0
votes
1answer
57 views

Calculating a cubic spline goes wrong

I am trying to solve a old exam and really stuck at the cubic splines. We have the function $f(x) = \cos^2(\frac{x}{2})$ and the points $x_0 = \frac{\pi}{2}$, $x_1=0$ and $x_2 = \frac{\pi}{2}$. ...
0
votes
0answers
15 views

estimate an upper bound for the error of an interpolation polynom

The task is to estimate the error of an interpolation polynom $p(x)$ to an function $f(x)$. The sampling points are $x_0 = -1,\ x_1= 0,\ x_2=1,\ x_3=3$ So i already calculated the polynom which does ...
0
votes
0answers
8 views

Changing the order of the elements of the divided difference Polynomial Interpolation

Apparently this is rather trivial but I don't understand why what I've highlighted in green is correct.
0
votes
0answers
17 views

Find the hermite interpolating polynomial

$$\begin{array}{ccc}x&f(x)&f'(x)&f''(x)\\0&1&\frac12&0\\1&2&1&-\end{array}$$ Find the interpolating polynom using divided difference table with the given ...
0
votes
1answer
48 views

How obtain a (accurate) function from this graph with these points?

I need obtain the function from 0 to 20 from this graph: I have the even numbers in the {x, f(x)} format: {0, 0}, {2, 1.8}, {4, 2}, {6, 4}, {8,4}, {10,6}, {12,4}, {14,3.6},{16,3.4}, {18,2.8}, ...
0
votes
0answers
34 views

How can I cleverly use the error term of polynomial interpolation?

Let $f(x):=x^2$. We're interested in the closed form of the error $|I(f)-T_n(f)|$ where ...
0
votes
0answers
25 views

Derivation of composite Gaussian quadrature error formula

I am working on studying for the Numerical Analysis qualifying exams. One of the questions I am stuck on is the following: Derive the error term for the composite Gaussian quadrature rule with ...
3
votes
1answer
28 views

Optimal way to find derivative - numerically

Suppose we are given points $x_0,x_1,x_2$ evenly spaced points $(x_0-x_1=x_1-x_2)$, and $u(x_1),u(x_2),u(x_3)$ Where $u$ is some function. Find the best way to approximate $u''(x)$ using only the ...
1
vote
1answer
28 views

Integration Rule Exact Degree

Given the integration rule $Q(x) = \alpha_1f(0)+\alpha_2f(1)+\alpha_3f'(0)$ for interpolating the integral $\int_0^1f(x) dx$ , I need to find $\alpha_1,\alpha_2,\alpha_3$ values s.t Q has exact degree ...
0
votes
1answer
31 views

Why does the interpolation error go to zero if we increase the number of sampling points?

This question is motivated by polynomial interpolation. We know that for $f\in C^{n+1}[a,b]$ and $a=x_0<\dots<x_n=b$ holds $$\| f - p_n \|_\infty \leq \frac{1}{(n+1)!} \| f^{(n+1)} \|_\infty ...
1
vote
0answers
22 views

What is the derivative of $\frac{f^{(3)}(\xi(x))}{6}$ at $x=x_0$

The error of interpolating polynomial is $$ E_n(x)=\frac{(x-x_0)(x-x_1)\cdots(x-x_n)}{(n+1)!}f^{(n+1)}(\xi(x)) $$ The derivative of $E_n(x)$ is $$ ...
0
votes
0answers
64 views

Using Lagrange polynomial to obtain the Second Derivative Midpoint formula

The Second Derivative Midpoint/Central Formula is $$ f^{\prime\prime}(x_0)=\frac{f(x_0-h)-2f(x_0)+f(x_0+h)}{h^2}-\frac{h^2}{12}f^{(4)}(\xi) $$ I tried to get this formula using Lagrange polynomial. ...
0
votes
1answer
24 views

Interpolation based on $n$ uniformly distributed points

We are given $n+1$ uniformly distributed points in the segment $[0,1]$: $x_i=\frac{i}{n}$, $i=0,1,...,n$ and a function $f(x)=e^{-x}$ $P(x)$ is the interpolation polynomial of $f(x)$ where ...
0
votes
0answers
34 views

Interpolation using four nodes

Suppose there are four points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ my target is to interpolate any point $x_I$ between $x_2$ and $x_3$. Is there any Interpolation method which gives linear ...
0
votes
1answer
35 views

Sum of Lagrange polynomial

I have to calculate $\sum_{i=0}^n x^k_i*l_i(x)$ for $k=0,1,2,...,n$ . I've proved that $\sum_{i=0}^n l_i(x) = 1$, but I cannot figure out how it may help me calculating $\sum_{i=0}^n ...
1
vote
1answer
33 views

Numeric Analysis Interpolation of $f(x) , f'(x) $

There is a problem i'm finding quite difficult to solve, i'd be grateful if anyone could point me to the solution : We want to interpolate the function $f(x)$ and it's derivative $f'(x)$ s.t ...
0
votes
1answer
77 views

