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
0answers
4 views

Significance of the complex conjugation symmetry of the DFT for real-valued input

For real-valued input $\mathbf{x} = (x_0, ..., x_{N-1})$ and its discrete Fourier transform (DFT) $\mathbf{X} = \mathcal{F}(\mathbf{x})$ we have that $$X_{N-k} = X_k^*$$ where * denotes complex ...
0
votes
0answers
6 views

method of undetermined coefficients and come up with a new quadrature.

I'm trying to solve some problems which is related method of undetermined coefficients to determine some weights and to come up with a new quadrature. for a function with $x_1=\frac{1}{3}$, ...
0
votes
0answers
11 views

Compute the condition number of the matrix and show for what $\Delta x$ it is singular

Given the laplacian $N \times N$ matrix \begin{align*} A=\frac{1}{(\Delta x)^2}\begin{pmatrix} 2&-1& & &\\ -1&2&-1& &\\ &\ddots&\ddots&\ddots&\\ ...
1
vote
0answers
5 views

Numerical Method for KdV travelling waves

Can someone please direct me to the best numerical method or some references for solving $$-cu_x + uu_x + u_{xxx} = 0$$ with periodic boundary conditions. This governs travelling waves of the KdV ...
0
votes
0answers
25 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? ...
3
votes
2answers
51 views

Efficiently solving many sets of linear equations without inversion or factorization

Suppose I have the normal set of linear equations $Ax = b$. If I can store and manipulate $A$ I have a variety of techniques available to me such as inversion, factorization, or an iterative method. ...
0
votes
0answers
15 views

Newton Method Variant with convergence of order 3

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be twice continuously differentiable for all $x$ in the neighborhood $U(\xi)=\{x\in\mathbb{R}:|x-\xi|<r\}$ of a simple zero $\xi$ of $f$ such that ...
0
votes
3answers
35 views

What does the function domain with letter C stand for?

I am reading a mathematics textbook on the subject of numerical analysis. In one theory the author says let us assume $f$ to be a function in $C^{n+1}[a,b]$. I understand that $[a, b]$ is the ...
1
vote
1answer
15 views

Accurate summation of mixed-sign floating-point values

Due to the finite representation, floating-point addition loses significant bits. This is particularly noticeable when there is catastrophic cancellation, such that all the significant bits can ...
2
votes
2answers
22 views

Reciprocal of a quadratic form

I am working with an expression of the form $$ \frac{x^TAx}{{x^TBx}}$$ and would like to simplify it. I understand that vectors do not have inverses, but viewing the bottom number as a 1 by 1 matrix, ...
0
votes
0answers
15 views

Big 'O' Notation - Taylor Series

Q) Use the Taylor Series Expansion to show the first derivative f '(x) can be approximated by $$-(3f(x) -2f(x+h) - f(x-h) / h ) $$ What is the precision? Now I found after using the Taylor ...
2
votes
2answers
71 views

Prove that $f:\mathbb{R}^2\to\mathbb{R}^2$ has a fixed point in a subset of $\mathbb{R}^2$

Let $f:\mathbb{R}^2\to\mathbb{R}^2$ by defined by $f(x_1, x_2) = \begin{pmatrix} \frac{1}{3}x^2_2 + \frac{1}{8} \\\ \frac{1}{4}x^2_1 - \frac{1}{6} \end{pmatrix}$ and let $D = \{ x \in \mathbb{R}^2 ...
4
votes
1answer
61 views

Any math competitions dedicated to calculations by hand (college level math)?

Most of the people consider hand calculations the thing of the past. However, I recently started thinking about it and there are many interesting ways to do basic arithmetics on large numbers, ...
0
votes
0answers
10 views

Find analytically the sequence of iterations $x_n$ for Newton's method applied to the function $f(x) = x^2$ with the starting point $x_0 = 1$.

Find analytically the sequence of iterations $x_n$ for Newton's method applied to the function $f(x) = x^2$ with the starting point $x_0 = 1$? How would you do this, we can't use the recursive ...
0
votes
0answers
17 views

reformulating a BVP into a system of first order ODEs

I need to convert the following bounded value problem $y'''+3y''+2y'^2-5y^2=1$ with conditions $ y(1)=1, y'(0)=1 $ and $y''(1)=0$ Into a system of first order Initial value problems in order to ...
0
votes
0answers
14 views

Connection between power iterations and QR Algorithm

I am seeking an intuitive understanding of why the QR Algorithm solves the symmetric eigenvalue problem. In class, and also in Golub and Van Loan, it has been suggested that there is somehow deep ...
0
votes
1answer
12 views

Error and residue in linear system

Consider the linear system $A x = b$ with a computed solution, what will be the relation between error and residue?
0
votes
0answers
9 views

