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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1answer
129 views

A strange quantum potential: $V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$

So I have a strange quantum potential I have been playing with: $$V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$$ where $\mu$ is the Möbius function. This is what ...
1
vote
2answers
2k views

Numerically integrating a function of 3 variables with respect to 2 variables in Matlab

Math people: I have Googled this question, tried some suggestions, and none of them worked. I browsed the Similar Questions here and didn't find what I wanted. I apologize if this is a duplicate. ...
1
vote
1answer
152 views

Iteration convergence.

How can I solve this problem? Let $$x(n+1)=-\frac{\exp(x(n)/2)}{5}$$ be a given sequence. Prove using the Banach contraction principle that this sequence converges to some fixed point $X$ with ...
4
votes
2answers
8k views

Numerical solution to x = tan (x)

I needed to find, using the bisection method, the first positive value that satisfy $x = \tan(x)$. So I went to Scilab, I wrote the bisection method and I got $1.5707903$. But after some reasoning I ...
0
votes
1answer
238 views

Chebyshev rational approximations to $\cos x$

How can we construct all the Chebyshev rational approximations of degree $3$ for $f(x) = \cos(x)$. So, I note that we first get the Taylor series of $\cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{24} ...
0
votes
4answers
299 views

loss-of-significance error

Reduce loss of significance error in the following equation by re-arranging terms: $f_1(x) = \frac{1- \cos(x)}{x^2}$ , assuming $x$ is near $0$. Let $f_2(x)$ be the function rewritten to reduce loss ...
2
votes
1answer
62 views

Possibility of Unboundedness in Least Squares Minimization

Suppose we have the quadratic minimization problem \begin{equation} \min_x \frac{1}{2} x^TPx + q^Tx +r \end{equation} We know that when $P$ is symmetric positive semi-definite, but the optimality ...
0
votes
1answer
909 views

Proving the relative error of division.

The problem says to show that the relative error for division on a computer is $$Rel(\frac{x_{A}}{y_{A}})=\frac{Rel(x_{A})-Rel(y_{A})}{1-Rel(y_{A})}$$ $$\approx Rel(x_{A})-Rel(y_{A})$$ provided ...
0
votes
1answer
234 views

absolute stability / inequality

i want to find the amountof $\theta \in[0,1]$ where it is absolute stable whith $y'=\lambda y$ ,$\lambda \in \mathbb C$ or $\lambda \in \mathbb R$ for $$u_{j+1}=u_j+h[\theta f(t_j,u_j)+(1-\theta ...
0
votes
1answer
90 views

Rate of convergence

Suppose that $p_{1},p_{2}>0$ and that $F_{i}(h)=L_{i}+O(h^{p_{i}})$ as $h \to {0^ + }$ for $i=1,2$. What are the rates of convergence of $F_{1}(h)F_{2}(h)$ for various values of $L_{1},L_{2}$?
0
votes
2answers
126 views

Numerical Analysis. Please, help with MATLAB.

Use the initial approximation $(p_0,q_0)=(-0.3,-1.3)$, and compute the next three approximations to the fixed point using a) Fixed-point iteration and equations : $p_{k+1} = g_{1}(p_k,q_k)$ and ...
0
votes
1answer
119 views

numerical integration for N datapoints

I understand why Simpson's Rule is better than the trapezoidal rule for 3 datapoints (because under the assumption that the function is smooth, a parabolic approximation is going to be better than a ...
1
vote
0answers
45 views

Characteristic-Galerkin convergence rate

I am reading the following article: Characteristic-Galerkin and Galerkin/least-squares space-time formulations for the advectiondiffusion equation with time-dependent domains by O. Pironneau, et al. ...
2
votes
1answer
4k views

Orthogonal polynomials and Gram Schmidt

How can we use the Gram Schmidt procedure to calculate $L_0,L_1, L_2, L_3$, where ${L_0(x), L_1(x), L_2(x), L_3(x)}$ is an orthogonal set of polynomials on $(0, \infty)$ w.r.t. the weight function ...
1
vote
1answer
88 views

Large system of ordinary differential equations

I am trying to find a large system (>20) of coupled ordinary differential equations in order to approximate them numerically on the computer and check the efficiency and effectiveness of various ...
1
vote
3answers
46 views

inequality with one number and a sum of numbers

Let $x_1, \ldots, x_n $ be non-zero real number, such that $\sqrt{x_1^2 + \cdots + x_n^2}=1$. Show that for any $i = 1, \ldots, n$, $$|x_i| \leq \sqrt{\frac{x_1^2 + \cdots + x_n^2}{n}}= ...
2
votes
1answer
96 views

Hermite-Gauss Quadrates Error Bound

Using Hermite-Gauss Quadrates to approximate the integral $I = \int_0^\infty e^{ -x^2} f(x) \, dx $, the error is given as $$ E = \frac{m!\sqrt{\pi}}{2^m (2m)!} f^{(2m)}(\theta) $$ with $0 < \theta ...
2
votes
1answer
118 views

