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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3answers
46 views

How many numbers can a typical computer represent?

I couldn't find this elsewhere so I thought I'd give it a try to figure out exactly how many numbers a typical desktop computer can represent in memory. I'm thinking about this in the context of ...
0
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0answers
10 views

Calculate the order of error for the (summed) Midpoint rule?

I'm reading a comparison of the summed rectangle and and midpoint rules for estimating the value of an integral. The midpoint rule: $\displaystyle\int_a^b \! f(t) \, \mathrm{d}t \simeq f(a + h/2)h$ ...
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0answers
19 views

Compute the experimental order of convergence

Compute the experimental order of convergence for a root finder with errors in 3 consecutive iterations of $10^4 , 10^7 ~\text{and}~ 10^{14} $ I'm having trouble understanding can someone give me a ...
0
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0answers
27 views

Applicantions of Newtons Method for $f(x) = \dfrac{e^x}{x^2+1}$

I´m study some applications of calculus and see that question: If $f$ and $g$ are real functions differentiable such that $g'(x)\neq 0$ for all $x$. $a)$ Show that $f(x)$ and $g(f(x))$ has the same ...
2
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0answers
20 views

Boundary Conditions for a Finite Difference Approximation of a Sixth Derivative

I am trying to use a finite difference scheme to numerically solve sixth order parabolic equations such as \begin{equation} u_t = u_{xxxxxx} \end{equation} with symmetry conditions \begin{equation} ...
1
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0answers
20 views

What is the difference between perturbation theory and numerical analysis?

What is the difference between perturbation theory and numerical analysis? Both subjects are trying to obtain the approximate answer. What are they study specifically?
1
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0answers
19 views

System of PDE's with unknown functions

So by messing around with some stuff in my own research I came across this problem and I have no idea how to proceed. I suspect it may have something to do with solving systems of PDE's but I could be ...
0
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0answers
15 views

Newton Divided Difference Table , if $f''(x)$ is given

Here is the problem: Suppose $f(0)=1$, $f(1)=f'(1)=f"(1)=0$, and $f(3)=16$. Compute the Hermite interpolation polynomial $P$. How would the divided difference table look for this, in order to ...
1
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0answers
28 views

How to efficiently solve a series of similar matrix equations using the LU decomposition

This is the problem I'm dealing with: Let $\sigma_1,\dots,\sigma_n \in \mathbb{R}$ and $b_1,\dots,b_n$ be column vectors of length $n$. Consider the system $$ (A - \sigma_jI)x_j = b_j, \quad ...
3
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2answers
30 views

Existence and uniqueness of weights for the rule $\int_a^b f(x) \ = \ \sum_{0 \leq k \leq n} w_k f(x_k)$

I want to establish this statement: If $a<b$ and $\{x_0,x_1, \cdots x_n\} \subset \mathbb{R}$ distinct, then there is one and only one set of weights $\{w_0, \cdots w_n \} $ such that $\int_a^b ...
0
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0answers
10 views

Differential Algebraic eqn using Adomian Decomposition Method [on hold]

Refer to research paper http://www.gbspublisher.com/ijpamsv3/ijpamsv3n1_10.pdf, in example 1 i am unable to understand how author has calculated u1,0 =14xsinx+sinx-xcosx. Can somebody plz ...
2
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1answer
50 views

Derivation of power method

POWER METHOD Let $x_0$ be an initial approximation to the eigenvector. For $k=1,2,3,\ldots$ do Compute $x_k=Ax_{k-1}$, Normalize $x_k=x_k/\|x_k\|_\infty$. Then $\|x_k\|_\infty$ approaches the ...
0
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1answer
33 views

Prove $\frac{dy}{dx}$ is approximated by $\frac{y(x+h)-y(x-h)}{2h}$ to $O(h^2)$

I tried to solve it by truncating the Taylor series expansions for $y(x+h)$ and $y(x-h)$ but I couldn't find a way to relate it to the derivative. I wasn't sure where the appropriate place to truncate ...
0
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0answers
29 views

Finding the proper g(x) for fixed point iteration on $2\sin{\pi x} + x = 0$

After spending over an hour trying to get this problem I realize my trig is weak. I found: Fixed point iteration .Numerical method. The selected solution is informative, but lacking detail to really ...
1
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0answers
29 views

Higher-order difference quotients

The Mean Value Theorem for Divided Differences says that if $f$ is $n$ times differentiable, and $x_0< x_1 < \dotsb < x_n$, then there is a point $\xi\in (x_0, x_n)$ such that $f[x_0, x_1, ...
0
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0answers
8 views

What's the best way to recognize a shape o a function with N-points

I've many shapes with points in theirs countours, how is the best way to recognize a shape? I think the DTF is available but i don't know whether this is the optimal way. P.S. I think if i will ...
-3
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0answers
26 views

What method I have to use (INTERPOLATION) and how please! [closed]

