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

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Identifying the iteration Scheme

The iteration $x_{n+1} = \frac{1}{2} (x_{n} + \frac{2}{x_{n}}) , n \ge 0$ for a given $x_{0} \ne 0$ is an instance of fixed point iteration for $f(x) = x^2 - 2$. Newton's method for $f(x) = x^2 ...
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1answer
12 views

Matrix for Mixed Boundary Value Problem

My friend and I have been working on numerical solving the following equation $$-u'' = f$$ with $x \in [0,1] $ , $ u'(0) = 0$, $u(1) = 0$. Analytically, we found the eigenvalues and eigenfunctions ...
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0answers
23 views

Singular matrix with homogeneous Neumann boundary conditions?

I'm numerically solving a PDE on a rectangular domain using a 9-point central difference scheme and purely homogeneous Neumann boundary conditions. Should the resulting matrix be singular? I thought ...
1
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1answer
18 views

Rate of convergence for quadratures

How can I find the observed (practical) order of convergence of a quadrature? I remember the formula $\frac{|x_{n+1}-x_{n}|}{|x_{n}-x_{n-1}|^q}$ but does this work here aswell? This formula gives me ...
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0answers
39 views

Numerically solve an equation using Python

How could I (numerically) solve this equation for $\alpha$ given $x_i$ (these are known) ? $\sum_{i=1}^N\frac{1}{x_i-\alpha} = \frac{2N}{\sum_{i=1}^{N}(x_i-\alpha)^2}\sum_{i=1}^{N}{(x_i-\alpha)}$ In ...
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0answers
29 views

Implicit system differential equations

I came across a system of differential equations in the form: $\newcommand{\D}[1]{\frac{\mathrm{d}#1}{\mathrm{d}x}}$ \begin{align} f_1(x,y,z)\D{y}+f_2(x,y,z)\D{z}&=f_3(x,y,z),\\ ...
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0answers
19 views

Richardson Extrapolation, numerical analysis [on hold]

Show how to use Richardson extrapolation employing ${x}_{n}$ and ${x}_{2n}$ if $$L={ x }_{ n }+{ a }_{ 1 }{ n }^{ -1 }+{ a }_{ 2 }{ n }^{ -2 }+{ a }_{ 3 }{ n }^{ -3 }+...$$
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0answers
19 views

sturm-liouville differential equations

I need method to find the values of $\lambda$ $$1.-y''+x^2 y=\lambda y$$ $$2. -y''+|x| y=\lambda y$$ $$ 3. -y''+(x^2 +x^4) y=\lambda y$$ with initial condition $y(0)=1,y'(0)=0$
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1answer
33 views

How to prove coercivity

I have a problem in understanding how to prove if a function is positive or negative coercive. I understood the definition of coercivity, which is: $$\lim_{||x|| \to +\infty}f(x) = +\infty$$ However, ...
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0answers
24 views

Gauss- Legendre and Simpson's rule for a double integral in MATLAB

I'm trying to resolve $$f(x,y)=\int_{0}^{1}\int_{0}^{2}e^{-xy}dxdy$$ with Gauss- Legendre quadrature and Simpson's rule, but being honest I have no idea how to write it on MATLAB. So, that's my ...
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0answers
6 views

Finite element method - master element

What is supposed to be a quadratic master element with 3 degrees of freedom? I think I have to consider a 3D case, with x,y,z directions...is it? And I think it is quadratic because I have to go ...
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0answers
13 views

Solution of ode system (morphogenesis) has one value once large once small

I solve Turing's morphogenesis with code available in following question: Solve Turing's morphogenesis with other method than Euler's The problem is: the pictures are nice, however there ...
2
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1answer
34 views

Prove existence of divergent sequence in Newton's method

Given $ f(x)=x^3-1$, how to prove that there exists a sequence of initial values $x_{0,1}>x_{0,2}>x_{0,3}>...$ where $x_{0,1}=0,x_{0,2}=-2^{1/3}$, such that the sequence produced by Newton's ...
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1answer
31 views

Household reflector or transformation

Let $A\in\mathbb{R}^{n\times k}$, $n\geq k$, and $rank(A) = k$. Consider the use of Household reflectors, $H_i$, $1\leq i\leq k$, to transform $A$ to upper trapezoidal form, i.e., ...
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0answers
16 views

Solve Turing's morphogenesis with other method than Euler's

I'm currently using following code to solve Turing's morphogenesis: ...
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0answers
40 views

Numerical methods and KKT in NLP

I am studying numerical methods and NLP. I started with gradient based methods, newton methods and KKT conditions. I found the following sentence: A local minimum is found by solving KKT conditions, ...
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0answers
41 views

