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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Computing splines using Hermite interpolants

The form for a cubic Hermite interpolant has the form $p_i(x)=a_i+b_i(x-x_i)+c_i(x-x_i)^2+d_i(x-x_i)^3$ according to the following conditions: $p_i(x_i) = y_i$ $p_i(x_{i+1}) = y_{i+1}$ ...
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26 views

Spline interpolation explanation

I'm trying to learn about spline interpolation, and I'm struggling to understand what h_i^3 is in the second and third equation. I don't understand how they derived that equation. I'm trying to ...
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1answer
29 views

Newton method for interpolation of polynomials

I have a function $f(x) = \frac1{1+25x^2}$, and 7 equally spaced nodes on the interval $[-1, 1]$ My points are $(-1, 1/27), (-0.6, 11), (-0.3, 4.25), (0, 2), (0.3, 4.25), (0.6, 11), (1, 1/27)$. I'm ...
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Find a cubic hermite interpolation of the function

Let be $I=[0,2]\subset\mathbb{R}$ and $f:I\to\mathbb{R}$ with $$f(x):=\frac{x^2-5}{-x^3+x^2-4}.$$ Define a polynomial $p$ using the cubic Hermite interpolation method with the grid points ...
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63 views

Can I make this numerical integration continuously differentiable?

Suppose I have the discrete values $f(x_i)$ for every $x_i$ greater than some $\varepsilon$, and I want to numerically calculate the following integral: \begin{equation} n = \int_\varepsilon^\infty ...
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50 views

Derivative of Lagrange interpolating polynomial

I'm using a textbook titled "Numerical Analysis" by Richard L. Burden, 9th edition. I'm having a problem with a particular derivation The Lagrange interpolating polynomial is given by $$f(x) = ...
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Error of Lagrange Interpolating polynomial is zero for polynomials of low degree

Let $f$ be a function defined on the interval $[x_0 −h, x_0, x_0 +h]$, and $f \in C^3[x_0 −h, x_0 +h]$. Let $h$ be the Lagrange interpolation polynomial of $f$ at the nodes $x = x_0 − h$, $x_0$, $x_0 ...
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Degree of Lagrange polynomial to satisfy a given error

Consider the function $f(x)=e^{x-1}$ on the interval $0\leq x\leq1$. For all $N=1,2,...$ we have the uniform partition $x_i=ih$, $i=0,1,...,N$, $h=1/N$. We find the Lagrange polynomial for $x_i$. If ...
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1answer
42 views

Interpolation polynomials

Let $p_k$ be the polynomial of degree $\leq k$ such that $p_k(x_i)=y_i$ for $0\leq i \leq k$. Prove that $p_k=p_{k-1}$ if and only if $p_{k-1}(x_k)=y_k$. I'm a first year PhD student and I ...
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44 views

Let $f(x)=1/x$ and prove that $f[x_0,x_1,…,x_n]=\prod_{i=0}^nx_i^{-1}$.

Let $f(x)=1/x$ and prove that $f[x_0,x_1,...,x_n]=\prod_{i=0}^nx_i^{-1}$. I'm sure how to approach this or even how/why we need $f(x)=1/x$. Any solutions or hints are greatly appreciated.
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1answer
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Error formula when using a polynomial interpolation

You have a function f that has a continous $(n+1)th$ derivative. And you use a polynomial $p_n$ to interpolate the function at points $(x_0,f_0), (x_1,f_1) ... (x_n,f_n)$. Then the error for the ...
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0answers
33 views

Interpolate inside cuboid / plane that intersects a point

I am trying to implement a trilinear interpolation algorithm for cuboids. Please excuse my lack of math jargon, this is not my area! The cuboids can be rotated in any dimension in 3D space. No two ...
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2answers
88 views

Find a polynomial of lowest possible degree

Find a polynomial $f$ of lowest possible degree such that $f(x_{1})=a_{1}$, $f(x_{2})=a_{2}$, $f'(x_{1})=b_{1}$, $f'(x_{2})=b_{2}$ where $x_{1} \neq x_{2}$ and $a_{1}, a_{2}, b_{1}, b_{2}$ are given ...
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1answer
29 views

Computing Two-Norm for interpolation of functions

$$ f(x) = x^3, p(x) = (3/2)x^2 − (1/2)x $$ The two-norm of f(x) - p(x) is: $$( \int_0^1 (f(x) - p(x))^2 dx )^{1/2} $$ p(x) interpolates f(x) at $$x=0, x=1/2, x=1$$ The result of the two-norm ...
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22 views

Determine Hermite interpolation polynomial and evaluate the appropriate error.

