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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2answers
24 views

Why is it problematic for a direction to be a linear combination of another in Powell's Method?

This is the algorithm for Powell's Method, as presented in a book, Numerical Methods in Engineering with Python. The section in the book goes on to explain that, when not dealing with quadratics, the ...
2
votes
2answers
39 views

What does it mean for the hessian to have a value?

The hessian matrix is the matrix formed by taking the second derivatives of some vector $X$. $$\nabla^2 X = H$$ In determining whether a function at critical point is a minimum or maximum, we test ...
2
votes
1answer
32 views

Best step in Runge Kutta 4

I'm simulating radioactive decay of an atom and I have a system of 6 differential equations that i'm solving numerically with runge kutta 4. I have to simulate that decay for about 40 days which ...
5
votes
2answers
314 views

Numerical analysis, differential equations, complex analysis for statistics

I am a student of the Statistics Department. And now I can choose a subject from such list: numerical mathematics (analysis), differential equations, complex analysis, real analysis. Can you please ...
6
votes
0answers
52 views

Stiff Nonlinear Differential Equations

As far as I know, the concept of stiffness is hard to define rigorously, but there are plenty of handwavy descriptions and motivating examples in the literature when it comes to linear differential ...
2
votes
0answers
25 views

FEM for a 1D heat equation system

I want to know how to implement the (nonhomogeneous) initial boundary value problem for a heat equation; $$u_{xx}=u_t ~~~x\in (-1,1),~t\in(0,1)$$ $$u(0,x)=u_0(x)$$ $$u(t,-1)=f(t), ~u_x(t,1)=0$$ Many ...
0
votes
0answers
16 views

Fourier transform of wave function and momentum of particle

Let $$\Psi\left ( x,t \right )$$ represents the wave function of a particle at some position x and time t. $$\Psi\left ( x,t \right )=\frac{1}{\pi \sqrt{2a}}\int_{-\infty}^{\infty}\phi\left ( k ...
0
votes
1answer
52 views

Derivation of the order condition for the Implicit Runge-Kutta method

I know how to derive the order condition for the explicit Runge-Kutta method by Taylor expansion, but do not know the implicit one. For instance, we list the two-stage implicit Runge-Kutta method for ...
0
votes
0answers
10 views

Numerical method to fit arbitrary 3D curve by distributing perturbing elements on a 2D grid

I am looking for help in choosing and possibly implementing an appropriate algorithm or method to solve the following problem: I have a surface that has a property $A(r)$ that I want minimized. I ...
1
vote
1answer
31 views

residue equation for the denominator in a Padé approximant for $e^{-x}$

I had success in computing the roots numerically for the Bessel polynomial $\theta_n(x) = x^ny_n(1/x)=\sum\limits_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\frac{x^{n-k}}{2^k}$ by using this residue equation I ...
0
votes
1answer
41 views

ODE implicit method

Determine the coefficient of an implicit, one step, ODE method of the form $$x( t+h ) = a x(t) + b x'(t) +c x' ( t+h )$$ so that exact for polynomial of high degree as possible. What is the order of ...
1
vote
0answers
52 views

Levenberg-Marquart with Hessian and gradient

I am minimizing a sum-of-squared-differences function using the Levenberg-Marquardt method. The off-the-shelf numerical implementations I have have looked through (MATLAB, Numerical Recipes in C ...
1
vote
1answer
40 views

How are polynomials graphs approximated?

Say I have the data: $x=[ 1, 2, 3.3, 4, 5.5, 8, 9, 10.2, 11, 45 ]$ $y=[ 9,27,64,91,164,330,462,540,630,10218]$ The data is subjective though. How would one approximate a valid polynomial for this ...
0
votes
0answers
15 views

How to classify this integro-differential equation?

I have a system of three coupled integral equations for three unknowns $j(t), \bar{x}(t)$ and $\lambda(t)$ to be solved between $t=0$ and $t=T$ (b.c. are $\bar{x}(0)=x_0$ and $\bar{x}'(T)=0$): (1): ...
4
votes
1answer
114 views

Finding integer solutions to $y^2=x^3+7x+9$ using WolframAlpha

I am an unconditional admirer of WolframAlpha and for this reason I want to let the people of this error (or is it really the fault of mine?). If I'm not mistaken, I would be very happy to contribute, ...
0
votes
1answer
23 views

How do I perform Gram-Schmidt on floating point vectors with epsilons in them?

Let $\epsilon$ be a small positive number such that $1+\epsilon$ and $3+2\epsilon$ are machine numbers but $3+2\epsilon + \epsilon^{2}$ is computed to be $3 + 2\epsilon $. Now, let the (classical) ...
1
vote
1answer
17 views

Which Properties of a Natural Cubic Spline does the following function possess and not possess

I need to determine which of the properties of a natural cubic spline the following function possesses or does not possess: $$f(x) = \begin{cases} (x+1)+(x+1)^{3}, & x \in [-1,0] \\ ...
0
votes
0answers
33 views

Way to verify a least-squares solution without actually solving for $x$ and $y$?

