Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.

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0
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1answer
21 views

Euler's method, non-existent $y(x)$ function

I'm trying to approximate the solution to this equation using the Euler's method: $$y'(x) = 3-\tan(x) \cdot y(x), y(2) = 4$$ When solving for the step of $0.2$, I don't know what to calculate when ...
4
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1answer
93 views

The accuracy from left to right and that from right to left of the floating point arithmetic sums

Question 1 Show that floating point arithmetic sums $$s_n=\sum_{k=1}^n\frac{1}{k^2} = 1+\frac{1}{2^2}+\frac{1}{3^2}+\dotsb+\frac{1}{n^2}$$ with accuracy $\mathcal O(n)\epsilon$ from left to ...
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2answers
24 views

Step size in Euler's forward method

I came across the following question. Kindly let me know if there is any generic solution to this type of question. $$ \frac{d^2 y}{dt^2} + 3 \frac{dy}{dt} + 2y = f(t) $$ Where $f(t)$ is an impulse ...
0
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0answers
17 views

Runge-Kutta 4 in polar coordinates

How is the Runge-Kutta method implemented on this differential equation: $$ \frac{d^2 \theta}{dt} = -\frac{g}{l} \theta $$ (pendulum motion) which is in polar coordinates? Let: $c = \frac{g}{l}$ ...
1
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1answer
17 views

Numerically solving the equation of a simple pendulum with Runge-Kutta.

I am trying to solve the equation $\dfrac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \dfrac{g}{L} \sin{\theta} = 0$ using Runge-Kutta. I have alread split it into the following equations ...
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0answers
25 views

Determining the minimal number of terms to use in a sum to approximate a number given a tolerance

In page 33-34 of Numerical Analysis by Burden & Faires an algorithm was given to compute the minimal value of $N$ for which $$|\ln{1.5}-P_N(1.5)|<10^{-5}\tag{1}$$,where ...
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1answer
25 views

Velocity Verlet method: How to calculate acceleration

The velocity Verlet method algorithm is as follows: Calculate: $$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{v}(t)\, \Delta t+\tfrac12 \,\vec{a}(t)\,\Delta t^2$$ Derive: $\vec{a}(t + \Delta t)$ from ...
2
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1answer
46 views

In common tongue, what is the differences between sparse and dense matrices?

What are the differences with sparse and dense matrices in practice, so as to offer some insight to new learners on a more intuitive level. Obviously everyone knows about the dictionary definition of ...
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0answers
25 views

Numerical integration of product of two functions

My question is in the context of spectroscopy/determination of excited state lifetimes. I have to numerically integrate the following integral using MS Excel $$ \int_0^{\infty} F(\lambda) \epsilon ...
5
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4answers
225 views

Which is bigger: $(n!)^{n!}$ or $(n^{n})!$? [closed]

To be honest I haven't spent a whole lot of time thinking about this other than the drive back home, and I won't have much time to think about it for a while due to shit-happening. So i thought I'd ...
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2answers
15 views

Is there a simple formula for the volume of an oblique triangular pyramid?

I have the xyz coordinates for 4 points in space that are not co-planar. These points form a triangular pyramid. Taking any 3 points as the base, the 4'th point will practically never be over the ...
2
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0answers
31 views

Error analysis for Runge Kutta, how to take Big O of 2 variables?

For the standard 4th order Runge Kutta: where the system is assumed to be smooth (so that the RHS has no discontinuous points) $\mathbf{y'} = \mathbf{F}(t,\mathbf{y})$ $\mathbf{y(t_0)} = ...
0
votes
1answer
22 views

Understanding the definition of an Interface

Im starting to learn about modeling of moving interfaces and am feeling daft about the basic definition itself: Given an $n$ dimensional space $\Omega$ an interface $\Gamma$ is a co-dimension 1 ...
2
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1answer
11 views

How to find the value of 'c' using Trapezoidal rule

If the Trapezoidal rule with single interval [0,1] is exact for approximating the integral $\displaystyle\int_0^1(x^3-c\ x^2) \ dx $, then what is the value of c ? I am confused with the word ...
3
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3answers
108 views

Practice job interview questions (sequence & powers) [closed]

I am practicing for a numerical test as part of a job interview. They sent me practice questions, some of which I am not able to figure out completely especially given the fact that I am NOT allowed ...
0
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0answers
29 views

Coding a bisection algorithm using matlab (numerical analysis)

To code the bisection algorithm. We assume that f at the two end-points a and b are of different signs. We will return the approximated solution when $b_{n}$ − $a_{n}$ < 2ε or when we have found ...
1
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1answer
15 views

Error Bound for Lagrange Interpolating

I have a question from a past exam paper, with no solutions provided, that I'm stuck on. Consider the set of interpolation nodes: $$\begin{array}{|l|cr} x & -1 & 1 & 2\\ \hline y & ...
0
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2answers
23 views

exponential regression for bacteria growth

I'm studying regression lines and curves, and I've learn the methods for working with curves of the types $ax^2+bx+c$ and $ax+b$ as well as $a\sin(x)+b\cos(x)$. Now I'm asked this: $$(0,32), ...
0
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1answer
39 views

