What is Interpolation: Definition and 154 Discussions
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing new data points within the range of a discrete set of known data points.In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate, i.e., estimate the value of that function for an intermediate value of the independent variable.
A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error.
Hi,
I have a 3D gridded ( Nx,Ny,Nz : integers, respectively, size of the grid in x,y and z direction ) which contains the charge distribution of an atom, say Hydrogen,
and I would like to simulate the charge density of another structure, in easiest case Hydrogen dimer. (H2)
To accomplish my...
Homework Statement
Suppose we are given f_0, f_1, f'_0 f'_1 of an unknown function. We want to integrate the region between x_0 and x_1 using a Newton-Cotes formula.
Homework Equations
Wat is the 'standard' way to solve such a problem?
The Attempt at a Solution
I started with the standard...
Hello everybody!
I'm dealing with a MATLAB script that is made up in this way:
I've this external loop
% ALTITUDE [ft]
h_ft = 0:200:10000;
% TEMPERATURE [°C]
T_C = 0:1:50;
for mm = 1:length(h)
for nn = 1:length(T_0)
The aim of this loop is to calcuate a velocity, so at the end I...
Hello everyone,
I am using the bicubic kernel described here ( http://en.wikipedia.org/wiki/Bicubic_interpolation#Bicubic_convolution_algorithm ) to interpolate my image after applying some transformations.
I an using the matrix kernel described here with a = -0.5.
Now, what I also...
Homework Statement
My questions concern interpolation and a method to assess if the interpolations I have made could be refined in regards of ill condition numbers etc.
How can I make this interpolation better? I have read about interpolation techniques in Matlab but I don’t know which one to...
Problem: We want to calculate a polynomial of degree N-1 that crosses N known points in the plane.
Solution A: solving a NxN system of linear equation (Gauss elimination)
Solution B: construction from Lagrange basis polynomials.
One of my professors said that the first solution is...
I've run into a problem programming a computer simulation. I have a discrete grid of values and a point an the middle with a value and I need to take the value and put it into the grid. I've been working on the 1 dimensional problem, and I can't get it. I know it's something simple I'm missing...
Hey guys,
I got this assignment last week and have been doing every bit of research I can to try to figure out what I am supposed to be doing. I have found a lot of cubic spline interpolation (csi) write-ups including one on here but we don't have a book in this class to help understand how to...
Let x_{0}, x_{1}, \cdots , x_{n} be distinct points in the interval [a,b] and f \in C^{1}[a,b].
We show that for any given \epsilon >0 there exists a polynomial p such that
\left\| f-p \right\|_{\infty} < \epsilon and p(x_{i}) = f(x_{i}) for all i=1,2, \cdots , n
I know \left\|...
I don't need any help for the engineering maths I'll be doing after but if I anyone could advise me on how to write a program that would take a set of x and y values and then give me back the y values at certain intervals on the x-axis I would be very grateful. I've got colossal amounts of data...
Here's a question for the COMSOL whizzes:
I'm using the COMSOL 3.5a Diffusion Toolbox. I have a 1D reaction-diffusion model that is controlled by a catalyst.
The spatial distribution of catalyst along the 1D space is independent of the other chemical species and I have data for the...
Homework Statement
I have been trying to learn how to use linear interpolation to find out the data needed from the thermodynamics chart, however, the book that is assigned does not even talk about it. Could someone please explain how linear interpolation is used? Thanks
Homework Statement I need to find a "Lagrange basis" corresponding to the function space spanned by the basis (1, x^2).
Homework Equations I have been told the Lagrange polynomial is of the form
(x-x_1)...(x-x_(k-1))(x-x_(k+1))..(x-x_n) / (x_k-x_1)...(x_k-x_(k-1))(x_k-x_(k+1))..(x_k-x_n)...
Hi,
I'm doing a project in medicine, I've used MATLAB to create 3d graphs using a vector with 5 fixed values as X, a vector with 7 fixed values as Y. As Z I have 5*7 measured values.
I need to interpolate my graph to "smoothen" the lines in the graph, but I am totaly lost, I have no idea how...
