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

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2answers
25 views

show a derivation of this recurrence relation

One of the approximations for $\sqrt3$ is $x_{n+1} = \frac{x_{n} + 3}{x_{n} + 1}$. I can see clearly why this is true, since if the sequence converges, $x_{n}$ approaches $x_{n+1}$ (then multiply ...
0
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1answer
9 views

Show that in Line Search Methods the “steepest descent direction” is the one along which the objective function decreases most rapidly

I want to verify the claim, that the steepest descent direction $-\nabla f(x^k)$ is the one along which $f\in C^2(\mathbb R^n)$ decreases most rapidly. Therefore, I considered the Taylor expansion ...
1
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0answers
24 views

Abel-Plana formula for $\zeta(s)$, is this integral approximation correct?

I wrote a computer program to calculate values for $\zeta(s)$. I was scanning for something that would calculate complex values for $\zeta(s)$. I found the following approximation under the Integral ...
1
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0answers
14 views

Computationally inexpensive method to find a rotation which minimizes the norm of two tensor difference.

So I have two matrices ${\bf T}_1$ and ${\bf T}_2$, they are tensors in the sense that they can be built as $${\bf T} = \sum_{\forall i} a_i({\bf v_i}{\bf v_i}^T)$$ with positive real weights $a_i$ ...
4
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0answers
19 views

Complex Chebyshev Polynomials

Chebyshev Polynomials can be used to compute a very nearly minimax polynomial approximation of an analytic function on $[-1,1]$. Is there a complex analog that can compute a nearly minimax polynomial ...
1
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0answers
16 views

Runge-Kutta methods with strictly positive Butcher tableau

An explicit $s$-staged Runge-Kutta method for an autonomous ODE $\dot y = L(y)$ can be written as $$ k_i = L\left(y_n + \tau\sum_{j=1}^{i-1} a_{ij} k_j \right)\\ y_{n+1} = y_n + \tau\sum_{i=1}^s b_i ...
1
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1answer
51 views

How do I numerically solve this type of differential equation? (Wave Equation)

I'm trying to solve the wave equation numerically. I'm brand new to this and what I'm basically trying to accomplish is simulating a plucked string with fixed endpoints. How do I find the $h(x,t)$ ...
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0answers
23 views

Example of absolutely continuous function whose integral can't be computed exactly

I'm reading up on AC functions (I need the background of such functions for my BSc degree thesis) but I only come across theorems, lemmas etc. I have two questions: I've read conflicting things in ...
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1answer
17 views

Finding the condition number of an iterative method.

I'm trying to find the condition number on the function $A$ for the iterative method below however I'm struggling to begin. $$p_{n+1}=p_n-A(p_n)\frac{f(p_n)}{f'(p_n)}=g(p_n)$$ In particular, the ...
0
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1answer
30 views

Fixed Point for finding a root

I need to solve this equation (find $\lambda$) using numerical methods: $\displaystyle N_0e^{\lambda}+v\frac{e^{\lambda}-1}{\lambda}-N_1 = 0$ All other terms are constant and known. N0 = 1000000; v ...
0
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1answer
19 views

If $A^k$ consistently approximates $\nabla^2f(x^k)$ with $x^k\to x^*$ and $\nabla^2f(x^*)$ regular, then the $A^k$ are regular, too

Let's call $\left\{A^k\right\}\subseteq\mathbb R^{n\times n}$ a consistent approximation of $\left\{B^k\right\}\subseteq\mathbb R^{n\times n}$ iff ...
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0answers
22 views

Integrate multi-variable autonomous ordinary differential equations using Runge Kutta 4

I have a first-order ordinary differential equation (ODE) of the form: $$ \mathbf{\dot{y} = A\cdot y+B\cdot u} $$ where $\mathbf{y}$, the state variable, is a $7\times 1$ vector; $\mathbf{u}$, the ...
0
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2answers
69 views

How to solve this integration?

I want to solve this $$\int_0^w (b/x)^{a+1} e^{(cx-(b/x)^a)} dx$$ where $a$, $b$, and $c$ are arbitrary positive real numbers. Do i have to solve it numerically? I have no clue to solve this ...
11
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1answer
91 views

Why is this approximation of polynomial root so accurate?

I have an engineering problem where I have to find the smallest positive real root of a polynomial in $x$: $$Ax^5+Bx^3 - C = 0$$ Instead of solving numerically, I want simple approximative formulas ...
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0answers
21 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....
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0answers
11 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
21 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
29 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 ...
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0answers
33 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
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1answer
43 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 ...
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0answers
50 views
+100

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 ...
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0answers
25 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
44 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
15 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
17 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
27 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 ...
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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
44 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
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1answer
34 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 ...
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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
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2answers
50 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: ...
3
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0answers
50 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
54 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
55 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
29 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 ...
8
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0answers
61 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 ...
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0answers
18 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 ...
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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
40 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
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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 ...
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0answers
24 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 ...
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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 ...
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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
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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
18 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 ...