Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points.
0
votes
1answer
68 views
How to find the equation from data?
I know things about linear/quadratic fittings etc.
I'm just wondering, if i know a set of data for value e.g.
z=[-2.563 -0.1932 -0.1502 -0.1102 -0.836 -0.5234]
and l=[1 2 3 4 5 6]
m=[6 5 4 3 2 1]
I ...
1
vote
1answer
86 views
Interpolation error for the exponential function
I'm studiyng for my exam of scientific computing, specifically to the subject of interpolation techniques, I'm stuck with this problem:
How many equally spaced nodes must be taken to interpolate the ...
3
votes
1answer
83 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. ...
1
vote
1answer
95 views
Physical meaning of spline interpolation
I remember that when I took my Numerical Analysis class, the professor said the spline interpolation take its name from a kind of wood sticks used to draw curved lines. Also Wikipedia say that the ...
0
votes
1answer
56 views
How to determine the function factors in order to fit the curve?
I am trying to calculate innovation and imitation factor by ovserving the usage of specific service among the population.
After going through an overview of the paper I wrote the fuction in matlab:
...
11
votes
2answers
163 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} + ...
2
votes
3answers
170 views
How can I find out 2 unknowns in a cubic equation?
I need to give a bit of a background first, so please bare with me. I have a set of values that represent servo motor position values. By default I end up with a large set of values and I'd like to ...
1
vote
2answers
42 views
found a function from equations and inequality?
I'm Software engineer and I'm having little issue solving this problem let's called H.
Well I'm looking for the mathematical expression of the function f(x) based on 3 equations and one inequality. ...
0
votes
1answer
241 views
Function to represent curve between two 2D points
I need a (simplest) function that interpolates values in range from predefined point $A$ to $B$ with rules:
it must be smooth curve
direction near $B$ must be the same as predefined $D$ vector
...
0
votes
0answers
19 views
Multivariate interpolation and inversion
I have tabulated data in the form of $x$, $y$, $f(x,y)$, and $g(x,y)$, where both $f$ and $g$ are monotonic in $x$ and $y$. I have working interpolation schemes for:
Getting $x(y,f)$ and $x(y,g)$
...
0
votes
1answer
199 views
Reconstruct Control points in a Bézier Curve?
I have a curve that I know is a (non-periodic) Cubic Bézier Curve (because I constructed it as such). I stored each ordered pair in the curve, but not the control points. Is it mathematically ...
0
votes
3answers
73 views
Which Interpolation should I use to create a curve?
I'm pretty weak in the field of mathematics, but a strong programmer. I am looking for a mathematical solution that, given two points on a line will give me a curve between them, including those two ...
2
votes
0answers
92 views
Runge function error second factor
I'm currently learning about the Runge function. On Wikipedia, I read the following:
Consider the function:
$ \dfrac{1}{1+25x^2}$
Runge found that if this function is interpolated at ...
1
vote
1answer
171 views
Finding an interpolating polynomial and natural cubic spline for a given accuracy
I'm trying to make an exercise but I don't know how to start. Is there somebody that can give me a hint so that I can start with the exercise. The exercise is:
Consider the function $f(x) = \sin(x)$ ...
0
votes
1answer
241 views
Resolve a thin plate spline function
I am trying to keep the outline of an object in a video. So I have the coordinate of the outline of the object in the image $t$ and after computing the optical flow I have the coordinate in the image ...
0
votes
1answer
533 views
How to perform simple linear interpolation on a data set
With the following data set, what is the best way to interpolate the data for each time.
...
0
votes
0answers
106 views
Should n points always be interpolated by n+1 degree polynomial?
I'm studying interpolation and I see that if you have 2 points you use a 3rd-degree polynomial and likewise a 6th degree polynomial for five points. Is this a general formula, and if so, what is it ...
1
vote
1answer
106 views
How to write this in sigma notation?
Newton's formula for interpolation is
$$P(x)=c_1+c_2(x-x_1)+c_3(x-x_1)(x-x_2)+c_4(x-x_1)(x-x_2)(x-x_3)+\cdots$$
I prefer sigma notation, when it is possible. Can this be written in sigma notation?
6
votes
1answer
185 views
Is there a name for these polynomials?
Given $t \in \mathbb{R}[0,1]$, consider the following set of polynomials:
$$
\left[-{\left(t - 1\right)}^{2} t, {\left(t - 1\right)} {\left(t^{2} - t
- 1\right)}, -{\left(t^{2} - t - ...
2
votes
0answers
132 views
linear interpolation error estimate for non-smooth function
Suppose I have two points $x_1,x_2$ between which I would like to have a linear interpolation $P_1$. I know the value of the function $f$ at $x_1,x_2$. The error at any point between the two will be ...
0
votes
1answer
259 views
Numerically find the intersection point between two sets of data.
I'm looking for an efficient way to determine if two paths (sets of x,y coordinates) intersect at a point.
Input - (x,y) from a Mercator Projection (longitude,latitude) coordinates
Output - ...
2
votes
3answers
138 views
Deriving an equation that satisfies many points
Say I have a collection of points, for example the following:
(1, 167), (2, 11), (3, 255), etc
Is it possible to construct an equation that satisfies all of ...
1
vote
2answers
2k views
Given four points on a cubic function curve, how can I find the curve's function?
Say I have a curve
$$y = ax^3 + bx^2 + cx + d.$$
I don't know $a$, $b$, $c$ or $d$, but I do know the $(x,y)$ values of four points on this curve.
How can the values of $a$, $b$, $c$ and $d$ be ...
1
vote
1answer
31 views
Finding gradual values
I'm writing some code for a pressure level sensor for propane tanks. The manual provides me with the following table with the caption:
"Best accuracy will be obtained using the calibration data in ...
0
votes
1answer
106 views
Spline with varing tension, selection of tension factor
I need to perform a special interpolation, using that kind of basis :
$$\varphi_{i,j}(x) = a_i + b_ix + c_i(\cosh(\tau\ x) - 1) + d_i(\sinh(\tau\ x) - \tau\ x)$$
where the $a_i$, $b_i$, $c_i$ and ...
3
votes
2answers
484 views
Need a formula for a quadratic spline
I'm trying to reproduce some results from a paper and I need an explicit formula for a specific quadratic spline to do so.
The problem is, I've only got a plot of it. The quadratic spline is from ...
0
votes
1answer
105 views
Derivative of a function defined by the divided difference of another function.
Given a function $f$ of class $C$ $^{n+2}$ in an interval $[a,b]$ and $x_{0}=a<x_1<x_2 ... <x_n = b$ a subdivision of $[a,b]$ into $n+1$ points. Given another function $g$ defined in the ...
2
votes
1answer
128 views
Why is this a linear interpolation?
Let $J_{k,n}$ be the dyadic partition of $[0,1]$, i.e. $n\in \mathbb{N}_0,k=1,\dots,2^n$, $J_{k,n}:=((k-1)2^{-n},k2^{-n}]$ and we denote with $\phi_{n,k}$ the Schauder functions over $J_{k,n}$, i.e. ...
1
vote
1answer
96 views
Interpolation of vectors with quadratic polynomial
I have following points (-|b-a|,a), (0,b), (|c-b|,c) with a, b and c as two-dimensional vectors. These should be interpolated component-by-component with a second-degree polynomial p.
My problem now ...
0
votes
1answer
67 views
Sagar an Payne stress-strain relationships and Boussinesq aproximation in Matlab
I have to do 2 problems in Matlab, and the Math course is not my favourite one. However, I have tried to resolve the first problem, based on another exercise, but I'm pretty sure it's wrong. Can you ...
0
votes
1answer
59 views
Fixing end derivatives up to the second order when interpolating points
I would like to interpolate a set of points in the real plane $(x_i,y_i), \ 1\leq i \leq n$ with specified end derivatives up to the second order. That is finding $f \in ...
1
vote
2answers
215 views
What mathematical function would do this: if $x = 0$ then $y = 0$ but if $x > 0$ then $y = 1$?
$x = 0$, $f(x) = 0$
$x = 1$, $f(x) = 1$
$x = 2$, $f(x) = 1$
$x = 3$, $f(x) = 1$
...
There have been so many times I could have used this at different programming problems but I always resorted to ...
0
votes
0answers
109 views
Cubic spline with derivative end conditions
I want to plot in Matlab the cubic spline with derivative end conditions equal to the exact values calculated with differentiation.
function is: $f(x) = sin^2x$
0
votes
1answer
555 views
Need to understand question about not-a-knot spline
I am having some trouble understanding what the question below is asking. What does the given polynomial $P(x)$ have to do with deriving the not-a-knot spline interpolant for $S(x)$? Also, since ...
1
vote
3answers
278 views
Polynomial regression interpolation? [duplicate]
Possible Duplicate:
Writing a function $f$ when $x$ and $f(x)$ are known
I'm not versed in mathematics, so you'll have to speak slowly...
If I want to fit a curve to the points,
...
1
vote
1answer
60 views
Proof that infinite functions can fit a table of numerical values
Suppose while conducting experiments, I measure a finite number of variables with some constants like temperature, etc. We get a table of finite number measurements (numerical values to some decimal ...
0
votes
1answer
80 views
Is it posible to interpolate convex hull in 2d space
I have $n$ points (in this example $11$) and I need to interpolate them in such a way that I have a function $f(t) \rightarrow (R, R)$ where $t \in [0; 2\pi]$. It can be parametric curve, but I need ...
0
votes
0answers
118 views
the natural cubic spline
Construct a natural cubic spline that interolates the function defined by f(x)=1-x^4 on[0,1] using nodes at 0 1/2 2/3 and 1
I don't really understand this question, I reviewed the notes then I'm ...
1
vote
1answer
71 views
Lagrange form and differences
For a function f and distinct points $\alpha$, $\beta$, $\gamma$; what is
meant by $f[\alpha,\beta,\gamma]$?
Find the Lagrange form for the polynomial $P(x)$ that interpolates
$f(x) = ...
1
vote
0answers
89 views
Extending Hermite polynomial interpolation
Working with the definition of Hermite polynomials
$x_0,\ldots,x_n$ are distinct in $[a, b]$, $f''(x)$ is continuous on [a, b], then
$$H_{2n+1}(x)=\sum_{j=0}^{n} [f(x_j)H_{n,j}(x)] +\sum_{j=0}^{n} ...
1
vote
1answer
209 views
Lagrange Coefficients in Maple
I'm trying to compute Lagrange coefficients in Maple. Having found the $n$ roots of a Lagrange polynomial, I want to calculate the $j$-th coefficient:
$$L_j(x) = \prod_{{i=0}\atop{j \neq ...
1
vote
1answer
177 views
Piece-wise linear interpolating polynomials
Somebody please help me to obtain piece-wise interpolating polynomials for the function $f(x)$ defined by the below data:
$x=1$, $f(x)=3$; $x=2, f(x)=3$; $x=4, f(x)=21$; $x=8, f(x)=73$
I know the ...
1
vote
2answers
292 views
Determine the coefficients of an unknown black-box polynomial
Let $p$ be a polynomial of known degree $n$:
$$p(x) = a_0 + a_1 x + \ldots + a_n x^n$$
Suppose we have a magic black box that can evaluate the polynomial for us. How could one then determine the ...
2
votes
1answer
103 views
Divided difference coefficient of product of two functions
For any function $f$ and distinct reals $x_1,\ldots,x_n$, denote by $f[x_0,\ldots,x_n]$ the coefficient of $x^n$ of the minimal polynomial interpolating $f$ at $x_0,\ldots,x_n$.
Let $f$ and $g$ be ...
1
vote
3answers
103 views
Polynomial interpolation $n+1$ distinct points
How would you show that $p(x)= \sum\limits_{i=0}^n b_i(x-c)^i$ is equivalent to
$p(x)=\sum\limits_{i=0}^n a_ix^i$ by expressing the $a_i$ in terms of $b_i$ and $c$?
Also we know that the polynomial ...
0
votes
0answers
50 views
Interpolation with mixed types on range
I have the following process: an item of certain type (there are about 20) has real value. After going though processing (which I don't have any information about) changes its value. It may be smaller ...
3
votes
1answer
135 views
Polynomial interpolation of the residues of a rational function
Let $g(z) = a\prod_{i=1}^N (z-\lambda_i) \in \mathbb{Q}[z]$ be square-free. At each root $\lambda_i \in \mathbb{C}$, let $r_i$ denote the residue $\mathrm{Res}_{\lambda_i} 1/g(z)$. Let $I_g(z)$ ...
1
vote
1answer
271 views
Construct / find the simplest function based on data
Let's say I have these 7 natural numbers (all between 0 and 255):
255, 23, 45, 32, 87, 52, 146
How can I find a function F(x) that, once computed, gives me back ...
1
vote
1answer
218 views
Interpolation error
Working with a homework problem where I'm to derive an estimation of the interpolation error, and compare it with the actual error. This part is ok and I'm done with it. But while working with this in ...
2
votes
0answers
76 views
explicit error bounds for Multivariate interpolation
I want to interpolate a function of $d$ variables over a Cartesian grid, using multivariate interpolation, while characterizing interpolation error in terms of bounds on partial derivatives of the ...
