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

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1answer
328 views

Newton's Method - iteration formula

The iteration formula $$x_{n+1}=x_n-(\cos x_n)(\sin x_n)+R\cos^2x_n$$ where $R$ is a positive constant is obtained by applying Newton's method to some function $f(x)$. What is $f(x)$? What can this ...
2
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3answers
642 views

Newton's Method - Why is there slow convergence with a high multiplicity

I'm using a calculator to observe the sluggishness with which Newton's method converges with $f(x) = (x-1)^8$. I let $x_0 = 1.1$. Clearly it's taking forever to get to the root $x=0$. I'm not ...
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1answer
436 views

Application of Newton-Raphson method

Is Newton-Raphson method applicable to solve rational equations? or any equation except polynomial equations.
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0answers
59 views

Inequality in H-curl function space

Define a function space V , $$ V:=\{\mathbf{v} \in \mathbf{L}^{1+\alpha}(\Omega), \mathbf{curl}~\mathbf{v} \in \mathbf{L}^2(\Omega)\}, $$ equipped with graph norm $$ \|\mathbf{v}\|_{V} := ...
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1answer
159 views

Bilinear Coons patches

Suppose that we have two bilinear Coons patches which share a common curve. Studying on Farin I find that, generally, these two surfaces join with C^0 continuity along that common curve, but I don't ...
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1answer
67 views

Condition Number of Polynomial (Condition Number = 0)

I'm calculating the condition number of a polynomial equation $$ y = (x-2)^{9} $$ for this equation, the Jacobian is equal to ...
2
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1answer
78 views

Why does this relative error work?

Assuming $p$ is the exact solution and $p_n$ is a numerical approximation. My question is that why most of the numerical analysis books using $$ \frac{|p_n-p_{n-1}|}{|p_n|} $$ to approximate the ...
2
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1answer
251 views

Solving Hamilton-Jacobi-Bellman equations numerically?

I've been told that HJB equations can be solved numerically. I know very little about the subject, could someone provide a couple of comments or a reference (ideally, one that is accessible for a ...
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3answers
2k views

Rate of convergence for sequences

I have numerically determined that the sequence $\{f_x\} = \frac{\sin(x^2)}{x^2}$ approaches $1$ (as $x$ approaches $0$) faster than the sequence $\{g_x\} = \frac{\sin^2(x)}{x^2}$. However, I am stuck ...
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3answers
149 views

If I know that a polynomial (of order $k \gt 2$) has at most $1$ positive real root - can I find that easily?

[update 2] Urgghh - the time-consumption really stems only from the construction of the h-order polynomial. The time for finding the roots (only 10 to 20 times Newton-iteration because of my nice ...
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1answer
106 views

Computing monthly loan payments when interest is 0%

I'm writing a Javascript program to display a mortgage amortization from a user input form that asks for typical things such as loan amount, interest rate, etc... A lot of sites, such as this one, ...
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1answer
64 views

Precision of $\sinh x$ at $x \approx 0$

What can we do to increase the precision of $\sinh x$ at $x \approx 0$? I tried to use conjugation , but it gives me $e^{2x} - e^{-2x}$ which is again the subtraction of "similar" numbers at $x ...
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1answer
43 views

Significant digits question. compute $\sin x$

Why if $x$ is a machine number on a $32$-bit computer that satisfies the inequality $x > \pi 2^{25}$, then $\sin x$ can always be computed with no significant digits? Thank you.
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1answer
389 views

What is a function that is discontinuous, yet the bisection method converges?

Also what about a function for which it diverges? I was thinking f(x) = 10 + 1/x would work for where it converges, but can't think of one where the bisection method diverges at near a root. Thanks ...
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1answer
53 views

Kirchoff's current law - unable to understand (possibly) simple transformation

I suspect it must be something very simple but for the life of me I cannot figure out how from the below (which models a circuit of two resistors connected in parallel) ...
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0answers
151 views

Matlab.Compute $f(x) = \sin(x) + \cos(x)-1$

Write a procedure to compute $f(x) = \sin(x) + \cos(x) - 1$ The routine should produce nearly full machine precision for all $x$ in the interval $[0, \frac{\pi}{4}]$ Hint: $\sin^2 \theta = ...
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1answer
618 views

Newton's Method Annuity Due Equation

I'm having a really hard time with a homework problem I've been assigned for my numerical analysis class. It's supposed to be a Newton's method question, but I don't see how to use the method here: ...
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1answer
234 views

Problem with Newton's Method

I am trying to solve this problem: Use Newton's method to find the intersection points of the two circles defined by $$x^2 + y^2 = 2, \quad (x-1)^2 + (y+1.5)^2 = 1.$$ I used this code in ...
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2answers
63 views

Which is the correct relative error?

