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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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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13 views

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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15 views

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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2answers
217 views

Accurate floating-point linear interpolation

I want to perform a simple linear interpolation between $A$ and $B$ (which are binary floating-point values) using floating-point math with IEEE-754 round-to-nearest-or-even rounding rules, as ...
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1answer
179 views

Upper bound for the error magnitude

For the function $f(x) = \mathrm{e}^x$ on the interval $[0,1]$, by using polynomial interpolation with $x_0 = 0$, $x_1 = 1/2$, and $x_2 = 1$, find the upper bound for the magnitude $$ \max_{0 ...
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25 views

Given two entire functions $f_1,f_2$ without common zeros, prove that one can find some entire functions $g_1,g_2$ such that $f_1g_1+f_2g_2=1$ [duplicate]

The question Let $f_1,f_2$ be some entire functions without zeros in common, so for every $z∈ℂ$ we have $|f_1(z)|^2+|f_2(z)|^2≠0$. Prove that there exist two entire functions $g_1,g_2$ such that: $$ ...
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3answers
63 views

Polynomial interpolation

I need to find the polynomial of degree $3$ with respect to these conditions: $$\begin{cases} p(0) = 1\\ p(1) = -1\\ p'(0) = 1\\ p''(0) = 0 \end{cases}$$ How do I deal with the condition on ...
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1answer
30 views

Approximating Fresnel integrals with standard functions

I would like to approximate the Fresnel S and Fresnel C with standard functions. I've started with the $ S(x) $ function: $$ approxS(x) = sgn(x) * \left ( sgn(x)* \left ( \frac{ \sin( \frac{x^2}{2} ...
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1answer
18 views

Fourier series on incomplete data [closed]

Given a periodic function that's only partly specified, e.g.: $$f(\theta)=\begin{cases}1 & \text{if } \cos(\theta)>a\\ -1 & \text{if } \cos(\theta)<-a\end{cases}$$ Obviously the ...
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1answer
1k views

Plane fitting using svd

I am trying to get a best fit plane in a 3d space of points. I am using an svd as described in http://stackoverflow.com/questions/10900141/fast-plane-fitting-to-many-points. If I use the data provided ...
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3answers
46 views

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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0answers
20 views

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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2answers
2k views

Spline interpolation versus polynomial interpolation

What is the difference, if any, between spline interpolation and piecewise polynomial interpolation?
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0answers
11 views

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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1answer
50 views

Is the following described $p(x), q(x)$ the same interpolation polynomial?

Suppose we are given an odd number of data points $x_i$ and the corresponding values $f_i=f(x_i),i=1,...,n+1$($n$ is even), which are symmetric about the origin, i.e for each $x_i$ there is a $j$ such ...
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0answers
7 views

Roll forward a payment

If I earned $100 per month from Jan 1, 2016 to April 30, 2016 how do I determine my projected 2016 salary if I am assuming an annual trend rate of 7.8 % starting May 1st? I would think it would be ...
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11 views

interpolation preserving boundedness property

I'm trying to construct interpolation for a function $m$ such that \begin{equation*} 0\leq m(x)\leq 1,\quad\forall x\in\Omega\subset \mathbb{R}^1. \end{equation*} I tried to use ...
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1answer
76 views

Equivalent condition for interpolation polynomial

Let $(x_1,y_1),...,(x_n,y_n)\in \mathbb{R}^2 $, where $x_i\neq x_j$ if $i\neq j$. Let $p$ be a polynomial such that $$\det\begin{pmatrix} p(x)& 1 & x & x^2 &\dots & x^n \\ ...
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1answer
28 views

Second Order Accurate Interpolation

On a grid I am having the values of a physical quantity say for example Temperature, at the E,W,N,S and P node all of them being calculated using a second order discretization scheme. I want a second ...
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0answers
69 views

Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
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1answer
17 views

Some doubts on Simpsons Rule by the Method of Undetermined Coefficients

There is this note about Quadratic Interpolation by Simpsons Rule that I don't quite understand how to get the LHS. Simpsons Rule by the Method of Undetermined Coefficients We seek an approximation ...
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1answer
22 views

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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19 views

Is there a standard name for interpolation parameters?