Solving a system of ODE that arose in solving Burgers' equation

Consider the Burgers' equation $$\partial_t u = \alpha u\partial_xu$$ Intend to solve this using Fourier Galerkin method. So When I convert this into $N$th Fourier partial sum, I get a system of ODE's ...
0
votes
1answer
32 views

$\kappa(B^{-1}A)=\kappa (AB^{-1}) = \frac{\lambda_{max}(B^{-1}A)}{\lambda_{min}(B^{-1}A)}$

Prove or disprove if $A,B$ are symmetric positive definite (s.p.d) matrices then operator 2-norm condition number $\kappa(B^{-1}A)=\kappa (AB^{-1}) = ...
1
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1answer
26 views

Does matrix norm change under an equivalence transformation? [closed]

Consider $||.||_2$ matrix norm. Let A, B be symmetric matrix, is ||A|| and $||BAB^{-1}||$ equal? Thank you in advance.
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0answers
40 views

How can I calculate this integral?

How can I numerically calculate the integral below from the following values? E : complex electric field vector at each point H : complex magnetic field vector at each point x : coordinate along x ...
0
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0answers
27 views

What does $\int \alpha(\alpha-1)f''(\alpha)d\alpha = f''(\phi)\int_0^1\alpha(\alpha-1)d\alpha$ has to do with taylor's theorem?

I'm reading a bad taken photo of the notes of my classmate, and there's an exercise that asks to calculate $\int_{0.2}^{0.4}\sin(x) dx$ by the fixed length trapezoid rule. Then there's this written ...
0
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1answer
42 views

What is the enclosed volume of an irregular cube given the x,y,z coordinates of the 8 corners?

I have the xyz coordinates of 8 points that forms an irregular-shaped cube. This is an animation, so the cube is undergoing periodic or cyclical shape-change. The co-planarism of any group or set ...
0
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1answer
25 views

Finding relation between inputs and outputs

Is anyone able to find the rule? If possible, can you explain how and what's the name of that kind of rule? Input Output 0,18 0,316 0,3 0,432 0,32 0,4536 0,33 0,4644 0,34 ...
0
votes
1answer
18 views

Implementing a function whose representation has a singularity.

$\newcommand{\R}{\mathbb R}$ Suppose I want to calculate the value of a continuous function $f\colon(a,b) \to \R$, with $a,b\in\R$, where there are functions $g,h\colon (a,b)\to\R$ such that for ...
0
votes
1answer
20 views

Relative Error Newton's Method

So I've been preparing for an exam by looking over old exam papers and I came across this question: a) Using Newton's method, show that the iteration equation to find the cube root of $R$ can be ...
1
vote
1answer
76 views

Find $||\cdot||_2$ norm of block matrix

Compute the square norm $||\cdot||_2$ of matrix : \begin{bmatrix} O & I_n & \dots & O \\ I_n & O & \ddots & \\ \vdots & \ddots & \ddots & I_n ...
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1answer
24 views

Using Stirling's Approximation to Find Maximum

How can we use Stirling's approximation, $$n!\approx\sqrt{2\pi n}(n/e)^n$$ to find the size/location of the max term in, $$\sum_{n=0}^\infty\frac{x^n}{n! }$$ for any $x>0$. I started off by ...
4
votes
1answer
78 views

Is there a name for this piecewise cubic interpolation kernel

I went looking for a way to do piecewise cubic interpolation, like natural cubic splines, but: expressible as a convolution of data points with a piecewise cubic kernel; and still C2-continuous ...
0
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0answers
24 views

Optimization with trace and eigenvalues

Let $M \in \mathbb{R}^{n \times n}$ be a symmetric matrix with given eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n$ with $\vert\lambda_1\vert > \vert ...
0
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0answers
8 views

Lax-Wendroff: Computing $\partial_{t}^{3}u$ given $u_{t}=a_{1}u_{xx}+a_{2}g(t)u_{x}^{2}$

I want to use the Lax-Wendroff method to $u_{t}=a_{1}u_{xx}+a_{2}g(t)u_{x}^{2}$, for g smooth function and $a_{i}$ constants. . That requires computing $u_{ttt}$ in terms of derivatives in x. Is ...
2
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0answers
48 views

The convergence of the fixed-point iteration for solving a cubic equation

I have a third-grade polynomial of the form $Ax^3+Bx+C$ and I want to find its roots. I cannot use Gauss to guess the first root and it is not trivial, so I try this: $0=Ax^3+Bx+C$ and for a given ...
1
vote
1answer
30 views

Numerical Approximation of a Differential Equation

I have the differential equation that models the velocity of a falling object: $$ \frac{dv}{dt}= \frac{c}{m}v^2 - g $$ Where: c= drag coefficient = constant m = mass g = acceleration due to ...
1
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0answers
30 views

Compute Function to Full Machine Precision [duplicate]

If we want to evaluate $$f(x)=\frac{e^x-1-x}{x^2}$$ then we have to observe its large relative error as $x\to 0$. My question is that how can we find a method so that we can compute $f(x)$ to full ...
0
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1answer
36 views

Prove or disprove if $µ_0(Bx, x) ≤ (Ax, x) ≤ µ_1(Bx, x), ∀x ∈ R^n$, then $κ(B^{−1}A) ≤ \frac{µ_1}{µ_0}$

Let $A, B ∈ \mathbb{R}^{n×n}$ symmetric. Show that conditional number $$κ(B^{−1}A) ≤ \frac{µ_1}{µ_0}$$ holds, if $B ∈ \mathbb{R}^{n×n}$ is a symmetric positive definite matrix satisfying $$µ_0(Bx, x) ...
0
votes
0answers
16 views

Hilbert Matrix, Gaussian Elimination with varying pivot strategies, and computation error.

