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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12
votes
2answers
178 views

Generalization of $\frac{x^n - y^n}{x - y} = x^{n - 1} + yx^{n - 2} + \ldots + y^{n - 1}$

I thought about a generalization for the formula $$\frac{x^n - y^n}{x - y} = x^{n - 1} + yx^{n - 2} + \ldots + y^{n - 1}$$ It can be written as $$\frac{x^n - y^n}{x - y} = x^{n - 1} + yx^{n - 2} + ...
10
votes
1answer
74 views

For a fixed degree, is there always a Lagrange polynomial below the original function?

Let $x_1<x_2< \ldots <x_n$ be $n$ real numbers, and let $y_1,y_2,\ldots,y_n$ be real values to be interpolated. Let $r\leq n$. For any $I\subseteq \lbrace 1,2,\ldots,n\rbrace$ of cardinality ...
9
votes
4answers
2k views

Polynomial fitting where polynomial must be monotonically increasing

Given a set of monotonically increasing data points (in 2D), I want to fit a polynomial to the data which is monotonically increasing over the domain of the data. If the highest x value is 100, I ...
8
votes
6answers
2k views

What is the pattern to this sequence?

$$0, 1, 3, 13, 51, 205$$ More specifically, $$(0,0)\quad(1,1)\quad (2,3)\quad (3,13) \quad(4,51)\quad (5,205)$$ I have tried using the interpolation feature in Grapher.app and Wolfram Alpha, but ...
7
votes
2answers
179 views

Negative value of $\sqrt[3]{20}$

Given $f(x)=\sqrt[3] x$, find an approximation of $\sqrt[3]{20}$ using Lagrange interpolation method. $x_0=0$, $x_1=1$, $x_2=8$, $x_3=27$ and $x_4=64$ $f(x_0)=0$, $f(x_1)=1$, $f(x_2)=2$, ...
7
votes
2answers
342 views

A Curious Binomial Sum Identity without Calculus of Finite Differences

Let $f$ be a polynomial of degree $m$ in $t$. The following curious identity holds for $n \geq m$, \begin{align} \binom{t}{n+1} \sum_{j = 0}^{n} (-1)^{j} \binom{n}{j} \frac{f(j)}{t - j} = (-1)^{n} ...
7
votes
1answer
274 views

interpolating the primorial $p_{n}\#$

The primorial $p_{n}\#$ is given by the product $p_n\# = \prod_{k=1}^n p_k$ (where $p_{k}$ is the $k$th prime) -- is there a natural (a la the gamma function $\Gamma(z)$) way of interpolating it for ...
6
votes
3answers
883 views

Is it possible to have a rule which generates: 2, 4, 6, 8, 10, 12, 14, 16, -23?

This is on Lagrange Interpolations . . . Is it possible to have a rule which generates the sequence: 2, 4, 6, 8, 10, 12, 14, 16, -23? The hint that he gave us is to use Summation Products, the only ...
6
votes
4answers
287 views

Are there smooth analogs to polynomial splines

Is possible to construct infinitely differentiable functions that interpolate through arbitrary points, the way polynomial splines do? If so, do they have a name and is there an algorithm for ...
6
votes
1answer
261 views

Cauchy Integral Formula for Matrices

How do I evaluate the Cauchy Integral Formula $f(A)=\frac{1}{2\pi i}\int\limits_Cf(z)(zI-A)^{-1}dz$ for a matrix ...
6
votes
1answer
4k views

What is the difference between natural cubic spline, Hermite spline, Bézier spline and B-spline?

I am reading a book about computer graphics. It is confusing about the various splines and their algorithms. What is the difference between natural cubic spline, Hermite spline, Bézier spline and ...
6
votes
1answer
193 views

Is there a name for these polynomials?

Given $t \in \mathbb{R}[0,1]$, consider the following set of polynomials: $$ \left[-{\left(t - 1\right)}^{2} t, {\left(t - 1\right)} {\left(t^{2} - t - 1\right)}, -{\left(t^{2} - t - ...
5
votes
2answers
182 views

$f(x)=1/(1+x^2)$. Lagrange polynomials do not always converge. why?

Let $f(x) = \frac{1}{1+x^2}$. Error of Interpolation with Lagrange polynomials for $n+1$ points is given by $$ e(x)=f(x)-P_n(x)=\frac{f^{(n+1)}(\eta_x)}{(n+1)!}\prod_{i=0}^n (x-x_i) $$ Carl Runge ...
5
votes
1answer
946 views

Lyapunov's Inequality for Weak-Lp Spaces

Let $(X,\mu)$ be a measure space. Suppose that $0 < p_{0} < p < p_{1} < \infty$ and $\frac{1}{p} = \frac{1-\theta}{p_{0}} + \frac{\theta}{p_{1}}$ for some $\theta \in (0,1)$. If $f \in ...
5
votes
1answer
354 views

Determining Coefficients of a Finite Degree Polynomial $f$ from the Sequence $\{f(k)\}_{k \in \mathbb{N}}$

Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ ...
5
votes
1answer
2k views

Natural cubic splines vs. Piecewise Hermite Splines

Recently, I was reading about a "Natural Piecewise Hermite Spline" in Game Programming Gems 5 (under the Spline-Based Time Control for Animation). This particular spline is used for generating a C2 ...
5
votes
1answer
644 views

What is the math used in Excel's GROWTH function?

