Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.
0
votes
0answers
10 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, ...
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votes
2answers
36 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
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0answers
10 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
35 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)
...
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0answers
15 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 ...
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0answers
31 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 ...
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0answers
9 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 ...
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0answers
12 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 $x$ ...
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0answers
23 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 ...
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1answer
36 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 ...
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votes
2answers
99 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 ...
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3answers
46 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
25 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 ...
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vote
4answers
60 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
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1answer
18 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 ...
