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

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3
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3answers
3k views

what is “kink”?

Pleas tell me that what a "Kink" is and what this sentence means: Distance functions have a kink at the interface where $d = 0$ is a local minimum.
3
votes
1answer
509 views

How do iterative methods applied to the companion matrix of a polynomial $p(\lambda)$ relate to $p$ itself?

A few days ago, I had a vague question in my mind about "matrix methods" for finding roots of a polynomial. Now I can ask at least a semi-precise question, thanks to the post How to calculate complex ...
2
votes
1answer
58 views

How are these two methods for numerical differentiation connected?

I've read a book about Numerical Differentiation, and I found this formula: coefficient derivative I think this might be connected to the derivative of Lagrange Interpolating Polynomial, if I'm not ...
2
votes
1answer
44 views

Partial Derivative of Newton's Divided Difference

Let $x_0<x_1<...<x_n$, and let $f$ be continuously differentiable. Show that $$ \frac{\partial}{\partial_{x_i}} f[x_0,x_1,...,x_n]=f[x_0,x_1,...,x_i,x_i ,x_{i+1},...,x_n] $$. I have the ...
2
votes
3answers
153 views

Numerical estimates for the convergence order of trapezoidal-like Runge-Kutta methods

I have to calculate approximations of the solution with the method $$ y^{n+1}=y^n+h \cdot [\rho \cdot f(t^n,y^n)+(1-\rho) \cdot f(t^{n+1},y^{n+1})] ,\quad n=0,\ldots,N-1 \\ y^0=y_0 $$ for various ...
2
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1answer
65 views

Is the following scheme for generating $p_n=(1/3)^n$ stable or not. $p_n=(5/6)p_{n-1}-(1/6)p_{n-2}$.

Is the scheme for generating $p_n=(1/3)^n$ stable or not? $$p_n= \frac{5}{6} p_{n-1} - \frac{1}{6}p_{n-2}.$$
2
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1answer
515 views

Quick way of finding the eigenvalues and eigenvectors of the matrix $A=\operatorname{tridiag}_n(-1,\alpha,-1)$

Matrix $A=\operatorname{tridiag}_n(-1,\alpha,-1)$ has the eigenvalue: $\lambda_i=\alpha-2\cos(i\theta),$ $i=1,\dots,n$ and the corresponding eigenvectors are: ...
2
votes
1answer
682 views

butcher tableau runge kutta methods

Hi I have had a go at this question- am i heading in the right direction? it would be much appreciated if someone could me Write the Butcher Tableau for the 1-stage $\theta$ method: $$U^n ...
1
vote
0answers
14 views

IVP Using Numerical Methods

Suppose that $y(t)$ is the exact solution of the ivp $$y'(t)=f(t,y(t)), y(0)=y_0$$ and $u(t)$ is any approximation to $y(t)$ with $u(0)=y(0)$. Define the error $e(t)=y(t)-u(t)$. How can I show ...
1
vote
1answer
65 views

Newtons Modified Method

In class we derived the convergence rate for Newton's method for a function such that the root was not simple. And we found it to be linear convergent However our professor then went on to a modified ...
1
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1answer
47 views

Runge-Kutta methods: step size greater than 1

The local truncation error of Runge-Kutta 4 is $O(h^5)$, while those of RK1 is $O(h^2)$. I wonder then what happened when the step-size of RK methods is greater than 1: will the accuracy be improved ...
1
vote
1answer
301 views

How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
1
vote
2answers
120 views

Numerical Approximation of Differential Equations with Midpoint Method

I want to proof that the local truncation error of the Midpoint Method is $d_{k+1}=O\left(h^{3}\right)$ Approach The local truncation error is defined as: ...
1
vote
2answers
47 views

Taylor Series approximation

Let $f(x) = (1-x)^{-1}$ and $x_0=0$. Find the $n$-th Taylor polynomial $P_n(x)$ for $f(x)$ about $x_0$. Find a value of $n$ necessary to approximate $f(x)$ within $10^{-6}$ on $[0,0.5]$. I am ...
1
vote
1answer
2k views

Evaluate derivative of Lagrange polynomials at construction points

Assume, that we have points $x_i$ with $i=1,...,N+1$. We construct the Lagrange basis polynomials as \begin{align} L_j(x) = \prod_{k\not = j} \frac{x-x_k}{x_j-x_k} \end{align} Now according to my ...
1
vote
2answers
666 views

Method of False Position (Regular Falsi) - Pros/Cons

Could anyone provide and explain some drawbacks and benefits of the method of false position against say newtons method. I know one of benefits is that it doesn't require the derivative and one of ...
1
vote
1answer
609 views

How to construct a parametric cubic B spline?

If I am given n+1 control point Pi(xi,yi), Po .... Pn , how do I construct a parametric relationship to draw a curve ? From what I understand , a parametric relationship is that you can express x and ...
1
vote
1answer
204 views

Non-negative matrix factorization with a regularization term

Given a matrix $X_{M*N}$ with non negative values, I would like to factorize it into $U_{M*K} , V_{N*K}$, where both of them have also non-negative values. Additionally I would like to have a minimum ...
1
vote
1answer
223 views

How to make recursive computation of $I_n=\int_0^1 \frac{x^n}{x+\alpha}\,dx$ stable?

This is homework, so please only hints. We see that the recurrence $$I_n=\frac{1}{n}-\alpha I_{n-1}$$ can be used to compute the value of the integral $I_n=\int_0^1 \frac{x^n}{x+\alpha}\,dx$ ...
1
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0answers
175 views

Differential equation with random variable

How can I derive analytically or compute numerically the solution to following differential equation $$ dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X $$ where X is a random variable (e.g. from a ...
1
vote
1answer
1k views

How to fit non-linear matlab data?

