0
votes
0answers
20 views
Are high dimensional cubic interpolation and cubic spline the same?
Hi I want to implement a 2d bicubic interpolation method. I checked the official matlab implementation code of imresize and interp2, and surprisingly found that bicubic interpolation method is ...
1
vote
0answers
21 views
Multigrid Interpolation and Restriction operators
I have a question about the restriction and the interpolation operators of a Multigrid algorithm.
Let those be given:
The full weighting restriction stencil (in 2D): $\frac{1}{16} \left[ ...
5
votes
0answers
57 views
Runge's phenomen: interpolation error using Chebyshev nodes oscillates
We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
2
votes
1answer
48 views
Interpolation with degree restriction
(using $f[x_1, ... , x_n]$ to denote the forward difference operator)
I have a polynomial $P(x)$ interpolating $5$ points $x_0, ... , x_4$ and $2$ derivative values $x_0, x_3$ across an evenly spaced ...
0
votes
0answers
25 views
Sinusoidal interpolation
I am new to the concept of interpolation, and know only how to interpolate with a polynomial function. What if you suspect a sinusoid will be a more accurate fit? What method should be used?
0
votes
1answer
70 views
B-Spline Interpolation/Approximation
I've got a couple of probably very simple questions, yet some googling didn't bring up what I was looking for.
First what I want to do: I have a grid, and the gridpoints are function values. I want to ...
0
votes
1answer
25 views
nth degree interpolating polynomials
Given five ${x}$ points and for four of them I know ${f(x)}$ values.
So I have to use interpolating polynomial to estimate unknown ${f(x_i)}$
${ \displaystyle f(x) = \sum_{j=1}^{n} P_j{(x)} }$ with
...
1
vote
1answer
46 views
+50
What do you call generating a function out of a graph?
In many physical phenomena, laws an relations to their variables are somehow interpolated (example by statistical data or samples) and then an approximate set of functions are generated to work ...
0
votes
1answer
64 views
finding derivative at intermediate point of known data set
I have a function $y = f(x)$, $ x \in [0,1] $ and $ y \in [0,1]$
Set of values $(x_i,y_i)$ are known for n points. I need to find derivative at point $x_{\zeta}$ such that $y(x_{\zeta}) = 0.5$
Now ...
0
votes
2answers
102 views
For n=3 Lagrange interpolation why is it equal to 1?
I'm studying Lagrange's formula for polynomial interpolation and I cannot seem understand why for $n=3$
$$L_0(x)+L_1(x)+L_2(x)+L_3(x) = 1$$
for all real x.
In my textbook it says as a hint that ...
0
votes
1answer
204 views
Interpolation polynomial
Consider the following table of values for a function $j_0(x)$:
$\begin{array}{c|ccccc} x & \delta_0(x)
\\ \hline ...
0
votes
1answer
127 views
Least-squares approximation polynomial
Consider the function
$\displaystyle f(x) = \frac{1}{\alpha (x-\beta)^2 + 1}$
in the interval $I = [-1,1]$. Set $\beta = 0$.
How do I get the expression for the least-squares polynomial, say $\tilde ...
0
votes
0answers
48 views
error of hermite monotone interpolation
There is a method pchip in Matlab, that is implementing monotone Hermite interpolation. I was not able to find the error analysis and was wondering if someone can point me to some paper. I also wonder ...
1
vote
2answers
166 views
Trapezoidal rule problem
In a trapezoidal rule problem I got following question:
"Evaluate the above integral using trapezoidal rule with five points."
My confusion is here what we take for the value of $n$ is it $5$ or ...
1
vote
2answers
77 views
Understanding proof of uniqueness in theorem on polynomial interpolation
There is a slight part of the following proof in my textbook which I don't quite get.
THEOREM
If $x_0, x_1,...,x_n$ are distinct real numbers, then for arbitrary values $y_0, y_1,...,y_n$, there is ...
1
vote
0answers
218 views
Interpolating polynomial with Chebyshev nodes
I am interested in constructing an polynomial that interpolates some known arbitrary function $f(x)$ over the domain $x \in [0,70]$. I want the polynomial to have degree 14 and so need 15 points.
...
0
votes
0answers
61 views
error of interpolating polynomial through 3 given points and given derivative in one point
What's the error of the interpolating polynomial $p$ which interpolates $f(x)$ in $(x_i,f(x_i))$ for $i=0,1,2$ and which has $p'(x_{i_0}) = f'(x_{i_0})$ for one $i_0 \in \{0,1,2\}$
We were given the ...