$L_2$ norm of trigonometric interpolation

Let $I_N f(x)$ be the trigonometric interpolation of discrete function $f(x)$ with Fourier coefficient $g_l$. How can I prove this relation: $$\|f||_2=\|I_N f(x)||_2$$
0
votes
2answers
11 views

Implementing Adams-Bashforth of order 2 (AB2) algoirthm

Assuming we are given the initial condition for an ODE such that: $$ \begin{cases} x' = f(x,t) \\ x(t_0) = x_0 \end{cases} $$ We are going to solve it numerically using AB2. We know that the ...
0
votes
0answers
5 views

CG as an orthogonal projection

I have heard that the Conjugate Gradient method can be viewed as an orthogonal projection onto the Krylov subspace $K(A,r_0)$, but I can't find a reference that deal with it in this way. Could you ...
1
vote
2answers
58 views

Fixed point iteration question

This image was taken from this youtube video https://www.youtube.com/watch?v=OLqdJMjzib8 Since only one of them converges, how do we know in advance which formula to work with?
0
votes
0answers
9 views

Approximating integral of Erf with certain available functions.

I am developing certain software that deals with symmetric 2D Gaussian densities. One of the most common operations in that software is integrating those Gaussians over various 2D shapes. These ...
1
vote
0answers
19 views

Why settle for Lagrange Interpolation when doing linear multistep ODE integration?

Say that we have some initial value problem: $y'(t) = f(t,y(t)) ; y(0) = y_0$ with $y_0$ and $f(t,y(t))$ known. If we use Euler's method to numerically approximate the first k points, then we have ...
0
votes
0answers
15 views

Help understanding a homework problem (Preconditioning matrices, numerical methods)

Below is a link to the problem (because I didn't want to have to go through the pain of TeXing it all out myself), the basic idea is we are supposed to be first showing that a specific matrix has a ...
0
votes
0answers
21 views

Solving traveling wave usin the shooting method

The spatially-dependent Hodgkin-Huxley equation for a cylindrical dendrite or unmyelinated axon: where $\frac{a}{2\rho}\frac{\partial^2V}{\partial x^2}$ is a diffusion term $a$ is the fiber radius, ...
2
votes
3answers
44 views

Numerical solution of an ODE system of equations using RK4

I have given an assignment to find the solution to the ODE system of equations as follow: $$\begin{cases} x_1' = x_1 + x_2 \\ x_2' = -3x_1 -10x_2 + x_2 ^2\end{cases}$$ With initial conditions: ...
2
votes
0answers
35 views
+50

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
1
vote
2answers
71 views

Solving equations like $xe^x = c$ via functional iteration

Yesterday I randomly thought of solving $xe^x = c$ via functional iteration (FI) after manipulating the equation into a form "$x = \cdots$" that gives the 'fastest' convergence rate regardless of the ...
0
votes
0answers
10 views

Comparing smoothness among approximations

We are interpolating a missing fragment of a 2D curve given a set of sample points. Our method generates several candidates of curve pieces to fill the missing part, but we want to select the solution ...
0
votes
0answers
20 views

How many Gauss points are required to provide exact value for the Gauss quadrature rule

How many Gauss points are required if the Gauss quadrature rule should provide the exact value of the integral $I=\int_{-1}^1f(x)dx$ for $f(x)=(x^2-1)^2$? I am really not sure what theorem to use to ...
0
votes
0answers
5 views

How to deal with the composite function in a numerical approximation problem?

Consider a quasilinear two-point boundary value problem: $$-(a(u)u'(x))' = f(x) , x\in (0,1)$$ with $a(u)>0$ and $u(0) = 0, u(1) = 0$. I am supposed to derive an algebraic system so that I can ...
2
votes
2answers
41 views

Show that the matrix $AA^T+\alpha I$ is positive definite, where $\alpha >0$ and $A$ is an $m\times n$ real matrix.

Show that the matrix $AA^T+\alpha I$ is positive definite, where $\alpha >0$ and $A$ is an $m\times n$ real matrix. So I need to show that $x^T(AA^T+\alpha I)x>0$ for all vectors $x$. I'm ...
0
votes
1answer
34 views

Simpson's rule is not producing better results than Riemann sums

I have to calculate RMS value $\sqrt {\int_0^T\frac 1T*f(t)^2dt} $ and I know from the maths that the Simpson's rule should provide better approximation of the definite integral than the Riemann sums. ...
0
votes
1answer
22 views

Some trivial but confusing terms about numerical integration

Some terminological questions about numerical integration: When a question states trapezoidal rule with 2 points, does that mean 2 subintervals or 3 subintervals? Since 3 subintervals have 2 points ...
1
vote
1answer
22 views

