1
vote
1answer
27 views

Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi ...
0
votes
2answers
44 views

Firstly what is an $O(h^3)$ formula? Also I am not quite sure how to answer the question?

The forward-difference formula can be expressed as $$f'(x_0)=\frac{1}{h}(f(x_0 +h)- f(x_0))-\frac{h}{2}f''(x_0) - \frac{h^2}{6}f'''(x_0) + O(h^3).$$ Use Richardson's extrapolation to derive an ...
2
votes
1answer
46 views

Checking if the Hessian is the derivative of the gradient

Suppose $f: \Bbb R^n \to \Bbb R$. I have a code that computes the gradient of $f$. I have another code that computes the Hessian of $f$ times a vector. Now I want to check if they are correct. ...
0
votes
1answer
9 views

Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in ...
2
votes
3answers
94 views

Intuitive Numerical Analysis Texts

Steven Strogatz has a great informal textbook on Nonlinear Dynamics and Chaos. I have found it to be incredibly helpful to get an intuitive sense of what is going on and has been a great supplement ...
1
vote
1answer
48 views

Euler's Numerical Method

Let $\eta(x;h)$ be the approximate solution furnished by Euler's method for the initial-value problem $y'=y, y(0)=1$. I proved that: $i) \eta(x;h)=(1+h)^{x/h}$; $ii) \eta(x;h)$ has the expansion ...
1
vote
0answers
31 views

Sequence and intermediate value theorem

For part (a), By intermediate value theorem, there exist c between 0 and 1 such that $f(c) = 0$ Now, I supposed that there also exist d between 0 and 1 such that $ f(d) = 0 $ and $ c \neq d $ I am ...
1
vote
1answer
122 views

Solve non-linear equations of 3 variables using Newton-Raphson Method iterms of c,s and q.

The three non-linear equations are given by \begin{equation} c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0 \end{equation} \begin{equation} s[2.001 *c + 835(1-q)]-2.001*c =0 ...
0
votes
1answer
31 views

How to write continued fraction as polynomial?

I have \begin{align} r(x)= 1 + \frac{x}{\frac{1}{2}+\frac{x-1}{-1+\frac{x+1}{1+\frac{x-1}{-1}}}} \end{align} for an interpolation problem, and I need to write $r(x)$ such that nominator and ...
0
votes
1answer
43 views

Quadrature Rule “is exact for polynomials of degree n”

Could someone kindly explain what "a quadrature rule is exact for polynomials of degree n" means? Here is what I understand about numerical (Newton-Cotes) quadrature rules: Suppose we want to ...
0
votes
1answer
19 views

Calculating the limit of an analytic function which gives a log answer.

I am trying to read through a paper and have gotten stuck at the following calculation several times. I've left if for a few days and came back to try it again four different times, but still no luck. ...
0
votes
1answer
47 views

Show that the iteration $x_{n+1} = x_n - 2\frac{f(x_n)}{f'(x_n)}$ converges quadratically to $x_*$ provided $x_0$ is sufficiently close to $x_*$

We have the following conditions for the above slightly-modified Newton's method iteration: $f$ is a real function of one real variable $f''$ is Lipschitz continuous $f(x_*) = f'(x_*) = 0$ I also ...
1
vote
0answers
35 views

how to prove this curious identity with the Chebyshev polinomials

we defined the Tm like this (where Tm are the Chebyshev polinomials) Then I showed this: And now I have no idea how to proove this: I also have to make the remark that I also proved that the ...
0
votes
0answers
29 views

Hermite interpolation with interior points

I am trying to solve the following problem: Given the conditions on a curve c(u) of degree 4 at the points -1, 0, 1 as: c(-1) = 4; c'(-1) = 4; c(0) = 6; c(1) = -4; c'(1) = -6; find the generalized ...
0
votes
1answer
49 views

Find the rate of convergence?

Given is the iteration $x_{k+1}=\frac{1}{11}(1-\cos(x_{k}))$ with $x_{0}\in (-\frac{\pi }{2},\frac{\pi }{2})$ without $0$. Check if the sequence converges to $x^{*}=0$ and find its convergence rate. ...
1
vote
1answer
98 views

Gaussian Quadrature -Deriving a Formula-

eThe following is an exercise in the problem section of the Gaussian Quadrature chapter. The theorem: Derive a formula of the form $$\int_{a}^{b} f(x)dx \approx w_0f(x_0) + w_1f(x_1) + w_2f'(x_2) ...
0
votes
0answers
24 views

Gaussian Quadrature Confusion

For the Gaussian Quadrature Thrm., why are we letting$ c_0$ and $c_1$ be $0$ in the example in the picture?
0
votes
0answers
55 views

