Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.

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Error of Chebyshev Interpolation

I am trying to find the error when using Chebyshev interpolation on the Runge function, but I am having trouble understanding how to do this. Specifically, I would like to use points for n from 10 to ...
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1answer
46 views

eigenvalues lesser than $1$ implies contractive map

Consider $(\mathbb R^n,d)$ where $d$ is the Euclidean metric. A map $w:\mathbb R^n\to \mathbb R^n$ is said contractive if there exists $0<s<1$ such that for every $x,y\in \mathbb R^n$ we have ...
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11 views

Numerical scheme for first order partial differential equation.

Does anyone know some numerical methods for solving such equations? $$ \begin{cases} V_t + \langle V_x,\, F(t, x) \rangle = 0\\ V(0, x) = h(x) \end{cases}, \quad 0 \leqslant t \leqslant T, $$ where ...
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13 views

Trapezoidal rule (differential equation) is not symplectic

Trapezoidal rule $y_n = y_{n-1}+\frac12h(f(y_n+y_{n-1}))$ is not symplectic. I have no clue to prove the claim. Can anyone give me some hints? Thanks for your time.
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2answers
105 views

Backward Euler's Method

This question was asked in CSIR. please help me to find out correct choice Let $y(t)$ satisfy the differential equation $$y'=\lambda y;y(0)=1$$. Then the backward Euler method for $n\geq 1$ and ...
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1answer
37 views

Analytical Solution for Second Order Linear PDE

I'm trying to solve the following PDE (derived from convective heat transfer in a fluid flow between two parallel plates) analytically and I'm not sure what path to take. I don't think separation of ...
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0answers
9 views

Efficient and stable computation of inverse CDF

What is the most efficient and numerically stable algorithm for computing the inverse CDF $F^{-1}(y)$ of a probability function, assuming that both the PDF $f(x)$ and the CDF $F(x)$ are known ...
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1answer
28 views

Error approximation of Simpson's rule

My calculus book states that the error of the Simpson's formula is equal to $$\frac{(b-a)^5 f^{(4)}(c)}{2880n^4}$$ for a $c \in [a, b]$, if the function has a continuous fourth derivative. Is this ...
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11 views

Mid-point quadrature error estimate for multivariate functions?

When using the one dimensional mid-point quadrature you can easily estimate the error by the second derivative of the function you want to integrate. Is it possible to get a similar estimate if you ...
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23 views

Prove Second-Order Explicit Runge-Kutta Method is Explicit Midpoint Rule

I'm doing a numerical analysis assignment and was asked to prove that a second-order explicit Runge-Kutta method is the explicit midpoint rule. I understand that this is a commonly proven fact but the ...
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1answer
27 views

Exact solution at t = 2 and x = 1 (PDE)

I need help in this question. I did the following steps: (a) I considered the characteristic $dx/dt = \frac{3t}{3x^2 +5}$ and solve it using the separation method and obtained $x^3 + 5x = ...
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1answer
19 views

Turning points of a weighted cosine basis sum

I'm doing some work with a cosine basis, where in the interval $[0, \pi]$ some function $f(x)$ is given by $$ f(x) = \sum_{n = 0}^{M} a_n \cos{\left( nx \right)} $$ For a given set of coefficients ...
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25 views

Recurrence relation of Chebyshev nodes

Doing a few practice questions, I came across a proof based question. How would one go around to solve this? "Prove the following recurrence relation:" ...
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0answers
15 views

Short term exact solution to semilinear heat equation $u_{t}=u_{xx}+u^{2}$ with periodic boundary

The semilinear heat equation $u_{t}=u_{xx}+u^{2}$ blows up in finite time, that depends on the initial condition. But I was wondering if it has an exact solution up to that time? The equation is ...
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19 views

Is it possible to design a strongly stable linear multistep method of order 7 which has stiff decay

I'm studying for a test and I'd like to know is it possible to design a strongly stable linear multistep method of order 7 which has stiff decay. I have no clue to verify the claim. Can anyone give me ...
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0answers
16 views

Is it possible to construct a consistent unstable one step method of order 2? why?

