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

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4
votes
2answers
956 views

How to obtain (prove) 5-stencil formula for 2nd derivative?

My question seems pretty easy. Prove the correctness of the following approximation: $$f(x)''= \frac{-f(x-2h)+16f(x-h)-30f(x)+16f(x-h)-f(x+2h)}{12h^2}$$ I rendered myself deeply saddened upon ...
0
votes
3answers
644 views

number with finite binary representation and infinite decimal representation

One can easily find numbers with finite decimal representation with infinite binary representation. (Like $0.3$ and $0.01010101..$) I assume there is an opposite case, meaning a number with finite ...
5
votes
2answers
167 views

Non-convex optimization: $\min ||y-Ax||_p$ for very small $p$ given that $||x||_2=1$

I need to find $x$ that minimizes the cost function $\|y-Ax\|_p$ when $p$ is close to $0$, subject to the constraint $\|x\|_2=1$ where $x$ and $y$ are vectors in $\mathbb{R}^n$ and $A$ is an $n\times ...
1
vote
2answers
163 views

wolfram mathematica, numerical integration, precision of a function/expression

I want to obtain the best numerical approximation (up to 10 decimal place would be ok for me) to an integral: $$ \int^{\infty}_{0} f(r)r^2dr $$ I am using the function $f(r)$, which is related to ...
1
vote
0answers
464 views

How to use Richardson extrapolation to derive modified Euler method?

For a given ODE $y'(t)=f(t,y)$, Euler's method is $$ y(t+h)=y(t)+hf(t,y(t)) + O(h^2) $$ It is said that by using Richardson extrapolation, we can improve it to $$ ...
1
vote
0answers
30 views

Density of finite element functions in $W^{1,p}(\Omega)$

I would like to know if the following statement is true: For each $u \in W^{1,p}(\Omega)$ and $\varepsilon > 0$ there exists a piecewise affine function $u_{\varepsilon}$ and a triangulation of ...
3
votes
1answer
1k views

Eigenvalues for LU decomposition

In general I know that the eigenvalues of A are not the same as U for the decomposition but for one matrix I had earlier in the year it was. Is there a special reason this happened or was it just a ...
2
votes
1answer
181 views

Proof of a lower bound of the norm of an arbitrary monic polynomial

In my course I have come across the following problem: The Chebyshev polynomial of degree $n$, $T_n(x)$, is defined on $[-1,1]$ by $T_n(x)=\cos n\theta$. Let $q_{n+1}(x)$ be any monic ...
7
votes
2answers
949 views

Big-O Interpretation

I have trouble understanding what the "Big O" notation, or asymptotic notation means. For instance, if you have $\sin(x)=x+O(x^3)$, what does this mean? Can anyone describe it in a simple way? I tried ...
1
vote
1answer
127 views

Evaluating the second fundamental form for a curve

I am (numerically) computing the second fundamental form for a curve $\gamma(t)$ embedded in a Riemannian manifold $(M, g)$. I would like to double check if what I am doing is correct. First, define ...
1
vote
3answers
303 views

Trapezoidal rule problem

In a trapezoidal rule problem I got following question: "Evaluate the above integral using trapezoidal rule with five points." My confusion is here what we take for the value of $n$ is it $5$ or ...
3
votes
0answers
52 views

finding the largest $p$ components of $x$

Given an $n \times n$ matrix $A$, and an $n \times 1$ vector $b$, the conventional way of computing an $n \times 1$ vector $x$ such that $x=Ax+b$ is to use the following iterations: ...
2
votes
0answers
29 views

Approximate a constant function with sequence of spline functions

Suppose that for a constant $c \in \mathbb{R}$ $$\sup_{t \in [a,b]}\Big|c- \sum_{l=1}^{m}a_{l}\ B_{l}(t;q)\Big|< \epsilon.$$ The $B_l$ form a B-spline basis of degree $q$ on the interval $[a,b]$ ...
3
votes
1answer
114 views

Iterative model fitting

I have a sequence of points $\{(x_k,y_k,z_k)\}$ and I need to fit some $2D$ model $P(x,y)$ that approximates $z$ in some sense. The $z_k$$'s$ are noisy samples of some $2D$ function $z_k = f(x,y) + ...
5
votes
3answers
971 views

Calculate Runge-Kutta order 4's order of error experimentally

The Problem Use the order 4 Runge-Kutta method to solve the differential equation $ \frac{\partial^2 y}{\partial t^2} = -g + \beta e^{-y/\alpha }*\left | \frac{\partial y}{\partial t} \right |^{2} $ ...
0
votes
1answer
134 views

