# 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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### Evaluate derivative of Lagrange polynomials at construction points

Assume, that we have points $x_i$ with $i=1,...,N+1$. We construct the Lagrange basis polynomials as \begin{align} L_j(x) = \prod_{k\not = j} \frac{x-x_k}{x_j-x_k} \end{align} Now according to my ...
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### Runge-Kutta methods: step size greater than 1

The local truncation error of Runge-Kutta 4 is $O(h^5)$, while those of RK1 is $O(h^2)$. I wonder then what happened when the step-size of RK methods is greater than 1: will the accuracy be improved ...
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### IVP Using Numerical Methods

Suppose that $y(t)$ is the exact solution of the ivp $$y'(t)=f(t,y(t)), y(0)=y_0$$ and $u(t)$ is any approximation to $y(t)$ with $u(0)=y(0)$. Define the error $e(t)=y(t)-u(t)$. How can I show that ...
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Can someone help me this question please, this is from past years exam.
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### estimating the error of $\sin(x) = x$ with Taylor's Theorem

I want to calculate the numerical error in approximating $\sin(x)=x$ with Taylor's Theorem. Furthermore, what values of $x$ will this approximation be correct to within $7$ decimal places? Here is ...
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### Conjugate Gradient Method and Sparse Systems

What is it about conjugate gradient that makes it useful for attacking sparse linear systems. Why would steepest descent be significantly worse? Please keep in mind that I am still trying to fully ...
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### Weak Formulations and Lax Milgram:

I have a question on how to put a PDE into weak form, and more importantly, how to properly choose the space of test functions. I know that for an elliptic problem, we want to start with a problem ...
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Is there an analytical way to know an interval where all points when used in Newton-Raphson will converge/diverge? I am aware that Newton-Raphson is a special case of fixed point iteration, where: $... 3answers 6k views ### fast algorithm for solving system of linear equations I have a system of linear equations,$Ax=b$, with$N$equations and$N$unknowns ($N$large) If I am interested in the solution for only one of the unknowns, what are the best approaches? for ... 1answer 463 views ### Is there a binary spigot algorithm for log(23) or log(89)? The Bailey-Borwein-Plouffe formula yields a binary spigot algorithm for π, and related formulas give the bits of log(2) and those of the logarithms of some other integers. I got stuck (over a year ... 3answers 264 views ### Exact result of a series using Euler-Maclaurin expansion. This is a variant of Exercise 64 in Chapter 9 of concrete mathematics. Prove the following identity \sum_{n = -\infty}^{\infty}' \frac{1 - \cos( 2\pi n k )}{n^2 } = 2 \pi^2 ( k - k^... 3answers 770 views ### Parallel lines divide a circle's area into thirds When I was young I came up with a geometry problem and drew it in a notebook: Suppose we have a circle with radius$r$and area$A$. Let two parallel lines be equidistant from the center of the ... 1answer 170 views ### Can we apply an Itō formula to find an expression for$f(t,X_t)$, if$f$is taking values in a Hilbert space? Let$U$and$H$be separable Hilbert spaces$Q\in\mathfrak L(U)$be nonnegative and symmetric with finite trace$U_0:=Q^{1/2}U(\Omega,\mathcal A,\operatorname P)$be a probability space$(W_t)_{t\...
For a differential equation like $\dot{x}=ax$ where $x$ is a function on an independent variable $t$, and $\dot{x}=\frac{dx}{dt}$, and $a$ is a constant, we define the time-scale $\frac{1}{a}$, which ...
### Digits of $\pi$ using Integer Arithmetic
How can I compute the first few decimal digits of $\pi$ using only integer arithmetic? By 'integer arithmetic' I mean the operations of addition, subtraction, and multiplication with both operands as ...