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. S

    MATLAB [matlab-fortran] translate interpolation

    hi all I am new learner on fortran. as a part of practice, i am trying to translate below equation into fortan. I have read about interpolation, but couldn't understand it well. Can anyone here help me to translate interp1 funtion into fortran90...thank you very much. for i=ja:jb...
  2. evinda

    MHB Interpolation with cubic splines

    Hello! :) I am looking at the proof of this theorem: Let $f \in C^{1}([a,b]),P:a=x_{0}<x_{1}<...<x_{n}=b $ uniform partition of $[a,b]$.Then there is exactly one function $s \in S_{3}(P)$ so that $s(x_{i})=f(x_{i}),i=0,...,n$ and $s,s',s''$ continuous at...
  3. D

    Trigonometric interpolation polynomial

    Homework Statement Let t_j=j/100, a_j=j, b_j=-j, for j=0,1,...,99. Define f(t)=\sum\limits_{k=0}^{99} (a_k\cos(2\pi kt)+b_k\sin(2\pi kt)) Determine the values of c_l, d_m for l= 0,...5, m=1,...,4, so that P(t)=\frac{c_0}{2}+\sum\limits_{k=1}^4 (c_k\cos(2\pi kt)+d_k\sin(2\pi kt))+c_5\cos(10\pi...
  4. S

    Coefficient Matrix Of Cubic Spline Interpolation.

    Homework Statement I'm trying to derive the coefficient matrix (a) of a parabolically terminated cubic spline. This is the matrix of coefficients ##a_i \rightarrow a_n## where n is the number of data points provided. With this matrix you can find all the other coefficients (b and c) that...
  5. G

    MHB Chebyshev Nodes for Interpolation

    Hello all, I'm not sure if this is the right sub-forum for this question, so please move my post if there is a better place for this. I have been studying interpolation of functions, and in particular, using Chebyshev nodes rather than equidistant points in the creation of an approximation...
  6. O

    Comp Sci [Fortran] making a more efficient bilinear interpolation

    Homework Statement I'm trying to write an efficient bilinear (2D)-interpolation, after reading some recipes, as a fortran-mex for Matlab that is used extensively throughout a long algorithm of solar image processing, and therefore is one of my main bottlenecks. I'm not a Fortran expert and...
  7. P

    MATLAB question re interpolation and approximating derivative.

    Hi, I'd sincerely appreciate it if someone were willing to review the few lines of MATLAB code below and indicate why they don't quite yield the expected output. Homework Statement I am asked to generate using MATLAB approximated values of f(x)=cos(x) at nodes x+h,x-h with random errors...
  8. P

    Interpolation polynomials and errors.

    Hi, Homework Statement A quadratic piecewise interpolation is carried out for the function f(x)=cos(πx) for evenly distributed nodes in [0,1] (h=xi+1-xi, xi=ih, i=0,1,...,πh). I am asked to bound the error. Homework Equations The Attempt at a Solution I believe the error in this case is...
  9. P

    Interpolation, proving P[x_0,x_1, ,x_n] = 0

    Hi, Homework Statement I am asked to show that P[x0,x1,...,xn] = 0 for P(x) \in∏n-1 without directly using the property that the divided difference over n+1 nodes is the nth derivative. The hint says to use the formula for the error. Homework Equations The Attempt at a Solution I...
  10. P

    Solving Interpolation Error w/ ≤ 5*10-8 Precision

    Hi, Homework Statement Interpolation for the function f(x)=cos(x) for evenly distributed values of x in [0,π] (h=xi+1-xi, xi=ih, i=0,1,...,πh) is carried out. Also known are f(xi). I am asked to determine the value of h so that the interpolation's error is ≤ 5*10-8. Homework Equations...
  11. E

    Best piecewise linear interpolation

    I have a time series of data that I want to interpolate using a piecewise linear function. The lengths of the linear pieces should be adaptive such that the maximum error of the points in the interval does not exceed a predetermined value Max_R. I guess the specification boils down to: Find a...
  12. S

    What is the Lagrangian Interpolation Formula for Approximating Functions?

