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

learn more… | top users | synonyms

0
votes
1answer
16 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 ...
1
vote
1answer
16 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 ...
0
votes
1answer
24 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 ...
0
votes
1answer
14 views

Need some help with applying specific boundary conditions to b-spline system of equations

I'm building a package for B-spline interpolation in Julia, and I've come across a boundary condition that I want to implement but can't wrap my head around how to do it (mathematically). Basically, ...
2
votes
4answers
46 views

Non-Piecewise interpolation over 3 points

I am trying to write an algorithm that interpolates between 3 values. The interpolation will be over the interval [0,1]. What I would like to do is: (Hopefully this makes sense) at x = 0, y = ...
1
vote
1answer
19 views

Should an interpolation coincide the original function on the given data points?

Suppose having a model $f(x)=y$ where $f$ is unkown. Moreover, suppose you have some data points for this model i.e. $(x_1,y_1), (x_2,y_2), \dots , (x_n,y_n)$. If one can find an approximate of $f $ ...
0
votes
0answers
23 views

Calculating B-Splines and dimension of spline space

I've got the following assignment: Let $S$ be the space of piecewise polynomials of degree $3$ on the intervall $[-1;1]$ with knots $x_i = -1+\frac{i}{2}, 0 \leq i \leq 4$. (a) Calculate a basis of ...
3
votes
1answer
63 views

Interpolation and mapping between scattered vectors in two unequally dimensioned spaces

Imagine two spaces: An ‘input’ space with dimension $m$. An ‘output’ space with dimension $n$. $m \geq n$ There are points in each of these spaces defined such that some characteristic is ...
0
votes
1answer
33 views

Finding an Entire function with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$

I am really stuck on a homework problem, which boils down to the following: We need to exhibit an entire function $f$ with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$. The only sorts of functions ...
4
votes
1answer
38 views

Problem involving polynomial and arbitrary continuous function

Let $f\in C^4[0,1]$ and $p$ a polynomial of degree $3$. Suppose: $$f(0)=p(0),\quad f'(0)=p'(0),\quad f(1)=p(1),\quad f'(1)=p'(1)$$ Show that for each $x\in [0,1]$ there exists $\xi\in [0,1]$: ...
0
votes
1answer
31 views

Fit $n$ Bézier paths to coordinates

I have a some coordinates $(X_i, Y_i)$ and I have to fit exactly $4$ cubic Bézier-paths to them (in other words, I have to find the 4 best fitting Bézier-paths, and by best fitting I mean that the ...
0
votes
0answers
21 views

Polynomial Interpolation and Secret sharing

Hypothesis: We define all values and polynomials over the field $\mathbb{F}_p$ for a large prime $p$ (e.g. 128-bit). My question is related to the "Shamir secret sharing" scheme in computer ...
0
votes
0answers
17 views

Root of interpolated polynomial when y-coordinates are permuted

Hypothesis: All values and polynomials are defined over a field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit) Suppose we have $n$ pairs of $(x_i,y_i)$. As we all know, given the ...
4
votes
1answer
80 views

Is there a name for this piecewise cubic interpolation kernel

I went looking for a way to do piecewise cubic interpolation, like natural cubic splines, but: expressible as a convolution of data points with a piecewise cubic kernel; and still C2-continuous ...
0
votes
0answers
14 views

Continuous/interpolating alternative to order of magnitude?

Define $\operatorname{magnitude}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rfloor }$ and $\operatorname{magnitude'}\left(x\right) = 10 ^ { \lfloor \log_{10} x \rceil }$ Currently I'm using this ...
0
votes
1answer
17 views

Find expression for $\sum_{k=0}^{n} l_k(0)x_k^{n+1}$

If the interpolation of $f(x)$ on the set of distinct points $x_0, x_1, \cdots x_n$ is given by $$\sum_{k=0}^{n} l_k(x)f(x_k).$$ Find an expression for $$\sum_{k=0}^{n} l_k(0)x_k^{n+1}.$$ I ...
1
vote
1answer
32 views

Bernstein Interpolation 2D

I am well aware of the equations for 1D Bernstein Interpolation. But I do not understand how to extend it to 2D. I am guessing that the equations in the following image would do for 2D Bernstein ...
0
votes
0answers
10 views

Divided difference acts on what space?

I'm currently trying to explain divided differences with a view to defining the Newton form of the interpolating polynomial. I'm using the definition: The $k$th divided difference of a function $f$ ...
0
votes
1answer
25 views

Is interpolating well-sampled data (Nyquist-Shannon theorem) a cheat?