Homework Statement
Suppose we have a signal given by
\[f(t) = rect\left(\frac{t}{64} - 1\right) + 3*rect\left(\frac{t-96}{64} - 1\right)\]
or in piecewise notation,
f(t) = \left\{
\begin{array}{c l}
0 & 0 \leq t < 32,\\
1 & 32 \leq t < 96 \\
0 & 96 \leq t < 128 \\
3 &...
Basically I've got to design and develop a software for computing a polynomial function involving a set of data points. I've got to use an algorithm based on the lagrange interpolation method. I know it should involve two loops inside the code.
What I've been told is that
"The input to the...
Homework Statement
If i have an interpolation formula, say:
y(t_i)=3+h\sum^i_{j=1}jy(t_j)
where i=0 to n
h=(a+b)/n
ti=i*h
and
when i=0, we have: y(t0)=0.
How would i write a code to find y(ti)?
Homework Equations
The Attempt at a Solution
function y=examp(a,b,n)...
Homework Statement
Hello, I have a set of data that is grouped into bins. The bin sizes increase logarithmically by 10% (1.1 x the previous bin).
Homework Equations
Bin
Mean
Radius Frequency
1.03 4924
1.10 9938
1.21 14009
1.32 12269
1.44 15813
1.58 18723
1.74 21471
...
I have a non-uniform grid within which velocity data is known at each grid point. This velocity data is to be interpolated to allow for evaluation of velocity at an arbitrary point inside the grid.
I am wondering about the correct approach to this problem. In actuality I want to evaluate the...
Hi everyone. I'm doing a microcontroller project where I'm using a 256 element array of 1 byte values to output a sine wave at varying frequencies. I'm using the top 8 bytes of a 16 bit "angle increment" value that's incremented by varying amounts as an index to the array of values, to control...
Hi everyone,
I have a question about interpolation methods. I am given two particles and I know their positions, velocities, accelerations, and jerks (time derivative of acceleration) at some initial and final time-values (t0 and t1, respectively). I want to find the minimum distance...
Homework Statement
Consider the use of cubic splines to interpolate a set of data. Suppose at some stage in the calculation we arrive at the following spline functions for two consecutive intervals
\tilde{f_{0}} = x^{3} + ax^{2} + bx + c over the interval -1 \leq x \leq 1
\tilde{f_{1}} =...
Hello, I'm having a really hard time doing some work in Matlab, I have a book but it just isn't making sense to me, the problem I have to do is in four parts so Ill just show the first part for now...
The following coordinates specify the
shape of a certain cars’ front fender. Interpolate...
Homework Statement
Let f be a contintuous real valued function on a closed interval [a, b]. The sup norm of such a function is maxx|f(x)|
Let pn be a polynomial interpolation of f determined by n points [xi, f(xi)] where xi Ε [a, b] for every i.
Suppose that the sequence {pn} is Cauchy...
Interpolation ?
A treatment plant consists of a primary sedimentation tank followed by a trickling filter that removes 80% of the influent BOD. The flow is then directed to a constructed marsh system that has an influent target BOD concentration of 30 mg/l before being discharged to a small...
Homework Statement
Hi m8s,
determine the max. step size that can be used in the function of F(x)=ex
∈ [0, 1] so the error in the linear interpolation less than 5*10-4
Homework Equations
The Attempt at a Solution
i used the Taylor expansion to get an eqn. so i be able to get the...
Homework Statement
Given points (x0,y0), (x1, y1)...(xn,yn) and derivatives at each of these points y'0, y'1...y'n : Write a program to find the interpolating polynomial (of degree 2n+1) and to evaluate it on given points xeval.
My question is what type of interpolation is this? I'm not...
Hi all,
http://www.bsodmike.com/stuff/interpolation.pdf"
I am going through some of my notes and quite a few books; they all skip the over the point I have marked with 3 red dots in the http://www.bsodmike.com/stuff/interpolation.pdf" .
\begin{equation}\label{eq:solution}\begin{split}...
I have to find a system of linear equations to determine a quadratic polynomial
p(x) = ax2+bx+c
Given the conditions:
p(1) = f(1)
p'(1) = f'(1)
p''(1) = f''(1)
where f(x) = xex-1
I know:
p(x) = ax2+bx+c
p'(x) = 2ax +b
p''(x) = 2x
f(x) = xex-1
f'(x) = ex-1 +...
