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Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various field. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

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1answer
28 views

Show that $x_{k+1} = x_k - \dfrac{2f(x_k)}{f'(x_k)}$ converges quadratically

Show that $x_{k+1} = x_k - \dfrac{2f(x_k)}{f'(x_k)}$ converges quadratically. Essentially this is Newton's method but with a $2$ infront. Let $r$ denote the root of $f$. Suppose $f(r) = 0$ and $f'(...
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0answers
6 views

Minimal proof of weak convergence of Euler-Maruyama method for SDEs

Assuming a basic understanding of Ito calculus and numerical methods, is there a relatively straightforward proof that the Euler-Maruyama numerical scheme converges to weak order 1? The papers by ...
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0answers
11 views

Upper and lower bounds are still true after this transformation?

Let $p,r\ge 2$ be fixed integers and $r=\sum_{i=1}^pr_i$ is a sum of positive integers. Assume we have constants $\lambda_i,c_i>0$ satisfying $$\sum_{i=j}^p\lambda_j=\sum_{j=1}^p\lambda_jc_j=1 \...
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0answers
18 views

How to derive 4th Runge-Kutta?

Just to the point, I have a homework that is deriving 4th Runge-Kutta. My teacher doesn't give me any hint except he talked about parameters (or variables maybe) with respect to time $t$ and i'm stuck ...
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1answer
24 views

Matlab Computation and Applied Modelling

Worked out so far; four equations from Newtons law of cooling: T=temp T*= environmental temp. S= salinity S*= environmental Salinity H= salt flux dT1/dt = k(T1-T1*) dT2/dt = -k(T2-T2*) dS1/dt = k(S1-...
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0answers
29 views

When integration of this polynomial is minimum

For which $x_k$ $$\int_a^b \prod_{k=1 }^n(x-x_k)$$ is minimum? It was given in my book that this is minimum at if we pick $x_j $ as roots of Chebyshev polynomials of second kind. But there is no ...
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0answers
34 views

Numerically stable computation of hyperbolic sine for $x \approx 0$

We are given numerically stable functions exp(x) and expm1(x) computing $e^x$ and $e^{-x}$. How compute numerically stably $\sinh(x)=\frac{e^x-e^{-x}}{2}$ for $x \approx 0$ using exp(x) and expm1(x)?
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0answers
7 views

Efficient Methods for Minimax over distributions

I am currently trying to find a method that is somewhat better than brute force to solve the following problem: Suppose there is a player A on a graph, located at node i, and they are trying to evade ...
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1answer
29 views

Newton's Raphson method - Absolute error

Can someone explain to me how the '$e_k$' section is evaluated? I know that the absolute error is $|x - x_n|$ but I can't understand how this values came up.
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2answers
33 views

Why Newton Coates is exact for polynomial of degree at most $n$

Let us suppose $x_1,...,x_n$ be $n$ nodes and we interpolate the functions $f$ with lagrange polynomial Then my book says $$\int_a^b f(x) dx=\sum A_i f(x_i), A_i=\int_a^b l_i(x)dx $$where $ l_i(x_j) ...
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0answers
15 views

Numerical Analysis: Show Lagrange interpolation polynomial of p is equal ot the polynomial p for all x and other Lagrange interpolation questions. [on hold]

Let $p$ be a polynomial of degree $n$ and $L_n$ be the Lagrange interpolation polynomial that interpolates $p$ at $x_0 < x_1 < \cdots < x_n,$ namely $L_n(x_i) = p(x_i),$ $i = 0, \ldots, n.$ ...
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0answers
20 views

Discrete numerical optimization with black-box target function

Let's say I have a function $f(x_1, \dots, x_n)$ of about 50 discrete variables mapping them to a natural number. Some are integers and so are orderable, but most are categorical features and don't ...
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2answers
41 views

Estimate how many iterations would be needed to determine the root to 16 decimal places? [on hold]

Consider the Newton-Raphson iteration scheme $$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ for the function $$ f(x) = \frac{1}{x} - a$$ where $a$ is a constant. Estimate how many iterations would ...
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1answer
35 views

Interval of convergence for $f(x) = x - e^{-x}$

Interval of convergence for $f(x) = x - e^{-x}$. We can take $g(x)=x-f(x)=x-x+e^{-x}=e^{-x}$. So now we have $g(x)=x$ when $x$ is a root of $f$. I now calculate $|g'(x)|\leq k < 1$. $g'(x)=-e^{-...
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1answer
51 views

Does the infinite iteration sequence of Newtons method always converge to a root?

