Questions tagged [lagrange-interpolation]

A method of generating a polynomial that crosses through a set of data. The degree of this polynomial is equal to the size of the data.

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Where to find proof for the remainder formula of the interpolation in two variables

Professor showed this result in the lecture without giving any proof (after proving the existence of the interpolating polynomial in two variables). I've been trying to prove it myself or find a book ...
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Showing recursive Formula for Polynomial Interpolation

Let $f : [a, b] \rightarrow \mathbb{R}$. Define $$ f[a_0, ..., a_n] := \frac{f[a_1, ..., a_n]-f[a_0, ..., a_{n-1}]}{a_n - a_0} $$ with $f[a]=f(a)$. Consider now $(m + 1)$ distinct nodes $x_j \in \...
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Lagrange Interpolation as generalized polynom

I need to write an algorithm that constructs a function of the form $f(x) = \sum_i^n q_i x^i$ that exactly goes through the points $p_i = (a_i, b_i)$. In general building such a function is not the ...
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Does there exist $f \in \mathcal{C}^\infty\left([a, b]\right)$ such that $M_n \rightarrow \infty$ but $\left\|f - p_n\right\| \rightarrow 0$?

Let $f \in \mathcal{C}^\infty\left([a, b]\right)$ and $n \in \mathbb{N}$. Let $p_n$ be the Lagrange interpolating polynomial for a partition of equispaced points $a = x_0 < x_1 < \cdots < x_n ...
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Does Lagrange interpolation at Chebyshev points solve the Runge phenomenon?

I recently came across the concept of the Runge phenomenon while studying numerical methods for special functions in the book "Numerical Methods for Special Functions" by Amparo Gil, ...
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Prove $\max_{|x|\leq \frac13}|f(x)-p_2(x)|\leq \frac{1}{2\cdot 3^{\frac{11}{2}}}\,\max_{|x|\leq \frac13}|f^{(4)}(x)|$ for Runge's function.

Let $f(x)=\frac{1}{1+x^2}$ be Runge's function. If $p_2(x)$ is the interpolating polynomial of $f$ regarding the nodes $-\frac13,\,0,\frac13$, prove that: $$\max_{|x|\leqslant \frac13}|f(x)-p_2(x)|\...
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How to prove the given statement based on the conditions related to interpolating polynomials used in Adaptive Backward Differentiation Formula.

Full disclosure: This was part of my homework that I couldn't solve completely. Also, I wasn't sure how to give a proper title to this question. Before going to the statement to prove, I think some ...
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error calculation in Lagrange interpolation

Construct Lagrange interpolating polynomials of degree two and three to approximate: a)$ f (0.25)$ yes $f(0.1) = -0.62049958, f(0.2) = -0.28398668, f(0.3) = 0.00660095, f(0.4) = 0.24842440 $ ...
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Error term refinement of Lagrange interpolation

The error in Lagrange interpolation is $$ E(x)\equiv f(x)-L_n(x)=\frac{f^{(n+1)}(\xi_x)}{(n+1)!}\prod_{j=0}^n(x-x_j) $$ where $f\in C^{(n+1)}([a,b])$ is the function to be inter-/extrapolated, $\{x_0,\...
Andras Vanyolos's user avatar
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Nonlinear ODE with (pseudo-)spectral method

Consider an ODE of the form $$c_0 u + c_1 u' + c_2 u'' \enspace = \enspace f(x,u) \quad .$$ I want to solve this ODE with spectral methods. To this end, let $\{ x_k \}_{k=0}^N$ be suitable nodes, e.g. ...
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Finding rational coefficients of a cubic polynomial that fits 4 data points that have been floored to an integer

I have 4 data points: (204, 5422892) (205, 5722486) (207, 6343357) (213, 8386502) I have information that these data points were generated with a cubic polynomial $y = ax ^ 3 + bx ^ 2 + cx + d$ with ...
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$\ \forall x_1,x_2,...,x_n \in \mathbb{R} (x_i\not=x_j)$ in the range of $[-1,1]$ prove:$\sum_{i=1}^{n}\frac{1}{\Pi_{k\not=i}|x_k-x_i|}\ge2^{n-2}$

