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Questions tagged [numerical-calculus]

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20 views

Advantages and disadvantages of the Golden-section search method

As I understand that the golden-section search is a zero-order line search method so it is a global method so in comparison with Newton's and the secant's method this is an advantage. But it has a ...
1
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1answer
31 views

First derivative approximation using function values in equidistant points

Given a function $f: [x_0,x_4] → \Bbb R$ and equidistant points $x_0, x_1, x_2, x_3, x_4$ so that $h=x_{i+1} - x_i > 0$. Normally I would do, $f'(x) \approx \dfrac{f(x+h)-f(x)}{h}$, but here I ...
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1answer
22 views

On highest degree of precision of numerical integration scheme that comes from interpolating polynomial

Let $x_1,...,x_n$ be distinct points in $[a,b]$ and $l_i(x):=\prod_{k\ne i}\dfrac {x-x_k}{x_i-x_k} $. Let $w_i=\int_a^b l_i(x)dx$. For every $f \in C[a,b]$, let $I_n(f):=\sum_{i=1}^n w_i f(x_i)$. ...
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0answers
35 views

How to improve the convergence of a stochastic differential equation?

I have a stochastic differential equation, i.e, $$ d\rho_t= \hat{A} \rho_s dt + \hat{B} \rho_s \nu dt + \hat{C}\rho_s\omega_{1t} dt + \hat{D}\rho_s \omega_{2t}dt \quad , \quad t>s $$ Here A, ...
2
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4answers
128 views

Finding general solution to DE subject to initial condition

How do we solve the following Differential Equation? $$2 x''' + xx'' =0$$ Subject to conditions: $$ x(0)=0$$ $$ x'(0)=0$$ $$ x'(\infty)=1$$ Is there any numerical method to solve it or some ...
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0answers
10 views

Solution of Transcendental equations, trigonometric ones

I am studying vibration of beams, with continuous properties; and I arrived to some kind of trascendental equations. The book I am using, (Chopra, 2014), says that one of the solution is the numerical ...
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0answers
28 views

backward propagation of errors

Suppose we have $L-1$ time intervals between $t_0$ and $t_{L}=T,$ i.e., $t_0<\ldots<t_l<\ldots<t_{L}=T.$ We have an error $e_{t_l}$ describing the absolute error difference between a real ...
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1answer
18 views

Bounding at Newton Divided Difference Formula

Let $a=x_0,x_1,...,x_n=b$ are $n+1$ points which are equally spaced in $[a,b]$. The distance between consecutive terms is $h= \frac{b-a}{n}$ and $x \in [a,b]$ Show that $ \biggr|\prod_{i=}^n (x-x_i) \...
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0answers
9 views

propagating error analysis

Tutor asked me a question about propagating error analysis: Here is the question: if for $t_0<t_1<\ldots<t_{l}<\ldots<T,$ $$e_{t_1}\leq e_1+e_2e_{t_{l+1}}+e_3,$$ and $$e_{T}\leq e_1+...
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0answers
45 views

How to program numerical integration $\int_t^{\infty}\frac{1}{(u+1)^2}du$?

I'm facing a problem when program the numerical integration of $\int_t^{\infty}\frac{1}{(u+1)^2}du$, I know that the true value is $\frac{1}{1+t}$, however, when I want to calculate the definite ...
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2answers
30 views

Obtaining expression for recursive sequence

Can someone suggest how to obtain an expression for $S[i]$ given that S[0] = 0, $S[i]=S[i-1]*(1-\gamma_i)^2 + \gamma_i^2$ where $\gamma_i = \frac{g+1}{g+i}$ EDIT: $g>0$ and $g$ can be assumed to ...
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1answer
12 views

Numerical Change of Variables

If I have $f(x)$ represented by 2 arrays. One array is the arrays of $x$ while the other is output from $f(x)$. So basically a numerical representation of the function. If I don't know what $f(x)$ is, ...
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1answer
24 views

Newton-Raphson on strictly convex function

Maybe someone can give me a hint here: question 1: Given a sequence {$x_n$} which is the Newton-Raphson sequence on some $f(x)$ s.t. $f(x)$ is strictly convex and $f'>0$. Let $\alpha$ be the ...
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1answer
23 views

Is it possible to evaluate error of numeric schemes by using integrals?

I'd just like to know if it is possible to evaluate the error of a numerical scheme by using it's solutions integral? To elaborate this. I have long transmission line, which I evaluate over time ...
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2answers
166 views

(Why) can we treat a function of a variable as another independent variable?