Maximum of $w(x)=\prod\limits_{i=0}^8(x-x_i)$

What is the maximum of $w(x)=\prod\limits_{i=0}^8(x-x_i)$ on the interval $[-1,1]$, with $\bullet$ equidistant nodes $x_i$, $(x_0=-1,x_8=1).$ $\bullet$Chebyshev nodes, $\displaystyle ...
0
votes
0answers
37 views

Convergence theorem (interpolation)

I am trying to complete a proof of the theorem which we have considered in my numerical analysis course. The tutor made a short sketch, but for me it was not very clear how we prove the statement of ...
0
votes
1answer
32 views

Use the zeroes of T3 to construct an interpolating polynomial

Use the zeroes of T3 to construct an interpolating polynomial of degree two for the function x^3 on the interval [-1,1] Okay, so I have been looking at Finding the zeroes using Chebyshev polynomials ...
0
votes
0answers
27 views

like Gauss-Chebyshev integration formula using Lagrange polynomials

Suppose that $L_k(x)$ is Lagrange Interpolation Polynomial for points $x=1,0,-1$. How to show that: $$\int_{-1}^{1}\frac{f(x)}{\sqrt{1-x^2}}dx=\sum_{k=-1}^1C_kf(k)+E$$ where ...
5
votes
2answers
182 views

$f(x)=1/(1+x^2)$. Lagrange polynomials do not always converge. why?

Let $f(x) = \frac{1}{1+x^2}$. Error of Interpolation with Lagrange polynomials for $n+1$ points is given by $$ e(x)=f(x)-P_n(x)=\frac{f^{(n+1)}(\eta_x)}{(n+1)!}\prod_{i=0}^n (x-x_i) $$ Carl Runge ...
0
votes
1answer
30 views

Interpolation of Polynomial using Lagrange

$f(x) = x^3 + 2x^2 + x + 1$. Find a polynomial of degree $4$ that interpolates the values of $f$ at $x = -2, -1, 0, 1, 2$. I was trying to use the Langrange algorithm, but I think i'm doing it ...
0
votes
2answers
23 views

Interpolation of Polynomial

Let $f(x) = x^3 + 2x^2 + x + 1$. Find the polynomial of degree $2$ that interpolates the values of $f$ at $x = -1,0,1$. I was able to do the an initial part of this problem (not written), but I ...
0
votes
1answer
22 views

Polynomial Interpolation - Bound on Error

Let the function $f(x) = \ln(x)$ be approximated by an interpoation polynomial of degree of 9 with 10 nodes uniformly distributed in the interval $[1,2]$. What bound can be placed on the error? I've ...
0
votes
2answers
64 views

How to obtain Lagrange interpolation formula from Vandermonde's determinant

Assume that we have An interval $[a,b]$ A function $f(x)$ that is continuous on $[a,b]$ $n+1$ distinct points $a = x_0<x_1<x_2<\cdots<x_n = b$ And $f(x_0),f(x_1),\ldots,f(x_n)$ Now we ...
0
votes
0answers
34 views

How to differentiate Lagrange Basis Polynomial?

How to differentiate Lagrange Basis Polynomial ? I don't know, if the term is correct, but the question is: If $x_0,...,x_n\in\mathbb R$ are pairwise distinct ...
0
votes
0answers
28 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; find the generalized ...
0
votes
0answers
37 views

How to use derivatives at points with interpolation

I am given given $n$ points with $x$ and $y$ values. I am also given the derivatives at each of these points. How can I use the derivatives to calculate or to improve my interpolation? I've been ...
1
vote
1answer
44 views

Lagrange Interpolation definition doubt

Based on some exercise which explains Lagrange Interpolation itself, I got some doubts: It introduces function $$f(x)=\frac{1}{x}$$ and given points $x_0=2$, $x_1=2.75$ and $x_2=4$ so the ...
2
votes
0answers
47 views

Lagrange's interpolation to solve for 0 of y(x)

I have the data composing of 7 elements x is from 0 → 3 incrementing by 0.5 y is from 1.8241 → -1.5427 I am supposed to use Lagrange's interpolation of three nearest neighbor data points. I am ...
1
vote
1answer
99 views

How can I find a non-negative interpolation function?

In numerical mathematics I have learnt about some interpolation methods, however today I've come across some sort of interpolation problem which I don't know how to solve or even work with: Let ...
0
votes
0answers
70 views

How to calculate the condition number and infinity norm of interpolating polynomial

Suppose I have a set of n+1 points: $\lbrace (x_i,y_i)\rbrace_{i=0}^n$ and the following polynomials that interpolate the above mesh. 1)Barycentric with 1st and 2nd kind of Chebyshev points 2) ...
0
votes
2answers
102 views

Linear Spline Interpolation

Can someone explain to me how linear splines work and what formulas are used. I can only seem to find information on cubic splines. Which I don't really understand either Specifically, if I were ...
0
votes
1answer
46 views

Show that Chebyshev nodes cannot be covered by an equispaced points

Given Chebyshev nodes on interval [a,b], show that we cannot find set of equispaced points ${y}_j$ in [a,b] st for all i there exist some j, ${y}_i={x}_j$ where Chebyshev nodes are defined ${x}_i = ...
0
votes
0answers
11 views

