Tagged Questions

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

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Discretizing Nonlinear Shallow Water Equations: Question on Flux Term

It's been suggested to me that the term $(hu)_x$ can be discretized as $\frac{(h(i+1,j)u(i+1,j)-h(i-1,j)u(i-1,j))}{2\Delta x}$. I don't see why this is written in this way. It's taking the derivative ...
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Growth rate of a function

I am having some trouble determining the growth rate of the function $m(n)=\inf\{m: \frac{1}{2^m}\le \frac{1}{n} m^{3/2}\}$. This comes up in problem 2.2.8 in Durrett's probability book. Essentially, ...
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How to integrate discrete data by Gaussian quadrature method

I'm trying to numerically integrate discrete data by Gaussian quadrature method. The file attached test.mat is a discrete data set taken from a finite-element mode ...
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A $C^1$-function, s.t. approximation by the Trapezoidal rule is more accurate than by Simpson's rule?

Find values $a, b \in \mathbb{R}$ and a function $f \in C^{1}[a,b]$, such that the approximation of $\int_{a}^{b} f(x)dx$ by the Trapezoidal rule $T(f)$ is better than the approximation by the Simpson ...
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Proving a contraction mapping is a Cauchy sequence

Let $\phi(x):[a,b]\rightarrow [a,b]$ be a continuous function. Show that if $\phi(x)$ is a contraction mapping on $[a,b]$ then the sequence $\{x^{(k)}\}$ defined by $x^{(k+1)} = \phi(x^{(k)})$ is a ...
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What is the necessary condition for ODE to have unique solution?

For the ODE: \begin{align} \dot{x}(t)&=f(x,t) \\ x(t_{0})&=x_{0} \end{align} If $\;\;f:\mathbb{R}^{n}\rightarrow{}\mathbb{R}^{n}$ is Lipschitz continuous on $\mathbb{R}^{n}$, then there exists ...
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Minimum error in floating point approximation of an elementary function.

I need a confirmation of a thing that probably is silly. Let $x$ a floating point number representable using $e$ bits for exponent and $m$ bits for mantissa, let $f$ a be an elementary function, you ...
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Uniqueness of interpolation polynomial.

I am new to numerical analysis and this is the first thing I came across. It says on my textbook that interpolation polynomials are unique and to prove that it was assumed that let there be two such ...
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Floating point numbers

In a certain computer represents numbers in base2, if the distance between 7 and the next largest floating-point number is $2^{-12}$. What is the distance between 70 and the next largest floating ...
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More precise trail function in Rayleighâ€“Ritz method

In order to obtain displacement field of an elasticity problem, say a plate structure, we approximate the solution using trigonometric series with unknown coefficients which satisfy the essential ...
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Approximating Geometric Brownian Motion numerically

I am trying to generate a numerical solution to the SDE for Geometric Brownian Motion. The stochastic process is given by $S_t = \exp(\sigma W_t + \mu t)$, and by Ito's lemma, we have that the SDE is ...
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Find parameters and node so that quadrature approximating integration will have maximum order

Find parameters $\alpha, \beta$ and node $c$ so that quadrature $Q(f) = \alpha f(a) + \beta f(b)$ approximating integration $\int_{a}^{b}f(x)dx$ will have maximum order. I don't know how to solve ...