$\ \forall$ $x_1,x_2,...,x_n$ $\in \mathbb{R}$ $(x_i\not=x_j)$ in the range of $[-1,1]$ prove : $$\sum_{i=1}^{n}\frac{1}{\Pi_{k\not=i}|x_k-x_i|}\ge2^{n-2}$$ my attempt : $$p(x) = \sum_{i=1}^{n}\left(...
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$f(x)=x^n+a_{n-2}x^{n-2}+...+a_1x+a_0\in \mathbb{R}[X]$. prove$\ \exists$ $i\in[1,...,n]$ so that : $|f(i)| \ge \frac{n!}{\binom n i}$

$f(x)=x^n+a_{n-2}x^{n-2}+...+a_1x+a_0\in \mathbb{R}[X]$. prove$\ \exists$ $i\in[1,...,n]$ so that : $$|f(x)| \ge \frac{n!}{\binom n i}$$ my attempt : i used lagrange interpolation and compared $x^n$ ...
farnood gholampoor's user avatar
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How many evaluations do you need to prove that a multivariate polynomial is the zero polynomial?

For a univariate polynomial $f$, you just need to prove that $f(x) = 0$ for $d+1$ distinct $x$ to prove that $f$ is the zero polynomial. But for multivariate polynomials, how does that work? How many ...
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How can I find a $n$-th degree polynomial for a line with $n-1$ predetermined points?

Essentially, I have a set of points which I want to create a polynomial that passes through a set of points. This is within the context of having set locations on a line to set times. It is preferred ...
Vivaan Singhvi's user avatar
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Polynomial or wave through aligned points [closed]

If I have the distinct points $(x_{1}, 0) ... (x_{n}, 0)$, A) What would be a simple polynomial (non-constant) passing through these points? And could I instead also write a simple continuous curve (a ...
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Different bounds for truncation error in linear interpolation

I am reading about the Lagrange linear interpolation for approximating a function. In the book, truncation error in linear interpolation is derived as follows : Suppose $f(x)$ is a function which is ...
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Possible dimension of vector space

Let $A=\{0,1,2,3\} \subset \mathbb{R}$ and $\mathbb{R}[X]$ be the set of all polynomials in one variable over $\mathbb{R}$. Let $V$ be the $\mathbb{R}$-vector space of functions $f: A \rightarrow \...
Pratik Rawal's user avatar
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How to use bilinear interpolation to interpolate a new optimized trajectory?

Sorry if this does not quite make sense as I am still wrapping my head around it as well. Currently, I have a vehicle with some initial position $x_0$. From this position, four optimal trajectories ...
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Linearly independent equivalent of lagrange interpolation

Im looking at lagrange interpolation and has been given that: $P(t) = \sum_{i=1}^n \; y_iL_i(t)$ Is equivalent to the following matrice equation: $$\begin{bmatrix} q_1 \\ q_2 \\ \vdots \\ ...
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Lagrange Interpolating Polynomials with custom way to add Local Maximum/Minimum.

Given this data table consists of 10 points: x | -3 1 2 4 5 6 9 11 12 14 y | 2 3 -2 0 -3 1 -1 0 -2 1 Construct a polynomial P(x) such that its graph passes through the points in the data table and P(x)...
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Showing that that a curve of degree $d$ by an affine combinations can also be expressed in terms of Lagrange polynomials of degree d

Im working with splines and interpolation and bit stuck on showing the following: Given $d+1$ points $(c_i)^d_{i=0}$ and parameters $(t_i)^d_{i=0}$ I have worked with and constructed the curve $q_{0,d}...
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For what $n$ can we find a degree $\leq n-2$ polynomial such that $P(i) \in \{0 , 1 \}$ for $i \in [n]$, but not all identical.

For what $n$ is the following statement true: There exists a choice of $ a_1, a_2, \ldots a_n \in \{ 0, 1 \}$, not all identical, such that there is a polynomial $F(x) \in \mathbb{R}[x]$ of degree at ...
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Showing that Lagrange Basis Polynomials sum to 1 [closed]

I am currently trying to explain why it is the case that when given a Lagrange polynomial with coefficients of all $1$ $P_3(x)$ $=$ $L_0(x)$ + $L_1(x)$ + $L_2(x)$ + $L_3(x)$ $=$ $1$, for all $x$. What ...
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Fundamental polynomials: do they form a base