I am trying to implement Microsoft Excel's GROWTH function in JavaScript. This function calculates predicted exponential growth by using existing data. What makes it tricky is that it must work with ...
5
votes
3answers
193 views

How to understand and create quaternions?

I have to multiply two quaternions to calculate a so called spherical linear interpolation between two $R^3$ coordinate systems within the interval $t = [0, 1]$. I understand how to do the ...
5
votes
0answers
220 views

Runge's phenomen: interpolation error using Chebyshev nodes oscillates

We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
4
votes
4answers
169 views

Profinite and p-adic interpolation of Fibonacci numbers

On the topic of profinite integers $\hat{\bf Z}$ and Fibonacci numbers $F_n$, Lenstra says (here & here) For each profinite integer $s$, one can in a natural way define the $s$th Fibonacci ...
4
votes
4answers
192 views

What are some “natural” interpolations of the sequence $\small 0,1,1+2a,1+2a+3a^2,1+2a+3a^2+4a^3,\ldots $?

(This is a spin-off of a recent question here) In fiddling with the answer to that question I came to the set of sequences $\qquad \small \begin{array} {llll} ...
4
votes
4answers
401 views

Proof of Schur's test via Young's inequality

I am able to prove the following generalization of Schur's test using the Riesz-Thorin interpolation theorem, however I have been stuck for days now trying to prove it using Young's inequality: Let ...
4
votes
2answers
81 views

Why is $L^{1} \cap L^{\infty}$ dense is in $L^{p}$?

It is mentionned that using the interpolation inequality $$\Vert f \Vert_{p} \leq \Vert f \Vert^{1/p}_{1} \Vert f \Vert_{\infty}^{1-1/p}$$ one can deduce that the space $L^{1} \cap L^{\infty}$ is ...
4
votes
1answer
84 views

Interpolation between iterated logarithms

I am investigating the family of functions $$\log_{(n)}(x):=\log\circ \cdots \circ \log(x)$$ Is there a known smooth interpolation function $H(\alpha, x)$ such that $H(n,x)=\log_{(n)}(x)$ for ...
4
votes
1answer
203 views

Polynomial interpolation

Let $P=[a,b]\times (c,d)$. Assume that we have given $n$ points $(x_1,y_1),...,(x_n,y_n)\in P$, such that $x_i\neq x_j$ for $i\neq j$; $i,j=1,...,n$. Does there exist a polynomial $f$ such that ...
4
votes
1answer
412 views

Hermite Interpolation of $e^x$. Strange behaviour when increasing the number of derivatives at interpolating points.

I am trying to understand Hermite Interpolation. Here is my pedagogical example. I want to approximate $f(x)=e^x$ on the domain $[-1,1]$ using Hermite interpolation. I choose the Chebyshev zeros ...
4
votes
2answers
54 views

How to non-linearly interpolate between 4 values

I'm looking for a non-linear way of interpolating between 4 values within a games engine. I have a unit square abcd. It has a different value for each edge ...
4
votes
1answer
736 views

Remainder term of Lagrange Interpolation Polynomial

Suppose $x_0,x_1,\ldots,x_n$ are $n+1$ distinct numbers in the interval $[a,b]$ and $f\in C^{n+1}[a,b]$. Then for each $x$ in $[a,b]$, there is a number $\xi$ in $(a,b)$ such that $$f(x) = P(x) + ...
4
votes
3answers
34 views

How to calculate the degree of Lagrange polynomial to satisfy a given error?

I need help. I have $f(x)=sin(x)$. If I want to use Lagrange polynomial to make an approximation of $f(x)$, what should be the degree of that polynomial if I work in the interval $[0,\pi]$, and the ...
4
votes
0answers
115 views

Could $4+2+4+2+4+2+\cdots = -1 $?

In physics classes, on this StackExchange and even in blogs the sum $1 + 2 + 3 + 4 + \cdots = - \frac{1}{12} $ has been under the microscope. Why does $1+2+3+\dots = {-1\over 12}$? The ...
4
votes
0answers
256 views

What is the maximum overshoot of interpolating splines in $d$ dimensions?