Proving error bound on Simpson's Rule, Numerical Integration

The approximation from "Simpson's Rule" for $\int_a^b f(x)\, dx$ is, \begin{equation} S_{[a,b]}f = \bigg[\frac{2}{3}f\Big(\frac{a+b}{2}\Big) + \frac{1}{3}\Big(\frac{f(a) + f(b)}{2}\Big)\bigg](b-a). ...
0
votes
0answers
19 views

Fourier series for absolute value of sin functiom

If we take the absolute value for sin function, then it becomes even. However, isn't period of this function pi? To find fourier series, 1.Even 2. period 2 pi. Can we just treat this function as ...
0
votes
0answers
10 views

Solving a matrix of ODEs with an invariant of the matrix as a variable coefficient

I have the following system of ODEs: $$ \dot{\mathbf A} (t) + c \thinspace I(\mathbf A(t)) \thinspace \mathbf A(t) = \mathbf 0,$$ where $I$ is, say, the second invariant of the symmetric matrix ...
0
votes
1answer
21 views

Transforming an integral to a different domain

For a given $v(x)$ with $x\in[0,1]$, use the variable transformation $x=g(\eta)=\frac{1}{2}\eta+\frac{1}{2}$ to transform the integral $I=\int_0^1v(x)dx$ to an integral over $[-1,1]$. My doubts: ...
1
vote
1answer
42 views

Numerically Solve a Second Order ODE with singular coefficients

I need to solve the following numerically: $$xy''+y'+xy=x$$ with initial conditions $y(0)=0$ and $y'(0)=1$. I need the solution for $x:[0, 10]$. I've written the ode as a system of first order odes ...
-3
votes
0answers
17 views

Estimate the number of candidates who obtained fewer than 70 scores.

In an examination, the number of candidates who obtained scores between certain limits are as follows: Scores $0—19$, $20—39$, $40—59$, $60—79$, $80—99$, Number of candidates $41$, $62$, $65$, ...
1
vote
0answers
19 views

Numerical stability of computational results

Let z be a function of a finite number of variables i.e. z=f(a,b,c,...). If we have the mathematical formula connecting z and the variables, we can determine how the value of z varies with a change ...
1
vote
0answers
11 views

How to build a matrix in MATLAB with the next characteristics?

Let $\lambda_1=\frac{k D_u}{2h^2}$ a constant value. How to generate a matrix in MATLAB with the next entries: $A= \begin{pmatrix} 1+\lambda_1 & -\lambda_1 & 0 & 0 & \cdots & 0 ...
0
votes
0answers
22 views

Numerical Analysis Stability or Condition

Im not sure if its the right place to ask this question since it may be more of an informatics problem and I apologize if its the case: I want to evaluate a certain function, $\frac{log(cos(x))}{x}$ ...
0
votes
0answers
20 views

Problem with code for numerical integration in matlab.

Hi I have problem with calculating this expected utility with means of numerical integration, using matlab. The matrix stock data is a 1000x8 matrix with columns representing 7 stocks (columns 2-8, ...
1
vote
2answers
34 views

Prove limit of a sequence in Newton's method

Given the $ f(x)=x^3+x-1 $, I have shown so far that $ f$ has a unique root $r\in(0,1)$ and that for the sequence $(x_{n}), n>=0$ produced by Newton's method we have $$\lim_{n\to\infty} x_{n}=r$$ ...
1
vote
1answer
18 views

Normal system of the least square method

I'm trying to show the following. $Pa$ is the approximation system of $y$. I want to show that finding the minimmum for the function $$f(a,y)=||Pa-y||_2^2$$ is equivalent to solve the normal system of ...
0
votes
0answers
11 views

Equation involving Bessel and Struve functions

I need to solve the equation $Z(\gamma) = r$ of the function $$Z(\gamma) = 1 - \frac{2}{\gamma} \left(J_1(\gamma) - i H_1(\gamma)\right),$$ where $J_1$ is the Bessel function and $H_1$ the Struve ...
2
votes
1answer
30 views

$LDL^t$ Factorization Algorithm to find a factorization of the form A

For $$ \begin{pmatrix} 4 & 2 & 2 \\ 2 & 6 & 2 \\ 2 & 2 & 5 \\ \end{pmatrix} $$ I found that $$ L=\begin{pmatrix} 1 & ...
0
votes
0answers
24 views

LU factorization Algorithm

How to show that the LU Factorization Algorithm requires $n^3/3-n/3$ multiplications/divisions and $n^3/3-n^2/2+n/6$ additions/subtractions?
0
votes
1answer
21 views

Numerical solution of ordinary differential equations, multistep method

I try to solve the following question, but I have no clue why we have $x'$ in the RHS: The formula $ x_{n+1} = (1-A)x_n + A{x_{n-1}} + \frac{h}{12}[(5-A)x'_{n+1}+8(1+A)x'_n + (5A-1)x'_{n-1}] $ is ...