First, is $P_n$ $n$ or rather $n+1$ dimensional real vector space of polynomials of degree at most $n$ ? Here, $n$ indicates that it should be $n$ but the basis $1,x,x^2,x^3,...,x^n$ has $n+1$ ...
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Prove that $P_{h}(f)$ satisfies $||f-P_{h}|| \leq Ch^{2}$

Consider a partition of the domain $\Omega=[a,b]$ $$ a=x_{1}<x_{2}<\cdots < x_{N}=b $$ with mesh size $h=\max\{x_{i+1}-x_{i}:i=1,...,N-1\}$. Let V be an inner product space, with inner ...
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How would a better mathematician than I complete this half-finished definition of what it means for a curve to be smooth?

Once upon a time, I was taught how to play connect the dots. Some years later, I was given pseudo-code for an algorithm which would compute a polynomial of minimum degree passing through some points. ...
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Understanding a proof for divided differences

suppose you are given the following proof of the statement: $[x_0, ..., x_n; f \cdot g]=\sum_{k=0}^n[x_0,...,x_k; f]\cdot [x_k,...,x_n; g]$. Some preliminary steps: Let $[x_0,...,x_n; f]=\sum_{i=0}^n\...
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Lagrange interpolation polynomial - error function

a) Calculate the Lagrange Interpolationpolynomial $p$ for the function $f(x) = \sin (\pi x)$ for the points $x_0 = 0, \ x_1 = 0.5 , \ x_2 = 1$. b) Then estimate the maximum interpolation error on the ...
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Prove a Vandermonde Determinant Formula

I seem to have stumbled upon a strange formula, and I'm not quite sure that I understand it. Consider the determinant standard Vandermonde matrix $${}^nD = \left|\begin{matrix} 1 & x_{0} & ...
Squircle's user avatar
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algebraic equivalent interpolation formulae

I'm approaching to finish reading this book by Ralston: A first course in numerical analysis book: but I'm still missing the reason for the solution of the exercise ${}^*22, $ part $d$. I understand ...
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Why must a polynomial of degree n-1 or lower have the following result?

I am given a homework problem that states: prove that for any polynomial $q$ of degree $n-1$ or lower that: $$\sum^n_{i=0}q(x_i)\prod_{j=0, j \ne i} (x_i - x_j)^{-1} = 0$$ Intuitively, I don't know ...
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(Determine coefficients) Backward Differtiation Formula for m $ = 2 $ using Lagrange polynomial

So I want to determine the coefficients regarding the BDF($2$). Note that I'm absolute beginner, so my attempt is not that great. The solution has to be: $ \frac{3}{2}y_{k+2} - 2y_{k+1} + \frac{1}{2}...
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find minimum value of $u$ for $\,u = \frac{ax^2+by^2}{\sqrt{a^2x^2+b^2y^2}}$

Find the minimum value of $u$ where $x^2+y^2=1\;$ and $\;u =\displaystyle{\dfrac{ax^2+by^2}{\sqrt {a^2x^2+b^2y^2}}}$
Kiran Kiran's user avatar
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Lagrange interpolation as two points get closer and closer

Suppose, you want to perform a Lagrange interpolation for $x_0$, $x_0+\varepsilon$ and $x_1$. What happens as $\varepsilon \to 0$? Here are the Lagrange-polynomials \begin{gather*} L_0(x)=\frac{(x-(...
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Issues with the Peano kernel in the Peano kernel theorem

I have a very simple question, because I somehow don’t get this definition. The version of the Peano kernel theorem that we had in class is: For all linear functionals $\mathcal{L}:C^{n+1}([a,b]) \...
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why is my lagrangian interpolation not approximating as nearly accurate as Newton's DD approximation?

