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

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3
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1answer
47 views

Non-linear system vs minimisation problem

If you have a non-linear system of equations which can be formally written as : \begin{equation} \begin{cases} F_1(\mathbf{x})=0\\ F_2(\mathbf{x})=0\\ \ \ \ \ \vdots\\ F_n(\mathbf{x})=0\\ \end{cases} ...
0
votes
0answers
11 views

Finding an analytic form of a function that satisfies asymptotic conditions

I have a family of functions that I obtain numerically. They depend on $x$ and parametrically also upon a certain parameter $L$. I would like to find an analytical form for this family of functions so ...
6
votes
0answers
58 views

fixed point iteration

I am trying to find the root of $f(x)=\arctan(x)$ by using successive iteration. There are some conditions to apply this in successive iteration . 1) The function has to be continuous. 2) ...
0
votes
0answers
16 views

Approximating an integral with a change of integral

(I have previously found out $x_1 = -\frac{1}{\sqrt{3}}$ and $x_2 = \frac{1}{\sqrt{3}}$ ) Approximate an integral using the 2-point rule, with an appropriate change of integral, to approximate ...
0
votes
1answer
21 views

How do I discretize a parabolic partial differential equation?

I have the following homework question: To keep my long sob story as short as possible, my awesome applied numerical methods teacher had a personal emergency and is replaced for the rest of the ...
1
vote
0answers
38 views

Name of this PDE: $\frac{\partial^2u}{\partial t^2}=\frac{\partial^3u}{\partial x^3}$

So I got an exercise to try some numerical methods on the following PDE: $$\frac{\partial^2u}{\partial t^2}=\frac{\partial^3u}{\partial x^3}$$ I tried to find some information about it, but I do not ...
2
votes
3answers
56 views

Why do I get a big relative error for my function? (Numerical Analisys - floating point)

When evaluating on the computer the following function: $$f(x)=\frac{x^2}{(\cos(\sin(x)))^2-1}$$ there is a big relative error for values $x\approx0$ (values very close to zero). I used the Taylor ...
0
votes
1answer
32 views

Advice to solve a system of 8th order univariate polynomials

I am struggling to solve a least square problem in which the tedious part is the initialization. Grid search methods are out of question. The initial problem I've stated my problem in a previous ...
1
vote
2answers
35 views

If root is very near to the max/min, what happenns with Newton Raphson method? Does it diverge?

If root is very near to the max/min, what happens with Newton Raphson method? Does it diverge? Or converges slowly? I know if some iteration involves a stationery point then we can not go further. But ...
1
vote
2answers
33 views

Newton-Cotes formula problem

Please help me to solve this problem... By the method of undetermined coefficients I found $a=c=1/6$ and $b=2/3$ and $\alpha=\gamma=2/3$ and $\beta=-1/3$. Also that both are exact for polynomials of ...
0
votes
0answers
18 views

(xλ)→(Ax−λxxxT−1) write down Newton's method for this equation

this question is taken from Rainer Kress (numrical Analysis). i coud not translate this question into newton's method form. Because there is matrix and vector. it is hard to take derivation of this ...
1
vote
1answer
28 views

The order of accuracy of the implicit Euler method is equal to $1$

I want to show that the order of accuracy of the implicit Euler method is equal to $1$. That's what I have tried: We have the initial value problem $\left\{\begin{matrix} y'(t)=f(t,y(t)) &, a ...
2
votes
1answer
26 views

Backward Euler method- How do we get the approximation?

Approximating $y'(t^n)$ at the relation $y'(t^n)=f(t^n,y(t^n))$ with the difference quotient $\left[\frac{y(t^{n+1})-y(t^n)}{h} \right]$ we get to the Euler method. Approximating the same derivative ...
1
vote
1answer
21 views

How does one bound computational error for a finite difference approximation of the second derivative?

I'm trying to wrap my head around ways to minimize total computational error (defined as a sum of the bounds on the truncation and rounding errors) by taking a differentiable function $f : \mathbb{R} ...
1
vote
0answers
33 views

Solving system of delay differential equations

Are there any numerical methods for solving systems of delay differential equations with time-dependent delays? For example, I have a system: $$\frac{dP_1}{dt} = f_1(t) P_2(t-\tau(t)) P_3(t)$$ ...
0
votes
1answer
25 views

Help with method(s) show an iterative method converges to a known fixed point

Are there any general techniques that can be used to show that an iterative method converges to a (known) fixed point?. In my current situation, I know the exact fixed point, but I am unaware of a ...
1
vote
0answers
54 views

How to solve this using computer.?