Is it possible to construct a consistent unstable one step method of order 2? why? I think the answer is no but I have no clue to prove it. Can anyone give me some explanations? Thank you in ...
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20 views

Quadratic polynomial interpolation from a transformation

Some modeling considerations have mandated a search for a function $$ u(x) = \gamma_{0}\exp(\gamma_{1}x + \gamma_{2}x^{2}) $$ where the unknown coefficients $\gamma_{1}$ and $\gamma_{2}$ are ...
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65 views

the second-order system of ordinary differential equations

I am learning PDE and can someone help me solve the following practice problem? I want to convert the following to a 4 x 4 first-order system $u'=f(t,u), u(0)=u_0$ and $u= [x, x', y, y']= [u_1, u_2, ...
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1answer
25 views

If $f(x)=e^{ax}$, show that $\Delta^n f(x)=(e^{ax}-1)e^{ax}$

Given as an assignment for Interpolation. I, first of all, doubt whether the question is correct or not, because $$\Delta f(x)=e^{ax+ah}-e^{ax}=e^{ax} (e^{ah}-1)$$ $$\Delta^2 f(x)=[e^{ax+ah} ...
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19 views

Do “interpolation polynomial” and “interpolating polynomial” mean the same?

I came across "interpolating polynomial" in linaer algebra book, and I could find an explanation to "interpolation polynomial". It seems there's no difference between "interpolation polynomial" and ...
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37 views

Solving a modified numerical heat equation

I'm having a bit of trouble finding a good numerical form for this modified version of the heat/diffusion equation and I was just wondering if I am tackling this question the correct way. Firstly, I ...
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1answer
33 views

Lagrange Polynomial Interpolation - Polynomyal Differences Depending Upon the Degree?

My question is simple: Give the table: | x |0|2|4|6| |f(x)|1|3|5|7| Why when calculating Lagrange Polynomial Interpolation for: | x |0|2| |f(x)|1|3| P1(x) = x+1 And when calculating Lagrange ...
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2answers
39 views

Big O-Notation Proof

I am trying to solve the following task: Use Taylor's Theorem to proof that for $f \in C^3$, the following holds: $$f'(x) - \frac{f(x+h)-f(x-h)}{2h} \in O(h^3)$$ I am not really used to this ...
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25 views

convergence of nonlinear PDE with parameter

We have a nonlinear PDE: $L(u^{\epsilon}, \epsilon)=f$. Here $u^\epsilon$ is the unknown function, $\epsilon$ is a parameter. When $\epsilon>0$, this is a hyperbolic PDE, and when $\epsilon=0$, ...
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1answer
28 views

If $y=(3x+1)(3x+4)\cdots(3x+22)$ prove that $\Delta^4 y=136080(3x+13)(3x+16)(3x+19)(3x+22)$

This is an assignment question from the topic 'Interpolation' I tried to begin this sum by calculating $\Delta^4=y_4-4y_3+6y_2-4y_1+y_0$
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19 views

Short-term stability

In numerical analysis, if we get a solution for a differential equation stable for a particular time (not for infinity), what we call this stability? Thanks
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2answers
36 views

Proof of Lagrange Polynomial

I am trying to prove the following concepts of the Lagrange Polynomial: $\sum_{j=0}^n L_j(x)=1$ $\sum_{j=0}^n x_j^m(x)L_j(x)=x^m, m \le n $ This is my work so far, but I am a little stuck on ...
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0answers
64 views

Ways to determine $\pi$ [duplicate]

I have read that it is possible to determine the value of a single digit, say the 874th of $\pi$. I know that it is a trascentental number, how is that possible? How many ways are there to determine ...
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3answers
74 views

$\sum_{n=0}^{\infty}a_n \cdot x^n=0$ why do we need $a_0=a_1=a_2=…=0$? prove it

To have $$\sum_{n=0}^{\infty}a_n \cdot x^n=0$$ for all $x$, we need $a_0=a_1=a_2=......=0$ I don't quite see the logic here, why only $(0,0,0,......)$ can satisfy this? Proof by contradiction: ...
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14 views

Number of function evaluation

During the implementation of a numerical scheme (Block form), I can determine the number of iteration needed for the solution to converge at each step. However, I have difficulty in determining the ...
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4answers
56 views

Explain how second order differential equations of the form $\ddot{y}+y=0$ exhibit osciallatory dynamics

I'm trying to build a skillset for research in computational neuroscience (and loving math even more as I go along) and have just jumped into the world of differential equations – very simple ones. ...
2
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1answer
35 views

When to use Newtons's, bisection, fixed-point iteration and the secant methods?

I've been introduced more or less to these methods of finding a root of a function (a point where it intersects the $x$ axis), but I'm not sure when they should be used and what are the advantages of ...
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2answers
109 views

Expressing $e^{-x}$ through a different series

I'm consider $e^{-x}$ with $x >> 0$. I want to express it in a series of the form $$ e^{-x} = \sum_{i=1}^{\infty} \frac{a_i}{x^k}$$ But analysis seems to resist it: Motivation: $ ...
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0answers
17 views

Lobatto quadrature

I have a question about Lobatto quadrature. Why the abscissas are the roots of the derivative of Legendre polynomial? Where should I get information about how to derive the abscissas and weights of ...
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1answer
61 views

Derive $f''(x) = \frac{f(x+2h) -2f(x+h) +f(x)}{h^2} + E(h) $ using taylor series expansions.