Linear system equations

I need to get, preferably by a numerical method, a solutions of: $$\left\{\begin{array}{lll} 2\sum_{i=1}^n b_{ik}x_i+x_{n+1}=0&\text{for}& k=1,2\ldots,n\\ \sum_{i=1}^n x_i=0 ...
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0answers
21 views

Computational complexity of conjugate gradient method for a positive semidefinite Hermitian matrix

Let us assume that we want to solve the linear system: $$\mathbf{A}\mathbf{x} = \mathbf{b}$$ with the conjugate gradient method. $\mathbf{A}$ is a positive semi-definite Hermitian matrix. The ...
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0answers
31 views

Solving system of nonlinear equations via iteration

I will give an example to illustrate the question: Assume I have the system: $$ xy + x + y = 7\\ x^2 + y^3 = 9 $$ and I want to solve for $x$ and $y$. It is a fairly common approach to rearrange ...
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1answer
22 views

Knot sequence for a natural cubic (B-)spline interpolant

say I am given $n+1$ data points $(x_i,y_i)$ with $0\leq i \leq n$ and $x_0 < x_1 < \dots < x_n$. I want to interpolate these with a natural cubic spline $s(x)$ ($C_2$ continuous at knots ...
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1answer
18 views

Elementary reflector $Q$ is orthogonal iff

Recall that an elementary reflector has the form $Q = I + \alpha xx^T\in\mathbb{R}^{n\times n}$ with $\|x\|_{2}\neq 0$. Show that $Q$ is orthogonal iff $$\alpha = \frac{-2}{x^Tx} \ \ \text{or} \ \ ...
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3answers
49 views

Finding a formula for a kth element in a sequence

I've setup a recurrence relation as part of a numerical analysis problem, and found that $$x_{n+1} = \frac{x_n+1}{2}$$ The notes then say that for $x_0=0$, it is easy to show that $$x_k = 1 - ...
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0answers
24 views

Non Linear Systems : Broyden's Method

I am trying to implement Broyden's method for solving systems of non-linear equations following these documents http://heath.cs.illinois.edu/scicomp/notes/chap05.pdf ...
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4answers
34 views

Pipe and Cistern

Pipes $A$ and $B$ can fill a tank in $9$ hours and $12$ hours respectively. Both pipes are opened together to fill the tank, but pipe $B$ is closed after some time. If the tank is full in $6$ hours, ...
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0answers
18 views

Modified Euler method for soving pair of simultaneous equations [closed]

dx/dt=xy+t,x(0)=0 dy/dt=x+t,y(0)=1 Solve by modified euler method for t=0.2 to 0.6
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1answer
90 views

Have I found ALL the solutions to this diff eq & boundary conditions?

If we find a solution to a differential equation and its boundary conditions, how can we know if we have found ALL the solutions? For example, let g(x) be a smooth continuous function of x: (Eq 1) ...
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0answers
12 views

Ask for reference convergence of implicit euler method for initial value problem with dissipative source term

I am considering the convergence of implicit euler method for solving the following initial value problem: \begin{cases} u'(t)=f(t,u(t)),t\in[0,T]\\ u(0)=u_0\in \mathbb{R}, \end{cases} where ...
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1answer
57 views

Can $\int_0^1 \frac{1}{x} e^{-x} dx$ be integrated?

I have an integral with a singularity at $x = 0$. $$\int_0^1 \frac{1}{x} e^{-x} dx$$ It's not a removable singularity so is it possible to perform the integration? For example could some complex ...
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1answer
29 views

using Euler's method to solve this question ($\frac{dv}{dt}=-kA$) [closed]

Suppose that a spherical droplet of liquid evaporates at a rate that is proportional to its surface area. dv/dt= -kA where V=volume (mm3), t =time (min), k =the evaporation rate (mm/min), and A ...
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0answers
21 views

Schur complement of a matrix $A$

Let $A\in\mathbb{R}^{n\times n}$ and its inverse be partitioned $$A = \begin{pmatrix} A_{11} & A_{12}\\ A_{21} & A_{22}\\ \end{pmatrix},\:\: A^{-1} = \begin{pmatrix} \tilde{A_{22}} & ...
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1answer
15 views

Computing $PAQ = LU$ using Gaussian elimination with complete pivoting

Suppose $PAQ = LU$ is computed via Gaussian elimination with complete pivoting. Show that there is no element in $e_i^{T}U$ i.e., row $i$ of $U$, whose magnitude is larger than $|\mu_{ii}| = ...
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0answers
53 views

About a geometric algorithm to compute $\sin$ based on the unit circle

In an old post I have found a user which claims to have a geometric algorithm to compute trigonometric  functions for an angle between $0^\circ$ and $90^\circ$ based on the unit circle. Here's the ...
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0answers
27 views

Interpolation, divided difference [closed]

The third divided difference of the function $f(x)=\frac{1}{x}$ for the points $\left (a,b,c,d \right )$ is equal to
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0answers
28 views

What does “order” exactly mean in numerical methods?