I need help with this interpolation method, please is a homework for tomorrow.. !Question1
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0answers
14 views

lagrange interpolation of a point

Let $f(x)=\sqrt[]{x}$ be our function. Let $P_n$ be the lagrange interpolation polynom of $f$ by $n$ points and $a$ be an element in the domain of $f$. Can rational points be chosen such that ...
-2
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0answers
28 views

Divided Differences for a function in $C_n[a , b]$ [on hold]

Let $f$ be in $C_n[a , b]$. Prove that if $x_0$ is in $(a , b)$ and if $x_1, x_2,\ldots , x_n$ all converge to $x_0$, then $f[x_0, x_1, \ldots, x_n]$ will converge to $f^{(n)}(x_0)/n!$. Please ...
1
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0answers
15 views

Solving many independent non-linear systems simultaneously

I'm working on solving lots of systems of nonlinear equations. Luckily, the non-linear equation is the same, but the parameters are different: $$ f(\vec{x}_0; c_0) = 0\\ f(\vec{x}_1; c_1) = 0\\ ...
0
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0answers
33 views

Calculators using Taylor polynomials?

I've always heard that calculators (TI-84's and the like) use Taylor polynomials to approximate trigonometric/exponential/etc functions. Do any of you know this for a fact?
1
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1answer
23 views

For what values of c will the iteration $x_{n+1} = g_c(x_n)$ converge to $\alpha_c$

Consider the equaction $x=g_c(x)\equiv cx(1-x)$, with c a nonzero constant. This equation has two solutions, and we let $\alpha _c $ denote the nonzero solution. What is $\alpha _c$?For what values of ...
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0answers
9 views

Below is the definition of A-Stable. However I am not sure how to apply the definition to show that the Backward Euler method is A-Stable.

We define a region $R$ of absolute stability for a one-step method as the region in the complex plane satisfying: $$R = {hκ ∈ C, |Q(hκ)| < 1}.$$ If $R$ contains the entire left half plane, the ...
2
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4answers
45 views

What’s the difference between Analytical and Numerical approaches to problems?

I don’t have much (good) math education beyond some basic university level calculus. What does “Analytical” and “Numerical” mean and how are they different?
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1answer
24 views

Alternate expression for finite summation

"How many arithmetic operations are required to directly compute $$y=1+x+x^2+...+x^{1023}$$ Use a formula for the sum to come up with an alternate expression for $y$, and show that only 10 ...
1
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0answers
23 views

Discrete Poincare Inequality

For the sake of this question, let $\Omega \subset \mathbb{R^2}$ be a regular domain. In variational problems involving the Sobolev space $W^{1,1}(\Omega)$ (or $BV(\Omega)$) one often uses the Sobolev ...
0
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0answers
23 views

Taylor approximation of $e^x$ on $[-1,1]$ [closed]

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 $e^x$. I'm completely stuck on this, any ...
0
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2answers
38 views

Elementary Numerical Analysis2

Just wanted to confirm my answer here again: Q: Evaluate $$p(x)=1-\frac{x^3}{3!}+\frac{x^6}{6!}-\frac{x^9}{9!}+\frac{x^{12}}{12!}-\frac{x^{15}}{15!}$$ as efficiently as possible. How many ...
0
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1answer
24 views

Prove error bound using Taylor's series Error term (Bound doesn't seem to make sense)

I have to prove that at least seven terms must be used in the Taylor series estimation of x - sin(x) in order for the error to be <= $10^{-9}$. This doesn't seem correct however. This series is ...
-1
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1answer
29 views

Polynomial Interpolation - Prove that if g interpolates the function f and if h interpolates f then the function interpolated f [on hold]

Prove that if g interpolates the function f at x0, x1, …., xn-1 and if h interpolates f at x1, x2, …., xn, then the function g(x) + (x0 - x)/(xn – x0) [g(x) – h(x)] interpolates f at x0, x1, …., ...
0
votes
1answer
25 views

I do not know the point at which this Taylor series was derived, can someone explain please?

I am required to derive Euler's method through Taylor's Theorem. I have been given the Taylor series for $y(t)$ as shown below. However I do not understand what point the Taylor series was derived. ...
0
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0answers
22 views

Is my step for showing euler method correct?

A step size by $\ \gamma$ follow by step size$\ \beta$ same as step size of$\ \beta$ follow by$\ \gamma$ using euler method Given $\ f'(t)=f(t,f(t)) $ $\ f(t+\gamma t)=f(t)+\gamma tf $ $\ ...
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0answers
14 views

Finding the range of $x$ in ${c_1}^x+\sqrt{\frac{\log(c_2)x}{c_3}}+\frac{\log(c_2)x}{c_4}\le c_5,$

Is there any way to find the range of $x$ that satisfies the following inequality: $${c_1}^x+\sqrt{\frac{\log(c_2)x}{c_3}}+\frac{\log(c_2)x}{c_4}\le c_5,$$ for $x>1$, $0<c_1<1$, ...
0
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0answers
13 views

Stuck on condition number derivation of the perturbed equation $(A + \Delta)\tilde{x} = b + \delta_b$

I've almost got what I want. We start with $Ax = b $ and $(A + \Delta)\tilde{x} = b + \delta_b$. What I have then is \begin{align*} \tilde{x} - x &= -A^{-1}\Delta\tilde{x} + A^{-1}\delta_b \\ ...
1
vote
1answer
20 views

What is the easiest way to check whether the function is Globally Lipschitz continous or Locally?

