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

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3answers
101 views

Does $\sqrt{\frac{2n}{n + 1}}$ have a global minimum, for large $n$?

Does the quantity $$\sqrt{\frac{2n}{n + 1}}$$ have a global minimum, for large $n$? Successive tries at WolframAlpha yield the following results: Minimize $\sqrt{\frac{2n}{n + 1}}$ for $n > ...
2
votes
1answer
97 views

Is WolframAlpha computing this radical correctly?

Is WolframAlpha computing this radical correctly? $$\sqrt{\frac{1}{1 + {10}^{-375}}}$$ When I double-check again, the inequality: $$\sqrt{\frac{1}{1 + {10}^{-x}}} > 1$$ leads to a ...
0
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1answer
44 views

numerical calculation on exponent of a matrix

I am using computer language (Matlab and mathematica) to compute the exponent of a matrix of the form $$B_n = \exp(n\hat{A}), \qquad n\in \{1, 2,3, \cdots, N\}$$ where $n$ is positive integer and N ...
2
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0answers
59 views

When solving PDEs is there an alternative to interpolation for out-of-grid point?

I'm numerically solving a PDE where the space domain is huge. So, I often need to interpolate to get out-of-grid points needed by the finite difference algorithm. As a result, I've a lot of numerical ...
1
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1answer
318 views

Fixed-Point Iteration method unable to converge to any of a function's infinte roots

An equation is given to me which has to be solved by direct iteration method: $$sin(x) = {x+1 \over x-1}$$ or $$f(x)=\sin(x)-{x+1 \over x-1} = 0$$ I follow the following procedure with reasons ...
1
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0answers
70 views

Order of a linear multistep method

Given a linear multistep method $\sum_{j=0}^{k}\alpha_jy_{n+j}=h\sum_{j=0}^k\beta_jf_{n+j}$ how to show using Taylor series expansion that the method is of order $p$ if and only if ...
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0answers
46 views

Appropriate numerical scheme for eigen value problem

I am looking for a numerical scheme which can easily handle the following eigen value problem I already had the analytical results of this problem, now I want to know how to treat this problem ...
1
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1answer
114 views

Find the best integer value possible using o- and O-notation for some series.

a. Consider the series \begin{equation} e^{\tan(x)}=1+x+\frac{x^{2}}{2!}+\frac{3x^{3}}{3!}+\frac{9x^{4}}{4!}+\dots\qquad (|x|\leq\pi/2) \end{equation} Retaining three terms in the series, estimate the ...
11
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1answer
266 views

Is there an efficient method for the calculation of $e^{1/e}$?

(I wonder whether this is appropriate for the Math StackExchange or whether it'd be better on Stack Overflow as it deals with computing, but I'm asking about mathematical details, not about ...
3
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1answer
248 views

How prove this “Trigonometric interpolation”?

let $f(x)$ show that $$f(x)\approx\sum_{i=0}^{2n-1}f(t_{j})L_{j}(t)$$ where $$t_{j}=\dfrac{\pi}{n}j$$ ...
1
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1answer
59 views

how to find the zero root of only one equation with two arguments by Newton-Raphson's method?

The two dimensional Newton-Raphson method is to find a zero root $(x_0,y_0)$ which satisfy \begin{array}{*{20}{c}} {f\left( {{x_0},{y_0}} \right) = 0} \\ {g\left( {{x_0},{y_0}} \right) = 0} ...
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0answers
201 views

Formula for intersection of “power” curve and parabola.

EDIT I have edited this question to make it more clear. I have spent quite some time trying to find this on Google, but haven't succeeded. I need the formula(s) to determine the intersection ...
-1
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1answer
277 views

Finding out the nodes from spherical symmetric equation for mathematica and then plotting it in gnu-plot

What I want to do is to draw a curve for the spherical partial differential equation for S at rho=0: \begin{equation} \frac{\partial^2S}{\partial \rho^2}+\frac{d-1}{\rho}\,\frac{\partial S}{\partial ...
1
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1answer
283 views

How do deal with a giant sparse matrices?

