Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.
0
votes
1answer
51 views
Infeasible start Newton's method
I am implementing infeasible start Newton's method from the information in the slides (slide 11 of the link) posted here. It requires us to calculate primal and dual Newton steps, denoted by, $\Delta ...
2
votes
1answer
43 views
Can we use (higher order) derivatives to help integration?
I'm wondering if it's possible to use derivatives to ease the evaluation of an integral. For instance, I know that to evaluate an integral with enough precision I need to evaluate it at $n$ points. ...
0
votes
1answer
19 views
Calculate variable in expression with Truncate function
I would like to know how to calculate MaxDigitWidth variable by using CharWidth and Pixels variables in the following expression:
...
3
votes
0answers
42 views
Determining algebraically a point of intersection.
A student I was tutoring posed the question: "I know how to solve
$$e^{-x} = \ln x$$
graphically, however how do you solve this algebraically?" I have been fiddling around with it for a while and I ...
24
votes
1answer
369 views
How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$
I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
3
votes
1answer
52 views
Finite difference implicit schema for wave equation 1d not unconditionally stable?
The wave equation 1D with constant density is defined as:
\begin{equation}
\frac{\partial^2 U}{\partial t ^2} = V^2 \frac{\partial^2 U}{\partial x ^2}
\label{eqa}
\end{equation}
And the implicit ...
2
votes
1answer
29 views
Problems where SPD linear system arises
I know some of the places where SPD linar systems arises such as elliptic PDEs and normal equations. Can I have a more comprehensive list of scientific applications which require solving SPD linear ...
1
vote
3answers
59 views
Numerical integration of nonlinear second order equation
I have (or, rather, a friend whom I'm trying to help has) a very messy differential equation, which I thought I'd try to solve numerically. However, I'm a little confounded as to what approach to use ...
0
votes
1answer
33 views
Convergence of Secant method
In Secant method -
$$x_{n} = x_{n-1} - f(x_{n-1})\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}$$
I have to choose 2 initial - $x_0$ and $x_1$ .
How could I know whether those 2 $x$'s going to ...
1
vote
1answer
44 views
Roots of a finite Fourier series?
In general, are there any clever tricks to help find the roots of a finite Fourier series? Presumably there aren't analytic methods, but can we use the fact that our function is a finite Fourier ...
1
vote
1answer
67 views
When is Newton's Method guaranteed to converge to a good solution (non-linear system)?
My knowledge of Newton's Method is partial. I am trying to understand what guarantees this method can give on the solution of systems of non-linear equations.
Specifically, I have a set of non-linear ...
0
votes
0answers
26 views
2nd order 2-step explicit Linear multistep method
I came across this question, i know how to carry out this question but my answer doesnt quite seem to correspond to the conditions.
the question states:
Construct a 2nd order 2-step explicit LMM ...
0
votes
0answers
30 views
linear multistep method- leapfrog
hi im having abit of trouble with this- im starting from the basics of Linera multistep methods and have come across this question:
write the leapfrog method down as a linear multistep method.
im ...
2
votes
1answer
60 views
Is the backward Euler method A-stable for non-linear equation?
I'm looking at the parabolic equation
$$
\frac{\partial u}{\partial t} - \frac{\partial}{\partial x}\left(\alpha(u)\frac{\partial u}{\partial x}\right) = f(x,t)
$$
I know the Backwards Euler method ...
2
votes
1answer
98 views
butcher tableau runge kutta methods
Hi I have had a go at this question- am i heading in the right direction? it would be much appreciated if someone could me
Write the Butcher Tableau for the 1-stage $\theta$ method:
$$U^n ...
0
votes
1answer
18 views
Reflection Lines
I am analyzing the problem of G^1 continuity between patches. I have found the statement: if the reflection lines on a surface are C^0 then the surface will be G^1. I would like to know the proof of ...
0
votes
0answers
49 views
Exact versus numerical approximation of an integral
Why does the following approximation work very well for small theta's but not for large theta's?
$$
I = \int_0^\infty x (1 + ...
0
votes
1answer
23 views
explicit ODE IVPs
Ive had a go at this question, just need pointing in the right direction.
A linear scalar ODE of the form
$$ \frac{du}{dt}=:d_tu=:\dot{u}=5tu+\sin(t) $$
$$ u(0)=1 $$
can be solved explicitly. ...
