(x_i)=f_i, i=0,1$. Can you say if the interpolating polynomial in this case is unique? Why? Can you ...
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1answer
27 views

How can I derive the second degree equation for a curve if I know the slope at two points and the y-intercept = 0?

How can I derive the second degree equation for a curve if I know the slope at two points and the x and y-intercepts = 0?
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22 views

Temerature and Strain images comparison & interpolation: finding offsets in X,Y coordinates systems.

I have two kind of 2D images from a Strain test, which show temperature and strain distributions in X and Y cordinate systems. One is thermal image, which gives temperature T values in a 200x300 ...
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10 views

interpolation with missing gradients

I am just wondering if anyone has encountered a similar problem before. Let's consider a differentiable function in 1 dimension for now. Suppose that this function is expensive to compute. In ...
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28 views

Error of derivatives of polynomial interpolant and best nodes with lobatto constraint

I know when interpolating a function $f(x)$ by a degree $n$ polynomial $P(x)$, the error is $$f(x) - P(x) = \frac{f^{n+1}(\xi)}{(n+1)!}\prod_{i=0}^n(x-x_i)$$ What does the right side hand look like ...
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1answer
50 views

Is the following described $p(x), q(x)$ the same interpolation polynomial?

Suppose we are given an odd number of data points $x_i$ and the corresponding values $f_i=f(x_i),i=1,...,n+1$($n$ is even), which are symmetric about the origin, i.e for each $x_i$ there is a $j$ such ...
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44 views

How many multiplications and additions will be needed to evaluate the interpolating polynomial at a point using the Lagrange formulas?

Consider the interpolating problem with $n+1$ points. How many multiplications and additions will be needed to evaluate the interpolating polynomial at a point using the Lagrange formulas? I think it ...
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1answer
27 views

C2 continuous interpolation on a 4-dimensional dataset

I am currently coding up a project where interpolations must be performed such that C2 continuity be preserved along the length of the whole set. The end result ought to look like a line (which will ...
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22 views

Parabolic interpolation in $k$ dimensions

I know the values of a smooth function of $k$ variables at $3^k$ points on a cube in $k$ dimensions (where $k=2,3$ or $4$). The central value is known to be the largest. I want to estimate the ...
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1answer
18 views

Trapezoid rule for finding coefficient

If we know that $\int_{a}^b t(x)=h \sum_{k=1}^2 dk * t(a+kh)+O(h^m)$ where $h=\frac{b-a}{3}$, how do we find the coefficient d1, d2 and m in the equation? Answer says that d1=3/2, d2=3/2, m=3 I ...
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1answer
28 views

Terminology: Spline interpolation

I have read two different definitions of Splines: A differentiable piecewise polynomial. A piecewise polynomial. If I build a piecewise polynomial using cubic polynomials, it's ...
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30 views

What is this graphed function with asymptotes at π/2 and -π/2?

I've come across this function with asymptotes at π/2 and -π/2, which crosses the axis at y=1. It doesn't seem to be polynomial or exponential—can anyone figure out what it is? The asymptotes and ...