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

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When $ e^x$ ~ $ e^{-2x}$ ? - Numerical analysis

For what $x$, $ e^x $ ~ $e^{-2x}$ ? And how one can change this expression to avoid significant digits loss? I am able to think only about $x =0$, but then both are equal and you lose nothing.
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1answer
705 views

Writing a MATLAB .m File to Generate a Plot of Absolute Error as a Function of h (step-size)

This is the question I am to solve: Given the function $f(x)=\ln(3x+1)$, compute approximations to $f'(0)$ using the centered 3-point formula: $f'(x_0)\approx\frac{f(x_0+h)-f(x_0-h)}{2h}$. ...
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1answer
243 views

Finding a Unit Vector v for a Matrix A such that the 2-norm of AV is equal to the 2-norm of A

I have been working on the following problem: Let A be the following 2x2 matrix: A = [1 1; 0 1] (MATLAB notation) Find the 2-norm of A and a unit vector v such that the 2-norm of Av = the 2-norm of ...
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2answers
127 views

Simple Polynomial Interpolation Problem

Simple polynomial interpolation in two dimensions is not always possible. For example, suppose that the following data are to be represented by a polynomial of first degree in $x$ and $y$, ...
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1answer
34 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)= ...
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0answers
41 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 ...
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2answers
244 views

Curves with “constant speed”?

I am new to the concept of curves. Let us a assume we have a simple function such as $f:\mathbb{R}^+\rightarrow \mathbb R^+\quad f(x) := \sqrt{x}$. (Or $f(x)=\exp(x)$ or a polynomial etc.). We can ...
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0answers
136 views

How to generate a random matrix which have given singular values?

I know one method: generate a random matrix, apply SVD decomposition, modify singular values, and then multiply those matrices back together. However, I'm wondering how random this method is. Since ...
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2answers
308 views

Determine the number of solutions of nonlinear system without solving.

$x^2-y^2+2y=0$, $2x+y^2-6=0$ I need to determine the number of solutions without solving it. There is a hint that a graph can help but I am still not sure how to go about this. Thanks
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1answer
35 views

Approximating an integral by evaluating the cumulated sum

I am using a cumulated sum to approximate an integral. My initial thought was that the integral in the interval from a to b by evaluating the cumulated sum at b and a, and subtract. When I do this, ...
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0answers
111 views

Analytical solution(root) for a tenth order polynomial?

is it possible to develop an analytical solution (root) for such a polynomial: $f(x)=\left(x^{10}-c_1^2\right)*\left(c_2-x\right)^2-0.2*\left(x^2-1\right)*c_1^2$ with $c_1$ and $c_2 >0$. Numerical ...
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1answer
30 views

element wise matrix operation problem

I am doing an element wise power calculation, and at a given point, I get a complex value out of real values! I have attached a screen shot from the debugging mode in Matlab So, one can see that the ...
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1answer
65 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 ...
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1answer
56 views

Simple Newton's method problem

Estimate the number of iterations of Newton's method needed to find a root of $f(x)=\cos(x)-x$ to within $10^{-100}$. The answer is $7$ iterations, but I have no idea how it was solved by my ...
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2answers
120 views

Give me a fun problem related to numerical methods.

I hope that this doesnt violate the rules since I need a problem instead of an answer. We have to make our own problem and present it in the class. First course in numerical methods using MatLab. ...
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2answers
9k views

Explanation and Proof of the fourth order Runge-Kutta method

Runge-Kutte 4th order method is a numerical technique used to solve ordinary differential equation of the form $dy/dx=f(x,y), y(0)=y_0$ It gives $y_{i+1}$ in the form $y_{i+1} = ...
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1answer
113 views

Calculation of integral with Bessel function

I have a trouble with to calculating (or bounding from above) the following integral: $$ \int_{-\infty}^{\infty}\left(\frac{J_2(x)}{x^2}\right)^p\, dx, \quad p\geq 1, $$ where $J_2(x)$ is a Bessel ...
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0answers
49 views

Is scalar product a well-conditioned operation?