I often use interpolations between two values in game / UI programming, for animations etc. - e.g. a linear interpolation: $$x = x_1 + a(x_2 - x_1)$$ Or a 'cubic' sigmoid type interpolation like: ...
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Interpolationspace for H^{-1}\cap H^{1}

For which $\theta\in[1,\infty)$ does hold $(H^{-1}(\Omega),H^1(\Omega))_{1-\frac{1}{\theta},\theta}=L^2(\Omega)$ if $\Omega$ is a bounded domain with smooth boundary and is three dimensional. I don't ...
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14 views

Fitting with initial conditions

i try to do some fits but i have ten initials conditions and i think it will be difficult to evaluate the sensitivity of my conditions. Do you know some methods which allow to know the sensitivity of ...
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1answer
563 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 ...
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0answers
29 views

The butcher array of explicit Runge- Kutta method

Just a quick question, for a family of explicit Runge-Kutta methods parametrized by order q, by applying up to $p-1$ passes of deferred correction to p steps of Euler's method. When $p=2$, should its ...
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0answers
35 views

Lagrange interpolation for ellipse

Consider the ellipse $$\frac{x^2}{4} + \frac{y^2}{2} =1$$ The line integral $I$ of the ellipse in the first quadrant is $$I=\int^2_0 \Big[ 1+(y'(x))^2 \Big]^{1/2} dx$$ Find the cubic polynomial ...
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0answers
25 views

Interpolate solution derived with Matlab PDE tool

I have tried to solve a bvp problem using Matlab. Matlab succesfully returned the result of the numerical procedure based on an internal finite element method. However I don't know how to interpolate ...
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1answer
29 views

Cubic Spline for a function

I have the function $f(x)=x^3$ and I need to find the cubic spline. The given points are: $\{-1, 0, 1\}$. What is the cubic spline for this function and what would a demonstration to this be? I would ...
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1answer
9 views

Adding two functions represented by a table of values with a different step size?

Let $f(t)$ be some numerically obtained $T$-periodic function represented by a table of values over one period or a set of points $(t, y)$ with a time step $\Delta t.$ Now let's change the ...
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19 views

Effect of location of nodes for interpolation

I've been doing some numerical experiments to see how the location of the interpolating nodes affects the performance of the interpolator. I am just curious about this because it seems like the ...
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2answers
110 views

How to find the formula of a function from its graph?

I've got some data points (X/Y coordinates) that were apparently created using a certain formula that I want to reconstruct now. I've only got those points, and I can plot them (e.g. like in this). I ...
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1answer
224 views

Interpolating Z-Values when given complete and incomplete XYZ pairs

I am building an application that works with PolyLineZ (ESRI Shapefile) data and rewrites outlying Z values. The minimum and maximum Z-values are defined by the user through the interface Let's take ...
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1answer
26 views

Calculate what function is approximated with the Lagrange Polynomial

I would like to find out what the sum estimates and prove that it estimates that function. $$\sum_{j=0}^ml_j(x)*x_j^k=?$$ From the Lagrange interpolation polynomials we know that $$l_k(x) = ...
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1answer
15 views

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) ...
2
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2answers
29 views

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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2answers
80 views

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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1answer
509 views

Newton backward interpolation in Mathematica

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
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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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1answer
33 views

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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0answers
63 views

Hermite interpolation with interior points

I am trying to solve the following problem: Given the conditions on a curve $c(u)$ of degree $4$ at the points $-1$, $0$, $1$ as: $c(-1) = 4$; $c'(-1) = 4$; $c(0) = 6$; $c(1) = -4$; $c'(1) = -6$; ...
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2answers
36 views

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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1answer
52 views

How does sinc interpolation work?

Every now and then I come across mention of sinc interpolation. Trying to read up on it, I have yet to get what it's about. I have done basic DSP work, have programmed stuff using FFT (using just a ...
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1answer
19 views

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 ...
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1answer
175 views

Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
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1answer
17 views

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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1answer
25 views

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 ...
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0answers
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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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1answer
55 views

How to choose new points after grouping/resampling?

I'm resampling a signal (which takes values [0,1]) of N samples (blu points) to one with N/5 samples, where (for each group of 5 samples) I store in two arrays the max and the min values of the ...