0
votes
0answers
49 views
Finite difference methods for nonlinear problems
How can we use the nonlinear finite-difference method with $h = 0.25$ to approximate the solution to the BVP $y'' = 2y^3$, $-1 \leq x \leq 0$, $y(-1) = 0.5$, and $y(0) = \frac{1}{3}$?
So, I ...
2
votes
1answer
42 views
How to use undefined value in Composite Simpson's Rule
I have to use the Composite Simpson's Rule to approximate the integral $\int_0^1 t^2\cdot sin(\frac{1}{t}) dt$. I've used the Composite Simpson's Rule, but when I work through the algorithm, one step ...
1
vote
1answer
59 views
Runge Kutta Graphical Analysis
Can someone represent the 4th order Runge Kutta method graphically when iterating from (xk,tk) to (xk+1,tk+1)?
Thanks
0
votes
0answers
25 views
Truncation error and difference method
I am stuck on the following question. I am not sure of how to calculate the truncation error for the difference method
any help would be appreciated thank you!
0
votes
1answer
40 views
Modified Euler method
I am revising the modified euler method and would appreciate some help with this question:
The equation is $$y'=\frac{2}{x}y+x^2e^x, y(1)=0$$
Use modified euler method to calculate $y(1.1)$ taking ...
0
votes
2answers
82 views
Nonlinear DE and Numerical System
I'm trying to investigate nonlinear system numerical methods. For the nonlinear DE x' = 2t(1+x^2). Use the value tan(1) = 1.557407724654....
a) how to find the explicit solution $x(t)$ satisfying ...
1
vote
1answer
40 views
numerical methods sketches
Can someone show graphically in the tx-plane of illustrating the process of moving from (tk,xk) to (tk+1, xk+1) in
-Euler's method
-Improved Euler's method
and RK4?
I understand the formulas but ...
0
votes
0answers
20 views
Help with operation count for Choleski factorization and Guass elimination
I'm having a lot of trouble finding the multiplication and division operation count for Choleski factorization (without pivoting) and the Gauss elimination.
I had a go at the Choleski factorization, ...
0
votes
2answers
86 views
Different methods and nonlinear systems
I'm trying to investigate nonlinear system numerical methods. So if we have a simple DE $x' = x$,
a) how to find the explicit solution $x(t)$ satisfying $x(0) = 1$?
b) how to use Euler's method to ...
1
vote
0answers
19 views
show that the remainder for midpoint rule is $\frac{(b-a)^3}{24}f''(\xi)$ for some $\xi\in[a,b]$ using hermite interpolation
i saw some proof using taylor theorem but cannot find one using hermite interpolation.
for newton-cotes quadrature rule with $n$ which is even, we have $n+1$ degree of precision, instead of $n$ when ...
1
vote
1answer
48 views
Heavily stuck on Newton-Cotes integration
For an integral of form $\int_{-2}^2 x^2e^xdx$, calculate the
Newton-Cotes quadrature and estimate the error for:
$n=1$ (Trapezoid rule)
$n=2$ (Simpson's rule)
$n=3$ (3/8 rule)
...
0
votes
0answers
22 views
Numerical simulation of impulsive differential equations
I am interested to know discretization techniques of impulsive differential equations and its numerical simulation in MATLAB. I discretized the system of impulsive differential equations in uniform ...
0
votes
0answers
36 views
Numerical methods and illustration
Can someone provide a picture in the $t$-$x$ plane that illustrates the process of moving from $(t_k,x_k)$ to $(t_{k+1},x_{k+1})$ in Euler's method, Improved Euler's method and Runge-Kutta 4?
I had ...
0
votes
0answers
18 views
Implicit Runge Kutta method is well defined
I need a hint or two on how to show that the implicit Runge Kutta method is well defined.
As a hint we are told to show that the implicit problem has a solution if ...
0
votes
0answers
13 views
Linear equation with prescribed precision of result
Let $x$ be an unknown real vector of size n.
Suppose we can find n vectors $v_i$, and are given the values $x^Tv_i$.
Then we can simply solve for $x$ by Gauss or some other method and determine $x$ ...
1
vote
0answers
36 views
van der pol and liapunov
i have attempted this question and done as much as i possibly could, any help regarding this question would be very helpful and appreciated.
a) show that the second-order differential equation for ...