Assume that a sequence $\{x_i\}_{i=0}^\infty$ of Newton's method $x_{i+1}=x_i-\frac{f(x_i)}{f'(x_i)}$ is convergent and set $x_\infty = \lim_{i\to\infty} x_i$. Furthermore assume that $f'(x_\infty) \...
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1answer
33 views

Explore divergence and convergence at any value of parameter $a$ [on hold]

Explore divergence and convergence at any value of parameter $a$ $$\sum_{n = 1}^{\infty} \left | \ln \left(\arctan\frac{1}{n}\right)-\ln \left(\tan\frac{1}{n}\right)\right |^{a}$$
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1answer
21 views

Finding the maximum absolute error

Given, $c=15300 \pm 100$. Then what is the maximum absolute error in $c^3$? My attempt: Let $u = c^3$, then maximum absolute error in $u$ is $\Delta u = \frac{du}{dc}\times \Delta c = 3c^2\times \...
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0answers
29 views

solution on the time domain becomes “periodic” after the inverse fourier transform

I was trying to solve european option pricing problem using Conv method (introduced by Lord in 2008 https://pdfs.semanticscholar.org/0632/460bd50b2151f74ac40028df4cc60e73a884.pdf). The final step of ...
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3answers
42 views

What is the maximum of the fourth derivative?

I am finding the error bound for the interpolation of the function $f(x) = cos(\pi x) + sin (\pi x) $ on the interval $[-1, 1]$. To do this I need the maximum of the absolute value of the fourth ...
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3answers
41 views

Solve the recurrence relation $y_{n+1}-2y_n+y_{n-1}-\frac h2(y_{n+1}-y_{n-1})-2h^2y_n=0$

I am solving the recurrence relation $$ y_{n+1}-2y_n+y_{n-1}-\frac h2(y_{n+1}-y_{n-1})-2h^2y_n=0. $$ In the question, it is suggested that this is related to the DE $y''-y'-2y=0$, but I have no clue ...
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0answers
25 views

Find rate of convergence to f'

This is an exercise with its solution from a course in numerical analysis that I can not understand at all. It's about convergence, Taylor expansion and the remainder theorem. Find at what rate for ...
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1answer
18 views

Extra terms in Fourier series estimate

Is there any reason why adding more and more terms to a Fourier Series representation of a function would cause any wierdness? I was under the impression that, in general, a smooth function can be ...
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1answer
40 views

Euler equation - Assignment Differential Equations and Numerical Methods

I don't know how to approach this question by numerical methods, any help will be appreciated: I need to solve the following differential equation using numerical methods formula I was given (I can'...
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2answers
50 views

Find the values of $\lambda$ for which the iterative method $x_{n+1} = x_n + \lambda (x_n^2 - 3)$ converges to $\sqrt{3}$.

Let $\phi(x) = x + \lambda (x^2 - 3)$, with $\phi'(x) = 1 + 2\lambda x$. As the title says, I want to find the values of $\lambda$ such that the sequence $x_{n+1} = x_n + \lambda (x_n^2 - 3)$ ...
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1answer
22 views

Calculating relative error in subtracting two approximate numbers

My numerical analysis textbook says: Let $u=x_1+x_2$ Then, absolute error in $u$ is $\Delta u= \Delta x_1+ \Delta x_2$ Relative error in $u$ is $\frac{\Delta u}{u}=\frac{\Delta x_1+ \Delta x_2}{x_1+...
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1answer
33 views

i need help in writing a Mathematica code in recursive programming style [on hold]

approximate the square roots x=Sqrt[a],a>0 The formula is given by xn+1=1/2(xn+a/xn)
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1answer
20 views

Mesh a circle with quadrilateral elements

I'm having some problems understanding an exercise. The domain $\Omega$ is the unit circle. Then it says: "To mesh it with quadrilateral elements, compose $\Omega$ from five mapped squares, one of ...
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0answers
11 views

Stability in the Finite Difference Method where we have source $f$

I was studying the finite difference scheme: $$u^{n+1}_j=\frac{u^{n}_{j+1}+u^{n}_{j-1}}{2}-\frac{c\delta t}{2 \delta x}(u^{n}_{j+1}+u^{n}_{j-1})+\delta t f^n_j.$$ for $n\in \mathbb{N}$ and $j \in \...
2
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1answer
52 views

Solve $x_{n+1}=Ax_n+\frac{B}{x_n^5}+\frac{C}{x_n^9}$

For $$x_{n+1}=Ax_n+\frac{B}{x_n^5}+\frac{C}{x_n^9}$$ a. Find $A,B,C$ that will give an optimal approximation (Highest order) for $\sqrt{2}$ b. Solve the system using LU decomposition, what is ...
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0answers
16 views

Showing that g is a contraction in the ∞-norm, by using the p-norm

My question is about, trying to show that the vector function $g = (g_1,\dots,g_n)$ is a contraction in the p-norm with $0 < L < n^{-1/p}$ and $1 \leq p < \infty$. After doing this, show that ...
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0answers
8 views

How to prove this statement concerning Machine Numbers and Primes?