Here is the set of data: $(X,Y)= (1,4.75), (2,4), (3,5.25), (5,19.75), (6,36)$ After I approximated to $3$rd order polynomial with Newton's interpolation, it modeled the true behavior of how $X$ and $...
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Newton's Divided Difference Formula without given data point

This is the original question: Let f(x)= sin((pi*x)/6) and P(x) a quadratic polynomial such that f(x) = P(x) at x=0,1, and 2. Find P(x) using Newton's Divided Difference Formula Since Newton's ...
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Simplifying $\frac{a^4}{(a-b)(a-c)} + \frac{b^4}{(b-a)(b-c)} + \frac{c^4}{(c-a)(c-b)}$ using Lagrange’s polynomial

I have a question of symplifying this expression $$ A=\frac{a^4}{(a-b)(a-c)} + \frac{b^4}{(b-a)(b-c)} + \frac{c^4}{(c-a)(c-b)} \tag{1} $$ where $a$, $b$ and $c$ are distinct nonzero real numbers. ...
Tom Howard's user avatar
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Convex optimization with Ridge regression

If this is the constrained version of ridge regression: min w∈Rd ∥Φw − y∥^2 s.t. ∥w∥2 ≤ s, (1) where Φ ∈ R n×d and y ∈ R n . Answer the following questions. •How can we prove that this is a convex ...
Srikanth Muttineni's user avatar
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Trouble trying to bound an approximation using Lagrange interpolation.

I know this has been asked various times but I do not understand any of the answers given yet. I'm working with the function $f(x)=e^{x}$ in the interval [-4,0] and I need to bound $|f(x)-Q_{n}(x)|$ ...
Alicia Garcia's user avatar
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Lagrange Polynomial to estimate the derivatives of a function

We have the following question for our homework. I'm completely lost on what to do. I have no idea how to compute the error of the derivative or how to proceed from there. I've tried googling Lagrange ...
pasha's user avatar
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Find Hermitian interpolation with Lagrangian polynomials

Hi I have an exercise that I cannot solve. Can someone help me? I have to find the solution to the following Hermitian interpolation problem p(x) under the conditions: $p(x_0)=-1, \ p'(x_0)=1 \ p(x_1)=...
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how to prove span of Lagrange basis functions is equivalent to${\rm span}\{1,x,\dots,x^n\} $

Suppose we have n+1 points $x_0, x_1 , \dots , x_n$ and the following is Lagrange basis function. $$ L_i(x):=\prod_{j=0, j \neq i}^n \frac{x-x_j}{x_i-x_j} $$ How to prove that $$ \operatorname{span}\{...
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Prove that $\sum_{1\leq i\leq n}\prod_{j\neq i} \frac{1-x_ix_j}{x_i-x_j} = 0$ if $n$ is even and $1$ otherwise

Let $x_1,\cdots, x_n$ be different real numbers. Prove that $$\sum_{1\leq i\leq n} \prod_{j\neq i} \frac{1-x_ix_j}{x_i-x_j} = \begin{cases} 0,&\text{if }n\text{ is even}\\ 1,&\text{if }n\text{ ...
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Bounding error term in Lagrange Interpolation

I've spent more time that I'd like to admit working out this bound. Online, I've seen that the remainder $R(x) = f(x) - L(x)$ in Lagrange interpolation is bounded by $$| R(x) | \leq \frac{(x_n - x_0)^{...
Leonardo Lovera's user avatar
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What is the most efficient algorithm to evaluate a polynomial of n degree at K points?

A brute force approach would be to evaluate each point for each of the terms of a polynomial, which will be $O(Kn^2)$. If we use logarithmic exponentiation to find each $x^i$ then it becomes $O(Kn \...
ishandutta2007's user avatar
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Is the following statement about polynomial approximation of non-differentiable functions true?

Statement $\forall L \in \mathbb{R}^+$, and $\forall f:\mathbb{R}\to\mathbb{R}$ for which $f$ isn't differentiable at point $a\in\mathbb{R}$ $f$ is infinitely differentiable everywhere else (on $\...
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I don`t understand this little incongruence about the error function of Lagrange interpolation polynomial.

Given $f \in C^{n+1}([a,b])$ and a set of $n+1$ points in $[a,b]$. And given $P$ the Lagrange interpolation polynomial, the error function is $f - P = \frac{f^{(n+1)}(\eta_x)}{(n+1)!}w_S(x)$ where $...
Gauss 'n Roses's user avatar
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1 answer
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Polynomial interpolation yielding integer coefficients

Let $y_1,\dots,y_n$ be integers. Conjecture:$\;$There exists $f\in\mathbb{Z}[x]$ such that $f(i)=y_i$ for $i=1,\dots,n\;$if and only if $(i-j){\,\mid\,}(y_i-y_j)$ for all $i,j$ with $1\le j < i\le ...
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