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

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2answers
77 views

Bound for $\left|\sin(x) +\cos(x)\right|$

I'm taking a numerical analysis class and i'm needing to bound $\left|\sin(x) + \cos(x)\right|$ quite often. So far i've been putting that this is always $\leq |1 + 1| = 2$. Is this the minimal bound? ...
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0answers
22 views

A sequence of intervals- Trying to find a fixed point -

This might be a trivial question but I couldn't come up with a clever trick,theorems or whatnot. Suppose $I_0=\left[\frac{1}{h_0},\frac{1}{l_0}\right]$ where $h_0=1$ and $l_0=\frac{1}{2}$. Given ...
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0answers
51 views

Alternative to the Gram-Schmidt Procedure for Orthogonalization

I was wondering if there is an alternative to the Gram-Schmidt procedure, which instead of being a successive orthogonalization scheme, would be non-successive (simultaneous)? In other words, is there ...
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2answers
47 views

Solving an algebraic equation for x

$(($ 3^$2\sqrt{3x})$/4$)$ $+3=$ 3^$\sqrt{3x}$ = $($ (3^${2}*{3x^{1/2}}$)/4$)$ $+3=$ 3^${3x^{1/2}}$ After simplifying: = ($3^{6x^2}$ $+ 3$)/4 $= 3^{3x}$ = $3^{6x} + 3 = 12^{3x}$ I tried ...
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3answers
76 views

root of the function $f(x)=\sqrt{2}-x$ by using fixed point iteration

How can I find the approximate value of $\sqrt{2}$ by using the fixed point iteration? I have tried $x-\sqrt{2}=0$, $x^2=2$, $x^2-2+x=x$, $g(x)=x^2-2+x$, $g\prime(x)=2x+1$ And i choose ...
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1answer
37 views

Numerical root finding for 5th degree polyomial

I have the equation $y^5 -ay -b=0$. I need to get a solution whether numerical or analytical. I heard $5$th order polynomials are not solvable analytically, so how can I get the root numerically. ...
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0answers
14 views

Definition of local truncation error

In the first reference of Wikipedia the local truncation error defined as $$ \tau_n = y(t_n)-y(t_{n-1})-hA(t_{n-1},y(t_{n-1}),h,f) $$ But in the second reference mentioned that $$ \frac{\tau_n}{h} ...
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1answer
23 views

Shared groceries expenses between roommates to be divided as per specific consumption ratio and attendance

My apologies if this question is in the wrong section. Couple of my roommates & I (total 5 people) share the groceries expenses. We record the purchases in an Excel sheet, and also have the ratio ...
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0answers
10 views

quadrature trapezoidal rule to solving integral

I want to solve an integral. when I used $n=50$ for quadrature trapezoidal rule, my answer is better than $n=100$. Is it true, or you think I have a mistake in my program?
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0answers
35 views

Local truncation error of Euler method

Wikipedia and this book say the local truncation error of Euler method is $O(h^2)$. But this book and A friendly Introduction to Numerical Analysis say it's $O(h)$. Which is correct? I have a similar ...
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1answer
50 views

Positivity of a stochastic process

I want to simulate the paths of a stochastic process $$ dS_t = r S_t dt + \sigma S_t dW_t$$ Using the Forward Euler method, we can write: $$ S_{n+1} = (1 + r \Delta t_n + \sigma \Delta W_{n}) S_n $$ ...
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0answers
11 views

Approximation of the ito SDE using backward Euler approximation

I have the stochastic SDE $ dX_{t}=a X_{t} dt+ b X_{t} dW_{t}$ I succeeded to formulate a forward Euler approximation to approximate it but I have some problems to derive the right backward Euler ...
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1answer
30 views

Use of taylor series in convergence

Homework problem here, would appreciate an explanation to the answer of this question. Problem: Find the rate of convergence of $$ \lim\limits_{h \to 0} \frac{\sin(h)}{h} = 0 $$ The book solves ...
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1answer
32 views

Question about “Magnifying” an inequality for big-oh analysis

Here's an example directly from my numerical analysis text book: $$ a_n = \frac {n+1}{n^2} $$ The goal is to find the convergence rate. So we know, $\lim\limits_{n \to \infty} \frac{n+1}{n^2} = 0$ ...
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1answer
28 views

Numerical integration of improper integral of the second kind

Peace be upon you, I had a simple and common question; but surprised after seeing no related results in search engine! I would like to know the numerical integration techniques for improper Riemann ...
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1answer
32 views

Prove that a function is in “Big-O”

Prove that $f(h)=h^2+5h^{17}$ is in $O(h^2)$. I don't understand this problem. Big O notation continues to befuddle me. I think that what I need to show is that there exists a constant $C$ such that ...
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0answers
38 views

Order of implicit trapezoidal (Heun's) method

What is the order of local truncation error (LTE) of implicit trapezoidal method? Is it $O(h^2)$ or $O(h^3)$? By integrating the $y^\prime(x)=f(x,y(x)),\, x \in [a,b]$, from $x_i$ to $x_{i+1}$ we can ...
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0answers
12 views

Stability of a system of ode

I have the ordinary differential equation $dX_{t}=A X_{t} dt $ where where $ X_{t} \in \mathcal{R}^{2}$ and the matrix A has two real eigenvalues $\lambda_{1} = 1 , \lambda_{2} = -10^{5}$. The ...
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0answers
17 views

How to find a separating hyperplane?