So, let $F=F(b,t,e_{min},e_{max})$ be a range of machine numbers, with b being a Prime Number. Now I have to prove: "Let $x,y$ be two machine numbers ($y\neq0)$ and let $q:=x/y$ be a normalized ...
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1answer
50 views

Richardson extrapolation in number theory

I found a text of Robert Israel about Richardson extrapolation http://www.math.ubc.ca/~israel/m215/rich/rich.html There is the formula $$Q_R=\frac{h_2^pQ(h_1)-h_1^pQ(h_2)}{h_2^p-h_1^p}+\mathcal O(h^...
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0answers
13 views

Estimate for a radial function

Let $$ \phi(x)=(\frac{1}{1+(|x|^2-1)^4})^\frac{\alpha}{8}\text{ if }|x|\geq 1. $$ Then $$ \phi(x)\leq\frac{c}{|x|^{\alpha}} $$ for some constant $C$ when $|x|\geq 1$. Can someone help me proving ...
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0answers
11 views

Precision / accuracy of quadrature formula

Consider $[a,b]\subseteq \mathbb R$. Construct a quadrature formula of maximum precision for $x_0=a, x_1 \in (a,b), x_2=b$. A quadrature formula $Q_n=\sum_{j=0}^n \alpha_jf(x_j)$ has precision $m$ if ...
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0answers
12 views

approximation of boundary conditions for the numerical solution of the three-dimensional heat equation

I am trying to solve a three-dimensional heat equation using an explicit scheme. I have an initial condition: $\ T|_{t=0}=0 $ and boundary conditions: $\ T|_{y=0}=T|_{y=1}=4x^2+sin\Pi z $ and $\...
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0answers
37 views

Numerically find intergral with cdf.

I need to find a value of this integral. $f(\tau) = \int_{b/\sigma}^\infty \dfrac{1}{\sqrt{2\pi}}*\exp(-\dfrac{y^2}{2})*\Phi(-\dfrac{1}{\sqrt{(1-r^2(\tau)}}(\dfrac{b}{\sigma}(\dfrac{\tau}{\tau_0}-1)+r(...
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1answer
36 views

Approximating $\sin(\frac{x}{2})$ by trigonometric polynomials in the uniform norm

Find a trygonometric polynomial of the form $$a_0+a_{1} \sin (x)+ a_2 \cos (x)$$ that best approximates the function $$\sin\left(\frac{x}{2}\right)$$ in the uniform norm on the interval $[-\pi,\pi]$. ...
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0answers
19 views

How to efficiently convert a Newton polynomial to its power form

I have been recently learning about polynomials and ways of presenting them and I have encountered a problem, which is converting a Newton polynomial to the power form in the efficient way. Newton ...
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0answers
14 views

What range of values is acceptable for matrix condition value (cond(A))? [closed]

Some say <1e5, but I encountered 1e32 with the comment 'Standard errors may be unstable'.
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2answers
42 views

How to find the correct iterative function for iterative method

We want to compute the root of $$f(x) = x^3-2x-5$$ Since $f(1.5)<0, f(2.5)>0$ the root must be within the interval $[1.5,2.5]$. 1) An intuitive iterative function would be $\phi(x) = 0.5 \...
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1answer
16 views

Semidiscretization and trapezoidal rule in PDE [closed]

enter image description here enter image description here How to get eq.(3) with spatial semidiscretization ? How to applying trapezoidal rule to get eq.(4)? anyone can explain to me? thank's.
2
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1answer
28 views

Simplifying ArcSine Function [closed]

I was wondering if there is a nice formula (or approximation) for $\arcsin(x)$ which is defined $[-1,1]$?
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0answers
25 views

the space $H^{-1/2}(∂T)$ [closed]

what is the space $H^{-1/2}(∂T)$ expressive?
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1answer
22 views

Divided differences equality

Is there some easy way to see that $$\frac{f[a,\frac{a+b}{2},b,x]-f[a,\frac{a+b}{2},b,\frac{a+b}{2}]}{x-\frac{a+b}{2}}$$ in the 2nd line is equivalent to $$f[a,\frac{a+b}{2},\frac{a+b}{2},b,x]$$ in ...
0
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2answers
28 views

Cubic Converging Functions with Newton

I have been tasked with attempting to find properties with the function f(x) that would make It such that using newton's method would converge to a particular root at least cubically. I don't exactly ...
0
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1answer
31 views

Analyzing convergence of finding roots for $f(x) = x^3 - a$ using Newton's method

Given $f(x) = x^3 - a$, we wish to find $f(x^*)=0$ as a way to calculate the cube root of a number. This can be done using Newton's method. So we have $x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}$. The ...
2
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1answer
30 views

Naive algorithm questions: Fourier Series

I'm having both conceptual and practical problems here. In a big picture sense, I get that a Fourier series is just an estimate of a function on some interval. That function is estimated by summing up ...
0
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0answers
7 views

Overall convergence rate from subproblems

Suppose an algorithm divides the full problem $A$ into two independent subproblems $a$ and $b$ with individual convergence rate: $$\epsilon_a=O(\frac{1}{n_a^2}),\ \epsilon_b=O(\frac{1}{n_b^2}),$$ ...
1
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0answers
15 views

Looking for numerical integration methods that only depends on endpoints.

I am looking for references for numerical methods of integration that only use information about the endpoints of integration (like the trapezoidal rule). I have a very unusual problem where I have a ...
1
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1answer
30 views

Find the order of $f'(x)\approx \frac{1}{12h}[-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)]$

Find the order of approximation of $$f'(x)\approx \frac{1}{12h}[-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)]$$ Use the expression to find approximation for $f''(x)$ $$-f(x+2h)=-(f(x)+2hf'(x)+\frac{4h^2}{...