# 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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### Approximating integral of Erf with certain available functions.

I am developing certain software that deals with symmetric 2D Gaussian densities. One of the most common operations in that software is integrating those Gaussians over various 2D shapes. These ...
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### Why settle for Lagrange Interpolation when doing linear multistep ODE integration?

Say that we have some initial value problem: $y'(t) = f(t,y(t)) ; y(0) = y_0$ with $y_0$ and $f(t,y(t))$ known. If we use Euler's method to numerically approximate the first k points, then we have ...
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### Help understanding a homework problem (Preconditioning matrices, numerical methods)

Below is a link to the problem (because I didn't want to have to go through the pain of TeXing it all out myself), the basic idea is we are supposed to be first showing that a specific matrix has a ...
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### Solving traveling wave usin the shooting method

The spatially-dependent Hodgkin-Huxley equation for a cylindrical dendrite or unmyelinated axon: where $\frac{a}{2\rho}\frac{\partial^2V}{\partial x^2}$ is a diffusion term $a$ is the fiber radius, ...
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### Solving equations like $xe^x = c$ via functional iteration

Yesterday I randomly thought of solving $xe^x = c$ via functional iteration (FI) after manipulating the equation into a form "$x = \cdots$" that gives the 'fastest' convergence rate regardless of the ...
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### Comparing smoothness among approximations

We are interpolating a missing fragment of a 2D curve given a set of sample points. Our method generates several candidates of curve pieces to fill the missing part, but we want to select the solution ...
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### How many Gauss points are required to provide exact value for the Gauss quadrature rule

How many Gauss points are required if the Gauss quadrature rule should provide the exact value of the integral $I=\int_{-1}^1f(x)dx$ for $f(x)=(x^2-1)^2$? I am really not sure what theorem to use to ...
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### How to deal with the composite function in a numerical approximation problem?

Consider a quasilinear two-point boundary value problem: $$-(a(u)u'(x))' = f(x) , x\in (0,1)$$ with $a(u)>0$ and $u(0) = 0, u(1) = 0$. I am supposed to derive an algebraic system so that I can ...
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### Show that the matrix $AA^T+\alpha I$ is positive definite, where $\alpha >0$ and $A$ is an $m\times n$ real matrix.

Show that the matrix $AA^T+\alpha I$ is positive definite, where $\alpha >0$ and $A$ is an $m\times n$ real matrix. So I need to show that $x^T(AA^T+\alpha I)x>0$ for all vectors $x$. I'm ...
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### Simpson's rule is not producing better results than Riemann sums

I have to calculate RMS value $\sqrt {\int_0^T\frac 1T*f(t)^2dt}$ and I know from the maths that the Simpson's rule should provide better approximation of the definite integral than the Riemann sums. ...
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### Some trivial but confusing terms about numerical integration

Some terminological questions about numerical integration: When a question states trapezoidal rule with 2 points, does that mean 2 subintervals or 3 subintervals? Since 3 subintervals have 2 points ...
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### Proving error bound on Simpson's Rule, Numerical Integration

The approximation from "Simpson's Rule" for $\int_a^b f(x)\, dx$ is, S_{[a,b]}f = \bigg[\frac{2}{3}f\Big(\frac{a+b}{2}\Big) + \frac{1}{3}\Big(\frac{f(a) + f(b)}{2}\Big)\bigg](b-a). \...
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### Fourier series for absolute value of sin functiom

If we take the absolute value for sin function, then it becomes even. However, isn't period of this function pi? To find fourier series, 1.Even 2. period 2 pi. Can we just treat this function as ...
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### Transforming an integral to a different domain

For a given $v(x)$ with $x\in[0,1]$, use the variable transformation $x=g(\eta)=\frac{1}{2}\eta+\frac{1}{2}$ to transform the integral $I=\int_0^1v(x)dx$ to an integral over $[-1,1]$. My doubts: ...
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### Numerically Solve a Second Order ODE with singular coefficients

I need to solve the following numerically: $$xy''+y'+xy=x$$ with initial conditions $y(0)=0$ and $y'(0)=1$. I need the solution for $x:[0, 10]$. I've written the ode as a system of first order odes ...
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### Numerical stability of computational results

Let z be a function of a finite number of variables i.e. z=f(a,b,c,...). If we have the mathematical formula connecting z and the variables, we can determine how the value of z varies with a change ...
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### Forward difference problem

How to compute $\Delta^{2}(cosx)$ ? I try using relation $\Delta =E-1$ where $E$ is shifting operator. Please need help.
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### Number of iterations for Gauss-Seidel

I am having some difficulty understanding the following solved problem: Question: Shouldn't we have $||T||^k_{\infty} ||e^{0}||_{\infty} \leq 10^{-6}$ instead? Where does the $5$ come from? And ...
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### Evaluating integral with a singularity.

I want to evaluate an integral numerically that contains one singularity. The software I use for this is Python. The actual integral I want to evaluate is quite long with a lot of other constants so I ...
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### Avoiding loss of significance without series.

How could the function $$f(x)=\frac{\sin x}{(x^2+1)^{1/2}-1}$$ be computed to avoid loss of significance? I know that $$f(x)=\frac{\sin x((x^2+1)^{1/2}+1)}{x^2}$$ But $x^2$ has a problem.... How to ...
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### Why is it that if a numeric method has quadratic rate of convergence then it can reach d digits of precision in logd iterations?

I was trying to understand why a method with quadratic convergence can get close to a good solution in $\log d$ iterations. Assume we have a method that has the property that the number of digits of ...
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### Numerical method with convergence greater than 2

It is a well-known fact that, for solving algebraic equations, the bisection method has a linear rate of convergence, the secant method has a rate of convergence equal to 1.62 (approx.) and the Newton-...
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### Weight Function in gaussian quadrature

My question is pretty simple, although I know of the properties that the weight function must follow , such as being well defined,positive,continuos and integrable on the interval . I do not know how ...
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### Method check: numerically calculate 1D integral of a 3D function

I have a function $f(r)$ where $r=\sqrt{x^{2}+y^{2}+z^{2}}$, $\forall x,y,z \geq 0$. I know the values of the function at many points, essentially I have a table of values with $r$ and the ...
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### Intersecting three rays and a sphere of known radius

So I actually solved this problem using an iterative solver, but it annoys me because as far as I can tell it should be possible to do it directly. I have three known 3D "rays" that all start at the ...
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### Solving a Linear IVP [closed]

I need help solving this linear Initial value problem: $$y'=-L(y(t)-\phi(t))+\phi'(t) \\ y(0)=y_0$$ where $\phi(t)=\cos(30t)$.