Given $B = \begin{pmatrix} 0.3 & 0 \\ 0 & 0.4 \\ \end{pmatrix}$, and $\pi = \begin{pmatrix}0.4\\0.6\end{pmatrix} $, I need to find the elements of the stochastic matrix (the rows sum to ...
1
vote
0answers
23 views

$A(\theta)-$ stable method, region of absolute stability

We have to look for numerical methods for the numerical solution of $\left\{\begin{matrix} y'(t)=f(t,y(t)) &, a \leq t \leq b \\ y(a)=y_0 & \end{matrix}\right.$ that have 'great' regions of ...
1
vote
1answer
32 views

Proof of an alternate Matrix Condition Number Representation

I'm currently looking over a section in my textbook on Matrix Condition Numbers and it's given the definition $cond(A) = ||A|| \cdot ||A^{-1}||$ but it's also equated this definition of a condition ...
2
votes
1answer
168 views

How can I solve $x$ from $\int_0^x e^{t^3}dt=4$?

I'm interested in numerical analysis but I don't have experience on it. I was wondering how one can solve integral equations numerically like $\int_0^x e^{t^3}dt=4$? I was thinking whether there is ...
0
votes
1answer
24 views

Finding the Tolerance of Adaptive Quadrature Estimations

So I'm learning about the Adaptive Quadrature "Algorithm" for estimating numerical integration and I am have trouble figuring out how you can approximate the error and actual value between two nodes. ...
1
vote
1answer
28 views

How many iterations of interval bisection would be requited to obtain a given level of accuracy?

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(x)=x^{5}+x^{3}+1$. Given that $f$ has a root between -1 and 0, how many iterations of interval bisection would be required to obtain the root to ...
1
vote
0answers
27 views

If the signal's frequency is multiples of the first harmonic frequency, transform method similar to DFT but use less number of samples?

Suppose that a continuous signal $f(t)$ has the first harmonic frequency $f_1$. $f(t)$'s frequencies that are not integer multiples of $f_1$ are known to have zero signal magnitude $|F(\omega)|$. This ...
1
vote
1answer
34 views

Numerical integration of function - result is another function

I'm new in integration and numerical integration. As i know to calculate definite integral you can use some methods, like rects, trapezes, Simpson's... etc. But is there a tool to make numerical ...
4
votes
4answers
180 views

How to compute a lot of digits of $\sqrt{2}$ manually and quickly?

After having read the answers to calculating $\pi$ manually, I realised that the two fast methods (Ramanujan and Gauss–Legendre) used $\sqrt{2}$. So, I wondered how to calculate $\sqrt{2}$ manually in ...
0
votes
1answer
14 views

Residue Calculation Using given Conditions

Assume $f(z)$ is holomorphic on a punctured domain (a is removed), and that $f(z)$ has a pole of order n greater than or equal to $1$. Need To Compute residue at $z=a$ of $f'(z)/f(z)$ How to look ...
2
votes
2answers
59 views

How to calculate the errors of single and double precision

We consider the initial value problem $$\left\{\begin{matrix} y'=y &, 0 \leq t \leq 1 \\ y(0)=1 & \end{matrix}\right.$$ We apply the Euler method with $h=\frac{1}{N}$ and huge number of ...
0
votes
1answer
17 views

Optimalization, plan comparision

Let's say there are two tariff plan options of a provider offering internet access and landline telephony. Option 1: DSL flatrate, landline flatrate : 29,95 \$ Option 2: DSL flatrate: 24,95 \$ , ...
0
votes
1answer
56 views

Solving Lotka-Volterra model using Euler's method

I am trying to solve Lotka-Volterra prey and predator model using Euler's method. Let $p$ be the prey density and $q$ is the predator density, thus: $$\frac{dp}{dt} = ...
1
vote
0answers
25 views

How to compute the natural spline interpolant

Given a set of points $\{x_i\}_{i=1}^N$, how would I compute the natural spline interpolant $s(x)$ $\mathbb{S}^2_3$ of a function $f(x)$?
1
vote
1answer
16 views

Interpolation question

I have a polynomial such that for i = 1, 2,..., N distinct nodes. How would I show that this possesses a unique solution and find a way of constructing it?
1
vote
1answer
57 views

Eigenvalue of the sum of a symmetric matrix and the outer product of it's eigenvector

I have a symmetric matrix $A$ with eigenpairs $(\lambda_k, v_k)$ with $k \in (1,..,n)$. A new matrix $B$ is made from an eigenpair $(\lambda_i, v_i)$ like this: $$B = A - \lambda_i v_i v_i^T$$ where ...
1
vote
2answers
64 views

Newtons's Method

If when you are using Newton's method and your results are just going back and forth between two values, say $0$ and $1$. It is $f(x)=x^3 -2x+2$ starting with $x=1$. What is the reasoning behind ...
0
votes
1answer
17 views

Prove $|\Delta y(x)| \leq \frac{\epsilon}{2}x(1-x)$

We have boundary value problem: $$y'' = g(x),\; y(0)=y(1)=0$$ and the solution: $$y(x) = \int_{0}^{1}G(x, \xi)g(\xi)d\xi$$ with: $$G(x,\xi) = \begin{cases} \xi(x-1) & \text{for } 0 \leq \xi \leq x ...
0
votes
0answers
96 views

