# Tagged Questions

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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### Cubic Spline Interpolation

My problem is to find a interpolating cubic spline to the points $$\left\{(0,0), \left(\frac{\pi}{2}, 1\right), \left(\pi,0\right), \left(\frac{3\pi}{2}, -1\right),(2\pi,0)\right\}$$ I did as ...
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### Least square polynomial interpolation

Given an arbitrary continuous function f(x), let Pn(x) be the polynomial of degree at most n that approximates f(x) in the least squares sense. Is it true that Pn(x) interpolates f(x) at n + 1 points? ...
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### Algorithm Identification

Background I'm currently working with a system that has a 4-dimensional function. Currently, an algorithm is used to speed up calculation of the final value via interpolation, and two of the ...
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### build function passing for specific points

I have to solve a problem very similar to this how-to-create-a-function-passing-through-given-points I need a function that draw a curve like the blue one in the picture here thus passing as ...
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### Multistep Method: Gear's Formula Interpolation

Please explain how to do this. How can we use Lagrange Interpolation to derive this formula? Thanks in advance.
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### Interpolate with smoothing parameter

I need to implement in C++ interpolation with smoothing parameter. To the non-familiar with this function: The smoothing parameter gets a value from 0 to 1. 0 brings absoulte linear interpolation (...
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### Why settle for Lagrange Interpolation when doing linear multistep ODE integration?

Say that we have some initial value problem: $y'(t) = f(t,y(t)) ; y(0) = y_0$ with $y_0$ and $f(t,y(t))$ known. If we use Euler's method to numerically approximate the first k points, then we have ...
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### Comparing smoothness among approximations

We are interpolating a missing fragment of a 2D curve given a set of sample points. Our method generates several candidates of curve pieces to fill the missing part, but we want to select the solution ...
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### Number of continuous derivatives of a piecewise quadratic polynomial

I've been trying to reason through the following problem: Suppose that we interpolate $n+1$ data points with a piecewise quadratic polynomial. How many continuous derivatives can this interpolating ...
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### SmoothStep: Looking for a continuous family of interpolation functions

Background: SmoothStep is a simple sigmoid-like function defined as S(x) = 3x^2 - 2x^3. It is monotonically increasing from (0, 0) to (1, 1), is rotationally symmetric over that interval, and has ...
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### n-order B-splines interpolation

I am wondering if the following statements are correct: (1) zero-order B-splines interpolation is equivalent to nearest-neighbor interpolation. $C^0$ continuity thus is not differentiable. (2) first-...
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### Numerically stable interpolation of circular curve

Suppose I start with two points $p_1, p_2 \in \mathbb{R}^3$. I want to interpolate along a circular arc between these two points, given normal vectors $n_1, n_2$ at each point. It's fairly ...
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### Quadratic polynomial interpolation from a transformation

Some modeling considerations have mandated a search for a function $$u(x) = \gamma_{0}\exp(\gamma_{1}x + \gamma_{2}x^{2})$$ where the unknown coefficients $\gamma_{1}$ and $\gamma_{2}$ are ...
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### Lagrange Polynomial Interpolation - Polynomyal Differences Depending Upon the Degree?

My question is simple: Give the table: | x |0|2|4|6| |f(x)|1|3|5|7| Why when calculating Lagrange Polynomial Interpolation for: | x |0|2| |f(x)|1|3| P1(x) = x+1 And when calculating Lagrange ...
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### Proof of Lagrange Polynomial

I am trying to prove the following concepts of the Lagrange Polynomial: $\sum_{j=0}^n L_j(x)=1$ $\sum_{j=0}^n x_j^m(x)L_j(x)=x^m, m \le n$ This is my work so far, but I am a little stuck on ...
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### How to interpolate elliptically

Given two orthogonal axes with different weightings along each axis, how do I interpolate elliptically between the two weightings? This is in 2d cartesian space. For example, axis1 might be Vector(2,...
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### NURBS Curves to Interpolate Points and Derivatives on a Surface of Revolution

Problem in Prose My starting point is a set of conic segments on a plane. Each of these conic segments interpolates between three points and known slopes on the two outer points. I want to find a ...
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### Alternatives to Shephard interpolation?

