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

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-1
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2answers
49 views

What is rule of this function?

I have these values.these are inputs and outputs of a function.I want to find rule of function.input is N. ...
0
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0answers
16 views

Minmax approximation

Let $f(x)=a_nx^n+....+a_1x+a_0, a_n\neq0.$Find the minmax approximation to $f(x)$ on $[-1,1] $by a polynomial of degree$\leq n-1 ,$and also find the error $\rho_{n-1}(f).$ This problem is from one of ...
6
votes
2answers
112 views

Exact result of a series using Euler-Maclaurin expansion.

This is a variant of Exercise 64 in Chapter 9 of concrete mathematics. Prove the following identity \begin{equation} \sum_{n = -\infty}^{\infty}' \frac{1 - \cos( 2\pi n k )}{n^2 } = 2 \pi^2 ( k - ...
1
vote
1answer
60 views

Solve quadric equation system

How to solve this? For given real and symetric matrices $A_1,A_2,A_3,A_4\in\mathbb{R}^{4\times4}$ find $x\in\mathbb{R}^4$ $$x^TA_1x=0$$ $$x^TA_2x=0$$ $$x^TA_3x=0$$ $$x^TA_4x=0$$
2
votes
0answers
46 views

Show that $\displaystyle\sum_{i=0}^{N-1}|\epsilon_i|\to0, N\to\infty$

Let $I_o=[t_0,t_0+T]\subset\mathbb R, T>0$, If $f\in C^0(I_0\times\mathbb R,\mathbb R)$ and satisfies the Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
4
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0answers
45 views

How many iterations of the Newton's method are needed to achieve a given precision

There is a formula for bisection method to estimate number of iterations that are needed to achieve a given precision (desired significant figures) in the interval $[a,b]$ $$ n\ge ...
1
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2answers
32 views

Given $f(x)= e^x - e^ax$ with roots $P$ and $Q$,$0<P<1<a<Q$ , show that $g_1(x) = e^x/e^a$ and $g_2(x)= a + \ln x$ have exactly two fixed points each.

I have a midterm tomorrow and while I was looking through old exams from my professor I stumbled on a problem for which I'm not able to see the solution. We want to find the rots of $f(x) = e^x - ...
0
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1answer
43 views

Rewriting partial differential equation

I have some trouble rewriting a partial differential equation, more specifically the heat equation in one dimension: $ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + f(x,t)\\ $ ...
2
votes
1answer
23 views

Find the largest value for $x_1$ in (0,1) such that $f(0.5)-P_2(0.5) = -0.25$ (interpolation)

I'm not really sure where to go with this problem and I'm hoping you can help. The problem states: Let $f(x) = \sqrt{x - x^2}$ and $P_2(x)$ be the interpolation polynomial on $x_0 = 0, x_1$, and ...
6
votes
4answers
184 views

Software, techniques and tricks of experimental mathematics to conjecture possible closed forms

It often happens that people conjecture possible closed forms of integrals, series, and so on starting from a numerical value calculated to very high precision. What are the techniques, tricks, ...
1
vote
1answer
12 views

Can anyone explain why reducing the stepsize h used in Euler's Method reduces the approximation of a function at a point?

Let $y'=t^{3}y^{2}$ where $y(0)=1$. Approximate $y(1)$ using Euler's method with h=0.25. I learnt online that reducing the step size h reduces the error of the approximation. Can anyone explain why ...
1
vote
1answer
35 views

Euler-Forward product rule

For a numerical approximation we use the Euler-Forward method, we have as definition $$ f'(x)=\frac{f(x+\Delta x)-f(x)}{\Delta x} $$ Now we have that $f$ is the product of two other functions namely ...
0
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0answers
19 views

Calculating availability of a system

Honestly I don't really know whether i should post this here or on cs.stackexchange.com! This is the question i have : Last year, a company providing web application services needed an ...
1
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0answers
44 views

PDE using $\theta$ method in Matlab

I'm trying to solve this problem numerically in Matlab: $ \left\{ \begin{array}{rl} \frac{\partial P}{\partial t} &= \frac{\partial^2 P}{\partial x^2} \ \ \ (\star) \\ P(x,0) &= 1 \\ ...
0
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0answers
33 views

MATLAB standard deviation

How do I calculate standard deviation using a for loop in Matlab? This is what I have but it seems too easy so I don't know if I am doing it correctly: ...
0
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0answers
29 views

Linear Interpolation adjacent nodes

I have a table of values which are $(x_i,y_i)$: (0.0,2.00),(1.0,2.1592), (2.0,3.1697),(3.0,5.4332), (4.0,9.1411),(5.0,14.406),(6.0,21.303). I am supposed to use linear interpolation between adjacent ...
0
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2answers
40 views

Evaluate the following function using as many significant figures as required to get a final result of 4 digits accuracy