    Homework Statement Consider the Lagrange Polynomial approximation p(x) =\sum_{k=0}^n f(x_k)L_k(x) where L_k(x)=\prod_{i=0,i\neq k}^n \frac{x-x_i}{x_k-x_i} Let \psi(x)=\prod_{i=0}^n x-x_i. Show that p(x)=\psi(x) \sum_{k=0}^n\frac{f(x_k)}{(x-x_k)\psi^\prime(x)} Homework Equations None...
  13. M

    Question regarding cross-linear interpolation

    This is not a homework problem. I am given the equation of two lines at different temperature gradients for (suction line temp,suction line pressure) => (x,y) At outdoor temp 95*F; y=.41667x+54.1667 At outdoor temp 85*F; y=.4231x+50.3846 I need to find suction line pressure for 72*F temp...
  14. J

    Interpolating R-134a Pressure/Temp: 600Kpa/25°C

    What does it mean when you are given a pressure and temperature of refrigerant and the values in the tables are not in the same row? Like 600 kpa and 25 degrees celsius. 600Kpa should be around 21.-- degrees.
  15. DrClaude

    Understanding Akima Bivariate Interpolation: A Missing Piece in the Puzzle?

    Anyone familiar with the bivariate interpolation method developed by Akima? I've been reading H. Akima, Commun. ACM 17, 18 (1974) and trying to implement his method, but he actually doesn't describe how to do the interpolation! The formulas for calculating the derivatives at each point are...
  16. A

    Solving Backward Interpolation Problem | Explanation and Solution

    Hi guys, I was solving this problem using backward interpolation https://dl.dropboxusercontent.com/u/49829206/1.PNG and I got this solution https://dl.dropboxusercontent.com/u/49829206/3.PNG but What I don't get is that how come is the the forth difference is constant while it not in real...
  17. B

    Derivation of Lagrange Family of Interpolation functions

    Folks, I am puzzled how the linear interpolation functions (see attached) were determined based on the following equation below ##\displaystyle...
  18. L

    Interpolation of Infinitely Many Points

    This sort of came up the other day: Given a sequence of monotonically decreasing points, a_n, such that a_n \rightarrow 0 does there exist an analytic f on ℝ such that f(n) = a_n ? I figured there should be some sort interpolation theory on this, but I haven't found anything...
  19. H

    How can I understand the equation for bicubic interpolation?

    The wikipedia article on bicubic interpolation appears to describe how to arrive at an equation for bicubic interpolation, but I cannot decipher it. Using the derivatives described I don't understand how to arrive at the relevant equation from the coordinates of the point and the values of the...
  20. M

    Interpolating Between Lines: Techniques for Finding Y-Values on a Graph

    Hi there. I have a problem regarding interpolation between lines. I have to say that I have never done it before and I can't find relevant info on the internet, so I am asking for your help. the graph in question can be found here: http://s9.postimage.org/kmj94nexb/physicsforum.png I do...
  21. E

    Estimating f(2) using Cubic Interpolation

    If a cubic function satisfies f(0) = -32, f(1) = 0, f(3) = 10 and f(4) = 0, use cubic interpolation to estimate f(2) I'm not sure how to approach this since I have only ever done quadratic interpolation and linear interpolation, is it just an extension of the lagrange interpolating...
  22. E

    Use linear interpolation to estimate sin 36 using as your 'known' values 0 & 60?

    here is the answer: 36/60 = x/.8660 60x = .8660 x 36 60x = 31.176 x = .5196 however, where does the value: 0.8660 initially come from? any help would be appreciated
  23. J

    Computational Physics (Making programs for interpolation and differentiation)

    Homework Statement Chapter 4 1. Write a program that implements the first order (linear) interpolation 2. Write a program that implemets n-point Lagrange interpolation. Trean n as an imput parameter. 3. Apply the program to study the quality of the Lagrange interpolation to functions...
  24. A

    Solving tri-linear interpolation parameters

    I'm trying to find the tri-linear interpolation parameters of a point C within a hexahedron of 3d vectors (C000, C100, C010, C011 etc) You could call this "inverse tri-linear interpolation" Ive used the same variable names as this wikipedia article...
  25. C

    Can Lagrange's Interpolation Be Used to Prove e^tD(f(x))=f(x+t)?