Someone point me in the right direction. I'm looking to do some heavy-duty manipulation of some really large and often very sparse matrices. Naturally, this problem overlaps programming heavily (I ...
2
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2answers
159 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: $v(x,0)=5x$ ...
3
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1answer
189 views

Ideas on matrix factorizations and/or transformations for $\ell_1$ minimization

I am starting with a typical $\ell_1$ basis pursuit problem: $$ \min_{\mathbf{x}} \Vert \mathbf{x} \Vert_1 \quad \mathrm{s.t.} \quad \Vert \mathbf{ERx} - \mathbf{y} \Vert_2 \leq \epsilon, $$ where ...
3
votes
3answers
303 views

Creating a degree $n$ Taylor polynomial for $\sqrt{1+x}$

I have been asked to produce a general formula for the degree $n$ Taylor polynomial for $\sqrt{1+x}$ using a=0 as the point of approximation. Given that ...
1
vote
1answer
60 views

How to distinguish the contribution of two variables in a large data set

I'm trying to solve a statistics problem with no formal training in the area, so please bear with me. I have a function with variables of pounds of product and number of figures which together give a ...
0
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2answers
78 views

Numerical Methods

Assuming I am given a Program which can calculate the value of a continuous, infinitely differntiable (we cannot calculate these derivatives), real, positive function of two real variables which has ...
4
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4answers
215 views

How calculate $\pi$ to an accuracy of 10 decimal places?

Let $a=3.00000000001234...$ (irrational number) If $\overline{a}=3.00000000001$ (approximation $11$ places) then $|a-\overline{a}|<10^{-11}$ Note that the reciprocal is not satisfied: If ...
4
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1answer
2k views

Can you calculate the accuracy of a calculator?

I have a phone with an inbuilt calculator. I love to play with calculators. So i did the following and the following was shown by the calculator. When I went in the scientific tab, and wrote $\pi$ it ...
1
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0answers
71 views

Solving the difference equation in the stability analysis of a multistep method.

So I am confused in going from a difference equation to a linear ODE. To make this concrete let's look at the second order Adams-Bashforth method we have: $$ Y_{n+1} = Y_n + h(\tfrac34 f_n - \tfrac12 ...
6
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2answers
2k views

Fast Matlab Code for hypergeometric function $_2F_1$

I am looking for a good numerical algorithm to evaluate the hypergeometric function $_2F_1$ in Matlab (hypergeom in Matlab is very slow). I looked across the ...
1
vote
0answers
103 views

Formula for monthly payment of mortgage

What is the formula for monthly payment of mortgage including Term, Interest Rate, Cost of Home Down, Payment Insurance, Property Tax, HOA Fee. I'm a programmer and want to add this functionality to ...
5
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0answers
117 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
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0answers
27 views

Lagrange finite elements [duplicate]

We consider in $\mathbb{R}^2$ the set of points $$\{M_1(-1,1),M_2(0,1), M_3(2,1),M_4(-1,0),M_5(1,0),M_6(2,0)\}$$ Let $\Omega$ a rectangular structure consisting of the heads $\{M_4(-1,0),M_6(2,0), ...
1
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0answers
99 views

Algorithm of projection

Suppose $S$ is a compact surface in $\mathbb{R}^{3}$ defined by a sufficiently smooth level set function $f$, that is, $S=\{s: f(s)=0\}.$ I am studying an algorithm that projects a point $x_{0}$on ...
2
votes
1answer
365 views

Integrating $\sin(n\theta(x))/\sin(\theta(x))$ for some function $\theta(x)$

I have an indefinite integral of the form: $$ \int \frac{\sin(n\theta(x)))}{\sin(\theta(x))} dx. $$ $\theta$ is a function of $x$ (and actually a complicated one). Is it possible to integrate it ...
3
votes
2answers
70 views

what is name of this numerical scheme for ode?

Let's have system of ODEs $$ \dot x(t) = A(t)x(t) $$ I came up with this numerical scheme: $$ x_{n+1} = e^{\frac{h}{2}A(t_{n+1})}e^{\frac{h}{2}A(t_n)}x_n $$ where $h$ is time step, $t_n = nh$ and ...
2
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1answer
179 views

Related to Applying Runge-Kutta Method

I have an initial value problem (henceforth IVP) as follows: $$\frac{d \Phi(t)}{dt}= A(t)\Phi(t)$$ subject to the initial condition $\Phi(t_0)=I$, where $\Phi(t), A(t), I$ are square matrices of same ...
2
votes
0answers
129 views

Solution of an implicit Fourier transform equation

How does one solve the following equation ($\hat{a}(k)$ denotes the Fourier transform of $a(x)$ and $q$ is real positive): $$\hat{a}(k)=f(k)\widehat{a^q}(k).$$ This equation appeared in some paper. ...
4
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1answer
535 views

Riemann sum error and the integral

It is a well known, that we have the following approximation error: $$ ...
1
vote
1answer
459 views

Numerical analysis Taylor's method question: Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$.

