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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52 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 ...
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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 ...
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2answers
164 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 ...
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1answer
118 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 ...
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1answer
438 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 ...
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1answer
110 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 ...
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1answer
189 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 ...
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2answers
162 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 ...
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154 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 ...
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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} ...
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961 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 ...
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1answer
371 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. ...
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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 ...
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2answers
178 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 ...
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1answer
266 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 ...
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2answers
407 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 ...
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1answer
252 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? ...
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1answer
439 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 ...
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0answers
179 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 ...
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1answer
207 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 ...
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995 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 ...
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4answers
210 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 ...
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2answers
93 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 ...
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1answer
344 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 ...
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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 ...
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2answers
615 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 ...
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1answer
97 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 ...
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1answer
560 views

Interpolation polynomial

Consider the following table of values for a function $j_0(x)$: $\begin{array}{c|ccccc} x & \delta_0(x) \\ \hline ...
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1answer
940 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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88 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 ...
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1answer
216 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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46 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 $u$.
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260 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, ...
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1answer
181 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
117 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 ...
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1answer
319 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 ...
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1answer
249 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 ...
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1answer
1k views

Find Y-Axis value from point on curve using X-Axis value

I think this maybe a basic question, but I'm not mathematically equipped to handle the details. The problem I have is that I will have a curve such as in the image below and I want to find the y-axis ...
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1answer
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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 ...
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2answers
404 views

Efficient Integer Interpolation

Let $n\in\mathbb N$. Let $\{{x_i\}}_{i=1}^{n}$ be $n$ positive real numbers. Can one think of a fast way to construct a function $f$ such that $f(x_i)=i$? (i.e. $f$ maps $\{{x_i\}}_{i=1}^{n}$ to ...
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1answer
115 views

Interpolation problem: scattered points => “full grid”

I have 2D matrix of size like 512x512 or so. I also have a set of points with coordinates within bounds of the mentioned matrix ([0..512,0..512] in this case then) and each such point stores a ...
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1answer
186 views

Knot placement for a natural cubic spline

I am trying to approximate a function via a natural cubic spline. Suppose I sample the function on a grid i.e. I know the value of the function at a fixed number of equidistant points, say on 200 ...
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3answers
330 views

Trapezoidal rule problem

In a trapezoidal rule problem I got following question: "Evaluate the above integral using trapezoidal rule with five points." My confusion is here what we take for the value of $n$ is it $5$ or ...
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93 views

variogramas kriging

In order to find the altitude of a surface, we use the following method. My question is: What is the name of this method? We have $$ A=\begin{pmatrix} a_{1,1} & .. & .. & ...
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1answer
828 views

Automatic calculation of the intersection of discrete curves

first of all, let me apologize for a poor math-english translation, I'll try my very best. I have the following situation: I have over 16.000 data files which I generated from a biometric ...
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2answers
1k views

Understanding proof of uniqueness in theorem on polynomial interpolation

There is a slight part of the following proof in my textbook which I don't quite get. THEOREM If $x_0, x_1,...,x_n$ are distinct real numbers, then for arbitrary values $y_0, y_1,...,y_n$, there is ...
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0answers
606 views

Interpolating polynomial with Chebyshev nodes

I am interested in constructing an polynomial that interpolates some known arbitrary function $f(x)$ over the domain $x \in [0,70]$. I want the polynomial to have degree 14 and so need 15 points. ...
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648 views

Trapezoidal Rule for Numerical Integration

If the trapezoidal rule approximates an integral with trapezoids, then I thought (and was tought in high school) that the formula is: $ \frac{h}{2}(f(x) + f(x + h))$ Where $h$ is the distance between ...
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1answer
147 views

Original Proof of Riesz-Thorin

Wikipedia says that Riesz proved the Riesz-Thorin theorem in 1926 without using any complex methods. Does anyone know where the original proof can be found? ...
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263 views

Spline interpolation on a growing series of data

In a project on which I am working, I have a computationally expensive process to calculate a value for a given point in time. As I am working with RF data at approximately 16MHz, this would be ...