1
vote
0answers
257 views
Trapezoidal Rule for Numerical Integration
If the trapezoidal rule approximates an integral with trapezoids, then I thought (and was tought in high school) that the formula is:
$ \frac{h}{2}(f(x) + f(x + h))$
Where $h$ is the distance between ...
1
vote
1answer
92 views
Finding the zeroes using Chebyshev polynomials
Use the zeroes of $\bar{T}_3$ and transformations of the given interval to construct an interpolating polynomial of degree 2 for $f(x)={ 1\over x}$ over the interval $[1,3]$
My biggest issue is ...
2
votes
1answer
300 views
Natural Cubic Spline S on [0,2]
A Natural Cubic Spline S on $[0,2]$ is defined by:
S(x)= $$S_0(x)=1+2x-x^3 \to 0 \leq x < 1 $$
$$S_1(x)=2+b(x-1)+c(x-1)^2+d(x-1)^3 \to 1 \leq x \leq 2$$
Find b,c and d
This question ...
1
vote
0answers
306 views
Finding error bounds for hermite interpolation
I am unsure how to find the error bounds for Hermite interpolation. I have some kind of idea but I have a feeling that I am going wrong somewhere.
$f(x)=3xe^x-e^{2x}$ with my x-values being 1 and ...
2
votes
2answers
549 views
Why is Lagrange interpolation numerically unstable?
Here is my understanding of the polynomial interpolation problem:
Interpolating by inverting the Vandermonde matrix is unstable because the Vandermonde matrix is ill-conditioned, so "difficult" to ...
1
vote
0answers
130 views
Polynomial 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 ...
1
vote
0answers
30 views
Constructing a sequence [duplicate]
Possible Duplicate:
Terms of a Sequence
Construct a sequence of interpolating values $Y_n \text{ to }f(1 + \sqrt{10})$, where $f(x) = (1 + x^2)^{-1}$ for
$-5 \leq x \leq 5$, as follows: ...
1
vote
1answer
87 views
Interpolation error for the exponential function
I'm studiyng for my exam of scientific computing, specifically to the subject of interpolation techniques, I'm stuck with this problem:
How many equally spaced nodes must be taken to interpolate the ...
2
votes
0answers
92 views
Runge function error second factor
I'm currently learning about the Runge function. On Wikipedia, I read the following:
Consider the function:
$ \dfrac{1}{1+25x^2}$
Runge found that if this function is interpolated at ...
1
vote
1answer
173 views
Finding an interpolating polynomial and natural cubic spline for a given accuracy
I'm trying to make an exercise but I don't know how to start. Is there somebody that can give me a hint so that I can start with the exercise. The exercise is:
Consider the function $f(x) = \sin(x)$ ...
0
votes
0answers
107 views
Should n points always be interpolated by n+1 degree polynomial?
I'm studying interpolation and I see that if you have 2 points you use a 3rd-degree polynomial and likewise a 6th degree polynomial for five points. Is this a general formula, and if so, what is it ...
2
votes
0answers
132 views
linear interpolation error estimate for non-smooth function
Suppose I have two points $x_1,x_2$ between which I would like to have a linear interpolation $P_1$. I know the value of the function $f$ at $x_1,x_2$. The error at any point between the two will be ...
1
vote
1answer
31 views
Finding gradual values
I'm writing some code for a pressure level sensor for propane tanks. The manual provides me with the following table with the caption:
"Best accuracy will be obtained using the calibration data in ...
0
votes
1answer
105 views
Derivative of a function defined by the divided difference of another function.
Given a function $f$ of class $C$ $^{n+2}$ in an interval $[a,b]$ and $x_{0}=a<x_1<x_2 ... <x_n = b$ a subdivision of $[a,b]$ into $n+1$ points. Given another function $g$ defined in the ...
1
vote
1answer
97 views
Interpolation of vectors with quadratic polynomial
I have following points (-|b-a|,a), (0,b), (|c-b|,c) with a, b and c as two-dimensional vectors. These should be interpolated component-by-component with a second-degree polynomial p.
My problem now ...
0
votes
1answer
566 views
Need to understand question about not-a-knot spline
I am having some trouble understanding what the question below is asking. What does the given polynomial $P(x)$ have to do with deriving the not-a-knot spline interpolant for $S(x)$? Also, since ...
1
vote
1answer
60 views
Proof that infinite functions can fit a table of numerical values
Suppose while conducting experiments, I measure a finite number of variables with some constants like temperature, etc. We get a table of finite number measurements (numerical values to some decimal ...