What is the easiest way to check whether the function is Globally Lipschitz continuous or Locally? Say,for example the function, $sin(x^2)$? Many thanks in advance.
1
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1answer
35 views

show the function has exactly one root

Given: Function: $f(x)=x^2-2x-3$ $[1,4]$ Question: Show that the function has exactly one root in $(1,4)$ My Answer: The function $f(x) = x^2-2x-3$ has one root in [1,4] $f(1)=1^2-2(1)-3=1-2-3=-4$ ...
1
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0answers
31 views

Explicit solution to a nonlinear equation possible here?

I am looking for a solution in $s$ to $$ \lambda -\frac{1}{s} +K e^t \log(\delta) \delta^s = 0 $$ Mathematica is not best pleased with this equation. If the equation were $$ 0- \frac{1}{s} +K e^t ...
2
votes
0answers
38 views

Moment Generating Function of a Beta random variable.

After getting some excellent help on this problem in the statistics SE, I am reformuluating my question. Let me know if I should just delete it and ask a new one. Let $V$ be a $Beta(\alpha,1)$ ...
2
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1answer
45 views

How to simplify this complex integral? [closed]

How to approximate this integral as a function of a and b? $$\int_0^\pi\int_0^{2\pi}\sqrt{(a-b\sin\varphi\cos\theta)^2+(b\cos\varphi)^2+(b\sin\varphi\sin\theta)^2}d\theta d\varphi$$ where a and b ...
0
votes
1answer
23 views

Comparing the order of convergence $\mathcal{O}( h^2 |\log(h)|)$

I don't have any intuition in judging how fast a term of the order $\mathcal{O}( h^2 |\log(h)|)$ is decreasing as $h \to 0$, so i tried comparing it with terms of the form $\mathcal{O}( h^\alpha )$ ...
0
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1answer
5 views

How can I determine the base of the following numbers for the operations to be correct?

Given: 24)A + 17)A = 40)A How can I find the base of the following number (A) so the operations are correct? NOTE: I am not sure what topic this would fall under. Hence sorry for any misplaced ...
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0answers
21 views

Numerical analysis Taylor 1/(1-x)

How does accuracy depend on the degree of the Taylor Polynomial and the distance from the point its being expanded about (say $x=0$). So I'm considering the function $f(x) =\frac{1}{1-x}$ centered at ...
2
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0answers
81 views
+50

numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$ In this linear PDE: \begin{cases} B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...
0
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0answers
5 views

approximation of a complex valued real rational function

all. I am now struggling with a approximation problem. Suppose we have a matrix-valued measure $\mathrm{d}\Lambda(\omega)$, with compact support $[a,b]$, then its Cauchy transform is a well-defined ...
0
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0answers
34 views

Taylor Polynomial Accuracy [closed]

How does accuracy depend on the degree of the Taylor Polynomial and the distance from the point its being expanded about (say $x=0$)? I'm considering the function $1/(1-x)$ centered at $0$. I have ...
0
votes
3answers
31 views

Show a continous function is bounded on a closed interval

For a homework problem, I need to show a function $\pi + 0.5\sin(\frac{x}{2})$ is bounded on the interval $[0,2\pi]$. I'm having trouble conceptualizing a good way to do this though. Can anyone help? ...
0
votes
1answer
26 views

Tridiagonalize matrices with Householder transformation

I know that it is possible to tridiagonalize symmetric matrices by using a Householder trafo. I also found that we can get any matrix to Hessenberg form by using Householder trafos, but I still don't ...
0
votes
1answer
27 views

Graphical estimate of convergence rates?

I am studying some numerical optimization methods, but I am not an expert in numerical maths. Question: If the convergence rate is linear, then the logarithm log(x_n) of the distance x_n to the ...
0
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0answers
13 views

If the coefficient matrix of a system of DEQs is defective , does this have consequences for its numerical solvability

I have a DEQ system $\dot{x}=Ax$ where A is defective. What happens if I naively plug A into a numerical scheme e.g. explicit Euler? What happens if A is time dependant and changes between defective ...
1
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0answers
15 views

Is there a numerical method to restore symmetrizability and row-stochasticity in a matrix?

I have a radiation-transfer matrix F that had been computed using some raytracing Monte Carlo method. From physics it is clear that it must be row-stochastic :$\sum_{j=1}^{N}F_{ij}=1$ because energy ...