I am a chemist, so I have little experience in the field of math. My program is that I have a set of points (approx. 20000) in some larger dimensional space (like 10-20 dimensions), and I want to be ...
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### Deriving a tridiagonal system for cubic spline interpolation

Can anyone explain how $B_{i-1} = 1/4$ and $B_{i+1} = 1/4$ were chosen in line 6 of the picture, just above the matrix? I'm trying to understand cubic splines but this result seems like it came out ...
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### Vector/Multidimensional version of Newton Divided Difference

newton divided difference polynomial (NDDP) finds an y=f(x) relation by interpolating a polynomial, is there a y=f(x,z) version for n dimensions? Any help appreciated.
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### Derivation of linear interpolation?

Anyone know a good derivation of the linear interpolation: $$\frac{y-y_0}{x-x_0}=\frac{y_1-y_0}{x_1-x_0}$$ Wikipedia gives one, which I don't understand.
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### Find 2 datapoints too interpolate sin(x)

I have the function $f:\left [ 0,\pi \right ]\rightarrow \mathbb{R}, x \mapsto \sin(x)$. How can I choose two points $x_0, x_1 \in \left [ 0,\pi \right ]$ for my polynomial interpolation such that I ...
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### Interpolation by splines: how to set up the equation system for finding the coefficients of the spline (in a B-spline basis)

Problem. I want to interpolate a function $f$ in some equidistant points $x_0<x_1<x_2<x_3<x_4$ using a quadratic spline. My attempt. I assume that we can use the interpolation points as ...
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### comparison of piecewise linear interpolation, cubic interpolation, cubic spline interpolation?

What are the advantages and disadvantages of piecewise linear interpolation, cubic interpolation, and cubic spline interpolation? I know that piecewise linear interpolation is not smooth and may not ...
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### Why do I have the wrong ratio for this linear interpolation question

$$f(x)=5x^3-8x^2+1$$ There is a root between x=1 and x=2, use linear interpolation (using similar triangles) to find the root correct to 1 dp So I tried doing this: $$f(1)=-2$$ $$f(2)=9$$ so the ...
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### Method of Undetermined Coefficients (Vandermonde) on data points?

I know how to interpolate using Vandermonde and obtain p(x) if the data points are given as something like p(-1)=1 p(0)=0 p(1)=1 But what if derivatives, p'(x) ...
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### The error formula for the Romberg integration [closed]

I am just wondering if there exists a error formula for the Romberg integration. Since it just applies Richardson extrapolation to Trapezoidal rule, is its error formula the same as that of the ...
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### An example of hermite interpolation [closed]

I found this example on wikipedia. What I don't understand is that on the right hand side, the column starts with $-10$. Why isn't the column $-10, -4, 4, 10$? Why do the numbers in the middle place ...
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### Radial Basis Function RBF Gaussian based Interpolation

Based on short description below (an image), how do I find the highlighted f function value? I understand that it is a value associated with the vertex, sorry I am not a good math student to ...
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### Distance between a map and a point using bicubic interpolation

I have an image (i.e., a two dimensional regular grid) with pixel values that represent elevations. To interpolate points of the surface described by the grid, I use bicubic interpolation. The image, ...
Let $f$ $\in$ C([0,1]), $f(0)=0$ and $\epsilon > 0$. Prove there exists a polynomial $p$ such that $p(0)=f(0)=0$, $p´(0)=0$ and $||p-f|| < \epsilon$ . The norm is sup-norm I Know that by ...
With chord method find real root of the equation $x^3 - 2x+1-{e^x\over2} = 0$ accurate to $0.001$ I can not perform first condition in the method of chords