Infinite Order Fourier Series and Infinite Series with Piecewise Function

Given $$ f(x) = \left\{ \begin{array}{ll} 4\pi^2 & \quad x = 0 \\ x^2 & \quad 0<x\le2\pi \end{array} \right. $$ First, compute the infinite ...
0
votes
0answers
31 views

Method of undetermined coefficients for finite difference approximations

I'm reading over my text, and the first mention of deriving the coefficients states: "Suppose we want a one-sided approximation to $u'(x)$ based on $u(x), u(x-h)$, and $u(x-2h)$ of the form: ...
0
votes
0answers
75 views

How many terms to use in a Taylor series for local truncation error

So from my understanding for a finite difference approximation, you're supposed to expand the series "about" the point $x$, e.g., $$u(x+h) = u(x) + h \ u'(x) + ...
0
votes
0answers
31 views

Divided differences And B-spline

Can someone help me? I want to prove that: $$f[x_0,\ldots,x_n] = \frac{1}{n!} \int_{x_0}^{x_n} f^{(n)}(t)B_{n-1}(t) \, dt$$ where $B_{n-1}$ is a B-spline of degree n-1 for the data points ...
0
votes
1answer
29 views

Understanding why an IVP has a solution, using uniqueness and existence theory

Given the existence and uniqueness theory: If $f$ is Lipschitz continuous over some region $D$, then there is a unique solution to the initial value problem (IVP): $u'(t) = f(u,t), \hspace{5mm} ...
3
votes
2answers
45 views

Determine best possible Lipschitz constant

I'm slightly confused by a homework problem here...I've been given the function: $ f(u) = log(u) $ With the bounds: $ 2 \leq u \lt \infty $ Now I thought I understood what the Lipschitz Condition ...
2
votes
0answers
51 views

Expected error due to the tablemakers' dilemma

[note: to me, this does not seem like a question for m.se, but on mathoverflow it has been retroactively closed, with very little indication of why or what might be corrected... and waiting for ...
1
vote
0answers
27 views

Kummer U function: U(a,a+1/2,z)

Is there any way to simplify $U(a,a+1/2,z)$ or relate it to any other common special functions such as the incomplete beta function or the incomplete gamma function? Here, $U(a,b,z)$ is Kummer's U ...
2
votes
1answer
80 views

SOR and Gauss-Seidel Method - Confusion

Can anyone explain to me the SOR Method for finding the root(s) of a function? Its supposedly very similar to the Gauss-Seidel method. The Gauss-Seidel method, from my understanding, is similar to ...
0
votes
0answers
107 views

Gauss Hermite quadrature on finite interval

I would like to approximate an Integral of the type $I = \int_{x_l}^{x_u} f(x) w(x) dx$ where $w(x) = \frac{1}{2\pi}e^{-\frac{1}{2}x^2}$ and $f(x)$ is only defined on the Intervall $D = [x_l, \, ...
0
votes
0answers
14 views

Rate of Convergence of matrices

Given a symmetric-"pentadiagonal" matrix $(n\times n)$ such as: $$\begin{bmatrix} 1 ...
2
votes
0answers
32 views

Question on numerical quadrature and precision

I am studying numerical analysis and I came across this question: Let $P_{n+1}(x)\in\Pi_{n+1}$ be orthogonal to $\Pi_n$ relative to a weight function $w(x)\geq 0$ on $[a,b].$ Denote by ...
1
vote
1answer
53 views

Richardson's Extrapolation

Use Richardson's extrapolation to find a 3 point 2nd order approximation of f '(x). I'm not sure how to go about to start this, i'm not the best when using richardson's extrapolation.
1
vote
2answers
31 views

Showing a limit for the mean value property

I need to understand a proof and i have a problem with the limit process at a specific point in this proof. I want to know, what is needed to get the following result: Let $f \in C^1(G)$, $\bar{x} ...
0
votes
1answer
71 views

Fredholm integral equation of first kind

I want to solve the Fredholm integral equation of first kind: $$ \int_L K(x,y)U(y)dy = f(x) $$ in these equation the function $U(y)$ is the unknown and the so-called kernel $K$ and the right hand side ...
0
votes
1answer
20 views

Modified Euler's method overestimator

Does the Modified Euler Method always overestimate the true solution values? Here is the modified Euler Method: $w_0=\alpha$ $w_{i+1}=w_{i}+\frac{h}{2}[f(t_i,w_i)+f(t_{i+1},w_i+hf(t_i,w_i))], ...
0
votes
1answer
42 views

Convergence of iterative method

Assume that iterative method: $x_{k+1}=F(x_{k})$ where $(k=0,\,1,\,2,\,...)$ converges to $\alpha$ which is root of $f(x)=0$ equation. Prove that if $F(\alpha)=\alpha$, $ ...
2
votes
1answer
59 views

Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$.