I know about support vector machine, and it's quadratic programming approach which delivers the best separating hyperplane. My question is: is there a relatively simple algorithm to find a ...
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0answers
18 views

Adam's Moulton 1 step method

This is part of a question I'm stuck on, http://i.imgur.com/o6VAxmE.png?1 I've already done the top bit which is to derive the equation. And for the second I've tried to sub stuff in but I don't know ...
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0answers
21 views

Halley's method question

Can anyone help me with this http://i.imgur.com/GuA1CGo.png?1 please, so far I've done a lot of rearranging but not really got anywhere. I've somehow managed to get rid of any Xs to the power of p but ...
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1answer
53 views

To find the inverse of an implicit function

I have a function $t(f)$ here: $t(f) = T(sin(2\pi f/B)/2\pi + f/B) $ for $[-B/2 \le f \le B/2]$. $B$ and $T$ are constants. How to find the inverse of this function that is $f(t)$ using numerical ...
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1answer
56 views

What is the rate of convergence of $\lim_{h\rightarrow 0}\dfrac{\sin(h)}{h}=1$?

I am having trouble understanding the concept of finding the rate of convergence. My homework question is a more beefed-up version of the following: Find the rate of convergence for ...
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0answers
8 views

Numerically doing trapezoid rule and richardson extrapolation using haskell.

so im trying to numerically to solve some integrals using trapezoid and richardson methods. Ive implemented the code as followed in haskell. ...
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0answers
58 views

Solving ${c_1}^x+\sqrt{\frac{\log(x)x}{2}}+3\log(x)x \le c_2$

Is there any way to solve $${c_1}^x+\sqrt{\frac{\log(x)x}{2}}+3\log(x)x\le c_2,$$ for $x>1$, $0<c_1<1$, and $0<c_2<<1$? Thanks
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0answers
5 views

The influence of time interval in the Newmark-beta method on the results

I have tried to solve an ODE using the Newmark-beta method. The results changed a lot when the time step changed and the results well exceeded the expected value. How these results came into being? ...
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1answer
36 views

The order of convergence and the asymptotic error constant of the sequence $p_n=g(p_{n-1})$

Let $g(x)=0.5(x+a/x)$. Determine the order of convergence and the asymptotic error constant of the sequence $p_n=g(p_{n-1})$ toward $x=a^{.5}$. This is a problem in our homework in the class ...
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0answers
17 views

Show that A is positive-definite matrix if and only if ${x^{(k)}}$ converge

Let A be invertible Hermitian matrix and $b \in \mathbb{C}$ Also N is invertible matrix such that $N+N^*-A$ is positive definite. In order to solve $Ax=b$ consider the iteration ...
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1answer
42 views

Finding the fixed point of $\exp(-x^2)$ numerically [closed]

Consider the function $g(x)=e^{-x^2}$. With a starting approximation of $p_0=0$, use the iteration scheme $p_n=e^{-p_{n-1}^2}$ to approximate the fixed point on $[0,1]$ to within $5 \times ...
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1answer
39 views

Proving that $\exp(-x^2)$ has a unique fixed point on the interval $[0,1]$

Consider the function $g(x)=e^{-x^2}$. Prove that g has a unique fixed point on the interval [0,1]. So, our teacher did not go over this section, but assigned it for homework and I have no idea ...
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0answers
14 views

How to simulate Coupled Brusselator System

How would I go about simulating the attached system? I've tried using a simple forward Finite Difference method but it seems like it's unstable. Any help or suggestions?
2
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1answer
24 views

Sum of Lagrange basis polynomials

Let $L_i(x)$ be Lagrange basis polynomials for $n+1$ points $(x_0,y_0),\ldots, (x_n,y_n)$. How do you prove that $\sum_{i=0}^n (x-x_i)^pL_i(x)=0$ for $p\leq n$?
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0answers
21 views

Solving a non-linear set of equations with non-exact constant values

I have a set of nonlinear equations which are related to a physical system. The constants of these equations (light hand side values-LHS) are determined through some measurement methods. It means that ...
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0answers
20 views

Multivariate Appoximation

I have a mathematical model for a complex system which I would like to approximate it. My idea is to run this complex model once and produce some outputs, and then fit a polynomial for these outputs. ...
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1answer
24 views

Find the roots of a polynomial in Matlab

I have a polynomial $f$ of order 15 and I want to find its roots. For solve(f==0), the answer is ...
2
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1answer
58 views

Fixed-point iteration for $x=\tan(x)$

I've been trying to find an accurate $g(x)$ in order to find a solution for $x=\tan(x)$ in the interval $[4,5]$. However, no matter what, all of them end up converging to zero which is not the answer. ...
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2answers
43 views

Sampling from the diamond: $|x_1|+\ldots+|x_n| \le 1$?

