Questions tagged [numerical-calculus]

This tag is for various question on numerical calculus / numerical analysis which concerned with all aspects of the numerical solution of a problem, from the theoretical development and understanding of numerical methods to their practical implementation as reliable and efficient computer programs.

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Numerical integration of an integrand including the dirac delta function

I have the following, $$\int_0^{2\pi} d\xi \, R(\xi) \, \delta[\mathbf{r} - \mathbf{R}(\xi)]$$ I need to evaluate this numerically on a series of grids. The dirac delta function serves to interpolate ...
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Find value for variable in a definite integral with given result of definite integral (numerical integration)

I need to compute a definite integral for example: $\int_{0}^{\frac{\pi}{2}} k*cos(x) \,dx$ with variable k = 1 this definite integral is equal to 1. $\int_{0}^{\frac{\pi}{2}} cos(x) \,dx = 1$ The ...
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A bound for the error $|(x-1)\ln x-P_3 (x)|$ in using $P_3(x)$ to approximate $(x-1)\ln x$ on the interval $[0.5, 1.5]$

I was wondering if someone could tell me how we can find a bound for the error $|(x-1)\ln x-P_3 (x)|$ in using $P_3(x)$ to approximate $(x-1)\ln x$ on the interval $[0.5, 1.5]$. I computed $P_3 (x)$ ...
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What computer methods are used to quickly calculate the $\zeta$-function (if any)?

So I can think of how I could compute $\zeta(\sigma + it)$ in principle. We can take $\zeta(\sigma + it)$ for $\sigma>1$ by the usual $\sum_{n} n^{-(\sigma + it)}$ summation. We can use the ...
48 views

Newton's method in higher dimensions

To calculate the inverse of a quadratic matrix A, we could solve the equation $F(X):=X^{-1}-A=0$. I need to show that if X is invertable, then $DF(X)(\Delta X)=-X^{-1}\Delta XX^{-1}$ where DF(X) is ...
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find $a,b,c$ such that integration is minimum. [closed]

find $a,b,c$ such that $\int_{0}^{\pi/2}\left|ax^2+bx+c-\cos x\right|^2dx$ is minimum. My idea is using Polynomial interpolation, but i can't solve.
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Round to n decimal places

So, let's say we want to evaluate $\pi$ using $\sum\limits^\infty_{n=0} \frac{4(-1)^n}{2n+1}$ (or any other series) correct to four decimal places. (i) Do we compare the series result to 3.1415 or 3....
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Using a successive double integration by parts show that the integral of $f(x)$ between $x_i$ and $x_{i+2}$ is equal the following expression [closed]

$f(x)$ between $x_i$ and $x_{i+2}$ is equal to this expression :" /> Show using a successive double integration by parts show that the integral of $f(x)$ between $x_i$ and $x_{i+2}$ is equal to this ...
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How to determine error value needed to attain certain decimal precision with numerical integration

I need to use numerical integration to integrate a function $f(x)$ from $a$ to $b$ correct to $k$ decimal places. The two methods I am interested in are the Trapezoidal rule and Simpson's rule. I'm ...
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Derive the Legendre differential equation given the recurrence relation

Given the 3-term recurrence relation $$(n+1)P_{n+1}=(2n+1)xP_n-nP_{n-1},$$ prove that $$(1-x^2)P''_n-2xP'_n+n(n+1)P_n=0.$$ I tried differentiate from both sides, but how can I get rid of the different ...
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No. of significant figures in absolute value w.r.t true value and relative percentage error

To find the no. of a significant figure in absolute value $= 0.05411$ with respect to true value $= 0.05418$ and the relative percentage error. Here's my solution: But I ain't sure whether it's ...
I am trying to solve an IVP problem consisting of the following equations: $y_1'=Ay_2$ $y_2'=-Ay_1$ $y_3'=0$ The analytical solution considering \$y(0)=\left[\begin{matrix}Csin(\theta)\\ 0\\ Ccos(\...