Suppose to sample a signal $s(t)$ with bandwidth $B$ with a sampling frequency $f_c$. Suppose also that the number of sample collected is $N$ (the duration of the signal acquisition is then $T = ...
0
votes
0answers
11 views

Monotonicity of 1D interpolation with signed errors

Suppose that we're given $\textbf{x} = (x_i,y_i)_{i = 1}^N \subset \mathbb{R}^2$ such that $x_i$ are distinct. There are a number of well known ways (see e.g. Implementation of Monotone Cubic ...
1
vote
2answers
41 views

How to find the polynomial given its factors? (A bit typical one here)

I recently saw this problem (and I should have paid more attention to my middle school maths classes). Find a 3 degree polynomial of $x$ which is $0$ when $x=1$ and $x=-2$, $4$ on $x = -1$ and $28$ ...
0
votes
2answers
37 views

Polynomial interpolation of $n+1$ points but ensure last coefficient is a certain number?

I have $n+1$ data points $(x,y)$, and I want to create an interpolating polynomial as described here https://en.wikipedia.org/wiki/Polynomial_interpolation. However there is a twist, I want to ...
1
vote
1answer
43 views

Is there an equation for the exact line of best fit?

Is there some sort of equation/formula that can be used to find the exact values of $m$ and $b$ in $y=mx+b$ of any data points for the line of best fit? I want to be able to do this manually, not with ...
1
vote
1answer
26 views

Finding the equation of the line through X number of points?

I have a line graph with $8,000,000$ points. The X axis goes from $0$ to $7,999,999$ in increments of $1$ and the Y axis is either a $0$ or a $1$. There are no fractions on either axis. Is there an ...
1
vote
1answer
29 views

cubic spline interpolation - derivative known -

I at the moment trying to understand how to apply the interpolation method stated above. I have been given a start and end position, and for both position i know what their slope is. $\dot{X_a} = ...
0
votes
1answer
55 views

Determine equation from graph

Background: I'm working on a script to read/parse a file generated by a piece of software I use to create music mixes. One aspect I'm having difficulty with is translating the volume value from it's ...
1
vote
1answer
29 views

Spline terminology

I am reading up on splines and as a beginner I have a basic question - Does it make sense to say - "I will fit a cubic b-spline to the data". As b-spline is just a representation of spline in terms ...
0
votes
0answers
17 views

Linear interpolation from perspective-correct interpolators

This question is trying to approach this problem from a mathematical perspective. I have some value $u$ that I want to interpolate linearly, as $(1-a)u_0+a u_1$. However, I can only use ...
3
votes
2answers
76 views

Smoothest function which passes through given points?

I am trying to interpolate/extrapolate on the basis of a known collection of (finitely many) points. I'm wondering if there is a way to formalize this intuitive notion: find a 'smoothest' function ...
0
votes
1answer
20 views

error bound for polynomial interpolation with derivative matching

We all know the following formula for the maximum error (evenly spaced) polynomial interpolation: $|f(x) - p_n(x)| \leq \frac{h^{n+1}}{4(n+1)} \max_{x\in [a,b]} f^{(n+1)}(x)$ where $p_n(x)$ is the ...
3
votes
3answers
370 views

Linear interpolation with two points

Question: $p(x)$ is the linear function that interpolates $\sin(x)$ at $0$ and $\frac{\pi}{2}$. And I need to show that $\ |p(x) - \sin(x)|\le\ \frac{1}{2}(\frac{\pi}{4})^2$ My attempt: $\ ...
2
votes
0answers
31 views

How to use piecewise quadratic interpolation?

I'm attempting to get the hang of quadratic interpolation, in MatLab specifically, and I'm having trouble approaching the process of actually creating the spline equations. For example, I have 9 ...
1
vote
0answers
28 views

Catmull-Rom: spline and filter

On this website, the author gives this definition for Catmull-Rom splines (slide 10): $$catmullRom(t) = \frac{1}{2}\left\{\begin{array}{ll} t^3 + 5t^2 + 8t + 4 & \text{if } -2 \le t \lt -1\\ ...
0
votes
0answers
20 views

Estimate the error of interpolation/extrapolation

I'd like to know how we can estimate the error of interpolation. For example, let's consider Lagrange interpolation. We don't know anything about the real function (it could be an algebraic or a ...
0
votes
0answers
13 views

Dynamic learning for efficient interpolation (combinatorics)

Please regard this as soft question and reference request. Suppose I have to test out a combination of certain candidates. For example a combination of two signals at side A and side B. Assume we ...
1
vote
1answer
36 views

Newton 3d grade polynomial , simpson 3/8

Hello guys and sorry for my bad English , i have the following homework i should composite Newtons polynomial interpolation 3d grade , Simpsons 3/8 method with matlab ! But i have some trouble, i ...
0
votes
1answer
33 views

Interpolation with logaritmic function

I want to interpolate with the function $$f(x) = a\ln(x+b)+c$$ That is, I assume some sort of logarithmic relationship, but there might be an offset. I assume that I need 3 datapoints, as there are ...
0
votes
2answers
38 views

Which interpolation method for complicated, smooth curves?