Let $\left(x_1, \ldots, x_n \right)$ be a point in $\mathbb R^n$. Sample uniformly at random from the diamond $$ |x_1|+\ldots+|x_n| \le 1. $$ In $\mathbb R^2$, one way is to sample the square, then ...
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0answers
13 views

How to express the quantity using significant figures to imply the stated error?

Express the quantity using significant figures to imply the stated error. $$1.77 \pm 0.06$$ I tried to factor out a $6$ and got $6(.295\pm0.01)$ but then when I try to make $.295$ have an error of ...
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1answer
18 views

Ordinary Differential Equations divergence of successive approximations

I searched this and couldn't find my question on here so here it goes: This is an example from a text I'm reading and I was hoping someone could shed some light on my misunderstanding. Let ...
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1answer
21 views

difference equation-numerical analysis

Assume $E$ is the difference equation. We have that $E^3(E-2)=0$ The solution should be $x=(a_1\ a_2\ a_3\ 2a_4\ 4a_4 … 2^n a_4…)$ But what is the basis of the solution space? I think it should be ...
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1answer
14 views

min/max in front of a sigma sign (Numerical Analysis)

I have an assignment in my Numerical Analysis class that involves an approximation of an integral with a sum. The sum looks to me like a Riemann sum, but I don't know what the "min" in front of the ...
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1answer
37 views

Numerical evaluation of Hurwitz zeta function

Is there a way to evaluate numerically the Hurwitz zeta function $$\zeta(s,a) = \sum_{n=0}^\infty \frac{1}{(n+a)^{s}}$$ that is more efficient (i.e., quick and precise) than simply explicitly adding ...
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4answers
149 views

How can I accurately compute $\sqrt{x + 2} −\sqrt{x}$ when $x$ is large?

How can the values of the function $f(x) = \sqrt{x + 2} −\sqrt{x}$ be computed accurately when $x$ is large? I have tried using Matllab. I am not able to understand when $x$ will be large.
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0answers
18 views

“Big-O” notation with a Taylor Series Expansion

Use a Taylor's expansion to rid the expression $1-\cos x$ of subtractive cancellation for $x$ small. Use a $\mathcal{O}(x^5)$ approximate. I understand Taylor series and I know that the expansion of ...
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0answers
25 views

stability of FTCS scheme for parabolic equation

Can you suggest any method for stability analysis of FTCS scheme for the the following parabolic equation ? D.E: $u_{t}=a(x,t)u_{xx}+f(x,t,u)$, $0<x<1$, $0<t<T$, $T>0$ BCs: ...
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0answers
51 views

Proof of theorem about iterative methods

How do I prove that if $A$ is a tridiagonal (or block tridiagonal) matrix then the corresponding $P_J$ and $P_G$ iteration matrices for the Jacobi and Gauss-Seidel methods satisfy that if $\lambda$ is ...
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1answer
31 views

Bounding volume of catmull-rom splines

I need to compute a 2D "spherical" bounding volume for the part of a catmull-rom spline $S(t)$ with four control points $P1$, $P2$, $P3$ and $P4$ in the domain $0 \le t \le1$. The purpose is to reduce ...
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0answers
27 views

The condition number of a function

Compute the condition number of $f(x_1, x_2)=x_1 + x_2$ I really don't see where is my mistake. Here is my solution: The condition number of a function $f$ is given by $\frac{\left\Vert x ...
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1answer
48 views

Solve system of first order ODEs when values are a matrix

I have a set of first order ODEs such that $$ \frac{d \Psi}{ds} = H(s)\Psi$$ H(s) is an 8x8 matrix. I would like to find $\Psi(b)$ where $b$ is 10, for instance. The initial value for $\Psi(a)$ is ...
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1answer
55 views

Find the coefficients in quadrature formula on $[0,1]$ with the nodes at $1/4$, $1/2$, $3/4$

In my worksheet I was given a question about numerical integration that says: Find the formula for $\int_{0}^{1}f(x)dx=A_{0}f(\frac{1}{4})+A_1f(\frac{1}{2})+A_2f(\frac{3}{4})$ I suppose the goal ...