About a theorem by Faber in interpolation theory

I am looking for a proof of this theorem: For any table of nodes there is a continuous function $f$ on an interval $[a,b]$ for which the sequence of interpolating polynomials diverges on $[a,b]$. ...
2
votes
2answers
2k views

Newton's Interpolation Formula: Difference between the forward and the backward formula

I was taught that the forward formula should be used when calculating the value of a point near $x_0$ and the backward one when calculating near $x_n$. However, the interpolation polynomial is unique, ...
1
vote
2answers
34 views

Subtlety about the definition of B-splines

I came across the following definition for the zero'th order B-spline $$b_0(x) = \left\{ \begin{array}{lr} 0 & |x|>1/2\\ 1 & |x|<1/2\\ 1/2& |x|=1/2. \end{array} ...
2
votes
0answers
90 views

lagrange interpolation question here

We have the function : $f(x)=\cos(x) + \sin(x)$ and $x_0=0, x_1=0.25 , x_2=0.5, x_3=1$ a)Find Lagrange polynomial for this function. c)Find the real approximation error. d)Find the limit of the ...
1
vote
2answers
80 views

Polynomial Interpolation

My professor gave the following question as a practice for study guide. Any assistance in terms of helping me to solve this would be much appreciated. Suppose that $f$ is continuous and has ...
1
vote
1answer
62 views

How to use any function interpolation method to create two functions …

I need help desperately on this. I have been working on it for a while. Use any function interpolation method studied in the course to create two functions $x(t)$ and $y(t)$ on $0 ≤ t ≤ 1$ so ...
1
vote
2answers
65 views

Polynomial Interpolation and Error

I have numerical analysis final coming up in a few weeks and I'm trying to tackle a practice exam. Assuming $p(x)$ interpolates the function $f(x)$, find the polynomial $p(x)$ that satisfies the ...
0
votes
1answer
453 views

Cubic Spline Interpolation practice

Going over practice problems for our final exam. I'm stuck on a problem involving cubic splines. In fact, I don't even know where to begin. I need to find the natural cubic spline $S(t)$ at $t_0=0, ...
0
votes
1answer
65 views

Use Lagrange interpolation to prove $\max_{x\in[a,b]}|f(x)|\leq\frac{(b-a)^2}{8}\max_{x\in[a,b]}|f''(x)|$

Suppose $f\in C^2([a,b])$ and $f(a)=f(b)=0$,use Lagrange interpolation to prove $$\max_{x\in[a,b]}|f(x)|\leq\frac{(b-a)^2}{8}\max_{x\in[a,b]}|f''(x)|$$ I tried to use the theoretic error to prove ...
1
vote
1answer
446 views

How to evaluate Newton's Divided Difference Polynomial in MatLab with an unknown degree?

I already have the code that finds the coefficients for the polynomial, but how do you find a value for the polynomial if given an x coordinate in MatLab code?
1
vote
1answer
46 views

Other way to write Lagrange's form (with derivative)

Prove that we can write polynomial $L_{n}\in\Pi_{n}$ which is interpolating function $f(x)$ in $n+1$ nodes $x_{0},\,\ldots,\, x_{n}$ in following form: ...
0
votes
0answers
158 views

Hermite-Birkhoff interpolation for finding a polynomial

So I have to consider $p \in\mathscr{P}_3$,where $\mathscr{P}_3$ is the set of polynomials of degree $3$, such that $p(0)=1, p'(0)=1, p(2)=1, p'(1)=2$ using the Hermite-Birkhoff interpolation with the ...
0
votes
1answer
97 views

How to calculate a trigonometric interpolation polynomial

I have the following $2 \pi$-period function f: $$ f(x) = \left \{ \begin{array}{l l l} x: & 0 < x < 2 \pi \\ \pi: & x = 0 \end{array} ...
1
vote
0answers
23 views

$M(x)$ and $L(x)$ interpolate $f(x)$ on $n+1$ points. Show, that $f(x)$ lies between $L(x)$ and $M(x)$

We have $n+2$ points $x_0 \lt x_1 \lt x_2 ... \lt x_{n+1}$. We have two polynomials - $L$ and $M$. $L(x)$ interpolates $f(x)$ on points $x_0,...,x_n$ and $M(x)$ does so on $x_1,...,x_{n+1}$. The ...
0
votes
1answer
26 views

Show $\Delta^mp(x) = 0$ when $p(x) \in P_n$

Show that $\Delta^mp(x)= 0 $ when $p(x) \in \mathscr P_n$ and $m\ge n+1$, where $\mathscr P_n$ is the set of polynomial of degree $n$ and $\Delta^m$ is the operator for the $m$-th forward ...
0
votes
1answer
32 views

Proof that $f[x_{i0},…,x_{ik}] = f[x_0,…,x_k]$

i try to show, that $f[x_0,...x_k]$ is a symmetric function of $x_i$. What means, that for a permutation $x_{i0},...,x_{ik}$ of numbers $x_0, ...,x_k$ applies: $$f[x_{i0},...,x_{ik}] = ...