QR fatorization for tridiagonal matrices

Let $$A = \left[\begin{array}{rrrr} \delta_1&\gamma_2 & &0 \\ \gamma_2&\delta_2 &\ddots & \\ &\ddots &\ddots &\gamma_n \\ 0 & &\gamma_n ...
3
votes
1answer
39 views

prove that $|\lambda(H) - \lambda(B)| \leq \sqrt{||(C^HC)||_2}$

Let A, B be Hermitian square matrices and $$H = \left[\begin{array}{rr}A & C \\ C^H & B\end{array}\right]$$ Show every eigenvalue $\lambda(B)$ of B, there is an eigenvalue $\lambda(H)$ of H ...
2
votes
1answer
19 views

psuedo inverse of a matrix counter-example

I have proved $A^{\dagger}A = I$ for a $m$ by $n$ matrix with $m\geq n$ and $\text{rank}(A) = n$ I am trying to find a counter example which shows that $AA^{\dagger} \not= I$ but to no avail. The only ...
1
vote
1answer
20 views

Proof that strictly tri-diagonally dominant matrix has an inverse

We are given the following theorem of which we need only know the result. Theorem Suppose an $n\times n$ matrix $A= (a_{ij})$ is tri-diagonal with $a_{i,i-1}a_{i,i+1} \neq 0$, for each ...
1
vote
2answers
51 views

Show $y(x) = \int_{0}^{1}G(x, \xi)g(\xi)d\xi$

We have boundary-value problem$$y'' = g(x)\; y(0)=y(1)=0,$$ Show that:$$y(x) = \int_{0}^{1}G(x, \xi)g(\xi)d\xi$$ with $$G(x,\xi) = \begin{cases} \xi(x-1) & \text{for } 0 \leq \xi \leq x \leq 1 \\ ...
1
vote
1answer
24 views

Find a bound for the error in this interpolation (Interpolation inside)

Consider the piecewise constant interpolation of the function $f(x) = ln(x)$ , $10 ≤ x ≤ 11$ , at points $x_i = 10+ih$, where $h = 0.1$. Thus, our interpolant satisfies $v(x) = ln(10+ih)$, $10 + ih − ...
0
votes
2answers
41 views

Sine function definition formula

I just learned about the definition of the sine function. $$ \frac{e^{ix}-e^{-ix}}{2i}$$ I wanted to make an accurate implementation of this function. When I plugged $e^{ix}$ when $x = 5$ on my ...
1
vote
0answers
24 views

Continuity of partial derivative notation

Given a function $f(x,y):X \times Y \rightarrow \mathbb{R}$ where $X\subset \mathbb{R}^3$ and $Y \subset \mathbb{R}^3$. Let say I want to impose additional regularity. Assume: $f(x,y) \in ...
0
votes
1answer
21 views

Proportion unitary method

3 kitchen assistants can prepare 10kg of vegetables in 30 minutes. How long would it take 15 assistants to prepare the same amount?
1
vote
2answers
72 views

How to find roots of 8th degree equation

I have tried to solve this 8th degree equation & calculated one approximate real root using Newton-Raphson method it is $x=1.340775827$. How to find other real roots of this equation ...
0
votes
0answers
25 views

Finite Difference method for nested derivatives

I'm moderately experienced with finite difference methods, but I'm hoping that somebody has better intuition than I do. Suppose I have the following expression which I want to evaluate via finite ...
1
vote
1answer
19 views

“Symmetric” numerical computation of second derivative

When numerically computing a first derivative, it is better to use $$f'(x) \approx \frac{f(x + \Delta x / 2) - f(x - \Delta x / 2)}{\Delta x}$$ than to use $$f'(x) \approx \frac{f(x + \Delta x) - ...
0
votes
0answers
16 views

Prove an equation involving newton forward difference formula

I have to prove the following equation: I have substituted: $\Delta f(i) = f(i+1) - f(i) $ $\Delta ^2 f(i) = f(i+2) - 2f(i+1) + f(i) $ $\Delta ^3 f(i) = f(i+3) - 3f(i+2) + 3f(i+1) - f(i) $ ...
2
votes
1answer
49 views

Construct a symmetric Poisson matrix from $\nabla \cdot (\rho\nabla p)$

Let $\rho$ and $p$ be 2D scalar fields. How do I construct a symmetric Matrix $A$ so that is satisfies \begin{equation} Ap = \nabla \cdot (\rho\nabla p) \end{equation} in finite differences? I am ...
0
votes
0answers
24 views

the rank of QR decomposition

I saw this in a paper, where one has a QR decomposition $C=QR$ ($C\in R^{m\times r}$, $Q\in R^{m\times r}$ is column orthogonal, $R\in R^{r\times r}$, $m>r$). However, under the condition that the ...
1
vote
1answer
37 views

Newtons Divided Difference practice problem

I haev a midterm tomorrow in Numerical Analysis and I was attempting to work my way through a practice problem. However I get stuck. I notice we have 2 points and can find a degree 1 polynomial using ...