Consider cubic splines $s( x, y )$ which interpolate values $y = \{ y_0, y_1, \dots,y_n \}$, on the uniform grid $\{ 0, 1,\dots, n \}$. Fix $s''(0) = s''(n) = 0$ (natural splines). How big can ...
3
votes
3answers
134 views

Stuck with integral

Having this: $\int x\sqrt{1-x^2}dx$ Substitution: $t = 1-x^2$ $dt = -2xdx => dx=\frac{-2x}{dt}$ So: $$\int x\sqrt{1-x^2}dx = -\int x t^\frac{1}{2}\frac{2x}{dt} = -\int \frac{2x^2 ...
3
votes
2answers
1k views

Why is Lagrange interpolation numerically unstable?

Here is my understanding of the polynomial interpolation problem: Interpolating by inverting the Vandermonde matrix is unstable because the Vandermonde matrix is ill-conditioned, so "difficult" to ...
3
votes
2answers
462 views

Spline interpolation versus polynomial interpolation

What is the difference, if any, between spline interpolation and piecewise polynomial interpolation?
3
votes
1answer
3k views

linear interpolation in 3 dimensions

Say that I have 2 points in 3 dimensional space specified in Euclidean coordinates $p_0(x_0,y_0,z_0)$ and $p_1(x_1,y_1,z_1)$. How would I go about finding the coordinates of an unknown point that ...
3
votes
2answers
2k views

Implementation of Monotone Cubic Interpolation

I'm in need to implement Monotone Cubic Interpolation for interpolate a sequence of points. The information I have about the points are x,y and timestamp. I'm much more an IT guy rather than a ...
3
votes
2answers
2k views

How to calculate interpolating splines in 3D space?

I'm trying to model a smooth path between several control points in three dimensions, the problem is that there doesn't appear to be an explanation on how to use splines to achieve this. Are splines a ...
3
votes
1answer
225 views

Fritsch and Carlson is non-linear?

I was reading about this interpolation method and saw that it was mentioned that the algorithm is non-linear. What does that exactly mean? I am confused because I don't get what is "non-linearity" in ...
3
votes
1answer
2k views

2D array downsampling and upsampling using bilinear interpolation

I am trying to understand how exactly the upsampling and downsampling of a 2D image I have, would happen using Bilinear interpolation. Now I am aware of how bilinear interpolation works using a 2x2 ...
3
votes
1answer
43 views

Embedding of Weak Lebesgue Spaces

My question is analogous to the embedding $L^p\subset L^q(\Omega)$, for $p>q$ and for a bounded $\Omega$. In weak $L^p$ spaces, that is, $L^{p,\infty}$, does such an inclusion hold for arbitrary ...
3
votes
1answer
64 views

DPLL Algorithm $ \rightarrow $ Resolution proof $ \rightarrow $ Craig Interpolation

I really need help here for an exam that I got tomorrow .. Let's say I got a bunch of constraints: $ c1 = { \lnot a \lor \lnot b } \\ c2 = { a \lor c } \\ c3 = { b \lor \lnot c } \\ c4 = { \lnot b ...
3
votes
2answers
130 views

How to fit a function that depends on several nominal and one real variable?

I have data that map several nominal variables and one real parameter into a real value. For example: ...
3
votes
1answer
174 views

Fitting a surface to 2D measurements

I am looking for a way to fit a surface given a set of measured data $(x, y) \mapsto z$. A typical example would consist of anywhere between $10$ and $30$ measurements spread evenly over a disc. ...
3
votes
1answer
1k views

MATLAB Hermite interpolation

Anyone know where I can find the Hermite interpolation algorithm in MATLAB. Which Hermite interpolation algorithm solves this? I need to calculate a polynomial. Example: ...
3
votes
1answer
25 views

Is the Lagrange polynomial integer-valued for points with consecutive integer x-values?

What I'm really wondering is, does Lagrange polynomial interpolation have an answer for every question of "what's the next integer in this sequence"? Does it define an infinite integer sequence to ...
3
votes
1answer
78 views

Sigmoid function with separate control of derivative at 0 and sharpness of bend.

For a physical relationship, $f(x)$, I'm trying to model, i have a fairly good determination of some of the boundaries, such that $ f(0) = 0$ $f'(0) = B$ $ \lim_{x\to\infty} f = A $ So far, what I ...
3
votes
1answer
568 views

Is this algorithm an example of exponential interpolation?

We have an algorithm I'm trying to get my head around. The original author is gone and away and the whole thing seems to generally work, but I'd like to verify that it's working correctly. (And ...
3
votes
1answer
119 views

How can I add a curve to my otherwise linear values?

I have built an interactive map for the Web that transitions smoothly from lon/lat point to lon/lat point. The duration of the transition is calculated dynamically and depends on the distance between ...
3
votes
1answer
73 views

Linear, Bi-linear or better

I have been writing some code to do some interpolation of 2D data on an irregular grid. So far what I have done is: Triangulate the known points using Delaunay. Find the vertices of the triangles ...
3
votes
1answer
80 views

Nadirashvili surface

I'm referring to the article of N. Nadirashvili "Hadamard's and Calabi-Yau conjectures on negatively curved and minimal surfaces". In the proof of proposition 4.3 author use a theorem of Walsh. Now ...