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

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33 views

Is this integral in its most simplified form?

The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$ it can ...
0
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2answers
14 views

How to solve this using gauss jordan method?

I am trying to solve the following equation using gauss jordan method but unable to solve due to the type of equations.At the end i am getting unwanted zeros in 2nd and 3rd row.Here is my work... ...
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1answer
24 views

Is it possible to restore the missing entry by Newton forward divided difference method?

I've only seen the similiar problem but there are some entries on higher degree given.
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0answers
8 views

Integration by parts applied to weak form of boundary value proble

In my finite element textbook the proof for strong and weak form equivalence is determined as such: $$\int_0^1w_{,x}u_{,x}dx = \int_0^1wfdx + w(0)h$$ Integrating by parts and making use of the fact ...
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0answers
42 views

What's the Fibonacci number sequence? In other words, which pattern do Fibonacci numbers have? In other words again, what are their properties? [on hold]

I want to know how to use the Fibonacci numbers to make a sequence, but first, please explain to me what the Fibonacci numbers are. I'm very curious about hearing your answers and I'm sorry if this ...
1
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0answers
12 views

minimize the total cost of transportation [on hold]

can anyone help me to solve this question to minimize the total cost of transportation,how to use Vogel’s approximation method
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0answers
19 views

Machine Floating Point Theorem

Completely stuck on this floating point question. Let $x \in \mathbb{R}$ have the following floating point representation: $$ x = (-1)^s[0.a_1a_2\dots a_ta_{t+1}\dots]\cdot \beta^e $$ [Where $\beta$ ...
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0answers
15 views

Determine error in Neville's Algorithm calculation

I've been mulling over this problem for a while and I don't even know how to start it. The book is hopelessly vague. The problem states Neville's Algorithm is used to approximate $f(0)$ using ...
0
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0answers
10 views

Testing numerical solvers with analytic solution to Ornstein-Uhlenbeck SDE?

I have an SDE I want to solve numerically that is fairly close to the Ornstein-Uhlenbeck process: $$ dx_t=θ(μ−x_t)dt+σdW_t $$ which has analytic solution $$ ...
0
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1answer
20 views

Is it possible to solve pde with 2 Neumann boundary conditions (Gaussian Elimination)?

I have the following equation: $$ \nabla^2u = f $$ over $\Omega: [0,10] \times [0,10]$ where boundary conditions: $$ \left\{ \begin{array}{ll} \frac{\partial u (0,y)}{\partial x} = 0 \\ ...
2
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1answer
30 views

Estimation of superexponential integral

I was wondering if anyone could give as precise an estimate as possible for the integral $$ \int_0^b e^{-a e^{-x^2}}\, dx, $$ where $a$ is positive. It is not related to any special functions as far ...
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1answer
43 views

All fixed points of a function are globally stable or unstable.

I am analyzing the function $\lambda \sin( \pi x)$ for $x \in [0,1]$ for a paper I am writing. I know that all fixed points of this function are either globally stable or unstable but I am not sure of ...
3
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3answers
49 views

Approximation of Natural Logarithm using arithmetic.

A friend of mine posed this question to me a couple days ago and it's been bugging me ever since. He told me to take the square root of 5 twenty times, subtract 1 from it, and then multiply it by ...
1
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0answers
64 views

Taking the Fourier transform of a Hankel function

Considering the following inverse Fourier transform $$ f(t) = -\alpha \int_{-\infty}^{\infty} F(\omega)H_0^{(2)}(k(\omega) \beta) \exp(+j\omega t) d\omega$$ where $F$ is an arbitrary function and ...
0
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0answers
11 views

Adams-Moulton and BDF methods

whats are the differences between Adams-Moulton and BDF methods. which one is better and which one computes the solution faster? i think adams moulton is a better method as it can get to the solution ...
0
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0answers
17 views

Numerical method of lines for solving PDEs

Could you please advise some literature about the numerical method of lines (MOL) for parabolic PDEs? It is a method of solving PDEs with discretizing only by space but not by time. A system of ODEs ...
0
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0answers
19 views

Uniqueness of a differential equation

Let $I_o=[t_0,t_0+T]\subset\mathbb R$, where $T>0$, $f\in C^0(I_0\times\mathbb R;\mathbb R)$ and satisfying Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
5
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1answer
85 views

Difference table for interpolation

For calculating divided (fraction) difference table for interpolating the points $(x_i, f_i)$, $i=1,2,...,n$; by using a polynomial with degree lower or equal to $n$, $n(n-1)/2$ fraction was used. I ...
0
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1answer
45 views

What's the formal difference between analytical and numerical?

While trying to wrap my head around differential equations in a practical way, I found a quite enlightening phrase about it Solving a differential equation can be done in three major ways: ...
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2answers
46 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. ...
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0answers
12 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 ...
2
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1answer
56 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
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1answer
46 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
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0answers
40 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 ...
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0answers
14 views

What is the theory behind numerical integration such as adaptive quadrature and laplace approximation? [on hold]

I am trying to understand the theory behind the numerical integration. How it is done and what it results? Thanks !!!
4
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0answers
41 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 ...
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0answers
76 views

Here are some number questions I want you to answer. What are the answers? [closed]

STARTING QUESTION: 1. I am a number. Multiply me by 4 and then subtract 3. The result is 45. What was I before? I am a number. Divide me by 1 and subtract -2. The result is 60. What was I ...
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2answers
27 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
32 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
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1answer
19 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 ...
4
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4answers
95 views
+50

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
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1answer
11 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
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1answer
31 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 ...
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0answers
19 views

Runge-Kutta method for solving (2x2) system of odes

I need to implement a 4th order RK-method (Runge-Kutta method) to the very well know non-linear system of odes: predator-prey system or Lotka–Volterra equations. The explicit system is: $$dx/dt = ...
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0answers
13 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 ...
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0answers
26 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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1answer
19 views

Bisection Method: Example of a Special Case [closed]

If $[a_0, b_0], [a_1, b_1],\dots [a_n, b_n],\dots$ is the sequence of intervals in a bisection method, give an example in which $a_0$ = $a_1$ < $a_2$ = $a_3$ < $a_4$ = $a_5$ < $a_6 \dots$
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0answers
25 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: ...
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0answers
25 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
26 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
22 views

modifying plot in matlab

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0answers
29 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 ...
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1answer
23 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
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0answers
455 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 ...
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2answers
32 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
34 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
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1answer
42 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
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2answers
32 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
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1answer
20 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
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0answers
14 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 ...