I'm doing a project for my Numerical Analysis class about computational error related to Gaussian elimination, gaussian elimination with partial pivoting, and gaussian elimination with scaled partial ...
0
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0answers
19 views

Finite difference scheme for piecewise domain heat equation

We have piecewise heat equation $u_{t}(x,t)=\left\{\begin{matrix} D_{1}u_{xx}(x,t) &x\in(0,1) \\ D_{2}u_{xx}(x,t) & x\in (1,2)\end{matrix}\right.$, with IC: $u(x,0)=2x+1$ and BC : ...
1
vote
1answer
39 views

Numerical integration of a system of stiff ODEs starting at a singular point

Good afternoon, I have a system of $3$ highly non linear differential equations, which I have to integrate form a starting singular point $x^1=[1,1,1]$, and theoretically I have to arrive to an ...
2
votes
1answer
48 views

Given $f(x) = \sqrt{x^2+1}-x$, determine the value of $f(42.545)$ in 4-digit rounding arithmetic

Suppose $f(x) = \sqrt{x^2+1}-x$ is to be computed in 4-digit rounding arithmetic. (i) Determine the value of $f(42.545)$ in 4-digit rounding arithmetic, and the relative error of this value. ...
1
vote
1answer
19 views

Establishing inequalities on condition numbers

I want to establish the inequality that $$\frac{\text{cond}_{2}(A)}{n} \leqslant \text{cond}_{1}(A) \leqslant n\cdot\text{cond}_{2}(A)$$ where the subscript indicates the matrix norm used, with ...
0
votes
1answer
18 views

Modelling free fall with Euler

I am trying to model free fall with air resistance using Euler method. I am using python and I was wondering if this is correct. $v_k = v_{k-1}-(g+ \frac{k}{m}v_y^2) \Delta t$ $x_k = x_{k-1}+v_{k-1} ...
0
votes
1answer
16 views

Find expression for $\sum_{k=0}^{n} l_k(0)x_k^{n+1}$

If the interpolation of $f(x)$ on the set of distinct points $x_0, x_1, \cdots x_n$ is given by $$\sum_{k=0}^{n} l_k(x)f(x_k).$$ Find an expression for $$\sum_{k=0}^{n} l_k(0)x_k^{n+1}.$$ I ...
0
votes
2answers
17 views

Error analysis on numerical solutiol of an equation

Say I am solving an equation numerically -- the derivatives in the equation I find by a finite difference scheme with an accuracy of the grid spacing $h$. Does this imply that the final solution I ...
2
votes
1answer
35 views

Explaining error in a floating-point calculation

I'm working through a numerical analysis text and came across this question. The function $f_{1}(x,\delta) = \cos(x + \delta) - \cos(x)$ can be transformed into another form, $f_{2}(x,\delta)$, ...
0
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1answer
38 views

How to solve underdetermined differential equations numerically?

Consider the following differential equation $-\dot{x}(t)^2+\dot{y}(t)^2+x(t)^2\dot{z}(t)^2=0$, [edited: or a more complicated one $-\dot{x}(t)^2+x(t)^{1/3}\dot{y}(t)^2+x(t)^{2/3}\dot{z}(t)^2=0$] ...
-1
votes
1answer
142 views

Exact computation of n choose k

For what n>=1 can the binomial coefficient "n choose k" be computed exactly in IEEE double precision arithmetic? For what n>=1 will intermediate factorials overflow?
2
votes
1answer
65 views

Nodal Lines of the Eigenvalue problem $\Delta u=\lambda u$

I have really enjoyed performing the method of separation of variables to identify the eigenfunctions and nodal lines (the set of points for which each eigenfunctions vanishes) of the 2-D wave ...
0
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0answers
11 views

4th order Runge-Kutta Scheme for Stochastic Differential Equations

In the book "Introduction to Stochastic Differential Equations" by Thomas C. Gard, they talk about higher-order runge-kutta type schemes. The SDE in question is a general Ito SDE of the form: $$dX = ...
0
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0answers
26 views

Approximate $f(0.05)$ with its third Taylor polynomial evaluated at $0.05$ using four-digit rounding arithmetic.

Given $f(x)=\frac{e^x - e^{-x}}{x}$, Approximate $f(0.05)$ with its third Taylor polynomial evaluated at $0.05$ using four-digit rounding arithmetic. Ok so i'm a little confused here. I see 2 paths ...