Floating point register and relative errors

Let's pretend I have a five places floating-point register. I have to represent the following numbers: $256.786$ and $256750000$. I can represent the first number as $2.568 \cdot 10^2$. The relative ...
0
votes
0answers
248 views

Numerically find minimum of 2 dimensional surface

I have a large equation that essentially takes two variables and returns a real number. I know that the two values are real numbers between 0 and 10. Is there a standard way to numerically find the ...
1
vote
1answer
68 views

Combining error terms from two Taylor expansions

When deriving the five-point differentiation formula as shown in this book, the IVT was used to combine $ f^{(5)} (\xi_1) $ and $ f^{(5)} (\xi_2) $ into one error term, $ f^{(5)}(\tilde{\xi}) $ As ...
0
votes
1answer
152 views

Relative error plot in mathematica

I was plotting the relative error of the $e^{11/12 -n}n^{n+1/2}$ approximation to the factorial as $n$ gets larger and larger and at after some very large value of $n$ mathematica gives this plot: ...
4
votes
1answer
2k views

How is the Taylor expansion for $f(x + h)$ derived?

According to this Wikipedia article, the expansion for $f(x\pm h)$ is: $$f(x \pm h) = f(x) \pm hf'(x) + \frac{h^2}{2}f''(x) \pm \frac{h^3}{6}f^{(3)}(x) + O(h^4)$$ I'm not understanding how you are ...
0
votes
1answer
350 views

Steady-state solution of an ODE

This is the problem given: I am not entirely sure what my Professor expected from an answer, but it seems I am to find the coefficient, angular frequency, and phase of the non-homogenous solution ...
0
votes
1answer
331 views

machine numbers in IEEE single precision

Is the following numbers machines numbers on the IEEE single precision system? $10^{304}$ $2^4+2^{27}.$ What do I have to do to know whether they are machine numbers on IEEE single precision?
7
votes
2answers
384 views

Stirling's Series Derivation

I was reading this paper and at the top of page 9 it says that as $n\to\infty$, $$\left(1+\frac{1}{n}\right)^{n+1/2}e^{-1}\left(1+\frac{a_1}{(n+1)}+\frac{a_2}{(n+1)^2}+\cdots ...
0
votes
2answers
99 views

Finding eigenvalues of sparse integer matrix

I need to find eigenvalues of a sparse matrix with integer coefficients. I understand in general this is not done by explicitly computing the characteristic polynomial due to numerical instability, ...
4
votes
5answers
241 views

Improving Newton's iteration where the derivative is near zero?

I'm implementing a root-solver for finding x coordinates of a function f(x), after I have an y-coordinate. The function is periodic, roughly sinusoidal with constant amplitude but non-linearly ...
0
votes
1answer
74 views

Anisotropic equations

Someone was giving a talk about modeling tumor growth in 3D, after which someone asked the question: "Are all of your equations anisotropic?" It sounded like he was referring to inclusion of unknown ...
1
vote
1answer
60 views

Help regarding a weird Matrix

Hi I have a matrix of the following form arising by discretization of a system of PDEs. I am working to get the invertibility of the Matrix. Can some one help me or at least give me some reference on ...
0
votes
1answer
81 views

Is it common in numerical analysis to make a change of variable when the condition number is high?

I want to approximate a function $f(x)$ on [a,b]. The condition number for computing a function $f$ at a point $x$ is defined by $$\kappa=\frac{x f^{\prime}(x)}{f(x)}$$ For my function $f$ this ...
1
vote
1answer
1k views

How to Convert a Number to Roman Numerals

i don't How to Convert a Number to Roman Numerals Using mathematical equation. if M=1000,D=500,C=100,L=50,X=10,V=5,I=1 then how convert any decimal number to Roman Numerals? such as if 1952 then its ...
0
votes
1answer
158 views

Application of Backward euler method

Information Let $y'(t)=f(t,y(t))$ and $y(0)=y_0$ The backward euler method together with the center rule is given by: $y(t_k)=hk$ where $h\in (0,\frac{1}{K})$ is the step size. Recursion: ...
2
votes
1answer
831 views

Optimizing integral functionals using Matlab

I am looking for some bibliography regarding solving integral optimization problems numerically (preferably using Matlab). I want to solve problems of the type $$ \min_{r \in A} \int_a^b ...
1
vote
2answers
1k views

How did people calculate numerical values of transcendental and trigonometric functions?