I want to Verify the spectral radius $r(A) = \lim_{n\rightarrow\infty}||A^n||^{1/n}$. Where $A$ is a matrix. I have a proof that involves Jordan Blocks. The proof is long and involved but it not ...
3
votes
1answer
91 views

Why below sequence is diverge?

This problem maybe simple for you,but i dont know that why below sequence is diverge?please help me about this: why $‎\lbrace\mid x_{k}\mid‎\rbrace$ with below definition is diverge? $x_{k+1}:= ...
1
vote
1answer
33 views

Let $y_1,…,y_k$ be the roots of $q$. Why is $q(x)\prod_{i=1}^n(x-y_i)$ only positive or only negative.

I'm trying to understand this exercise: Well, my teacher told me that I need to suppose $q$ has $k<n$ different roots in $(a,b)$. So we have the roots $y_1,...,y_k$ of $q$. Then if I set $p(x)= ...
0
votes
0answers
33 views

Check for the value of x for function f is ill conditioned

I have two examples: $$ i) \ f(x) = \sqrt{1 - x^2} $$ $$ ii) \ f(x) = \sqrt{x^2 + 1} - x $$ I must check the value of x for which the calculation of the values ​​of the function $ f $ is ill ...
0
votes
1answer
58 views

$a + b = a$ in machine precision [closed]

I have the following statement: "If $a + b = a$, then $b = 0$" may not true with the floating point operations. Actually, if $|y| ‎< (\varepsilon / B) |x|$, then $fl(x+y) = x$, where ...
0
votes
1answer
73 views

Positive real number has a finite number of binary when is in form $ m/2^n $

Prove that positive real number $ ( x \in \mathbb{R} \ x > 0) $ has a finite number of binary if and only if when is in form $ \frac{m}{2^n} $, where $ m, n \in \mathbb{N} $ I found this solution: ...
1
vote
1answer
97 views

contraction point?

This is an interesting question I saw in a book online: Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence ...
2
votes
1answer
54 views

Convergence and Constant sequence?

Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence $x_0 x_1 x_2...$ given by $x_n = g(x_{n-1})$. converges ...
1
vote
1answer
191 views

Prove that $ \ln[e(2/e)] $ is a fast way to calculate $ \ln2 $

Consider formula $ (*) \ln(x) = \sum_{k=1}^{\infty} (-1)^{k-1}\cdot \frac{(x-1)^k}{k}$. If you calculate $ \ln2 $ with error less then $ \frac{1}{2} \cdot 10^{-6} $ we need more than two milion ...
1
vote
0answers
36 views

What are the weights of the quadrate formula with weight function $x\mapsto (1-x^2)^{-1/2}$

I'm trying to solve this numerical analysis exercise: I was able to prove everything until the part marked in red. I think I need to use this: So we get an exact result for $T_0$: ...
1
vote
1answer
53 views

Show that $\varphi_{j+1}(x)-C_j x \varphi_j (x) = \sum_{k=0}^j \alpha_{jk} \varphi_k (x)$ where $\{\varphi_j \}$ is a syst. of orth. polynom.

This is a homework exercise. I'm only asking for hints, please don't give a full solution. This is the exercise: This is my attempt to solve this problem: If $C_j$ is chosen to be equal to the ...
1
vote
1answer
33 views

Taking the derivative of $n$ products

I'm reading my numerical analysis book, but I don't understand this step: I'm assuming that this $l'$ must be $l_0'$, as there is no $l$ defined anywhere. If you want, you can read the text above ...
0
votes
0answers
72 views

Small symbols behind parantheses

I am currently reading the following paper where the author uses constantly small symbols after parantheses, but I do not know what this means. I am particularly interested in equation (23), so you ...
2
votes
2answers
138 views

Solving $v_{t}+v(x,t)v_{x}=0$ with initial condition

This problem comes from an undergraduate course in PDE. The first question of the problem was to solve the following PDE: $v_{t}+v(x,t)v_{x}=0$ with the following initial condition: $v(x,0)=5x$ ...
5
votes
0answers
102 views

Do there exist solutions for this equation?

We know that solutions exist for equations of the following variety: $$ye^y=x \iff y=W(x)$$ Where W is the Lambert W function. We can augment the problem slightly, and ask if there exist solutions ...
1
vote
6answers
605 views

smooth functions or continuous

When we say a function is smooth? Is there any difference between smooth function and continuous function? If they are the same, why sometimes we say f is smooth and sometimes f is continuous? Please ...