Which is the correct relative error? $$ r_1=\frac{|p_n-p|}{|p_n|} $$ or $$ r_2=\frac{|p-p_n|}{|p|} $$
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0answers
158 views

Non-linear ODE for which backward Euler becomes unstable?

One way to solve initial value problems of the type $\dot{x} = f(x), \; x(0) = 0$ numerically is to use the backwards Euler method $x_{n+1} = x_{n} + \Delta t f(x_{n+1}), \; n = 1,\ldots \\ x_{1} = ...
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0answers
120 views

how to choose point spacing to approximate a parametric curve using line segments?

Suppose I have a parametric equation for a curve $\vec{r} = f(t)$, which I wish to draw using line segments between some set of points at times $t_0, t_1, t_2,$ etc. If I want to achieve a given ...
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1answer
512 views

Avoid cancellation errors for approximating the derivative of the sine function

The function $f_1(x_0,h) = \sin(x_0+h) - \sin(x_0)$ can be transformed into another form, $f_2(x_0,h)$, using the trigonometric formula: $$\sin(\phi)-\sin(\psi) = 2\cos\left(\dfrac{\phi ...
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1answer
56 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 ...
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1answer
164 views

Bounding a bicubic polynomial

My actual situation is working with bicubic polynomials, (that is, polynomials of the form $\sum_{i=0}^3 \sum_{j=0}^3 a_{i,j} x^iy^j$) defined on the unit square $[0,1]^2$ (actually these are ...
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1answer
67 views

Approximating the optimal value of a function involving a Gaussian integral

Consider the following function $$ f(\lambda) = \alpha (1+\lambda^2) + (1-\alpha)2\int_\lambda^\infty (x-\lambda)^2 \phi(x) dx $$ where $\alpha \in (0,1)$ and $\phi$ is the standard normal probability ...
2
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1answer
131 views

Eigenfunctions of Laplacian on sphere - numerical approach

Consider a Laplacian: $$\hat L=\frac1{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac1{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2},$$ where ...
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1answer
263 views

solving bessel equation numerically.

Assuming there's an equation (bessel) and I'm told to solve numerically. This means, to solve this type of equation, we must convert the equation to a system of first order ODE's by letting $z=y'$ and ...
0
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1answer
57 views

Numerical analysis the use of Fast Fourier Transform

If I am going to use the pseudo-spectral method to solve $$ u_t = -u_{xxx} - 6uu_x, $$ how do I set up the RHS? Would it be $-\left[\mathcal{F}^{-1}\left[(ik)^3\mathcal{F}(u(x))\right] + ...
0
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1answer
853 views

Matlab fast summation

I was wondering whether there is a faster way to evaluate this double sum in matlab: $$\sum_{n=1}^{\text{max}} \sum_{m=-n}^{n} f(n,m).$$ Cause I am currently doing this with a foor loop over n and m ...
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2answers
941 views

Matlab for loop output into a vector

this is where im at right now, the only thing i need is to be able to stop the procedure if error is below tolerance. ...
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3answers
266 views

Law of Cosines for very acute angles, round-off error

We have $$ c^2 = a^2 + b^2 - 2ab\cos(\gamma) $$ If $a \approx b$ and $\gamma$ is very small, then the above formulation has quite a bit of round-off error. Is there a better formulation that would ...
6
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3answers
132 views

Why is $(1+x)^2$ less accurate than $ (x+2)x+1$ for small $x$?