I know that back in the Stone Age, people used tables on this thing called paper to look up values for functions like $\sin$ and $\ln$. But how did the guys who wrote the tables calculate those ...
2
votes
0answers
149 views

Check for Ill Conditioned matrix

How can I efficiently check if a tridiagonal system with 1's in diagonal is ill-conditioned or not ? The common way is to get the ratio of largest and smallest singular values and see if its greater ...
1
vote
0answers
29 views

Problem with the convergence of a Nystrom algorithm

I programmed a Nystrom Algorithm specifically for my problem: This is the exact equation i want to solve: $y′′=(w2−e∗cos(t))∗sin(t)−b∗y′$ And this is my algorithm ...
1
vote
0answers
121 views

Successive Approximation

If I'm given the following parameters, Let the function $f(x)$ be four times continuously differentiable and have a simple zero ξ. Successive approximation $x_n, n = 1,2,...$ to ξ are computed from ...
1
vote
2answers
4k views

Convergence of Bisection method

I know how to prove the bound on the error after $k$ steps of the Bisection method. I.e. $$|\tau - x_{k}| \leq \left(\frac{1}{2}\right)^{k-1}|b-a|$$ where $a$ and $b$ are the starting points. But ...
1
vote
1answer
623 views

Using permutation matrix to get LU-Factorization with $A=UL$

Let $Q$ be the $n$x$n$ permutation matrix $$Q= \begin{bmatrix} 0&0&...&0&1\\ 0&0&...&1&0\\ .& \\ .&\\ .&\\ 0&0&...&0&0\\ ...
3
votes
1answer
98 views

least square problem

Let $1<p<\infty $.We define the space: $L_{V}^{p}(-1,1)=\left \{ f:(-1,1)\rightarrow \mathbb{R}:\int_{-1}^{1}\left | f(x) \right |^{p}V(x)dx<\infty \right \}$ We define the norm: $\left \| ...
1
vote
1answer
339 views

Fixed point iteration “Simple iteration” - restating a problem

Okay, here's the question: Suppose we want to find a solution to $\frac{1} {2} e^{x} - x = 0 $ on the interval [0,1]. Show how to restate this problem as a fixed point problem. This is the first ...
0
votes
1answer
80 views

Iterarions count in Newton's method;

How many iterations must I do for getting $n$ signs after floating point in calculating square root by Newton's method P.S Sorry for my bad English. Please mention to me where I've done mistakes. ...
3
votes
1answer
262 views

Understanding accuracy of Newton's Method

In a numerical analysis book I'm reading it says that using the Newton error formula we can find an expression for the number of correct digits in an approximation using Newton's Method. Here's the ...
2
votes
3answers
3k views

The mode of the Poisson Distribution

Lately, I am doing an investigation on Stirling's formula and its applications. So I thought I could use it to prove that the mode of the Poisson model is approximately equal to the mean. Of course, ...
3
votes
1answer
104 views

Euler's Method question

I have this differential equation $y'=\frac{y(\sin t)}{t}$, $y(0)=2$, and $h=\frac{1}{4}$. The first set of values, the inital, are $(0,2)$. For the next iteration would it be ...
2
votes
0answers
65 views

Initial Conditions for Finite Difference of PDE

I am having trouble with figuring out what my initial conditions should be for a simple finite difference algorithm I wrote in Matlab. Specifically, I'm trying to show that the regular 1-Dimensional ...
1
vote
2answers
6k views

Proving the inverse (if any) of a lower triangular matrix is lower triangular

The inverse of a non-singular lower triangular matrix is lower triangular. Construct a proof of this fact as follows. Suppose that $L$ is a non-singular lower triangular matrix. If $b \in ...
4
votes
1answer
649 views

How to solve an equation using Newton's method with and without backtracking?

Lets assume I have this equation: $$\log(e^x+e^{-x})=2x+5,\quad x \in (-50,50).$$ As always we have to pick a starting point to solve this by Newton's method, but how can i know for what initial ...
2
votes
1answer
107 views

Dividing a range into major and minor divisions

I'm drawing a graph, and need to annotate the x axis. The x axis is dimensioned in days -- eg, 11/25, 11/26, 11/27... The number of of days on the x axis may range from about 10 to perhaps 100 or ...
1
vote
1answer
168 views

Proof of GMRES convergence

I am working on a homework, where we have to proof that GMRES finds an exact solution of $Ax = b$ in at most m steps (with A being an $m \times m$-matrix). The proof is split up into several steps, ...
1
vote
0answers
203 views

Numerical integration of binomial pdf with respect to a conditional probability

How do you numerically integrate the following loss function with respect to $\Phi(f_t)$, given $N_t=50$, $d_t=0$, $\rho=0.2$ and $\pi=0.01$? $P(D_t=d_t)=\bigl(\begin{smallmatrix} N_T \\ d_t ...
2
votes
1answer
265 views

Calculating the gradient without knowing the function

I have to develop an optimizer for a simulation. There are 7 reference values $$ r_1, r_2,\ldots,r_7 $$ (certain values which are expected to show up) and 7 corresponding actual values $$ ...