Local truncation error for the forward-difference method

I need to show that if $\gamma=\frac{K\tau}{h^2}=\frac{1}{6}$, then in the explicit forward-difference method $\frac{w_{k,j+1}-w_{k,j}}{\tau}-K\frac{w_{k+1,j}-2w_{k,j}-w_{k-1,j}}{h^2}=0$ the local ...
1
vote
2answers
177 views

Using Taylor Series for $\sin x$ and $\cos x$ to derive $\cos{(x-a)}$ and $\sin{(x-a)}$

My professor had this problem on our last problem set but got rid of it as it was "more involved" than he thought but I am still curious as to how it would be done (Its good that he ditched it because ...
0
votes
1answer
401 views

Find a point on the curve where the curve transitions from linearly increasing to exponential growth?

I've a set of 2-d points. Let's say x and y. I start from x = 0 and increment by 1 and for each increment I record the value of y. So, y is a function in x. If i plot the graph of x v/s y, initially y ...
0
votes
2answers
204 views

Why does the internal rate of return have no analytic expression?

I know that the IRR can be computed using iterative methods, but why is this necessary? What makes it impossible to give an expression for IRR? How would you prove it to be impossible?
1
vote
2answers
201 views

Differential to Difference equation with two variables?

For the following information : $$\frac{dx}{dt} = -10x+3y$$ $$\frac{dy}{dt} = 2$$ How do I convert this to a difference equation ?? I want to use a simple discretisation technique (first order) ...
1
vote
0answers
84 views

new method for solving ODE

Can you please help me with this question? In order to build the new method for solving ODE $y '(x)=f(x,y(x))$, we use an integration formula: $\int_{a}^{b}f(x)dx\approx (b-a)f\left ( \frac{a+b}{2} ...
3
votes
1answer
235 views

Optimization problem with ratio objective

I need to solve the following optimization problem $$ \text{maximize} \quad \frac{(a^T x)^2}{x^TBx+c^T|x|} \quad \text{subject to} \quad \|x\|_1=1 \quad (\text{or alternatively} \quad c^T|x|=1), $$ ...
6
votes
2answers
547 views

Convergence of fixed point iteration for polynomial equations

I'm looking for the solution $x$ of $$x^n+nx-n=0.$$ Thoughts: From graphing it for several $n$ it seems there is always a solution in the interval $[\tfrac{1}{2},1)$. For $n=1$, the ...
0
votes
2answers
462 views

For n=3 Lagrange interpolation why is it equal to 1?

I'm studying Lagrange's formula for polynomial interpolation and I cannot seem understand why for $n=3$ $$L_0(x)+L_1(x)+L_2(x)+L_3(x) = 1$$ for all real x. In my textbook it says as a hint that ...
0
votes
2answers
193 views

Iteration for solving x=g(x).

$g(x) = \frac{x^2}{3}$ $P=3$ $p_0 = 3.5$ 1)Graph $g(x)$, the line $y=x$, and the fixed point P (DONE) 2)Using the given starting value $p_0$, compute $p_1$ and $p_2$ (THE ANSWER MIGHT BE $p_1 = ...
0
votes
1answer
472 views

Proof of convergence of a telescoping series

Show that the telescoping series below converges if and only if the $\lim_{j\to\infty} c_j$ is defined and finite. $$\sum_{j=1}^{\infty} c_j - c_{j+1}$$ Not really sure where to start for ...
0
votes
1answer
38 views

Numeric Differentiation of Analytic funtion

Can anyone validate if my understanding regarding numeric differentiation is correct?? $z = f(x,y)$ is an analytic function. $$\frac{\partial z}{\partial x} = \frac{f(x+h,y)-f(x,y)}{h}$$ ...
4
votes
2answers
239 views

Bounding $x^TAx$ when A is not a symmetric matrix

If $A$ is a real and symmetric $n\times n$ matrix, then we know $\lambda_{min}||x||^2\le x^TAx \le \lambda_{max}||x||^2\ \forall x\in \mathbb{R}^n$ where $\lambda_{max}$ and $\lambda_{min}$ are the ...
0
votes
1answer
94 views

How to speed up the convergence?

I should you Aitken's method and the following formula : $T_n = S_n - \dfrac{{A_{n+1}}^2}{A_{n+1}-A_{n+2}}$ in order to speed up $S_n=\sum_{k=1}^{n} (0.99)^k $ Please, help!
1
vote
2answers
127 views

Numerical Analysis. Newton-Raphson formula.

Attempt: a) $g(x) = x- f(x)/f’(x) = x – [(x-2)^4]/[4(x-2)^3] = (2-x)/4+x = (3x+2)/4$ So, $p_k = [3p_{k-1}+2] /4$ b) p(1)=2.1; for j=2:5, p(j) = (3*p(j-1)+2)/4 end p = ...
1
vote
2answers
381 views

Numerical Analysis. Bisection method.