I've known that accuracy is based on the amount of roundings (or multiplications) that occur, but from what I can tell, both equations will require the same amount. My first thought was to related ...
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0answers
41 views

Gauss-Seidel Smoother for Problems with jumping coefficients

for a project I am working on a "robust" multigrid implementation, i.e. a implementation which achieves fast convergence even for problems with discontinious coefficients. My first problem is: $ - ...
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4answers
659 views

Damped oscillation fit

We have some measurement data like this: The expected behavior of the data is a damped oscillation: $$y=a e^{d*t} cos(\omega t+\phi) + k$$ Where: $t$ Current time $y$ Current deflection ...
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1answer
34 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 ...
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2answers
445 views

LU decomposition by hand

Can someone show me a step by step solution to calculate the $LU$ decompisition of the following matrix: $A = \begin{bmatrix} 5 & 5 & 10 \\ 2 & 8 & 6 \\ 3 & 6 ...
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1answer
228 views

Computational efficiency using Gaussian elimination

Assume it took 2 seconds to solve an equation Ax=b for x (where A is a 3×3 matrix and b is a 3×1 matrix) using Gaussian elimination, how much longer would it take to: a) use Gaussian elimination to ...
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1answer
361 views

Problem related to fixed point iteration

How can I use fixed point iteration to solve $x^2 = 3$ using $g(x) = x^2 + x - 3$ to find the numerical value of the solution $x = +\sqrt{3}$. What happens? Then I use $g(x) = (x + 3/x)/2$. For which ...
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0answers
36 views

Show that $g'(p) \approx ({p_n-p_{n-1}})/({p_{n-1}-p_{n-2}})$

Suppose the sequence {${p_n}$} is generated by the fixed point iteration scheme $p_n = g(p_n-1).$ Further, suppose that the sequence converges linearly to the fixed point $p.$ Show that $g'(p) ...
2
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1answer
129 views

Density of smooth functions in fractional Sobolev space

I am reading a paper on the analysis of numerical methods, and am confused about a statement made. I am working in fractional Hilbert spaces, but I don't think that this has much bearing on the answer ...
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2answers
88 views

Conversion of single precision double into a floating point

How do I convert $0$ $00000001$ $00000000000000000000000$ into a floating point number? Apparently the sign is + I've started with $00000001$ which is 1 in decimal. Then I applied formula ...
0
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1answer
90 views

Roundoff Error by a machine number

What is the roundoff error when we represent $2^{-1} + 2^{-25}$ by a machine number? (Note: this refers to absolute error, not relative) Please, help
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2answers
379 views

Is there a mean value theorem for higher order differences?

The standard mean value theorem tells us $\frac{f(x+h)-f(x)}{h} = f'(c)$ for some $c$ between $x$ and $x+h$. Rewriting this, we may see it as $\frac 1h\Delta_h f(x) = f'(c)$. This makes me wonder if ...
2
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2answers
177 views

Lipschitz Condition…

I am independently studying Numerical analysis and came across a question for which I am stuck at. Assume that $g(x)$ is differentiable. Show that if $|g'(x)|<1$ over $[x_0-p, x_0+p]$, then $g(x)$ ...
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2answers
594 views

Prove that $g(x)=e^{-x^2}$ has a unique fixed point on the interval [0,1]

Hello I need help with this question, Prove that $g(x)=e^{-x^2}$ has a unique fixed point on the interval [0,1]
1
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1answer
282 views

Newton-Raphson Method

just want somebody to help me verify if I'm doing using Newton-Raphson method correctly by checking the result of an equation. $f(x) = e^{x-1} + x^2 - 7$ I'm trying to find the zero of f(x) with ...
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0answers
258 views

FFT for Fourier Integrals of analytic Functions

So, I'm trying to implement a fourier-transform of an analytic function through DFT or FFT as I'm only interested in a certain frequency range. So far I've tested Numerical Recipes' FFT algorithm ...
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0answers
76 views

Lobatto quadrature rule

Where can one find explanation and rigorous theoretical introduction to Gauss-Lobatto, Gauss-Radau, Gauss-Kronrod rules. Also pros and cons of each method are of interest too.
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1answer
65 views

Numerical solution of a difference equation

I would like to solve in $f$ the equation $f(x) - f(x-d) = g(x),$ where $g$ is a given function and $d$ a given constant delay. We can assume $f(x) = 0$ for negative $x$. When $g$ is sampled (with a ...
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5answers
470 views

How to calculate $2^{\sqrt{2}}$ by hand efficiently?

I've been trying to calculate $2^{\sqrt{2}}$ by hand efficiently, but whatever I've tried to do so far fails at some point because I need to use many decimals of $\sqrt{2}$ or $\log(2)$ to get a ...