Determine Hermite interpolation polynomial and evaluate the appropriate error y:=f(x) ...
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4answers
136 views

Newton form vs. Lagrange form for interpolating polynomials

I'm just wondering, what are the advantages of using either the Newton form of polynomial interpolation or the Lagrange form over the other? It seems to me, that the computational cost of the two are ...
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1answer
20 views

$\bar x$ value for $f(\bar x) = 4.5$ with tabled values

The exercise asks for the $\bar x$ value as $f(\bar x) = 4.5$ using linear interpolation. I couldn't reproduce a table with MathJax, so I'll put the tabled with data as: $f(2) = 10$ $f(4) = 13$ ...
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33 views

Linear interpolation optimization

I want to interpolate any function $f(x)$ using only linear interpolation. So far I have found that the following equations do the trick pretty well. $$m(a,b,x)=\frac {f(b)-f(a)}{b-a}(x-a)+f(a)$$ ...
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27 views

Proof of Neville's algorithm

Let be $n\in\mathbb{N}$ and $(x_0,...,x_n)$ data points with the belonging functions $(f_0,...,f_n)$. Let be for $i=0,...,n$ and $i<j\leq n$ $$\begin{align*} p_i(x)=p_{i,i}(x) &= f_i\\ ...
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1answer
26 views

How to find the intersection of a line and a plane with interpolation ( given two points in the opposite side of the plane)

I have two points in the opposite side of a plane (P1,P2) in 3D space, and i know their signed distances to the plane(D1,D2). how can i use interpolation to calculate the point that is the ...
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1answer
19 views

How to resample translated grid to ensure consistent interpolation?

I have a grid of values, sampled at certain locations. I'd like to translate grid by some offset and resample it. The questions is: what should be the resampling and interpolation formulas such that ...
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49 views

When fitting a polynomial to data points, how to determine the reasonable degree to use?

I have wondered the following: Suppose that there is a set of data points $(x_i,y_i)$. Then I would like to know if it is more reasonable to assume if there is a polynomial relation of degree $m$ ...
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1answer
48 views

How does one derive Runge Kutta methods from polynomial interpolation?

In some numerical analysis classes, a neat way of deriving the Adams-Bashforth and Adams-Moulton methods is to approximate the function by a polynomial, and integrate the polynomial analytically over ...
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1answer
102 views

Find the approximation for the interpolation of $f(x)$ by a polynomial of second degree

Assume that $f(x)$ has a minimum in the interval $x_{n-1}\leq x\leq x_{n+1}$ where $x_k=x_0+kh$, $k$ being an integer. Show that the interpolation of $f(x)$ by a polynomial of second degree yields ...
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1answer
21 views

Should I use interpolation when finding median, and quartiles?

I am a S1 maths (Edexcel) AS student in the UK. My question: Say we have a stem-and-leaf diagram with 26 values. We want to find the lower quartile. To get the marks for our specification, we need to ...
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1answer
39 views

Why polynomial interpolation is considered as better than others?

Why polynomial interpolation is considered as better than others? In case of interpolation, the function $\phi(x)$ to approximate the unknown function $f(x)$ may be polynomial, exponential, ...
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RBF - triangle mesh interpolation - skinny triangles and incorrect results

I have triangle mesh, that I need to describe by RBF. I need to do this only locally on vertex neighborhood. All is working correctly if underlaying triangulation is reasonably regular. But if there ...
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3answers
49 views

Given the following data pairs, find the interpolating polynomial of degree 3 and estimate the value of y corresponding to x = 1.5.

So i was given this question Given the following data pairs, find the interpolating polynomial of degree 3 and estimate the value of y corresponding to x = 1.5. a) $(0, 1), (1, 2), (2, 5), (3, 10)$ ...
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2answers
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Does there exist smooth or $C^2$ function for some infinite given points $a(n)$?

I know that there exist some smooth function (polynomial) for finite numbers of values. The question is if there exists function (not necessary unique) which is twice differentiable and $f(n)=a(n)$?
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Finding an error bound for Lagrange interpolation with evenly spaced nodes

I know that the error bound for Lagrange interpolation is usually $$\frac{M_{n+1}}{(n+1)!}\max_{x\in[a,b]}|(x-x_0)\cdots(x-x_n)|$$, where $M_i=\max_{x\in[a,b]}|f^{(i)}(x)|$. I'm trying to find the ...
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Can monotone cubic interpolation be implemented explicitly in B-spline form?

I have been interpolating cubic splines to some data, but it is now clear that I need my curves to be monotonic. Wikipedia and StackExchange sources describe how to impose the monotonicity condition ...
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Does cubic spline interpolation preserve both monotony and convexity?