Hi there,
I am still trying to work out how to get a value of chi-squared from some data (see post below).
I have a model curve with 85 points ( a result of solving an ode, so I do not have the equation of the curve), and 307 data points plotted. In order to work out chi-squared by hand I...
Hi,
I've been given a problem to solve and I was hoping for some pointers.
To start with we have a surface which is originally represented as a Bezier patch, a peicewise set of bi-cubic bezier surfaces. It might be important to mention that the surface is "fairly two dimensional", the surface...
i have solved a differential equation with shooting methods, the output is in form of interpolate function, i can see it with a graphics, but i i would have an approssimate analitic form what kind of function i have to do ?
the equation give in output have this form...
Hi, I'm implimenting a method presented in
http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/GDC03.pdf
Part of the way through it it presents a linear interpolation method, which I wanted to expand to three dimensions, but I don't understand how it works in two dimensions...
Hi,
i have no idea how to start up my program on Newton interpolation. i have solved it manually but I am clueless on how 2 start my program. below represnts the input which i want 2 solve. every 4 lines represnts one problem. the 1st line- n,m where n is no of data points and m is no of...
Can anyone tell me what is the equation for First Order of Newton’s Interpolation?
The one of Interpolation I know is Lagrange’s Interpolation such as
p(x)= \frac {x-x1}{x0-x1} f(x0) + \frac {x-x0}{x1-x0} f(x1)
and so on ...
Thanks in advance
Attached below are two cubic spline tutorials:
1. Explanation of the classic tri-diagonal cubic spline formulation. Included are 2 example problems.
2. Extension to parametric cubic splines. Included are 2 example problems
.
:smile:
Homework Statement
Given a set of data points, derive the interpolation polynomial using divided differences.
Homework Equations
The Attempt at a Solution
My main question is: How am I supposed to do this if they don't give f(x)? Can I just look at the plotted data points and...
Homework Statement
If 0 < \alpha < n, define an operator T_{\alpha} on function on \mathbb{R}^n by
T_{\alpha}f(x) = \int |x-y|^{-\alpha}f(y)dy
Then prove that T_{\alpha} is weak type (1,(n-\alpha )^{-1}) and strong type (p,r) with respect to Lebesgue measure on \mathbb{R}^n, where 1 < p <...
I completely forgot how to do this. I have a thermo test coming up and i will be interpolating a lot of data. What is the fastes/easiest way to interpolate a point on a chart not given?
The problem is described here:
http://phstudy.technion.ac.il/~wn117066/Problems2.pdf
In question 1 I need to find the refractive indices for both polarizations at 760nm. I'm not quite sure how to go about this - which model of dispersion best fits Calomel?
(I'm done with the exercise...
I have a problem where I need to work out a value of a specific point that lies in the quadrilateral that is formed by four separate points.
For each point that makes up the quadrilateral, I know the X and Y coordinates and their value, Z. For the point I am trying to find the value for, I only...
Hello everyone
Here is my problem
lagrange interpolation polynomial across the points(x0,y0),(x1,y1) and (x2,y2) is given by y0L0(x) + y1L1(x) + y2L2(x)
where L0(x)=-x and L1(x)=x ^2 + x
Therefore L2(x) is given by
I tried it but i could'nt crack it
Hi,
Does anyone know how to design a 8x lagrange interpolation filter in matlab?
From what I understand, let say my input is
input = [1 2 3 4 3 2 1]
let say if I want to interpolate by 2, then I insert 0 between every
sample.
input_pad = [1 0 2 0 3 0 4 0 5 0 4 0 3 0 2 0 1]
then...
In my comp physics class, we've been introduced to both c++ and fortran languages. For instance, for our first assignment, I am not sure how to go about investigating the quality of interpolation points for i.e f(x)=sin(x^2) by using n-point langrange interpolation, where n is an input...
Could someone help me understand why interpolation rather than extrapolation should be used for calibration curves (with the exception of Standard additions). I know that extrapolation is less precise, but the book I've got doesn't over any more than that, and I think I need a little more detail...