I'm currently reading my numerical analysis textbook and something's bugging me. To get into it, let's take a look at the following differential equation; $$u'(x) = f(x, u(x))$$ In order to ...
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0answers
31 views

Using the Newton's method to find the global minimum of a 2D problem with a constraint

I am solving an optimization problem $$\min f(x_1,x_2)\\ \text{s.t.}~~~~ c(x_1,x_2)\leq 0$$ In my problem, there are two min and two max and I am looking for the global min. I know that with the ...
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0answers
24 views

limit of the ratio of two divergent integrals

I have to calculate two ratios of two pairs of proper integrals, for a range of parameters. I am stuck when it comes to calculating these ratios when the parameters are such that the integrands start ...
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1answer
28 views

How I can to prove the succession $x_n$ converge to 1? [duplicate]

Let $0<x_0<1$ if $x_{n+1} = sin(x_n)$ show that $\lim_{n\to\infty} \frac{x_n}{\sqrt{3}/n} = 1$
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2answers
44 views

Show that a series converge or diverge: $\sum_{n=1}^{\infty}\frac {(-1)^n(2n-1)!}{3^n}$

Check wether this converges or diverges $\sum_{n=1}^{\infty}\frac {(-1)^n(2n-1)!}{3^n}$ I have a couple of questions: Using Abel-Dirichlet criteria when you have $\sum_{k=1}^{n}b_k$ bounded and $a_n\...
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2answers
27 views

Convergence test for a series

Let S = $\sum_{n=1}^{\infty}{\frac{n^k}{((n^3+n)^{\frac{1}{3}}-n)}},\forall\space\space\space\space k\in\mathbb{Z}.$ Say if this series converge of diverge. My attempt: $$S=\sum_{n=1}^{\infty}\frac{...
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3answers
25 views

Determining coefficients of a parametrization of an epicycloid given a predefined arc length.

I am trying to determine the coefficient q in the parametrization of a epicycloid which gives me the arc length of 4.25. The parametrization can be glimpsed in my attempt of a solution in the ...
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1answer
24 views

How to understand the order of convergence $\|x_{k+1} - x\| \le C \|x_k - x\|^p$ (Convergence of a power function form)?

By definition, a sequence $x_k \in \mathbb{R}, k \in \mathbb{N}$ converges with order $p \in [1,\infty)$ to $x := \lim_{k\to\infty} x_k$ if \begin{align} \exists C \in [0,\infty): \forall k \in \...
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0answers
29 views

Numerical integration with delta functions?

I would like to integrate the following numerically: $\int\limits_{-\infty}^{+\infty} \int\limits_{-\infty}^{+\infty} \delta(x^2+y^2-1) dx dy = \pi$ I could replace the dirac delta function with a ...
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0answers
7 views

The iteration,which is only locall convergent,does it has at least 2 fixed point?

When I draw the figure of the iteration which is just locally convergent,it feels like that ,there must be at least 2 fixed points.So, I have the hypothesis Assume $g(x)\in C^{1}(\mathbb{R})$,the ...
0
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0answers
28 views

Evaluate an integral using three-point Gaussian Quadrature

Write a procedure for evaluating $\int_{a}^{b}f(x)dx$ by subdividing the interval into $n$ equal sub-intervals and then using the three-point Gaussian Quadrature formula modified to apply it to the $n$...
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2answers
46 views

Give an exact expression for the doubling time of the function y = 80 (1.4)^x/4, and quote an approximate numerical value.

Give an exact expression for the doubling time of the function y = 80 (1.4)^x/4, and quote an approximate numerical value. what I did: y = 80 (1.4)^x/4 ...
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3answers
54 views

Numerical methods for solving nonlinear equations

I have this equation $x^{5}+x^{3}+3=0$ and I'm supposed to find one root with a given accuracy using the secant method. I was wondering how I can localize a solution to an interval so I can apply ...
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0answers
25 views

Optimal Values for a Taylor Series with Multiple Parameters

I'm taking a (fairly basic, but graduate level) numerical analysis class this semester and came across the below question: Construct a Taylor Table for: $$a_0(u_{xx})_{j-1} + (u_{xx})_j + a_2(u_{xx})...
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1answer
56 views

How to find second derivative as output of MATLAB ode45?

I am using ode45 function to find numerical solution for my system of equations, where I have 4 equations and 4 variables, with command: ...
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0answers
72 views

Order of convergence of midpoint rule

A problem asks to integrate the function $f(x) = \frac{x}{1+x^4}$ on $[-1, 2]$ using the Midpoint rule and the Trapezoidal rule, which I did in MATLAB. Then it asks to determine the value of this ...
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0answers
38 views

3D Gauss-Hermite Quadrature

Is it possible to examine a 3D integral by using Gauss-Hermite quadrature type technique? I mean there might be an equation like this (with analogy to 1D Gauss-Hermite quadrature): $\int_{-1}^{-1} \...
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1answer
89 views

Numerical integration with arbitrary precision

I have to perform a numerical integral on $[0,1]$ with a complicated integrand (possibly with integrable singularities at $0$ and/or $1$) and I am looking for a platform (sagemath, maple, python, ...
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0answers
33 views

What does “smooth” mean? (Numerical Analysis)

I know a notion of smoothness for functions, say, in $\mathbb{R}^n$, which simply means of class $C^\infty$. But in studying Numerical Analysis I sometimes read the term smooth for discrete functions, ...
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0answers
25 views

Numerical integration on a part of the domain using quadratures

I have relatively simple question and I did not find the answer yet. Suppose I want to numerically integrate a function. Using simplest scheme with equal weights, it is easy: $$ \int_{0}^1f(x)dx=\...
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1answer
262 views

vpasolve and for loop Matlab

I am trying to solve numerically symbolic equation. But one variable in my equation is series of values zv=1:-0.1:0. So, according to my code: ...
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0answers
119 views

How to get derivative as result of Matlab ode45?