I'm reading a course and one of the exercises is about establishing whether scalar product is a well-conditioned operation. Here's their solution. They disturb each element of the vector by ...
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1answer
50 views

Constrained non-linear optimisation algorithm making use of problem structure

I have a problem that in some ways is quite simple and in other ways is quite hard. I feel that there is probably an algorithm out there that is better suited to solving my problem than the one I am ...
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1answer
198 views

Gaussian integration on triangles

I need to integrate by Gaussian rule on a triangle. Please help me. Thanks alot.
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1answer
352 views

Wave Equation Non-uniform string (PDE)

The wave equation in a non-uniform string is : $$ u_{tt} = c(x)^2 u_{xx} $$ $$ u(x,0) = f(x) = e^\frac{(x-\mu)^2}{2 \sigma^2} , \:\:u(0,t) = 0\:,\:\:u(L,t) = 0, \:\: u_{t}(x,0) = -cf'(x) $$ ...
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1answer
35 views

Proof about fixed point and sequences [closed]

Let g be a function of a variable that is a contraction on $X = [a, b]$, and $\{x_n \}^{\infty}_{n=0}$ the iteration sequence generated by this function. If $g \in C^{1}(X) $shows that: a)When $g' ...
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1answer
37 views

Least square problem and pre-Hilbert, Numerical analysis Homework

how to show the minimizing max has a solution. confusing about how to approve it
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1answer
649 views

Derive error term by using Taylor series expansions.

Using Taylor series expansions, derive the error term for the formula \begin{equation} f''(x)\approx \frac{1}{h^{2}}\left [ f(x)-2f(x+h)+f(x+2h) \right ]. \end{equation} I've tried it on my own ...
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1answer
52 views

$\forall$ ${i \in \{1,…n\} }$ $ a_{i}<u $ and $\nu<0.01$ prove that there exists \eta…

Here's my exercise: EDIT: $v=nu$, not $\nu$ (same Latex code but one is without ) $\forall n \in \mathbb{N}$ $\forall$ ${i \in \{1,...n\} }$ $ |a_{i}|\leq u $ and $nu<0.01$ and $u=2^{-t-1}$ prove ...
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1answer
34 views

Knowing that $b\leq\frac{a}{1-a}$ and $a<0.01$ show that $b \leq 1.01a$

I've been solving a problem in numerical analysis and to finish one of the exercises I need the following result. Knowing that $b\leq\frac{a}{1-a}$ and $a<0.01$ show that $b \leq 1.01a$. Now I ...
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1answer
31 views

Non-recursive way to present $ p_{0}=0$, $p_{n+1}=(e+1)p_{n}+e$ for some $e>0 \in \mathbb{R}$.

Is there a non-recursive way to present this function: $ p_{0}=0$ $p_{n+1}=(e+1)p_{n}+e$ for some $e>0 \in \mathbb{R}$. Or at least some estimation from the top would satisfy me.
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3answers
690 views

Can I approximate sine and cosine without derivatives?

Assuming I don't know derivatives (and Taylor series) can I manage to approximate sine and cosine of a generic given (rational) angle in radians?
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1answer
211 views

calculate velocity using parametric functions

if i have the following parametric functions where time is m/s : x = 8 t y = -5 t2 + 6 t and i want to find the initial velocity can i do the following: ...
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2answers
599 views

Explanation of Lagrange Interpolating Polynomial

Can anybody explain to me what Lagrange Interpolating Polynomial is with examples? I know the formula but it doesn't seem intuitive to me.
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2answers
876 views

Accurate Numerical Integration for unequally spaced data

I need to calculate numerical integration of unequally spaced data accurately. For equally spaced data, richardson extrapolation on romberg integral works quite well. ...
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1answer
120 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: ...
3
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1answer
364 views

How to calculate Bessel Function of the first kind fast?