I need to evaluate $$ f_5(0.2) = 5! \left[ e^{0.2} - \left( 1 + (0.2) +\frac{(0.2)^2}{2!}+\frac{(0.2)^3}{3!} + \frac{(0.2)^4}{4!} +\frac{(0.2)^5}{5!} \right) \right] $$ using as many as required ...
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0answers
24 views

modifying plot in matlab

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0answers
60 views

MATLAB linear interpolation

I'm trying to write a MATLAB program to do linear interpolation and to check its accuracy. I have to input $x_0$ and $x_1$ and then generate the data values using $y=e^x$. Then, for a variety of ...
1
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1answer
47 views

Lagrange interpolation: Getting a bound and finding the error

I am struggling to understand this: The problem asks me to find the lagrange error of the polynomial approximation given the nodes $x_0 = 1, x_1 = 1.25, x_2 = 1.6$ with $x = 1.4$ The function I am ...
3
votes
0answers
492 views

Mean Absolute Deviation for a Stable Distribution as a Function of the Tail Exponent

Consider the standard Lévy-Stable (or Alpha Stable) distribution $S(\alpha,\beta, \mu, \sigma)$ where $\alpha$ is the tail exponent, $1 \leq \alpha \leq 2 $. Picking the symmetric case with $0$ mean ...
1
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2answers
40 views

Literature and web sources for Computer Aided Geometric Design (CAGD)

Through the Numerical Analysis lecture I came across Bèzier curves, B-Splines and Spline Interpolation and found it very interesting. The title of the chapter was Computer Aided Geometric Design and I ...
1
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1answer
36 views

Fixed-point theorem restriction in numerical analysis

The Banach fixed-point theorem states that if $f:[a,b]\to [a,b]$ is $\lambda$-Lipschitz where $\lambda\in[0,1)$ is such that satisfies $|f(x)-f(y)|\leq \lambda |x-y|$ for every $x,y\in [a,b]$ (I'm ...
1
vote
1answer
49 views

Very confused with interpolating polynomials

I have a problem from my homework that I completely botched, and no matter what I do I end up with the wrong answer. Here's the problem: For a given function $f(x)$ let $x_0 = 0, x_1=0.6, x_2 = ...
1
vote
2answers
34 views

newton's formula

How is the number of iterations found using the Newton's formula? I tried $|P-P_n|<k^n\max\{P_0-a_1, b-P_0\}<TOL$ Can anyone help me with another formula in finding $N$ (the number of ...
0
votes
1answer
24 views

Iterative Scheme-Programming Matlab

I don't know if this is going to seem like a dumb question, I am new to this and to matlab, but I'm trying to construct an iterative scheme in MATLAB to compute $\sqrt(b)$ for a given b>0, and program ...
0
votes
0answers
16 views

for loop standard deviation

I'm very new to MATLAB programming and thus I doubt myself when doing things with matlab. I just wanted to confirm I am doing this correctly. I am supposed to complete this program: function ...
1
vote
1answer
43 views

function program for $e^x$

I'm Using this approximaion: $$e^x\approx \text{myFunc}(x)=\sum_{i=0}^{10}\frac{x^i}{i!}$$ I'm trying to write a function program to evaluate $e^x$ and have an error that is $\leq$ $10^{-7}$ which ...
0
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0answers
19 views

How to compare the speed of convergence of infinitesimal of the same order?

Given two iterated function $f(x)$ and $g(x)$. Both of them converge to A, that is, both $\{p_n=f(p_{n-1})\}_{n=1}^\infty$ and $\{q_n=g(q_{n-1})\}_{n=1}^\infty$ converge to A. And the order of ...
0
votes
0answers
66 views

Euler method inequality

Given the problem for $t\neq0$ and $t\le1$ $y'(t)=y^2(t)$ $y(0)=1$ Let $\mu>0$, and $\epsilon_n=\frac12(f(t_{n+1},y_{n+1})-f(t_n,y_n))$, such that $|\epsilon_n|\le\mu|y_n|$ is ...
1
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0answers
52 views

Elastica - numerical check

Following on from rmhleo's fantastic answer here, where he states that the deformation of an ideally elastic circle is a problem of the calculus of variations which may be solved with an ODE of the ...
0
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2answers
90 views

Quicker way to compare numbers without calculator

Question: Find the order of $(1/2)^{1/2}$, $(1/e)^{1/e}$, $(1/3)^{1/4}$ without using calculator. Extra constraint: You only have about 150 seconds to do it, failing to do so will eh... make you run ...
1
vote
1answer
34 views

A proof of Newton's Iteration for Finding Square Roots

I find this theorem in my textbook: Assume that $A>0$ is a real number and let $p_0>0$ be an initial approximation to $\sqrt{A}$. Define the sequence $\{p_k\}_{k=0}^\infty$ using the recursive ...
0
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0answers
24 views

How to estimate the error of a numerical multiple integration

I'm integrating over the wholes space the function $$f(\vec{r_1},\vec{r_2})=\exp{\bigg[-(r_{1\alpha}+r_{1\beta}+r_{2\alpha}+r_{2\beta})\bigg]} \cdot 1/r_{12}$$ where $\vec{r_\alpha}=(-R/2,0,0), ...
0
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0answers
33 views

Why is the order of the difference operator defined as $p$ rather than $p+1$ for the second order differential equation by multistep methods?