    Homework Statement Let D:R[x]->R[x]be the differentiation operator D(f(x))=f'(x),prove that e^tD(f(x))=f(x+t) for a real number t Homework Equations application of Lagranges interpolation The Attempt at a Solution i don't know how to begin or construct the proof here
  26. H

    MHB Interpolation and Error Bound

    Problem: Use the Lagrange interpolating polynomial of degree three or less and four digit chopping arithmetic to approximate cos(.750) using the following values. Find an error bound for the approximation. cos(.6980) = 0.7661 cos(.7330) = 0.7432 cos(.7680) = 0.7193 cos(.8030) = 0.6946 The...
  27. B

    Interpolation Functions and their derivatives

    Folks, How do determine whether the derivative of a quadratic interpolation function ##ax^2+bx+c## is continous/discontinous in the context of the following We have a a true solution approximated by 2 quadratic interpolation functions ie, The approximation function f_1(x)=ax^2+bx+c...
  28. Z

    Complicated Interpolation (Multivalued)

    Hi, I'm trying to interpolate a data set of classical trajectories, but I'm having trouble because the data is basically a multivalued function. The data maps out a periodic function: for every x value, there are about 7 y values. Does anyone have any advice as to how to approach this...
  29. K

    Multivariable interpolation

    Hello! I am wondering if it is possible to establish a relationship between three sets of points (x,y,z) by interpolating. Basically i need a function that takes x and y and gives me a z that matches the following points: 130 472 5 130...
  30. V

    Fastest interpolation method for attitude quaternions?

    I hesitated between posting this in the Mathematics forum or here, but since it's fairly applied, I chose this place. Sorry if it should've gone somewhere else. I posted another thread earlier (https://www.physicsforums.com/showthread.php?t=599737), about having trouble finding the quaternion...
  31. A

    Double Interpolation thermodynamics

    For Refrigerant 134a at T = 60C and v = 0.072 m3/kg, determine pressure in kPa. the v is in superheated region so I have a mini table look like in the picture I've added there I need to do a double interpolation but don't know how. the anser for p is: 3.63Kpa. Thanks
  32. R

    What are some techniques for 2-D interpolation and how do they work?

    Hi All, I'm working on coding some 2-D interpolation techniques. With wikikepdia's help, I've managed blinear interpolation. What other techniques are out there? Please tell me a little about how your favourite technique works. I've read about higher order methods, using more...
  33. S

    Lagrange interpolation formula

    Homework Statement (a) If x_{1},\ldots, x_{n} are distinct numbers, find a polynomial function f_{i} of degree n - 1 which is 1 at x_{i} and 0 at x_{j} for j \ne i. Hint: the product of all (x - x_{j}) for j \ne i is 0 at x_{j} if j \ne i. This product is usually denoted by \prod_{\substack{j...
  34. N

    Mathematica Mathematica: Interpolation Functions

    Hi I have a data set of the form: data = {{0, 0}, {1, 1}, {2, 2}, {3, 20}, {4, 1}, {20, 1}, {21, 1}, {22, 0}}; This data set is a probability density function (PDF), and I need to be able to integrate it from 0 to some x<22. I thought that I would use an interpolating function for...
  35. X

    Lagrangian interpolation of sin(x) in Python

    Homework Statement The polynomial pL(x) is known as Lagranges interpolation formula, and the points (x0; y0), . . . , (xn; yn) are called interpolation points. You will use Lagrange's interpolation formula to interpolate sin x over the range [0; 2pi]. Begin with n + 1 interpolation points...
  36. J

    MATLAB Simple matlab interpolation help

    suppose I have a function f(x) which I know increases as x. I'm trying to combine a for loop and if statement x(1)=a; x(2)=b; x(3)=(a+b)/2; for i=3:n if f(x(i)) > A x(i+1) = (x(i-2)+x(i))/2; else x(i+1) = (x(i-1)+x(i))/2 end end this method works for x(4), the rest come with...
  37. K

    Interpolation of pressure from one mesh to another mesh

    I have performed interpoaltion of applied pressure from one mesh (mesh of tetra elements) to another mesh (mesh of brick elements). I have done the interpolation. I want to introduce some checks if my interpoaltion is rightly done.. Please can anyone advise what could be the checks that...
  38. kaniello

    Interpolation for numerical integration

    Hallo, \Gamma(t) is a function that i can know only at dicrete points and appears in this integral: \int\Gamma(t) * tan (\Gamma(t)*t + \varphi) My question is now, which could be the best interpolation of \Gamma(t) that would allopw an exact integration? Thank you very much in advance
  39. K

    Interpolation help fitting curve to three points

    Hello, I have the following points that need to be fitted with a curve: (1,20);(2,4);(5,3) I'm wondering how to use the http://en.wikipedia.org/wiki/Lagrange_polynomial to do this. If possible, can these points be fitted with a cubic function? I tried to fit a cubic to this, but...
  40. T

    MATLAB. Creating a function involving interpolation.

