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

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-5
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0answers
16 views

What method we use when n = odd for evaluate the integral using simpson's rule ??? [on hold]

hi any one can tell me What method we use when n = odd for evaluate the integral using simpson's rule ??? plz help....
0
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0answers
4 views

Numerical evaluation of the Laplace operator near the singularity points in spherical coordinates

If one had to evaluate $\Delta Y_l^m$ numerically everywhere on the unit sphere, including the singularity points $\theta = 0,\pi$, how would they do it? Let's say $Y_l^m$ is a spherical harmonic. I'm ...
0
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0answers
18 views

Need help rearranging a function to avoid loss of significance!

I need to be able to change the equation f(x)= (sqrt(x^2+4)-2)/x in a way which removes the potential for loss of significance. My understanding is that a possible way to do this is to times the ...
0
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1answer
23 views

Show that if the leading principal minors of a nonsingular $n\times n$ matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization

I am stucked at this problem: Prove by induction that if the leading principal minors of an $n\times n$ nonsingular matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization. (The ...
0
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0answers
30 views

Show that for every $n\times n$ matrix $A$, there exist an $n\times n$ permutation matrix $P$ such that $PA$ has $LU$ factorization

I am stucked at this problem: Prove by induction on $n$ that for every $n\times n$ matrix $A$, there exist an $n\times n$ permutation matrix $P$ (a matrix obtained by rearranging the rows (or ...
4
votes
1answer
41 views

Splitting up a double integral

I need to compute the following integral: $$ 2\pi\nu^2\int^a_be^{x^2}\int_{-\infty}^xerfcx(-y)dydx, $$ where $erfcx(x)=e^{x^2}erfc(x)$, $erfc(x)=1 - erf(x)$, and $erf(x)$ is the error function. The ...
1
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0answers
26 views

Using Newton's Divided Difference formula, prove that

Using Newton's Divided Difference formula, prove that $$f(x) = f(0) + x\,\Delta f(-1) +\frac{x(x+1)}{2!} \, \Delta^2f(-1) + \frac{(x+1)(x)(x-1)}{3!} \, \Delta^3f(-2)+ \cdots$$ where $\Delta$ is ...
0
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0answers
28 views

Representable numbers in binary system but not in decimal system

I need some advices to solve this problem. What numbers are representable with a finite expression in the binary system but are not representable in the decimal system? What I've tried is to ...
0
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0answers
24 views

Pseudo-arclength continuation scheme

I have implemented a simple parameter continuation scheme to find the stationary solutions of a nonlinear problem at different parameter values. However, my scheme cannot handle bifurcations - it ...
0
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0answers
43 views

First develop the function $\sqrt{x}$ in a series of powers of $(x-1)$ and then use it to approximate $\sqrt{0.9999999995}$ to ten decimal places. [on hold]

First develop the function $\sqrt{x}$ in a series of powers of $(x-1)$ and then use it to approximate $\sqrt{0.9999999995}$ to ten decimal places. I'm stuck on how to do this problem. Any solutions ...
0
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0answers
14 views

Efficient ways to find a single root of a multivariate polynomial system to arbitrary precision

I am looking for a practical and efficient way to compute, to arbitrary precision, a single root of a multivariate polynomial system (over $\mathbb{Q}$). It seems like the fancy methods compute all ...
1
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0answers
16 views

implicit-explicit predictor-corrector scheme for non-linear parabolic PDE

I am working through a paper on implicit-explicit predictor-corrector scheme for non-linear parabolic PDE and having trouble understanding some concepts. I have this PDE: $\frac{\partial u}{\partial ...
0
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0answers
25 views

Which method do I use ? Interpolation

I have table of $5$ values (i.e abscissa and ordinates are given). I have been asked to find derivative at particular point and also second derivative at that value. That value is between my given ...
0
votes
1answer
45 views

Method of successive approximations to solve y'=y^2

(a) Show that all the successive approximations for the problem $y'=y^2$, $y(0) = 1$, exist for all real $x$. (b) Find a solution of the initial value problem in (a). On what interval does it ...
0
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1answer
40 views

For small values of $x$, the approximation $\sin(x)\approx x$ is often used. Estimate the error using this formula with the aid of Taylor's Theorem.