Let $f(x)=\tan^{-1}(x)$ Let $P_n(x)$ be the $n$th Taylor polynomial for $f(x)$ about $x_0=0$ Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$. Is ...
0
votes
2answers
83 views

Proving the continuity of a function at a given point - help needed

I have come across this question and am not sure as to how to go about finishing it. I have started off with working at out the limit at $x=2$ and this is $-4$. How then (or what do I use) to equate ...
0
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0answers
50 views

Estimate parameters in $y=y_{0}(1-\frac{t}{\tau})e^{-\alpha t/\tau}$

Given the function $y(t)$ with two independent parameters $\tau$ and $\alpha$ $$ y=y_{0}\left(1-\frac{t}{\tau}\right)e^{-\alpha t/\tau}, $$ We have two data points (experimental data) $ ...
1
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6answers
2k 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 ...
0
votes
1answer
61 views

Subdividing a Bézier patch

I have a tensor-product Bézier patch and I want to subidivide this adding a curve inside the patch, which creates two rectangular subpatches. I found that the following statement holds: "if we ...
1
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1answer
780 views

Numerical integration over a surface of a sphere

I am integrating a double integral in spherical coordinates over the surface of a sphere in MATLAB numerically. Although I have changed the relative and absolute tolerance I get the feeling that ...
0
votes
1answer
286 views

optimization of a non-differentiable, component-wise step function

I would like to estimate the (local) minimum of a function $c:R^N \mapsto R^+$ where: $c$ is only differentiable almost everywhere, there exists a component $j$, such that $\frac{\partial ...
2
votes
1answer
79 views

Local constant interpolation in $L^1$

I really hope that anybody of you can help me with the following question: Consider the set $U\subseteq L^1([0,1])$ of non-negative integrable functions with unit mass, i.e. $u\geq 0$, $\int_0^1 u\,dx ...
1
vote
1answer
99 views

Looking for a window containing the solution of an equation

I need to solve billions of times equations $\,f(x)=0\,$ with $$f(x) := \sum_{i=1}^N \frac {z_i}{c_i + x}$$ All $z_i$ are positive and add to $1$. Among the $N$ coefficients $c_i$, $M$ are negative ...
1
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0answers
70 views

When examining global error bounds for Euler method, can I rescale the domain limits?

I'm looking at provable global error bounds of the Euler method for the first time and I was surprised to find that the bound grows exponentially in the amount of time (the domain size) propagated ...
2
votes
1answer
119 views

What is a norm that can measure average oscillatory amplitude?

I am numerically computing the growth of an oscillatory instability in a fluid system. Suppose for simplicity that the function $f(x)$ [defined on a finite interval] has oscillations of different ...
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3answers
6k views

Modified Euler Method for second order differential equations

The question I am doing is asking me to carry out the Modified Euler method for a second order differential equation: Consider the following initial value problem: ...
2
votes
1answer
473 views

Optimizing trigonometric and nonlinear functions

First, Please, keep in mind that I'm a programmer not mathematician, and I have a fair mathematical background. I used optimization in Java to fit some observations to a trigonometric function, I ...
1
vote
1answer
424 views

Runge-Kutta method for multiple springs

If we have a spring attached to a wall with an object on the other side, the differential equations describing the system are: $$x'=v$$ $$v'=-\frac{k}{m}x-\frac{b}{m}v$$ Where: x is position of the ...
0
votes
1answer
144 views

norm and invertibility

I'm currently solving some problems on matrix norms and one of these is asking me to show if a matrix is invertible or not. I wento trough the solution and at the end there was written : $\|A\|>0$ ...
1
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1answer
162 views

Power iteration provably works if the matrix has a unique eigenvalue $\lambda$ and $\lambda>0$

Let $A$ be a $n\times n$ real matrix and $v_0 \in \mathbb R^n$ s.t. $||v_0|| = 1$. Define a sequence $(v_k)_k$ of $n$-dimensional real vectors by $v_k = A^kv_0 / || A^kv_0 ||$. Assume that $A$ has a ...
0
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1answer
38 views

Calculate the weigths of a quadrature with highest precision

How do I calculate the weights $H_0$ and $H_1$ so that the the precision of the approximated function is as high as possible? $$ \int_{-1}^{1} f(x)\, dx \approx H_0 f\left(-\frac{1}{2}\right) + H_1 ...
3
votes
1answer
68 views

Numerical solution to diffusion-like equation with changing sign

I am trying to numerically solve an initial value problem $$ \frac{\partial f}{\partial t} = \frac{1}{x} \frac{\partial^2 f}{\partial x^2}$$ where $f = f(x,t) \text{ for } x \in [-1,-1],\ t \in [0, ...