I have a question. Let's say i have a function $f(\cdot)$ such that $Dom(f) = [a,b]$. The function is at least of class $C^2$ and it is both strictly monotone and convex. My question is, does a ...
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1answer
46 views

Difference between varying a matrix or a variable over time

My question is: I have a Rotation_Matrix and I want to interpolate the rotations over time (from instant 0.0 to instant 1.0). I've done it with two approaches: For the first case, I extracted the ...
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1answer
26 views

Finding y-value in data point to determine coefficient on interpolating polynomial.

Let $P(x)$ be the interpolating polynomial for the data $(0, 0)$, $(0.5, y)$, $(1, 3)$ and $(2, 2)$. Find $y$ if the coefficient of $x^3$ in $P(x)$ is $6$. I tried finding the Lagrange interpolating ...
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1answer
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One parametric family that interpolates continuously between identity and natural logarithm on (0,1]

I am looking for a family of continuous functions $f_p$, $(0,1]\to\mathbb{R}$, and $p\in [1,\infty)$ that fulfill $$ f_1 \equiv \log(x) \\ \lim_{p\to \infty} f_p \to x$$ for $x\in (0,1]$. I ...
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23 views

Gauss Markov Theorem vs Least Squares

I am learning about the Gauss Markov theorem - I know I have not understood it correctly. Assuming the conditions are met, I was aware that there exists infinite solutions for a line of best fit.I ...
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42 views

Linear, Bilinear and B-spline interpolation

I've read that the linear interpolation isn't differentiable everywhere and it would be better to model a continuous-space image using quadratic or cubic B-spline interpolation because is ...
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2answers
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how to map series of coordinates onto a series of coordinates with different resolution

I have a set of target coordinates and a set of actually clicked coordinates which should be approximately the same, but not identical. The y coordinates are equal, however, the x-coordinates differ, ...
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Order of continuity of the cosine interpolant

I can't found any reference about the order of continuity of the well-known cosine interpolant: (1 - cos(t * PI) / 2; , where t is in the interval [0, 1] Anyone ...
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Fuzzy Logic Calculations for Non-Intersecting membership values

So i am not a fuzzy logic expert but wanted to see if what I am trying to do falls into something that I can solve using fuzzy logic. I have 3 categories A, B & C and for each category I have 4 ...
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1answer
21 views

Quadratic spline with the smallest sum of squares of derivatives

In quadratic spline interpolation, a formula is given and all we really need to do is calculate the values $$S_i'(t_i) = z_i$$ In quadratic spline interpolation specifically, one has a single degree ...
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1answer
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Spline interpolation degrees of freedom

When using cubic spline interpolation, we have to solve $n-1$ equations with $n+2$ unknowns. What we can do is set $z_0 = z_n = 0$, which gives the natural cubic spline. But could we also set some ...
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1answer
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Having trouble interpolating a polynomial using Newton's Method

Given the data: Find an interpolating polynomial using Newton's method. I am a bit confused on how to do this. I think it should be fairly simple.. Here is what I have so far: $p_0(x) = -1$ ...
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2answers
27 views

Interpolating polynomials

If I interpolate a polynomial of degree n with n+1 points, will I always get the polynomial itself back? if so, does it work for k < n+1 points?
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2answers
57 views

Cubic spline with clamped boundaries

I have a cubic spline interpolation problem to work through. I think I understand what is required of the question, but my biggest concern is the nature of clamped versus natural boundaries. All the ...
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17 views

Existence of a continously differentiable interpolator of a finite dataset

Formally, a function of several real variables f: Rm → Rn is said to be differentiable at a point x0 if there exists a linear map J: Rm → Rn such that $\lim_{\mathbf{h}\to \mathbf{0}} ...
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Other proofs of uniqueness of interpolating polynomial

I think that one of the well known proofs is this one: Let $f:[a,b]\to\mathbb{R}$ be a function and $P_n:[a,b]\to\mathbb{R}$ be the interpolating polynomial for $f$ on $[a,b]$. Let the nodes of ...
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Express interpolating polynomial as a linear combination of interpolating polynomials for subintervals

Given a set of points $(x_i, u_i), \; i = 1, \dots, n$ (for my application $n = 5$) one can construct an interpolating polynomial $$ P(x) = \sum_{i=1}^n u_i \ell_i(x), \quad \operatorname{deg} P = n - ...
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111 views

Difference between ordinary kriging and simple kriging with normalization

Simple kriging assumes known mean, which it seems can be induced by normalizing the data in each dimension around a mean of 0. What is the difference between performing simple kriging in this manner ...
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Polynomial Interpolation When part of $y_i$'s are Shuffled

Hypothesis: Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and ...