I just found the least squares solution of the system $\mathbf{x}A = \mathbf{b} = \begin{pmatrix} x & y \end{pmatrix}\begin{pmatrix} 3 & 2 & 1 \\ 2 & 3 & 2\end{pmatrix} = ...
1
vote
1answer
23 views

What does it mean to permute the subdiagonal entries of a matrix?

\begin{pmatrix}1 &0 &0& 0 \\ 2& 1& 0& 0\\ 3& 0& 1& 0\\ 4& 0& 0& 1 \end{pmatrix} can turn into \begin{pmatrix} 1& 0& 0& 0\\ 4& 1& ...
0
votes
1answer
45 views

Solving non-linear pde with newton method

I know that to solve a nonlinear pde, you either have to linearize or you have to solve it using Newton's method. I didn't find any clue or example about how to do it with Newton's method. Can any ...
1
vote
0answers
29 views

Godunov's method

Can anyone send me a link/refer me to some useful sources about Godunov's method? I need something simple and easy to understand, which doesn't require too much prior knowledge in PDE and associated ...
0
votes
0answers
24 views

Can anyone explain why newton-cotes formula is exact on all polynomials of degree less than n?

Newton-cotes formula based on a polynomial interpolation at n+1 nodes is exact on all polynomials of degree less than or equal to n. It seems like that if we integrate the polynomial interpolating at ...
1
vote
1answer
32 views

Non-linear SDE: how to?

$$ \newcommand{\mcl}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\avg}[1]{\langle#1 \rangle} \newcommand{\pth}[1]{\left( #1 \right)} \newcommand{\bck}[1]{\left\{ #1 \right\}} ...
0
votes
0answers
17 views

Typo in Atkinson?

This is problem 6.14 from Atkinson's Intro to Numerical Analysis. How can the two step method $y_{n+1}=\frac{1}{2}(y_n+y_{n+1})+\frac{h}{4}(4y_{n+1}'-y_n'+3y_{n-1}')$ be second-order? It doesn't ...
1
vote
1answer
41 views

Trying to find the approximate solution of this non-linear system

So, I'm trying to find an approximate solution for the non-linear system $$n·8.0000003=p-\sin(q),$$$$n·7.9999996=q-\sin(q),$$$$ n·16.0000003=p+q-\sin(p+q),$$ where $n,p,q>0$. Ive tried Newton ...
0
votes
1answer
17 views

tridiagonal block matrix

Let us consider a linear system of equations $$ Ax=b $$ Where $A$ is a block tri-diagonal matrix, which is given by $$ \begin{eqnarray} A=\left[\begin{array}{ccccc} A_{11} & A_{12} & \dots ...
1
vote
1answer
24 views

How to approximate $z=a^{-1/4}$ using Newtons method?

I need a function that isn't a polynomial and leads to a $f(x)$ that doesn't have a division by the iterate. The natural way of attempting this is by getting $x^4-\dfrac{1}{a}=0$ but this is using a ...
2
votes
1answer
46 views

Ways to generate triangle wave function.

I recently when searching for parameters on a unit cube in $\mathbb{R}^9$ (we all have our more or less peculiar hobbies, don't we?) found a practical reason to implement a triangle wave function ...
1
vote
0answers
28 views

Normalization in least-p'th minimax algorithm

In the book "Practical Optimization: Algorithms and Engineering Applications", the least-$p$th minimax algorithm is presented, for approximation of the minimax optimizer (Alg. 8.1): $Loss_x(k)$ = ...
1
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0answers
18 views

largest number machine number that this comp has, numerical analysis, mantissa

I am having trouble understanding the basics on how to calculate the m, s and c of the formula. I have looked in my textbook but not much of an explanation, also on the IEEE websites but nothing much. ...
1
vote
1answer
23 views

Use the Cholesky Theorem to prove the equivalence of two properties of symmetric matrices

Recall the Cholesky Theorem on $LL^{T}$-Factorization: Theorem: If $A$ is a real, symmetric, and positive definite matrix, then it has a unique factorization, $A=LL^{T}$, in which $L$ is lower ...
1
vote
0answers
47 views

Are two linear system equivalent?

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 ...
0
votes
1answer
20 views

Floating Point and machine error

Let $a$ and $b$ be to real arbitrary real numbers, show that the relative error that you made by computing $a^2b$ with floating point arithmethic is bound to $5\epsilon + O(\epsilon^2)$, with ...
3
votes
1answer
80 views

Gauss Quadrature Proof.

