# 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.

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1. ### C/C++ Interpolation of a rapidly oscillating function

I have an analytic function F(x,y,z) and grids in x,y and z. I would like to reproduce the values I get for F at a given x,y and z through carefully interpolating the values given in the grids I have in each of these variables. The problem is that in some part of the x,y,z phase space, namely...
2. ### Mathematica Mathematica Interpolation function error

Hello everyone, I am relatively new to Mathematica, and I am encountering an issue when trying to interpolate numerical data imported from an Excel file. Here are the steps I've taken: I imported the numerical data from an Excel file into Mathematica. I attempted to interpolate the data using...
3. ### Python Why backpropagation dominates neural networks instead of interpolation

Hi guys, I was learning machine learning and I found something a bit confusing. When I studied physics I saw the method of least squares to find the best parameters for the given data, in this case we assume we know the equation and we just minimize the error. So if it is a straight line model...
4. ### Creating Plots using Matlab For Interpolation

My code in Matlab for this practice question is: ( x = linspace(0,4,10); y = sqrt(x); plot(x,y,'-o'); hold on y2=polyfit(x,y,2); plot(x,y2,'--or') ) Is this the best way to do? My plots look nearly identical and are on top of one another but a later question asks to graph the error, so I am...
5. ### A Piecewise linear interpolation with uncertainties

Hello! I have a function ##y = f(x_1,x_2)##, and I would like to do a piecewise linear interpolation. However, both the dependent (##y##) and independent variables (##x_1, x_2##) have uncertainties associated to them (the uncertainty is the same for a given variable i.e. all ##x_1## measurements...
6. ### I Uncertainties from linear interpolation

Hello! I have a function of several variables (for this questions I assume it is only 2 variables), ##y = f(x_1,x_2)##. I want to learn this function using simulated data (i.e. generated triplets ##(x_1,x_2,y)##) and then use that function to get ##y## from measured ##(x_1,x_2)##. There is no...
7. ### Can I improve the sinc interpolation?

Hello everyone. I am working with mathematica, where I have developed a two-dimensional shannon interplation, just as can be seen in the slides 15 to 18 of this presentation. The code is as follows: savedX = Table[XposX = mat[[All, 1]]; YposX = mat[[All, 2]]; windXVal = mat[[All, i]]...
8. ### Chemistry Help with this interpolation (change in entropy while heating water)

Hello, everyone :). I try to resolve this common problem. But, when i got in the interpolation of state 2, the values not make the sense. I have 25 psia and 75 F, but, in the superheated water table, there are not values with 25 psia (only 20 psia and 40 psia). And, the temperature values...
9. ### Substitute PID Controls with a Polynomial Equation/Table?

So, I had a discussion with a friend of mine, neither of us are in controls but I was curious about an answer here. In a PID controller, we essentially take in an error value, do a mathematical operation on it and determine the input (controller output signal B) needed to the actuator to produce...
10. ### I Logarithmic scale - interpolation

Hi, knowing the coordinates of two points: ##(x_1,y_1)## and ##(x_2,y_2)## on a linear scale plot, I can use linear interpolation to get ##y## for a point of known ##x## using the formula below: $$y=y_1+(x−x_1) \frac{(y_2−y_1)}{(x_2−x_1)}$$ But how does it look like in the case of logarithmic...

45. ### MHB A PHP Function To Perform Nth-Order Lagrange Interpolation

The following Lagrange interpolation function is extremely useful. It can be used in just about any branch of science. I use it extensively in astronomical computations for such things as finding the dates and times of the seasons over thousands of years and phases of the moon at any given...
46. ### Cubic interpolation with matrix

Hello, I am trying to understand the slides in the PDF I posted. I am looking particularly at slides 20-24. I am so confused how the matrices are set up with two separate coefficient conditions. The context of these slides is that we are learning how to interpolate with cubic splines. What...
47. ### Can polynomial functions be determined in 3D using given points and coordinates?

If given three points ##P_0 = (x_0, y_0)##, ##P_1 = (x_1, y_1)## and ##P_2 = (x_2, y_2)##, the polynomial function ##f(x)## that intersect those points is ##f(x) = a_2 x^2 + a_1 x^1 + a_0 x^0##. where: ## \begin{bmatrix} a_0\\ a_1\\ a_2\\ \end{bmatrix} = \begin{bmatrix} x_0^0 &...
48. ### Interpolation with 2 variables

If given three points ##P_0 = (x_0, y_0)##, ##P_1 = (x_1, y_1)## and ##P_2 = (x_2, y_2)##, the polynomial function ##f(x)## that intersect those points is ##f(x) = a_2 x^2 + a_1 x^1 + a_0 x^0##. where: ## \begin{bmatrix} a_0\\ a_1\\ a_2\\ \end{bmatrix} = \begin{bmatrix} x_0^0 & x_1^0 & x_2^0...
49. ### Trigonometric interpolation of a sampled signal

Given N sampled points, using the FFT we can get the Fourier transform of those N points Xk. With N/2 the Nyquist frequency and X0 the DC value. Using the inverse we can then get back the original function we just measured. However if we would like more points then just the N we have measured...
50. ### Using Linear Interpolation to Find Interest

Homework Statement Homework Equations The Attempt at a Solution I understand how they calculated NPW but how did they use the linear interpolation method?