0
votes
0answers
119 views
the natural cubic spline
Construct a natural cubic spline that interolates the function defined by f(x)=1-x^4 on[0,1] using nodes at 0 1/2 2/3 and 1
I don't really understand this question, I reviewed the notes then I'm ...
1
vote
1answer
71 views
Lagrange form and differences
For a function f and distinct points $\alpha$, $\beta$, $\gamma$; what is
meant by $f[\alpha,\beta,\gamma]$?
Find the Lagrange form for the polynomial $P(x)$ that interpolates
$f(x) = ...
1
vote
1answer
211 views
Lagrange Coefficients in Maple
I'm trying to compute Lagrange coefficients in Maple. Having found the $n$ roots of a Lagrange polynomial, I want to calculate the $j$-th coefficient:
$$L_j(x) = \prod_{{i=0}\atop{j \neq ...
1
vote
1answer
180 views
Piece-wise linear interpolating polynomials
Somebody please help me to obtain piece-wise interpolating polynomials for the function $f(x)$ defined by the below data:
$x=1$, $f(x)=3$; $x=2, f(x)=3$; $x=4, f(x)=21$; $x=8, f(x)=73$
I know the ...
1
vote
2answers
295 views
Determine the coefficients of an unknown black-box polynomial
Let $p$ be a polynomial of known degree $n$:
$$p(x) = a_0 + a_1 x + \ldots + a_n x^n$$
Suppose we have a magic black box that can evaluate the polynomial for us. How could one then determine the ...
2
votes
1answer
104 views
Divided difference coefficient of product of two functions
For any function $f$ and distinct reals $x_1,\ldots,x_n$, denote by $f[x_0,\ldots,x_n]$ the coefficient of $x^n$ of the minimal polynomial interpolating $f$ at $x_0,\ldots,x_n$.
Let $f$ and $g$ be ...
1
vote
1answer
212 views
How to find the best interpolating function if we know $y(x_i)$ and $dy(x_i)/dx$
Imagine you are given a set of data points $\{x_i,y_i\}$, supplemented by a list of known first derivatives $\{y'_i\}$.
How would you construct an interpolating function $y(x)$ (which satisfies ...
6
votes
2answers
164 views
Negative value of $\sqrt[3]{20}$
Given $f(x)=\sqrt[3] x$, find an approximation of $\sqrt[3]{20}$ using Lagrange interpolation method.
$x_0=0$, $x_1=1$, $x_2=8$, $x_3=27$ and $x_4=64$
$f(x_0)=0$, $f(x_1)=1$, $f(x_2)=2$, ...
1
vote
2answers
262 views
Multivariate function interpolation
I have a (nonlinear) function which takes as input 4 parameters and produces a real number as output. It is quite complex to compute the function value given a set of parameters (as it requires a very ...
4
votes
1answer
312 views
Hermite Interpolation of $e^x$. Strange behaviour when increasing the number of derivatives at interpolating points.
I am trying to understand Hermite Interpolation. Here is my pedagogical example.
I want to approximate $f(x)=e^x$ on the domain $[-1,1]$ using Hermite interpolation.
I choose the Chebyshev zeros ...
3
votes
0answers
370 views
Computation of coefficients of Lagrange polynomials
For our homework we should write a program, that creates Lagrange base polynomials $L_k(x)$ based on a few sampling points $x_i$. Now i am eager to develop a formula to be able to compute the ...
1
vote
0answers
339 views
How to derive Hermite polynomial from a given data set?
The problem is asking to find a Hermite polynomial to predict the position of the car and its speed when t = 10s.
The Hermite polynomial formula is defined as:
$$H_{2n+1}(x) = f[z_0] + ...
0
votes
1answer
153 views
Is this Hermite interpolation correct?
Someone can explain this hermit interpolation algorithm with example?
Thank you,
...
2
votes
2answers
81 views
Interpolating polynomials
So I have this question on a homework and I just can't seem to figure it out.
Let $f \in C^4 [0,1]$ and let $p$ be a polynomial of degree $\le 3$ such that $p(0) = f(0)$, $p(1) = f(1)$, $p'(0) ...
2
votes
0answers
81 views
(Experimental) Can it be shown that this extension of the secant-interpolation has quadratic convergence?
Background: I needed some efficient but simple interpolation-methods aside of Newton's iteration because I want to have it in contexts, where the derivative of a function is not always known. So an ...
1
vote
1answer
248 views
existence and uniqueness of Hermite interpolation polynomial
What are the proofs of existence and uniqueness of Hermite interpolation polynomial?
suppose $x_{0},...,x_{n}$ are distinct nodes and $i=1 , ... ,n$ and $m_{i}$ are in Natural numbers. prove exist ...