Let $x_0<x_1< ... < x_n$ be the roots of an n+1 degree orthonormal polynomial $\phi_{n+1}$ with respect to the inner product: $$\langle g,h \rangle = \int_a^bw(x)g(x)h(x)dx$$ and $$p_n= ...
1
vote
1answer
20 views

Condition number of 2x2 nonsingular matrix

I am working on the problem that I have to show why the infinity-norm condition number and 1-norm condition number of 2x2 nonsingular matrix are equal. MY ATTEMPT: Since $I=AA^{-1}$, the condition ...
1
vote
0answers
30 views

Fast evaluation of an integral convolution with an “expanding kernel”

Suppose I have a 1-D integral convolution transform like this: $$ g(x) = \int_{-\infty}^{+\infty} dy\, f(y)\, K(x-y). \qquad (1) $$ Say the kernel $K(x)$ is a known analytic function, and say we have ...
1
vote
0answers
24 views

How are subtraction operations better conditioned than addition?

I was reading about error analysis in numerical methods, particularly about calculating the sum of arrays. I understand how naively summing all elements can lead to accumulation of error especially ...
3
votes
0answers
27 views

Numerical method for SDEs

I'm using a 4th order Adams predictor-corrector method to numerically solve a regular differential equation. Now I would be interested to be able to include a noisy term to the equation -as in the ...
0
votes
0answers
17 views

estimation of error - spline interpolation

I have two intervals: $[10, 12]$ and $[12, 14]$. On the $[10, 12]$ I constructed square polymonal. For the interval $[12, 14]$ I constructed cubic polynomial. And now I think how to estimate error ...
1
vote
1answer
71 views

How to solve the transcendental equation $\frac{\zeta'(\alpha)}{\zeta(\alpha)}=-\frac{1}{n}\sum_{i=1}^n\ln x_i$

How can I solve the transcendental equation: $$\frac{\zeta'(\alpha)}{\zeta(\alpha)}=-\frac{1}{n}\sum_{i=1}^n\ln x_i,$$ where $\displaystyle\zeta(\alpha)=\sum_{m=1}^\infty m^{-\alpha}\;\;\;?$ The ...
0
votes
0answers
20 views

Least square estimation using Euler discretization

I have a set of differential equations $\dot{x}_i = f_i(x, \theta)$, with $x = [x_1, \ldots, x_n]^\top \in \mathbb{R}^n$ and $\theta \in \mathbb{R}^m$. Measurements of the variables $x_i$ are ...
0
votes
1answer
43 views

How to derive $f^{'''}(x)$ using Taylor expansion?

I tried doing the difference between the taylor series expansions of $f(x+h)$ and $f(x-h)$, but it didn't reflect the answer that I should get using the third order central divided difference. The ...
0
votes
1answer
32 views

State the midpoint rule

State the Midpoint rule for: $$\int_c^dg(t)dt$$ with $m$ subintervals Is this how you state it? $$M_n=\frac{d-c}{m}[g(t_m)]$$ or $$M_n=\frac{d-c}{m}[g(t_1)+g(t_2)...g(t_m)]$$
0
votes
1answer
23 views

How to display character drawn by the Bezier curve

How can I draw/display a character based on Bezier equations? I have the plot equations: x(t)=3t-3t^2 y(t)=2-3t^2+2t^3 x(t)=3t-3t^2 y(t)=1-3t^2+2t^3 and ...
3
votes
2answers
54 views

There are exactly two values for which $x^2=x\sin x+\cos x$ holds

I'm asked to show there is exactly two values for which $x^2=x\sin x+\cos x$ I have no idea where to start but I was thinking about taking the difference $h(x)=x^2-x\sin x-\cos x$ to show that $h'(x) ...
0
votes
0answers
11 views

boundary conditions in closed domain

I want to solve the fourth-order Bernoulli equation in a closed domain. I do not know how to impose the 'closure' conditions at the ends when I numerically implement the equation. In particular, if ...
0
votes
1answer
35 views

Formulation of matrix for FEM

I am trying to use FEM to approximate the solution to the following BVP: $-\frac{d}{dx}[a(x)u'(x)]+b(x)u(x)=f(x)$, on [0,1] where $u(0)=0$ and $u(1)=1$. I am using the Galerkin method with hat ...
0
votes
2answers
21 views

Find the value, given the error formula for trapezoid rule

The global error of $\int f(x) \mathrm{d}x$, between two $x$-values by the trapezoidal rule is $-(1/12)h^3f''(ξ)$ a) $f(x) = x^3, x := [0.2,0.5]$ Find the value for $ξ$. Not really sure where to ...
0
votes
0answers
13 views

numerical solution of integral equation with unknown bound

I am reading a paper on High Harmonics Generation (HHG) and a Lewenstein model The paper is here. I would like to reproduce some results but I am stuck at the following problem. I have: ...
0
votes
1answer
33 views

Determine the number of correct digits in the number $x$ given its relative error $E_r$

Determine the number of correct digits in the number $x$ given its relative error $E_r$ (a): $x=0.4785, E_r=0.2\times 10^{-2}$ (b) : $x=386.4, E_r=0.3$ (c): $x=86.34, E_r=0.3$ For the problem ...