I am trying to understand the concept of order in solving numerical differential equations of the form $\frac{dx(t)}{dt}=f(t,x(t))$. Let's start from the local discretisation error at $t$: ...
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1answer
28 views

The Runge - Kutta method and two-body problem

Is it possible to get an approximation of the two body problem: $$\left\{\begin{array}{lll} x''(t)=-\frac{x}{(x^2+y^2)^{3/2}}, & x(0)=1-\varepsilon, &x'(0)=0\\ y''(t)= ...
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0answers
35 views

Complex Roots (Numerical Methods)

I was given the following question in my Numerical Method exam and I think it is related to Newton's Basis Polynomial, but couldn't solve it. Could anyone guide me to the solution? Show that for ...
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1answer
41 views

Numerical integration in Matlab (Gaussian 3 point quadrature)

Write a Matlab function that applies the Gauss three point rule to N sub-intervals of $[a, b].$ The input parameters should be the name of the function being integrated, $a, b,$ and $N$. Attempt: ...
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0answers
22 views

Truncation Error of 2-stage Runge-Kutta Method

I'm trying to derive the truncation error for the 2-step Runge-Kutta Method given by $$k_1 = f(x_n,t_n)$$ $$k_2 = f \left(x_n+\frac{2\Delta t}{3}k_1,t+\frac{2\Delta t}{3} \right)$$ $$x_{n+1}=x_n + ...
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0answers
40 views

Numerical method for solving equation with $u \frac{\mathrm{d}u}{\mathrm{d}x} + u$

I'm looking for a finite difference method to solve $$a(x) u \frac{\mathrm{d}u}{\mathrm{d}x} + u = b(x)$$ where $u(0) = c$. I tried to do a lagging convergence on the $u$ ie $$a(x) u^{(n)} ...
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0answers
14 views

Applying Boundary Condition to Finite Element Matrix

Several times now I have seen the following done without justification and I cannot figure out why it can be done: Consider the 1 dimensional "pde" $-u'' = f, u(0) = a, u(1) = b$ over $[0,1]$. We ...
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0answers
25 views

Trapezoidal Rule Mathematical Error

I want to find the absolute error for Trapezoidal rule numerical error,so I have this function: $\displaystyle f(x)=\frac{1}{1+x^2}$, the type of error is: $\displaystyle \epsilon \leq \left| ...
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0answers
18 views

Example of “no analytical solution”

Is there a good test for no analytical solution? How can I learn the difference between equations that have an analytical solution and the ones that need numerical methods ("unsolvables" in analysis)? ...
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0answers
17 views

Gauss Seidel - Finite Element Method

I am solving an equation using finite element method, and for that I have to use Gauss Seidel to invert a matrix. In Gauss Seidel I am using a "while" which breaks if the absolute error reaches the ...
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2answers
63 views

Approximate solution of a trigonometric equation using only pen and paper

I found an exam question that I managed to solve via calculator but not by using only pen and paper. Is there a solution to this? Prove that there is an $x$ satisfying $10x-9 = 9\sin x-10\cos x$ and ...
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1answer
15 views

Gauss transforms to factor $A = LU$

Consider a symmetric matrix $A$, i.e., $A = A^{T}$. Consider the use of Gauss transforms to factor $A = LU$ where $L$ is unit lower triangular and $U$ is upper triangular. You may assume that the ...
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0answers
34 views

Numerical methods for ODE: Implicit, explicit, stability, stiffness

Hy everybody! I am new to the subject "numerical methods for ODE". I read some basic literature but since most of the concepts and methods are new to me, I wanted to ask you, if you could give me ...
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0answers
27 views

Numerical methods for ODE: Taylor vs. Interpolation approaches

Hy everybody! I am new to the subject "numerical methods for ODE". I read some basic literature but since most of the concepts and methods are new to me, I wanted to ask you, if you could give me ...
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1answer
45 views

Gauss-Legendre three point rule

Use the change of variables $$x=\frac{a+b}{2}+\frac{b-a}{2}t,$$ to show that $$\int^b_a f(x) \ dx = \frac{b-a}{2} \int^1_{-1} f\left( \frac{a+b}{2} + \frac{b-a}{2}t \right) \ dt.\tag{1}$$Hence ...
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0answers
20 views

Solving Poisson Equation Finite-difference using Python

Hi I'm trying to compute numerically the solution to the next Poisson equation: $$ \dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^{2}u}{\partial y^{2}} = 4 $$ with the boundary conditions $$ ...
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1answer
27 views

Condition number for each variable

Condition number of a matrix tells us how viable it is to solve $Ax=b$ $$A= \begin{bmatrix} 1.001&1\\ 1&1 \end{bmatrix} $$ Is a matrix that would be difficult to solve numerically. However ...