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

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0
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3answers
712 views

solve the initial value problem ,by Taylor's method of order $N=3$

solve the initial value problem ,by Taylor's method of order $N=3$ $y'(t)=ty(t)+(1-t)e^t,0\le t\le 2,y(0)=1$ with an accuracy of $5 \times10^{-3}$ first we consider the taylor expansion of $e^x$ $ ...
2
votes
1answer
266 views

Error analysis of exponential function

By definition: $$ e^x = \lim_{n \rightarrow \infty} ( 1 + \frac{x}{n} ) ^ n$$ I am interesting in calculating the error $$\left | e^x - \left( 1 + \frac{x}{n} \right) ^ n \right|$$ for some fixed $n ...
0
votes
1answer
46 views

Derive the Simpon's Rule for numerical integration

I think I'm over thinking this because I'm coming up blank. Any help would be appreciated. Here is the question: Derive the Simpon's Rule for numerical integration in a interval $[x_{0}, x_{2}]$ ...
2
votes
0answers
42 views

Finding $5^{1/3}$ with Newton's method

I have this exercise in my book: Let $f(x)=x^3-5$ and we are looking for the solution $f(x^*)=0$, that is $x^*=5^{1/3}$, with Newton's method. So $x_{n+1}=x_n-\frac{x_n^3-5}{3x_n^2}$ and let ...
0
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1answer
47 views

Numerical iterative method for equation with $\cos(x)$

I am practising for the test of numerical methods and here I stumbled on the exercise I don't know how to solve: Show that equation: $x-0,4 \cos(x)=7$ has only one solution $x^*\in\mathbb{R}$ and ...
1
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1answer
878 views

Bisection Method for intersection of two functions

I know how to use the bisection method when finding roots, however I don't know how to use it for when two lines intersect, any help with this would be much appreciated
14
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1answer
267 views

Fastest curve from $p_0$ to $p_1$

I'm trying to solve a problem in path planning: Given points $p_0$ and $p_1$ and vectors $v_0$ and $v_1$, find a function $p(t)$ st. $p(0) = p_0$, $p(T) = p_1$, $p'(0) = v_0$ and $p'(T) = ...
1
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0answers
38 views

Determine which one more accurate approx $f''(x)$

Derivatives can be written 1.) $$f'(x) = lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ 2.)$$f'(x) = lim_{h\rightarrow 0} \frac{f(x)-f(x-h)}{h}$$ Also $$f''(x) = lim_{h \rightarrow 0} \frac{f'(x+h) - ...
0
votes
1answer
356 views

Applying two-point forward to two-point forward formula

What do you get when you apply the two-point forward finite difference formula for the first derivative of $f(x)$ to the two-point forward finite difference formula for the first derivative of ...
1
vote
1answer
36 views

A=N-P decomposition convergence to true solution

I'm taking a numerical analysis course and we're looking at an iterative method for solving $Ax=b$ where $A$ is a square matrix, with a $A=N-P$ decomposition with the formula $$Nx^{k+1}=Px^k+b$$ ...
2
votes
1answer
430 views

Prove $T_n(x)$ of Chebyshev Polynomial given the recurrence relation

Using the recursion formula for Chebyshev polynomials, show that $T_n(x)$ can be written as $$T_n(x)=2^{n-1}(x-x_1)(x-x_2)...(x-x_n)$$ where $x_i$ are the $n$ roots of $T_n$ The recurrence ...
3
votes
1answer
123 views

Transfrom a Legendre polynomial from $\int_{-1}^{1}\phi_j(x)\phi_k(x)dx $ into $\int_{a}^{b}\phi_j(t)\phi_k(t)dt$ given $t=\dfrac{1}{2}[(b-a)x+(a+b)]$

The Legendre polynomials satisfy $$\int_{-1}^{1}\phi_j(x)\phi_k(x)dx = \begin{cases} 0 &j\neq k\\\\ \dfrac{2}{2j+1} &j=k \end{cases}$$ Suppose that the best fit problem is given on the ...
1
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0answers
157 views

How do you numerically solve a multivariable ODE system with different time steps per state variable?

If you have a large multivariable ODE system, and certain processes occur at a much smaller time scale, how can you implement a solver that uses smaller time steps for state variables involved in fast ...
1
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3answers
37 views

Find a way to calculate $f(x) = x-\sqrt{x^2-\alpha}, \alpha << x.$

I thought about this but I could not come up with a way to calculate this. Any comment?? note:$x$ is much larger than $\alpha$
0
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1answer
63 views

How can I apply Runge-Kutta to evaluate integral?

I would like to evaluate cumulative normal (0,1)-distribution values using Runge-Kutta method but the problem is that I don't know how to apply the method. Namely, if I have that $y'(x)=e^{-x^2/2}, ...
1
vote
2answers
904 views

Newton's method for polynomial interpolation

I've seen that in Newton's method for interpolating polynomials, the coefficients can be found algorithmically using (in Python-ish): ...
1
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2answers
135 views

Fixed point iteration .Numerical method.

what is the fixed point for the following function? 2sin(pi x)+x=0 between [1,2] I have done it as follows but it doesn't converge. x=-2sin(pi x) and x=(1/pi)arcsin(-x/2) but both of these ...
1
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2answers
1k views

How to find upper bound on absolute error with composite trapezoid rule

Obtain an upper bound on the absolute error when we compute $\displaystyle\int_0^6 \sin x^2 \,\mathrm dx$ by means of the composite trapezoid rule using 101 equally spaced points. The formula I'm ...
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0answers
39 views

What is the error when approximating $L^2([0,1])$ by a finite dimensional space?

Let $X \subset L^2([0,1])$ such that $f([0,1]) \subset [-M,M]$, for some constant $M$, and any $f \in X$. By choosing a finite dimensional basis $V=\left(v_i\right)_{i=0}^n$, where each $v_i \in X$, ...
0
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1answer
88 views

Numerical integration problem

How can I solve integral equations of the form $$\int_{-3}^x e^{e^t}dt=3?$$ Is there for example Sage code for that kind of equations? Is there better method that evaluating numerically ...
3
votes
1answer
63 views

Proving an identity

We define $\|x\|_A^2:= x^TAx$ and $(x,y)_M := y^TMx$ for a symmetric positive definite matrix $A$ and an invertible matrix $M$. I want to show the following identity for the errors of Richardson's ...
1
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1answer
56 views

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.
1
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1answer
744 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: $f'(x_0)\approx\frac{f(x_0+h)-f(x_0-h)}{2h}$. ...
0
votes
1answer
254 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
132 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 $y$, ...
1
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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
44 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 ...
2
votes
2answers
262 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 ...
2
votes
0answers
150 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 ...
1
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2answers
337 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
0
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1answer
38 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, ...
1
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0answers
121 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 ...
0
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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 ...
0
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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 ...
0
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1answer
59 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 ...
0
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2answers
130 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. ...
8
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2answers
10k 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} = ...
5
votes
1answer
115 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 ...
2
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0answers
53 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 ...
0
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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
221 views

Gaussian integration on triangles

I need to integrate by Gaussian rule on a triangle. Please help me. Thanks alot.
6
votes
1answer
356 views

Wave Equation Non-uniform string (PDE)

The wave equation in a non-uniform string is : $$ u_{tt} = c(x)^2 u_{xx} $$ $$ u(x,0) = f(x) = e^\frac{(x-\mu)^2}{2 \sigma^2} , \:\:u(0,t) = 0\:,\:\:u(L,t) = 0, \:\: u_{t}(x,0) = -cf'(x) $$ ...
1
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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' ...
0
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1answer
38 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
2
votes
1answer
661 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 ...
1
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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 ...
1
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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 ...
1
vote
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.
5
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3answers
755 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?
0
votes
1answer
218 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: ...