We have: $f(x+2h) = f(x) +2hf'(x) + 2h^2f''(x) + \frac{4h^3}{3}f'''(\zeta_1)$, where $\zeta_1 \in (x, x+2h) $ $f(x+h) = f(x) +hf'(x) + \frac{h^2}{2}f''(x) + \frac{h^3}{6}f'''(\zeta_2)$, where ...
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0answers
30 views

LU Decomposition vs. QR Decomposition for similar problems

Suppose I want to solve the 2D Poisson equation with Neumann boundary conditions. The solution is non-unique up to an additive constant. I have previously asked a related question here for the 1D ...
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39 views

How to find numerically stable version of a function?

What's the general procedure for finding a numerically stable form for a function? Such as: $$\frac{1}{1+2x}-\frac{1-x}{1+x}$$ when $x≈0$ or $$ln(x)-1$$when $x≈e$ I've only seen examples, but ...
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1answer
38 views

How to solve angle from area of circular segment formula?

I know the radius $R$ of the circle and the area $A$ of the segment. How can I solve for central angle $\alpha^{\circ}$ in this (or some other) equation: $$A=\frac{R^{2}}{2} \left( \frac{\alpha ...
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1answer
20 views

Numerically solving advection equation

Consider the advection equation: $$Ut + aUx = 0, U(x, 0) = f(x)$$ on the interval $A=[-1 \space \space \space 1]$ The exact solution is a periodic version of U(x, t) = f(x − at). Take A = a = 1. ...
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1answer
22 views

Numerical analysis: what is the error term for the rule…?

The question goes: derive the error term for the rule $phi$ to approximate the third derivative of f(a). I have attached a screenshot I understand how to take the Taylor series in the hint, but the ...
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19 views

Newton Raphson's multiple zeros

Assuming Newton’s method is applied to a function with a zero of multiplicity q > 1. Show that the multiplicity of the zero can be estimated as the integer nearest to $q = ...
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1answer
28 views

Is there a way to study “how good” is the Newton's Raphson method applied to a function?

My question is as simple as that. When we're applying the fixed point algorithm, we can see if it's going to converge or diverge finding the derivative and checking if the absolute value of that ...
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1answer
27 views

Numerical derivative of function wrt natural log of variable (non-analytic)

The function that I am trying to evaluate is $$ \frac{d y }{d \ln(x)} $$ where $d$ is the derivative. However I have a set of data points for $x$ and $y$ with uncertainties. Now I think that this ...
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1answer
24 views

Richardson Extrapolation for Quintic Integration

In the picture above, what exactly is the question asking for? I know that the error in simpson's rule is to the order of h^5. Thus doubling the length increases error by 32x and so should I ...
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22 views

ODEs, seting up a non linear system and Newton-Raphson-Method

at the moment I study for an examen regarding ODEs. I try to solve a task but I'm not sure how to do it. Task: We should use the trapozoidal Rule $y_1 = y_o * \frac{h}{2}[f(x_0,y_0) + f(x_1,y_1)] $ ...
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1answer
84 views

Can $\int_0^1\frac{1}{t}e^{-t} dt$ be analytically or numerically integrated?

The following integral has a singularity at $t = 0$ as in this situation the exponential term becomes $1$ and it no longer dominates the $\frac{1}{t}$ term. $$f(x) = \int_0^1\frac{1}{t}e^{-t}dt$$ So ...
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1answer
26 views

Fast Rational Bézier Surface Evaluation Problem

I am currently writing a NURBS ray tracer. What I do is convert the NURBS into rational Bézier patches and then perform the intersection test using Newton's method. To do this fast (the ray tracer ...
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33 views

Iterative methods to find roots

I'm trying to do optional exercises for my numerical methods class. I'm stuck in this one right now: Consider the function $f(x)=-e^{-2x}+3x$. a) Prove that $f$ has an unique real root. ...
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1answer
25 views

Accuracy of Lax-Wendroff scheme

Consider the Lax-Wendroff scheme $$\frac{u_j^{n+1}-u_j^n}{\Delta t}+\frac{u_{j+1}^n-u_{j-1}^n}{2\Delta x}+\frac{\Delta t}{2}\frac{2u_j^n-u_{j+1}^n-u_{j-1}^n}{\Delta x^2}=0$$ for differential equation ...
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1answer
29 views

Richardson extrapolation - deriving methods for forward difference

I am rather stuck on a question from a textbook that I am practicing from which goes as follows: The forward difference formula can be expressed as $$f'(x_{0}) = \frac{1}{h}[f(x_{0} + h) - f(x_{0})] ...