I am using Matlab ode45 to solve system of first order differential equation presented with: ...
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1answer
106 views

Corrected trapezoidal method resources

I have been reading about the various numerical analysis methods for integrals, and I came across a mention of the "corrected trapezoidal rule", which I googled and found rather little about - mostly ...
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1answer
83 views

*Numerical* Convergence of the Babylonian Method?

I understand the sequence $x_{n+1} = \frac12\left(x_n + \frac2 {x_n}\right) $ converges to $ \sqrt2 $ algebraically. That is proved by means of fixed-point method or monotone convergence theorem and ...
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0answers
22 views

Stable, 2nd order multi-step method

I need to solve this exercise: Prove that every stable multi-step method, with at least second order, that verifies the condition: (1)$\operatorname{Re}( \rho(e^{\theta i})*\sigma(e^{-\theta i}...
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0answers
26 views

how to get time-speed function from distance-speed function, numerical?

I have speed function depends on distance. Actually, numerical connection. $v = v(s)$, so i have $[ s_i, v_i ]$ values. But I want to get speed function depens on time ( $v=v(t)$ ) time and distance: ...
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1answer
89 views

Numerical Differentiation - Finding h value

I am trying to understand one question's solution, but I didn't get that how they computed "h" in this particular question: Original Question Statement: With using the data on the table below, ...
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0answers
56 views

Newton's method for fraction

Let $a\gt0$. Start from a convenient equation and use Newton's method to deduce a method to approximate $\frac{1}{\sqrt{a}}$ without divisions. How is the starting value chosen? What is the ...
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1answer
29 views

Determine an interpolation polynomial

Let $$\:f\:∈\:C4\left[−1,\:1\right]$$ Determine an interpolation polynomial of minimum rank that satisfies the conditions: $$P\left(−1\right)\:=\:f\left(−1\right),\:P\left(0\right)\:=\:f\left(0\right),...
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0answers
18 views

Inverse of twovariate function with specific properties

I have two nonlinear functions of two variables $y_1=f_1(x_1, x_2)$ and $y_2=f_2(x_1, x_2)$. I would like to obtain their inverses $g_1$ and $g_2$: $$ x_1=g_1(y_1, y_2) \quad\mathrm{and}\quad x_2=g_2(...
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0answers
54 views

How to compute this integral over a 2D subspace?

Given the function $f(x,y)$ defined as $$f(x,y)\equiv\frac{x^\frac{p}{2}}{\left(1+x+y^2\right)^\frac{p+n}{2}},\ \left\{(p,n)\in\mathbb{Z}^2,x\in\mathbb{R}^+,y\in\mathbb{R} \mid p\ge-1,\ n\ge4\right\},$...
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0answers
122 views

Numerical analysis: Three point method

Hoping for some advice as to how to tackle this question from Numerical analysis. I assume some form of the three point formula is used, however the mid/end point formula seem to fail since there is ...
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0answers
96 views

Solving a Matrix-Differential equation with e.g. ode45

Solving a Matrix-Differential equation with e.g. ode45 derived from some matrix-valued function $E(t)$ where $E$ is a $m\times m$ Matrix, I get a ODE $$\dot E=-G(E)+...
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0answers
10 views

Error limit equivalence of a fixed point method

If we have an interactive method such as $x_{n+1} = \psi(x_{n})$, $\xi$ is a fixed point of $\psi(x)$ and $\frac{|x_{n+1} - \xi|}{|x_n - \xi|^k} = \lambda$,$k \geq 1$ and $\lambda \geq 0$, prove ...
1
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1answer
74 views

Bisection Method - Number of steps required proof [duplicate]

I am currently reading $\textit{Scientific Computing: An Introductory Survey}$ by Michael Heath. In Section 5.2.1 when talking about the Bisection Method it is said that the number of iterations $n$ ...
0
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1answer
48 views

Approximate the given integral using Gaussian quadrature

Consider the Gaussian quadrature formula for $n=2$ with weight function $w(x)=1-x^2$ and orthogonal polynomials on $(-1,1)$ given by $$P_0(x)=1, P_1(x)=x, P_2(x)=\frac{1}{4}(5x^2-1).$$ The formula ...