I have wrote a C++ code to calculate the first kind of Bessel Functon by its infinite series definition. I took the sum of the first 20 series as the value of Bessel Function, which is same as MATLBA ...
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0answers
59 views

Algorithm for finding power

I has been searching for a high precision library in PHP to do calculations like $$232323232323^{121212.2232323232}$$ etc (ie, with very large numbers, including decimals), but failed to get any. ...
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1answer
95 views

Chebyshev polynomials with non-negative constants

Please let help me solve the following problem that I encountered while engaging in my research. I'm dealing with a class of functions, in which each function has a unique series representation of ...
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1answer
129 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 ...
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2answers
154 views

Sequences and Contraction of a fixed point

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 $x_0, x_1, x_2,\ldots$ given by $x_n = g(x_{n-1})$. ...
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1answer
59 views

Writing Lagrange form of an interpolating polynomial

Write the Lagrange form of the interpolating polynomial of degree at most 2 that interpolates $f(x)$ at $x_0, x_1,$ and $x_2$, where $x_0<x_1<x_2$ I'm guessing I would start at (not sure how ...
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2answers
57 views

Ode equation how to solve them.

enter link description here I have function y' = |y|. Could you explain me how to solve Ode quation step-by-step without using complicated mathematical things. It is easy for me when i knew that y = ...
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2answers
100 views

Develop second-order method for approximating f'(x)

I am stuck on the following question: Develop a second-order method for approximating $f'(x)$ that uses the data $f(x-h), f(x)$, and $f(x+3h)$ only. Any hints/tips?? Thanks!
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1answer
84 views

Need Help! Recognizing types of errors: Truncation and Roundoff

I am a little unclear on the difference between the two. What exactly are they? As simplified as possible :) How can i recognize them and identify parts of formulas or algorithms that would give ...
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1answer
457 views

MATLAB. Secant Method test.

Test the secant method on an example in which $r$, $f'(r)$ and $f''(r)$ are known in advance. Monitor the ratios $e_{n+1}/(e_n e_{n-1})$ to see whether they converge to $- \frac{1}{2} f"(r) / f'(r)$. ...
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2answers
361 views

Alternative form for sinh(x)/cosh(x) .

I have the following expression. $$ \tanh (x) = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} $$ I know that they derive one from another , but how do I rewrite them in alternative forms ...
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1answer
118 views

Hermite Integration problem 1

Hey I am trying to calculate this problem of Hermite polynomial by hand, but I think it's way easier to do on Maple, so can anybody help me write the Maple code: Let $f(x) = 3xe^x - e^2x$. a) ...
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1answer
1k views

Solving system of equations by using simple iteration method

I have a problem: $$\begin{cases} \sin(x) + 2y = 2 \\ \cos(y - 1) + x = 0.7 \end{cases} $$ with margin of error 0.00001 And I need to solve this by using Fixed-point iteration method. Can someone ...
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1answer
71 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 ...
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1answer
361 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 ...
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1answer
693 views

Find an efficient algorithm to calculate $\sin(x) $

Suggest an efficient algorithm to determine the value of the function $ \sin(x) $ for $ x \in [-4\pi, 4\pi] $. You can use only Taylor series and $ +, -, *, /$. I know, that $$ \sin(x) = ...
4
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1answer
148 views

What is the upper bound on the error of a matrix multiplication

When both A and B are n x n upper-triangular matrices, the entries of C = AB are defined as follows: $$ c_{ij} = \begin{cases} \sum _{k=i}^ja_{ik}b_{kj} & 1\leq i\leq j\leq n \\0 & 1\leq j\lt ...
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1answer
44 views

I need to solve this equation numerically but don't have access to any software…

I need to solve this equation numerically, but I don't have access to any software: $$2^{1-t} \log \tfrac12 = a(t) \log a(t)$$ where $a(t):= 2^{-t} + \tfrac12$. The logs are natural logarithms. ...