For small values of $x$, the approximation $\sin(x)\approx x$ is often used. Estimate the error using this formula with the aid of Taylor's Theorem. For what range of values of $x$ will this ...
0
votes
1answer
33 views

Simpsons Method of Order 4

I am trying to solve an ODE using Simpsons Method of order four. I don't know the corrector to use whether implicit or explicit. I need to correct for $y(x_{n+2})$ and $y(x_{n+1})$. Please help me ...
0
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0answers
12 views

Solutions for the dependency problem

Currently I read about the dependency problem of interval arithmetic. Mainly it's the problem that in the equation $X-X$ for $X$ being an interval the following is calculated: $$X-X=\{x-y:x\in X, y\in ...
0
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2answers
41 views

How do i evaluate $ \bigtriangleup^{10}(1-ax)(1-bx^{2})(1-cx^{3})(1-dx^{4}) $

How do i evaluate $ \bigtriangleup^{10}(1-ax)(1-bx^{2})(1-cx^{3})(1-dx^{4}) $ where $ \bigtriangleup$ is forward difference operator. Now to evaluate this is impossible almost using definition which ...
1
vote
2answers
43 views

Find Taylor series for $f(x)=e^x$ at $c=3$. Then simplify the series and show how it could have been obtained directly from the series $f$ at $c=0$.

Find the Taylor series for $f(x)=e^x$ about the point $c=3$. Then simplify the series and show how it could have been obtained directly from the series for $f$ about $c=0$. Taylor's Theorem: ...
2
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0answers
43 views
+50

Name for kind of big O notation with leading coefficient

Context: As known the big O notation $O(f(n))$ describes a function $g(n)$ such that there is a constant $C \ge 0$ with $\limsup_{n\to\infty} \left|\frac{g(n)}{f(n)}\right| \le C$ (I assume that ...
2
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1answer
53 views

How many terms required in $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place?

How many terms are required in the series $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place? Here is what I have: $$e\approx ...
3
votes
2answers
52 views

Use Taylor's Theorem with $n=2$ to prove that the inequality $1+x<e^x$ is valid for all $x\in \mathbb{R}$ except $x=0$.

Use Taylor's Theorem with $n=2$ to prove that the inequality $1+x<e^x$ is valid for all $x\in \mathbb{R}$ except $x=0$. Taylor's Theorem: $$ f(x)=\sum_{k=0}^n{1\over ...
0
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0answers
27 views

How to derive the error when approximating divergence using the Gauss divergence theorem?

I am trying to derive the error for approximatively computing the divergence of a vector field $\mathbf{a}$. The Gauss divergence theorem states $\int_V \nabla \cdot \mathbf{a} dx = \oint_{\partial ...
6
votes
0answers
50 views
+50

Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]\cdot[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as ...
0
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0answers
19 views

Finite difference for variable coefficient with Neumann Boundary

The equations is the same as this post, but with respect to the Neumann boundary. The physically correct boundary conditions for this equation are \begin{equation} A(x)\frac{\partial u(x)}{\partial ...
1
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0answers
17 views

condition number with component-wise norm for the sample variance any help is appreciated! :)

I'm looking through some notes and came across the following two statements in the notes where the author states it can be shown that one leads to the next. I've tried to show this using the ...
-3
votes
0answers
17 views

what is formula to this eqution [(256)16]1/32+[(169)6]1/12 [closed]

how to solve this equation [(256)16]1/32+[(169)6]1/12 what is formula of this? What is the closed form expression for this? What is the right domain for this Hamiltonian 2? what is the right ...
0
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0answers
38 views

Is it possible for the Simpson's method to converge faster than Rombergs method?

I have the following integral: $\int_{0}^{100} \frac{x^{3/2}}{\cosh{(x)}}dx$ I am running code for the Simpson's method and Romberg method to evaluate the integral numerically and the results show ...
2
votes
1answer
36 views

Name for some kind of logarithmic norm/error

As known $(\mathbb R, +)$ and $(\mathbb R^{+}, \cdot)$ are isomorphic with $\exp:\mathbb R\to\mathbb R^{+}$ as an isomorphism. When I transfer the absolute value $|\cdot|$ on $(\mathbb R, +)$ via ...
0
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1answer
29 views

Explain instability in Numerics so that I can understand and answer this question that involves roots of a equation

I found this question in my math book: Instability. For small |a| the equation (x - k)^2 = a has nearly a double root. Why do these roots show instability? I read and belive I understood the ...
-1
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0answers
22 views

Can someone show me how to properly code the secant method with what I have been given? [closed]

Function: z(t)=sin((pi)*t)*exp(-t) Using the secant method matlab code (with while loop) find the location of the points where the given function passes through z = −0.5. Confirm that a tolerance of ...
0
votes
2answers
30 views

Why is the estimate of the order of error in Trapezoid converging to $2.5$?