What will happen if the bisection method is used with the function $f(x) = \tan(x)$ and a) $[3,4]$ b) $[1,3]$ Attempt: Check the signs of the function: $f(x) = \tan(x)$ a) $f(3)f(4) ...
0
votes
1answer
119 views

Polynomial proof Newton

Let $f(x)=a_mx^{m}$ + (lower degree terms) be a polynomial. Show that $$f[x_0,...,x_n,x] = \begin{cases} {degree}[m-n-1], & n < m-1 & \\ a_m, &n =m-1 \\ 0 & ...
1
vote
0answers
175 views

what is sub-linear convergence called?

What is sub-linear reciprocal convergence called? Would it be called parabolic convergence? For example, consider the series: ...
1
vote
2answers
895 views

Loss of significance error

Give exact ways of avoiding loss-of-significance errors in the following computations: a. $log(x+1)-log$, with large $x$ b. $\frac{1-cosx}{x^2}$, with $x\approx 0$ ...
0
votes
1answer
266 views

Loss of significance errors

Give exact ways of avoiding loss-of-significance errors in the following computations: a. $\tan x-\tan y$, with $x\approx y$ b. $\sin x - \sin y$, with $x\approx y$ I don't know how ...
0
votes
1answer
522 views

Interpolation polynomial

Consider the following table of values for a function $j_0(x)$: $\begin{array}{c|ccccc} x & \delta_0(x) \\ \hline ...
0
votes
1answer
229 views

butcher tableau for given algorithm

Given $y'(t) = f(t,y(t))$ and the following algorithm: $$y_{n+\frac{1}{2}} = y_n + \frac{h}{2}f(t_n,y_n)$$ $$y_{n+1} = y_n + hf(t_n+\frac{h}{2},y_{n+\frac{1}{2}})$$ We should show that this can be ...
1
vote
1answer
161 views

cubic spline verses cubic b-splines? [duplicate]

hi friends now a days am dealing numerical problem with cubic spline but am little bit confuse while using them because of term spline and b-spline. I just want to know in easy and simple words what ...
2
votes
2answers
444 views

Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes

I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...
0
votes
2answers
72 views

convergence of newton algorithm when looking for roots of $f(x) = xe^x$

I want to show the convergence of newton's algorithm when calculating the root of $f(x)=xe^x$ using an $x_0 \geq 0$. The resulting recursion is $$x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)} = x_k - ...
2
votes
1answer
84 views

newton: convergence when calculating $x^2-2=0$

Find $x$ for which $x^2-2=0$ using the newton algorithm and $x_0 = 1.4$. Then you get $x_{k+1} = x_k + \frac{1}{x_k} - \frac{x_k}{2}$. How to show that you need 100 steps for 100 digits precision? ...
1
vote
1answer
229 views

Proving numerical stability of an algorithm.

Let us define $\alpha=x^{2}-y^{2}$ and $\beta=2xy$ for $x,y\in\mathbb{R}$. In order to calculate the $\alpha$ and $\beta$, we use the following algorithm: $$p:=x-y;\quad q:=x+y;\quad \alpha=p\cdot ...
1
vote
1answer
567 views

Lagrange basis function

Let $x_0,...,x_n$ be distinct real numbers and $l_k(x)$ be the Lagrange's basis function. $\delta_n = \prod^n_{k=0}(x-x_k)$. Prove that: a. - $\sum^n_{k=0}(x_k-x)^jl_k(x)\equiv 0$, for ...
2
votes
1answer
295 views

householder transformation matrix

Hi could you help me with the following: Let A be the matrix $$\pmatrix{-2 & 1& 1 \\ -2& 2& 1\\2 &-2& 3 \\ }$$ with an eigenvalue $\lambda = 2$ and corresponding ...
2
votes
3answers
487 views

Changing $(1-\cos(x))/x$ to avoid cancellation error for $0$ and $π$?

I have to change this formula: $$\frac{1-\cos(x)}{x}$$ so that I can avoid the cancellation error. I can do this for 0 but not for $π$. So I get: $$\frac{\sin^2(x)}{x(1+\cos(x))}$$ which for $x$ close ...
1
vote
1answer
137 views

Conjugate gradient method

http://www.webpages.uidaho.edu/~barannyk/Teaching/hw7_Math432.pdf Can someone help with #2? This is NOT homework. This was a website for Fall 2011, but it is a very interesting question and I would ...
5
votes
2answers
1k views

Numerically stable extraction of Axis-Angle from Unit Quaternion

I am looking to extract an axis-angle representation from a unit quaternion. From the definition, a naive attempt might be: $ q = \begin{bmatrix} cos(\theta/2) \\ \omega_x \sin(\theta/2) \\ \omega_y ...
1
vote
1answer
129 views

Condition number of $a^2-b^2$

Can someone tell me how to count Condition number of $a^2-b^2$ or recommend a site where I can read about this. I know how to count Condition number of a matrix, but here I'm confused