    I have attached a pdf of the complete problem. I have been trying this for a few days and still can't figure it out. The book provided with this course does not give good information on this topic nor did the class lecture. y1 = interp1(x,y,x1) This is what I have so far: function y1...
  41. M

    Calculating heat req. by fuel in boiler, interpolation needed.

    1) Feed water enters a boiler at 172.52 degrees Celsius. The boiler produces super heated steam at 5600kpa and 472 degrees Celsius. The boiler efficiency is 82%. Find the Heat required by fuel. 2) I interpolated the value given for feed water and super heated steam. h@5600kpa 472 degrees...
  42. N

    Lagrange Interpolation and Matrices

    Homework Statement Prove I=T1+T2+...+Tk Where Ti=pi(T) Homework Equations T is kxk pi(x)=(x-c1)...(x-ck) is the minimal polynomial of T. pi=\pii(x)/\pii(ci) \pii=\pi(x)/(x-ci) To evaluate these functions at a matrix, simply let ci=ciI The Attempt at a Solution From lagrange interpolation...
  43. F

    I don't fully understand this question about cubic spline interpolation

    If each spline is given in the form of gi(x) = ai(x-xi)3 + bi(x-xi)2 + ci(x-xi) + di where i = 1 to N for N+1 data points. Then given that b1 and bN+1 are zero (because the second derivatives are zero at the endpoints, due to this being a natural cubic spline), then there are N-1...
  44. E

    Mathematica Working with functions defined by Interpolation in Mathematica

    Working with functions defined by "Interpolation" in Mathematica Hello, Just perhaps a simple question for a Mathematica expert: I have a function of two variables f(a,b) defined using Interpolation option in Mathematica. I am wondering how to determine the value of one of the variable...
  45. X

    Lagrange Polynomial Interpolation

    Homework Statement Find the polynomial p(x) of degree 20 satisfying: p(-10) =p(-9) = p(-8) = ...=p(-1) = 0 p(0) = 1 p(1) = p(2) = p(3) = ...p(10) = 0 Homework Equations L(x) := \sum_{j=0}^{k} y_j \ell_j(x) The Attempt at a Solution i tried using the formula above: a =...
  46. K

    MATLAB Matlab polynomial interpolation

    I have this function (1-6*x^2)^-1 and i want to polynomial interpolation (lagrange and spline) in 21 equidistant points [-1,1] I made this function x =linspace(-1,1,21); y = (1-6*x^2)^-1; z=[-1:0.01:1] c=polyfit(x,y,20) p=polyval(c,z) s=spline(x,y,z) plot(z,(1-6*x^2)^-1, z, p, z, s)...
  47. D

    Interpolation Method: Solving Missing Data

    hi i need some help with a problem I'm dealing with i have a text file of this format 01 08 2002 12 45 26.7 01 08 2002 13 00 27.8 01 08 2002 13 15 27.0 01 08 2002 13 30 28.2 the last being the temperature and i need to interpolate some missing elements using not only the values...
  48. G

    Linear interpolation between two surfaces

    Hi, I'm trying to create an interpolated volume from two surfaces. Let me explain exactly what I'm doing. I am trying to obtain a rough estimate for the temperature in a certain geographical area and at depth. I have the temperature of the rocks at the surface in the form T1=T(x,y,z) where z...
  49. B

    How can I accurately interpolate a C1 function with infinite second derivative?

    I have to interpolate a function between a small number of points n (say, 3-5), at which I know both the value of the function and its first derivative. Normally, this would be a good candidate for polynomial interpolation. The only problem is that at the first point the function is only...
  50. B

    Interpolation with knowledge of derivatives

    I want to interpolate a function between the points A, B, C. At A and C I only know the value of the function, but at B (lying between them) I also know the function's first and second derivatives. How would you interpolate between these points?
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