The integral in question is: $\int_{0}^{\infty} \frac{x^{3/2}}{\cosh{(x)}}dx$ I coded a program to compute $p$, an estimate of the order of the error for the Trapezoid method of numerical ...
1
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0answers
22 views

Von Neumann stability analysis of non-linear systems

The von-neumann stability analysis is based on the time and space discretisation schemes, what if the schemes are non-linear and too complicated to analyse. Is there a way to look at the matrices of ...
0
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0answers
16 views

Clarification of matrix equation needed in recursive least squares example.

I was looking at the answer to the post entitled "simple example of recursive least squares" and I would like to post a question concerning the matrix equation that is presented in the answer. First ...
4
votes
0answers
32 views

Numerically iterating the dynamics of a constrained Newtonian system

This question is about the dynamics (in classical mechanics) of a rigidly linked chain of $N$ point masses, see figure. Let us say that the masses $m_1,\ldots,m_N$ have initial positions ...
0
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0answers
17 views

How to choose grid for a numerical integral of complex function?

I need to numerically integrate a complex function $f(x)$ on R, i.e. to approximate $\int_{-\infty}^\infty{f(\xi)d\xi}$. Performance is crucial as the integration is repeated a high number of times ...
1
vote
0answers
24 views

Finding the Equations of Motion for the Leapfrog Integrator

I understand that the Leapfrog Integrator is used to find an integral for Newton's Laws of Motion and that the Equation of Motion are given by: $$\frac{dx}{dt} = v$$ and $$\frac{dv}{dt} = F(x) = ...
2
votes
1answer
30 views

System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
0
votes
0answers
58 views

Crank-Nickolson

I have these two equation $$ \frac{\partial q(t,x)}{\partial t} = \frac{\partial^2 q(t,x)}{\partial x^2} - \frac{L_1 a(t, x) q(t, x)}{1 + \frac{L_2}{L} (1 - q(t,x))}\\ \frac{\partial ...
1
vote
3answers
37 views

Absolute error in computing a sum [closed]

I couldn't solve this trouble, I hope you can give me some ideas. In computing the sum of an infinite series $\sum_{n=1}^{\infty}\,x_n$, suppose that the answer is desired with an absolute error ...
0
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3answers
51 views

4th-Order Runge-Kutta Method

I am struggling with this question regarding the 4th Order Runge-Kutta Method. I wish to find an approximate solution to the ODE: $$\frac{dx}{dt} = f(x)$$ Using the 4th Order Runge Kutta method: ...
0
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1answer
19 views

Asymptotic error expansion of global error for single step methods

My question refers to the proof of the following theorem, but it may suffice to just skip the theorem and continue with the problematic taylor expansion $(\ast)$: Let $f(t,y)$ and the single step ...
0
votes
0answers
20 views

Quantify difference between regularity and irregularity

I am solving an equation numerically on a 1D-domain using the finite-element method. I am solving it using two different domains, one regular and one irregular. Naturally, the solution varies slightly ...
1
vote
1answer
22 views

$L^2$ product of Chebyshev polynomials and Legendre polynomials

The following was a problem in a recent numerical analysis exam: Let $k \in \mathbb{N}\setminus\{0\}$. Prove or disprove: $$ \int_{-1}^{1} cos\left(k \operatorname{arccos}(x)\right) \cdot ...
1
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0answers
35 views

Inverse of $A^\top BA+C^\top DC$?

I'm numerically solving a system of equations of the form: $$Mx = b$$ where: $$M = A^\top BA+C^\top DC,$$ $B$ and $D$ are block-diagonal, $A$ and $C$ are $n\times m$ matrices with $m \leq n+3$. ...
1
vote
1answer
17 views

Improving the performance of eigs for a large spd Problem

I have two large (think around $100.000\times 100.000$), sparse, real symmetric and positive definite matrices $A$ and $B$ and I want to find the smallest generalized eigenvalue $$Ax = \lambda_{\min} ...
2
votes
1answer
67 views

Find quickest line of interception to a moving object

First, a visual illustration of the problem: http://tube.geogebra.org/m/1512793 The goal is to mathematically predict the direction in which the player need to run to intercept the ball as fast as ...
0
votes
1answer
62 views

Newtons Method to approximate inflection point

Here's a question from my tutorial which I'm having difficulties with. Consider the function $$f(x) =\frac{e^x}{( 1+ x^2)}$$ a) Show that $f$ has an inflection point at $x = 1$ My answer: $$f'' ...
2
votes
0answers
57 views

Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems

I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) : Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear ...