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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2
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1answer
71 views

Interpolation with degree restriction

(using $f[x_1, ... , x_n]$ to denote the forward difference operator) I have a polynomial $P(x)$ interpolating $5$ points $x_0, ... , x_4$ and $2$ derivative values $x_0, x_3$ across an evenly spaced ...
1
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1answer
70 views

Extension of Fourier Transform

We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
2
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1answer
140 views

Nontrivial expansion of a multivariate power series in form of a single variable series?

I am trying to interpolate a function defined over a three-dimensional real space: $$f: R^3\rightarrow R\\(x,y,z)\rightarrow f(x,y,z)$$ Let assume I have $N_1 N_2 N_3$ points in the space which form ...
1
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0answers
29 views

Is there a way to estimate the range of fitting coefficients from only the data?

Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain $f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$ where ...
1
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2answers
970 views

Easing function, constant velocity then decelerate to zero

I'm trying to write an interpolator for a translate animation, and I'm stuck. The animation passes a single value to the function. This value maps a value representing the elapsed fraction of an ...
2
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1answer
57 views

Is there a nice way to interpret this matrix equation that comes up in the context of least squares

So I am working on this problem with fitting a second degree polynomial of the form $y=a_1x^2+a_2x+a_3$ to four points using least squares. One of the parts of the problem is to write out the matrix ...
0
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2answers
127 views

Finding t in basic geometric interpolation

I have the following problem related to interpolation and was wondering if anyone had any ideas? In the following diagram, there exists known-points: A, B, C, D, E, unknown points F, G. I'm trying ...
3
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1answer
100 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 ...
2
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1answer
1k views

How does one interpolate between polar coordinates?

I'm finding that when I try to use the standard methods of interpolation in polar space, the result is not what I would expect. For example, when interpolating between the following polar coordinates: ...
1
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1answer
58 views

What are the explicit expression for this interpolation problem

We want to fit $f(x) = a_0 + a_1 *x + a_2 * x^2 + ... + a_n * x^n$ to the data $(x_i,f(x_i))$ for $i = 0 ... n.$ It will give rise to the following system $ A a = b $ Here $ a = [ a_1 a_2 a_3 ...
2
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1answer
297 views

About Generalized Hermite interpolation

I am currently studying numerical method. I understand that the regular way of cubic Hermite interpolation on arbitrary interval $[a,b]$ is: $ p(u)=\left( \begin{array}{c} 2t^3-3t^3+1\\ t^3-2t^2+t\\ ...
0
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1answer
119 views

Triangle $z$-index interpolation between the vertices

I got a $2$D triangle, each vertex has a $2$D coordinate with a $z$-index value (NOT a $z$ coordinate!). The $z$-index value indicates whether a vertex lays on, in front of, or behind your screen ...
1
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1answer
278 views

B-Spline Interpolation/Approximation

I've got a couple of probably very simple questions, yet some googling didn't bring up what I was looking for. First what I want to do: I have a grid, and the gridpoints are function values. I want to ...
1
vote
1answer
53 views

What is the way to determin how good a sequence will interpolate?

Say I have to sequences of numbers: $$[5, 10, 14, 21, 27, 31]$$ $$[1, 20, 21, 22, 30, 31]$$ Even though they both get to $31$ by the $6$th element, logic tells me that only the first one is a good ...
1
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0answers
87 views

What are the constants in the relationship between point density and median point distance?

Given $\rho$ particles uniformly distributed on a plane within a unit square ($\rho > 1$), each particle has another particle that is closest to it; the median of those nearest distances is called ...
4
votes
2answers
171 views

How to make 3D object smooth?

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal ...
0
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1answer
121 views

nth degree interpolating polynomials

Given five ${x}$ points and for four of them I know ${f(x)}$ values. So I have to use interpolating polynomial to estimate unknown ${f(x_i)}$ ${ \displaystyle f(x) = \sum_{j=1}^{n} P_j{(x)} }$ with ...
6
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1answer
458 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 ...
1
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1answer
117 views

interpolation question. From MyMaxScore free AP BC Calculus exam.

First, I disagree with the answer sheet. That is why I am posting this question here. The question: Part B, Question 3a. The answer given (Sorry for the picture being so small). I know ...
2
votes
1answer
203 views

A Question About Linear Interpolation

So lets say I have two points $A=(x_1, y_1, z_1)$ and $B=(x_2, y_2, z_2)$. $A$ and $B$ are each associated with some scalar value $K_1$ and $K_2$. $K_1$ is negative and $K_2$ is positive and all the ...
1
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2answers
181 views

Interpolating multivariable functions

Assume I have two functions, $f_1$ and $f_2$, that both depend on $x$ and $y$, so that $f_1(x,y)$ and $f_2(x,y)$. I don't know the exact functions, but know values of each function at some points ...
0
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2answers
157 views

What do you call generating a function out of a graph?

In many physical phenomena, laws an relations to their variables are somehow interpolated (example by statistical data or samples) and then an approximate set of functions are generated to work ...
1
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0answers
71 views

Analogue for finite sums of $\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g)$

Please help me to find an analogue for finite sums of $$\int_{a}^{b}fg=t\bar{f}\bar{g}(b)-t\bar{f}\bar{g}(a)+\int_{a}^{b}(\bar{f}-f)(\bar{g}-g) \tag{*}$$ where $ \bar{f}(t)=\frac{1}{t}\int_{a}^{t} ...
0
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1answer
1k views

Understanding Lagrange error.

Here is an example from my Numerical Analysis book (Burden & Faires). Trying to understand Lagrange error, but I do not understand the statements in bold. In example 2 we found the second ...
0
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1answer
391 views

How to do interpolation using the newton basis?

I have these data points. $f_i(x)= \{10, 11, 14\}$ $x_i= \{0, 1, 3\}$ then the basis functions are. $\pi_0 = 1 \\\pi_1=(x-x_0)=x \\\pi_2=(x-x_0)(x-x_1) =x(x-1)$ So the matrix will become. ...
1
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2answers
66 views

What is the equation stands for in geometry(intuitively)?

I am writing a bilinear interpolation method. This method can be abstract by solve the equation A*x = b, A is a 4x4 matrix below: $A=\begin{pmatrix} 1 &x_1 &y_1 &x_1y_1\\ 1 ...
1
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2answers
181 views

Finding a simple spline-like interpolating function

I am looking for a continuous function $y=f(x,\alpha)$ for the interval $0\le x \le 1$ such that $0\le y \le 1$ and $y(0,\alpha)=0$ and $y(1,\alpha) = 1$ and $y(\alpha,\alpha) = 1-\alpha$ and ...
0
votes
1answer
278 views

finding derivative at intermediate point of known data set

I have a function $y = f(x)$, $ x \in [0,1] $ and $ y \in [0,1]$ Set of values $(x_i,y_i)$ are known for n points. I need to find derivative at point $x_{\zeta}$ such that $y(x_{\zeta}) = 0.5$ Now ...
1
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2answers
421 views

Cubic spline interpolation not producing an interpolant with continuous first derivative consistently

I have coded a nice cubic spline interpolator following the basic methods laid out here http://people.math.sfu.ca/~stockie/teaching/macm316/notes/splines.pdf . My program reproduces the example laid ...
0
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1answer
263 views

How to determine a function from outputs

Given the following inputs and outputs to a function, how would I determine an algorithm to fit? ...
0
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1answer
456 views

formula for an upwards-sloping convex curve with known endpoints

For a project I am working on, I need a formula that can describe a curve between two known endpoints, where the curve will always be upwards sloping and always convex (or flat). There should be many ...
2
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0answers
186 views

B-Spline Definition

I'm currently working on my master's project. For this, I rely on one PhD-thesis in which I found a statement I do not understand. Unfortunately, the author hasn't answered to my mails yet, so I have ...
0
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1answer
229 views

Closest edge to a point

I have a tricky question and can't seem to find a good solution. Let's say I have 3 convex polygons (ABCD, ADEF, BHGC) that share 2 common edges, unknown J, and a known point I: I want to find the ...
6
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3answers
1k 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 ...
4
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4answers
216 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 ...
0
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2answers
98 views

Interpolating point outside quad

I'm trying to interpolate a destination point $(x,y)$ outside of a $4$ point convex polygon given a $4$ point source polygon with known values. It's sort of like inverse bilinear interpolation, except ...
3
votes
1answer
358 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 ...
1
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1answer
2k views

How would I create a exponential ramp function from 0,0 to 1,1 with a single value to explain curvature?

I need an exponential function that will take linear input from 0,0 to 1,1 and give me back an exponential shaped curve such that changes in X near the 0 point result in small increases in Y, but each ...
0
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2answers
670 views

For n=3 Lagrange interpolation why is it equal to 1?

I'm studying Lagrange's formula for polynomial interpolation and I cannot seem understand why for $n=3$ $$L_0(x)+L_1(x)+L_2(x)+L_3(x) = 1$$ for all real x. In my textbook it says as a hint that ...
3
votes
1answer
100 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 ...
0
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1answer
571 views

Interpolation polynomial

Consider the following table of values for a function $j_0(x)$: $\begin{array}{c|ccccc} x & \delta_0(x) \\ \hline ...
1
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1answer
988 views

Lagrange's basis function and Interpolation

Let $x_0,...,x_n$ be distinct real numbers and $l_k(x)$ be the Lagrange's basis function. $δ_n = ∏^n _{k=0}(x-x_k)$. Prove that a - $\sum^n_{k=0}x^j_kl_k(x) ≡ x^j$. for $j = 0,1,...,n$ ...
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1answer
90 views

monotone interpolation given two points and their derivatives

Is there an algorithm or implementation? I have searched the monotone cubic interpolation or monotone piecewise cubic interpolation. It seems that both the two methods cannot preserve the derivatives ...
1
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1answer
223 views

3D Numerical differentiation with spline approximation

I have three 3D matrices X, Y, and Z that define a matrix V of the same size over some region. The matrices are regularly spaced. I'm trying to compute the gradient of V. I have read that ...
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0answers
47 views

Interpolation Inequality in finite dimensional linear space

How can we prove - $$|\dot{u}(x) - \tilde{u_h}(x)| \leqslant h \left[\max_{0 \leqslant y\leqslant1}\ddot{u}(y)\right],$$ where $0\leqslant x\leqslant 1$ and $\tilde{u_h}$ is interpolant of $u$.
1
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1answer
266 views

monotone Hermite interpolation

Is there a reason why in particular it is so popular to use monotone cubic Hermite interpolation compare to say, quadratic? I understand the the order of the accuracy gets better with a higher degree, ...
1
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1answer
191 views

What is the motivation for this theorem on Polynomial Interpolation Error?

My textbook presents this theorem without any sort of introduction. It does cover using the Newton and Lagrange forms of the interpolation polynomial, so I've got that. Anyway, here's the theorem: ...
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1answer
119 views

Interpolation problem

I have the equation $f(x)=148x^4 + 3x^3 + 251x^2 + 56x + 157$. This equation gives us the points below. $(0,157),(1,101), (2,67), (3,4), (4,72)$ I want to interpolate this points in a $4$ degree ...
0
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1answer
339 views

Least-squares approximation polynomial

Consider the function $\displaystyle f(x) = \frac{1}{\alpha (x-\beta)^2 + 1}$ in the interval $I = [-1,1]$. Set $\beta = 0$. How do I get the expression for the least-squares polynomial, say $\tilde ...
1
vote
1answer
275 views

Lagrange interpolant degree 2

Consider the function $$f(x) = \frac{1}{\alpha (x-\beta)^2 + 1}$$ in the interval $I = [-1,1]$. Set $\beta = 0$. The Lagrange interpolating polynomial of $f(x)$ with degree $n=2$ for equally spaced ...