0
votes
1answer
90 views
Nonlinear Second-order ODE BVP with 4 boundary conditions
My Lagrangian comes out in this form when I impose spherical symmetry:
$$ φ''(ρ)+{3\overρ} φ'(ρ)+{4μ^4\over M^2} φ(ρ)-{4μ^4\over M^4} φ^{3}(ρ)-{μ^4\over2M} ϵ=0 $$
The following boundary conditions ...
0
votes
1answer
87 views
Two Dimension Heat Equation ADI Local Truncation Error
Given a two dimensional heat equation $\displaystyle \frac{\partial u}{\partial t}=K(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2})$ solved using ADI (the alternating-direction ...
5
votes
2answers
116 views
Newton's method, for which initial guesses it converges?
We've got a function: $ f: R \rightarrow R$ , $f(x) = x^3 - 9$.
Let $x^* $ be its root, which means $ f(x^*) = 0$. We want to find approximation for $x^*$ using a Newton's method.
There are two ...
1
vote
3answers
71 views
How to find the limit of a convergent matrix?
I'm trying to learn how to show a series of matrices is convergent and find the limit. However my numerical analysis books fail to mention this and I cannot find any relevant material online! Anyway ...
0
votes
1answer
30 views
$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta$ ,$\theta=$?
This question is a follow up question to this answer.
In the equation:
$$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta.$$
$a$ and $b$ are given. What is the best way to solve for ...
1
vote
4answers
65 views
Solving a set of 3 Nonlinear Equations
In the following 3 equations:
$$
k_1\cos^2(\theta)+k_2\sin^2(\theta) = c_1
$$
$$
2(k_2-k_1)\cos(\theta)\sin(\theta)=c_2
$$
$$
k_1\sin^2(\theta)+k_2\cos^2(\theta) = c_3
$$
$c_1$, $c_2$ and $c_3$ are ...
1
vote
1answer
29 views
Derivative of solution of ODE
I have a set of nonlinear differential equations with parameters.
$$\dot{\vec{x}} = F(\vec{x},\vec{\beta}) $$
where $\vec{x} \in \mathbb{R}^p$ and $\vec{\beta} \in \mathbb{R}^q$ ($p,q \in ...
0
votes
0answers
20 views
How to solve a simple B-spline with table function?
Determine the natural cubic B-spline for a function in a table form of:
$$x_i\|-1\|0\|1\|2 \\ y_i\|-3\|0\|1\|2$$
So I'm very new to cubic splines and am quite lost on what I should do. I ...
0
votes
0answers
33 views
Logarithmic accuracy
Does anyone know how the method of logarithmic accuracy works and what do I have to know about it (as far as applied Mathematics is concerned)? Any references, examples or guidelines would be ...
2
votes
0answers
40 views
Proving invertibility of matrices using banachs lemma
I'm studying for finals and trying to understand how you can possibly use banach's lemma for anything worthwhile, more particularly we have a bunch of sample questions that say it can be used to prove ...
4
votes
4answers
33 views
Generating a Monotonically Decreasing Sequence that adds to 1 for any length
I would like to generate a monotonically decreasing sequence whose elements will add to one, and generate this sequence for any order $L$. For instance, if $L=2$, then $\vec{s} = [ \frac{3}{4}, ...
0
votes
1answer
71 views
Nonlinear equation / iteration method
$$ 1 x^5-9.067 x^4+24.726 x^3-13.998 x^2-15.278 x+1.014 = 0 $$
Wolfram Result
but i need solve roots using "iteration method" (i don't know how it correctly called in English, but Newton's method is ...
0
votes
1answer
57 views
How can I minimize the following function
I have a simple function and want to minimize it using any method with precision of about 0.01. The domain of x is [-2,2]. The function is a simple $f(x) = x^2$ plus a triangle in the form of ...
0
votes
1answer
65 views
matrix with distinct bounded eigen values is bounded?
I am looking from the numerical methods perspective. I have a mapping $G$ that maps points in the numerical iteration to the new level. I would like to show its stability. For that I need to show that ...
0
votes
1answer
19 views
Simultaneous eigenfunction problems
I'm familiar with solving eigenfunction problems using finite difference methods and eigenvalue solver like Eigensystem[] in Mathematica. But now I've come across a problem where I have two ...
1
vote
1answer
114 views
Von Neumann Stability Analysis
I came across the following task recently:
Use the von-Neumann stability analysis to investigate the stability of the discrete form of $\frac{\partial c}{\partial x} = \frac{\partial^2 c}{\partial ...
