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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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### 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.
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### 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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### the space $H^{-1/2}(∂T)$ [closed]

what is the space $H^{-1/2}(∂T)$ expressive?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Overall convergence rate from subproblems

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