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

learn more… | top users | synonyms (2)

0
votes
1answer
13 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}| = ...
8
votes
0answers
130 views

Mathematical conjectures for which Numerical Search promising

I am fascinated by mathematical conjectures, especially those that are believed to be false. Now I wonder, are there some mathematical conjectures for which the numerical search would be promising? ...
0
votes
0answers
22 views

Linear system equations

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

Computational complexity of conjugate gradient method for a positive-semi definite 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 ...
0
votes
0answers
24 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 ...
1
vote
1answer
57 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) ...
5
votes
4answers
1k views

Can I approximate sine and cosine without derivatives?

Assuming I don't know derivatives (and Taylor series) can I manage to approximate sine and cosine of a generic given (rational) angle in radians?
0
votes
1answer
30 views

What does this notation of defining a function mean?

I have found this formulation there $f$ is a mapping, $f(x)(t) = t(x(t)+1)$ How am I supposed to correctly understand that? I mean is the left hand side the same as $f(x,t)$ or what is the idea of ...
0
votes
0answers
10 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 ...
1
vote
3answers
42 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 - ...
0
votes
1answer
16 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} \ \ ...
0
votes
0answers
16 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 ...
3
votes
1answer
859 views

Derive error term by using Taylor series expansions.

Using Taylor series expansions, derive the error term for the formula \begin{equation} f''(x)\approx \frac{1}{h^{2}}\left [ f(x)-2f(x+h)+f(x+2h) \right ]. \end{equation} I've tried it on my own ...
0
votes
4answers
32 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, ...
-1
votes
0answers
15 views

Modified Euler method for soving pair of simultaneous equations [on hold]

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
0
votes
2answers
62 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 ...
0
votes
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 ...
4
votes
0answers
99 views

Large system of nonlinear equations

I am trying to solve a problem, which I find quite hard, like, headache-hard. I have to solve the following set of $M$ nonlinear equations: $$F(X)=\begin{bmatrix}f_1 (X)\\f_2 (X)\\...\\f_M (X)\\ ...
0
votes
1answer
56 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 ...
-1
votes
1answer
28 views

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

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 ...
4
votes
2answers
682 views

Software for numerical solution of a non-linear ODE system?

I have been given a nonlinear system of ODEs which has arisen out of a colleague's engineering research: $$\begin{array}{rcl} \dot{x}_0&=&x_1\\ ...
0
votes
0answers
19 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}} & ...
1
vote
0answers
51 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 ...
2
votes
0answers
60 views

Are two linear system equivalent? [on hold]

Let $A$ and $M$ be square matrices of size $s$ and $n$ respectively, let $k_i \in\mathbb{R^n}$ be column vectors for all $i=1,\ldots,s$. Denote $K=\left[ \begin{matrix} {{k}_{1}} \\ \vdots ...
-5
votes
0answers
27 views

Interpolation, divided difference [on hold]

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

Solution to equations involving Plasma dispersion function

I am trying to solve an equation involving a complex argument for the plasma dispersion function as: $z = x + \iota y$, $ x = \omega / \sqrt2 k v_{Ti} $ $ y = \nu_i /\sqrt{2} k v_{Ti} $ $S[z] = ...
0
votes
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 $t$: ...
4
votes
2answers
554 views

Fixed-Point Iteration method unable to converge to any of a function's infinte roots

An equation is given to me which has to be solved by direct iteration method: $$\sin(x) = {x+1 \over x-1}$$ or $$f(x)=\sin(x)-{x+1 \over x-1} = 0$$ I follow the following procedure with reasons ...
0
votes
1answer
40 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: ...
0
votes
1answer
27 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)= ...
0
votes
1answer
45 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
vote
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 ...
0
votes
0answers
21 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 + ...
0
votes
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)} ...
0
votes
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 ...
0
votes
0answers
24 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| ...
0
votes
1answer
384 views

Runge Kutta Method Matlab code

So I have a programming assignment with the following instructions: Consider the nth-order differential equation $$Ax^n (t) = x ^{(n-1)}(t) + x^{(n-2)}(t) + ... + x(t)$$ where $A$ is a ...
0
votes
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)? ...
0
votes
0answers
16 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 ...
0
votes
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 ...
3
votes
1answer
57 views

$LU$ Factorization

Suppose the $A\in\mathbb{R}^{n\times n}$ is nonsingular and that $A = LU$ is its $LU$ factorization. Give an algorithm that can compute, $e_i^TA^{-1}e_j$,i.e., the $(i,j)$ element of $A^{-1}$ in ...
2
votes
0answers
32 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 ...
2
votes
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 ...
0
votes
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 ...
3
votes
1answer
24 views

Explicit Finite Difference Scheme For Approxating a p.d.e

$\frac{du}{dt} = \frac{d}{dx}[\frac{1}{x^2+1}\frac{du}{dx}]$ I am trying to approximate this pde with a finite difference scheme but I am confused with the d/dx. Do I just take the derivative of ...
0
votes
0answers
19 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 $$ ...
0
votes
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 ...
1
vote
1answer
47 views

Error for Trapezoidal Rule in multi-variable integrals

For one dimension integrals $\int_{a}^{b}f(x)dx $, we know the global truncation error goes like$\ \approx\mathcal{O}(h^2)$ where $h=\frac{b-a}{N}$ and N is the number of intervals. Also knowing how ...
8
votes
6answers
3k views

An alternative way to calculate $\log(x)$

How can I replace the $\log(x)$ function by simple math operators like $+,-,\div$, and $\times$? I am writing a computer code and I must use $\log(x)$ in it. However, the technology I am using does ...
1
vote
0answers
17 views

Finite difference method and division by zero problem with no flux boundary condition

I am trying to implement an angionesis model described by Anderson and Chaplin in 1998. The model is based on a set of PDEs defined on an unit square with the following no-flux boundary condition ...