Which interpolation method should I use for complicated "smooth" curves such as $\frac{sin(x)}{x}$ for $x>0$.
1
vote
1answer
41 views

Number of points needed for linear interpolation of sine in $[0,\frac{\pi}{2}]$ with given error bound

I want to get a set of equispaced points in $[0,\pi/2]$ and use piecewise linear interpolation generated by those points to fit the sine function. And I want to determine how many points do I need to ...
1
vote
1answer
32 views

Finding Roots of a Polynomial Represented in Point-Value Form

Consider we have $n$ pairs of $(x_i,y_i)$. We all know that given the $n$ pairs we can interpolate a polynomial of degree at most $n-1$. Also, it is clear if we want to find roots of a interpolating ...
0
votes
1answer
34 views

Interpolation and divergence of $n$-th derivative

i have a curiosity. Let's say i have a function $f \in C^{\infty}[a,b]$ such that there's a $x_0 in [a,b]$ that makes and $f(x_0) = 0$ and $a_n = f^{(n)}(x_0)$ diverges (i.e. $|a_n|=\infty$) could ...
1
vote
1answer
44 views

Signal processing : future values prediction

Let $f : \mathbb{R}^+ \rightarrow \mathbb{R} $ be a continuous function. Do you have some references (books or online resource) about techniques that allow to predict $f(x_{n+1})$, knowing $f(x_0), ...
2
votes
1answer
158 views

Approximating on a line

Say I have sampled some points in $[0,1]^2$ and evaluate a function $f(x,y)$ for them. I am interested in the behavior of $f$ along a single dimension. If the points were like ...
0
votes
1answer
31 views

Is it possible to interpolate $e^n$ in more than one way?

The most basic definition of exponentiation is repeated multiplication, $$e^n = e \cdot e \cdot \cdot \cdot \cdot e$$ $n$ times However, if $n$ is a rational number such as $2.4$, this ...
0
votes
0answers
17 views

Composite 3 point Gauss integration of f(x) over an interval

I'm attempting to express the composite 3-point Gauss formula for integrating f(x) over an interval [a,b]. $\int_{a}^b f(x) dx = \sum_{j=0}^n f(xj)*cj$ where $cj = \int_{a}^b lj(x)dx$ $lj(x)$ are ...
1
vote
0answers
24 views

Calculating Newton Form of Interpolating Polynomial

For the function $f(x)=\frac{1}{x}$, I am trying to calculate the Newton form of the interpolating polynomial for the points $x_0=2$, $x_1=3$ and $x_2=4$. I understand that the Newton form of the ...
0
votes
0answers
7 views

Taylor Remainder over an interval for polynomial interpolation

When attempting to find how big n should be so that $|e^x - p(x)| < 10^{-4}$ over the interval $[-1,1]$ using Taylor Remainder, what value should I be using for $x$ in $(x - x0)^{n+1}$? I'm using ...
0
votes
0answers
15 views

Convergence theorems of periodic and natural cubic Splines

I have this question on the topic convergence theorems of cubic splines, for cubic splines which theire first derivatives at the start and end points are equal to first derivatife of the $f$ function ...
0
votes
1answer
18 views

Find a quintic polynomial that takes the values given in the preceding problem and, in addition, satisfies $p(3)=2$

Consider the following table: $$\begin{matrix}0& &2& &-9& &3& &7& &5\\ &0&&2&&-6&&10&&17\\ ...
1
vote
1answer
31 views

Using the polynomial of lowest order that interpolates $f(x)$ at $x_1$ and $x_2$, derive a numerical integration formula for $\int_{x_0}^{x_3}f(x)dx$.

Using the polynomial of lowest order that interpolates $f(x)$ at $x_1$ and $x_2$, derive a numerical integration formula for $\int_{x_0}^{x_3}f(x)dx$. I know that we aren't assuming uniform spacing. ...