I am reading the book Discrete Variable Methods in Ordinary Differential Equations (1962) by Peter Henrici. I am confused about the accuracy definition in multistep methods for the second order ...
0
votes
1answer
16 views

Bound on numerical integral

I'm running a numerical integral over a function which I can only know on specific points as the problem is defined on a lattice. I've been using a trapezoidal method which has for known error: ...
0
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0answers
22 views

Why is the order of the difference operator defined as $p$ rather than $p+1$ for the second order differential equation?

I am reading the book Discrete Variable Methods in Ordinary Differential Equations (1962) by Peter Henrici. I am confused about the accuracy definition in multistep method for the second order ...
2
votes
1answer
29 views

Parallelism in Golub & van Loan's Jacobi algorithm for symmetric eigenvalue problems

In Matrix Computations by Golub and Van Loan (3rd edition, page 433) an algorithm is given for a parallel version of the classical Jacobi algorithm for solving a real symmetric eigenvalue problem. The ...
0
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0answers
28 views

Numerical solution of SDEs with fractional Brownian motion

I am trying to numerically solve some SDEs representing a nonlinear circuit (possibly chaotic) driven by noise: $$ dX = f(X) dT + \sqrt{P_{w}} dW + \sqrt{P_{f}} dC $$ where $X$ is my circuit state, ...
2
votes
1answer
182 views

Jacobian Matrix Requirement for Linear Approximation

It is my understanding that when searching for a linear approximation of a nonlinear function, using the Jacobian (matrix) could help. I did some reading, and read that there is a condition: the ...
2
votes
2answers
285 views

Interpolation polynomial Challenge

suppose $p(x)=x^k-x^t, k \neq t $ (k,t is a positive integer). function q(x) be a Interpolation polynomial from degree lower or equal n, to data $i=1,...,n+1, (x_i ,p(x_i))$. if ----------- then ...
0
votes
2answers
40 views

Taylor Approximation

For $f(x)=e^x$, find a Taylor approximation that is in error by at most $10^-7$ on [-1,1]. Using this approximation, write a function program to evaluate $e^x$. Compare it to the standard value of ...
1
vote
1answer
32 views

Approximating a real from some other reals.

Given a list of $n$ real numbers: $R=(r_1,r_2,\ldots,r_n)$ with $r_i < r_{i+1}$, and a target real number $t$, How can we find the subset of $R$ of size $k$ with a sum that best approximates $t$? I ...
0
votes
2answers
90 views

First-degree spline interpolation problem

I have been spending hours working on a problem from Cheney & Kincaid on first-degree spline interpolation problem but am heading nowhere. Any explanation, pointers or hints from you would be bery ...
1
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0answers
29 views

Strictly diagonal matrix

Suppose that matrix $A$ is strictly diagonally dominant, show that $||A^{-1}||_{\infty}\leq[min(|a_{ii}|-|\sum_{i\neq j}^n a_{ij}|)]^{-1}$.
0
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1answer
32 views

Bounding an approximation error of Bernstein polynomials

I have to show that for $ 0 \leq x \leq 1 $ \begin{align*}\sum|f(x)-f(k/n)|p_{nk}(x)\leq(2M/\delta^2)\sum_{k=0}^n (x-k/n)^2p_{nk}(x)\end{align*} Important to know is that $p_n(x)$ and $p_{nk}(x)$ are ...
0
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0answers
25 views

The weak formulation of Navier Stokes Equation

I come across this problem in the weak formulation in Navier-Stokes equation. In the book, it let $D(\Omega_T)=\{\vec{\phi}\in C_0^{\infty}(\Omega_T),div\vec{\phi}=0 $, where ...
1
vote
1answer
43 views

Condition number - proof

I have a problem with which I've been struggling for a while. It's probably not that difficult, but I seem to be stuck so... here we are. Let A be an invertible lower- or upper-triangular matrix ...
0
votes
2answers
33 views

Euler method(path s1s2=s2s1)

Given a differential equation $\frac{dy}{dx}=f(x,y(x)), y(x_0)=y_0$. What is the condition for function of f(x,y) such that the result of $y(x_0+S_1+S_2)$ by using Euler forward method, a step size ...
1
vote
2answers
50 views

The Mean Value Theorem

I am trying to understand why the following question is correct. Regarding to the MVT. Given: Find a number c satisfying the conclusion of the Mean Value Theorem $f(x)= x^{1/3}$, on $[-1,2]$ $f(x)= ...