I'm working on a problem in scientific computing namely fitting data to this equation $c(z) = 4800 + p_1 + p_2 \cdot z/1000 + p_3 \cdot e^{ -p4 \cdot z/1000}$ The data is in a background question ...
1
vote
1answer
760 views

If modulus of each one of eigenvalues of $B$ is less than $1$, then $B^k\rightarrow 0$

Let $B$ be a $n\times n$ matrix and let $X$ be the set of all eigenvalues of $B$. Prove that if $|m|<1$ then $\lim \limits_{k\rightarrow\infty}B^k=0$, where $m=\max X$. Thanks. Actually, there ...
1
vote
1answer
2k views

Solving Laplace's equation using finite differences

I having coding in MatLab to approximate solutions to Laplace's equation in 2D using finite differences. I was able to do it without much problem. I learnt about how to implement this using this: ...
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3answers
1k views

Computing the square root function with Newton's method

Show that Newton's method can be used to compute the square root function $\sqrt a$ using the formula $$x_{n+1} = \frac{1}{2}\left(x_{n} + \frac{a}{x_{n}}\right)$$ show that the error is $$\sqrt a ...
1
vote
1answer
436 views

Chebyshev polynomial question

Consider the Chebyshev polynomials $T_n(x), n = 0, 1, \ldots$ which are recursively defined by $$T_0(x) = 1; \quad T_1(x) = x; \quad T_n(x) = 2x\cdot T_{n−1}(x) − T_{n−2}(x)$$ for $n = 2, 3, \ldots$. ...
1
vote
2answers
10k views

Solving coupled 2nd order ODEs with Runge-Kutta 4

I'm having a hard time figuring out how coupled 2nd order ODEs should be solved with the RK4 method. This is the system I'm given: $x'' = f(t, x, y, x', y')$ $y'' = g(t, x, y, x', y')$ I'll use the ...
1
vote
1answer
211 views

Functional Iterations (numerical analysis)

(I'm opening a new post because the claim has been found in another book and now I have an exact description) I wish to prove this claim I found in a numerical analysis book : Assume that ...
1
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1answer
846 views

existence and uniqueness of Hermite interpolation polynomial

What are the proofs of existence and uniqueness of Hermite interpolation polynomial? suppose $x_{0},...,x_{n}$ are distinct nodes and $i=1 , ... ,n$ and $m_{i}$ are in Natural numbers. prove exist ...
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0answers
353 views

Integrating pde backwards in time

I need to solve a partial differential equation backwards in time. In other words, I have the discrete partial differential equation $$ \begin{align} p_\text{new} & = p_j + ...
0
votes
1answer
52 views

Convergence of the fixed point iteration for sin(x)

Sin(x) converges at x=0 as can be seen with a graphical illustration. When attempting the fixed-point iteration for sin(x) on a computer, the number gets closer and closer to zero but it does not ...
0
votes
1answer
28 views

Lagrange 2nd dregree interpolating polynomial

Given these set of points: I must find the interpolating polinomial and find the value $0.32$ Using the Lagrange formula, I did: $$P(x) = ...
0
votes
2answers
34 views

Find the first order system of linear equations

Regard the diff equation: $mϕ′′+aϕ′+(mg/L)ϕ=0$ $ϕ(0)=0.1$ $ϕ′(0)=0$ where $m=0.1,L=1,a=2,$ 1) Rewrite the second order diff equation as a system of first order linear equations. 2) What is the ...
0
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1answer
57 views

Dilating sinusoidal period at certain abscissa $x$?

Lets take a look at this function : $$f(x) = \sin\left(\frac{\pi x }{2}\right)$$ when $x$ tends to $1$ this functions get closer to $1$ by bigger values now look at this one : ...
0
votes
2answers
45 views

We have x=0.2234 and Δx =0.2*10^-3.Find how many accurate digits does the number have?

So we know that $Δx≤(1/2)*10^{-n}$ where n is the number of the accurate digits. Now I just have to replace $Δx =0.2*10^-3$=$(1/2)*10^{-n}$ and find n. But why have I been given x=0,2234 since I dont ...
0
votes
2answers
765 views

Trapezoid rule over trigonometric polynomials

The question is regarding trapezoid rule applied on trigonometric polynomials Here is the question Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
0
votes
2answers
932 views

Newton's method - determine accuracy in calculation

I have almost managed to solve a problem (I think), but I am a bit unsure if my procedure is correct, and my answer is not quite the correct one. Would appreciate any input! The problem is as ...
-2
votes
2answers
34 views

Use taylor series to arrive at the expression f'(x)=1/h[-3*f(x)/2+2f(x+h)-f(x+2h)/2]

I'm not really sure how to go about this.. any help is appreciated.
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votes
1answer
89 views

Can we conclude that this matrix is definite positive? [duplicate]

Let $A$ be a $n\text{-by-}m$ matrix. Suppose that columns of $A$ are linearly independent. Can we conclude that $A^TA$ is definite positive? Could you help me with proof? Thanks.
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votes
2answers
600 views

How to show that a root of the equation $x (x+1)(x+2) … (x+2009) = c $ can have multiplicity at most 2?

How to show that a root of the equation $$x (x+1)(x+2) ....... (x+2009) = c $$ can have multiplicity at most 2 , and to find the value of $ c $ for which this is possible. I proceeded by using the ...
9
votes
3answers
2k views

Finding the all roots of a polynomial by using Newton-Raphson method.

Is there a general formulation for finding all roots of a polynomial, especially the complex ones, by using the Newton-Raphson Method?
10
votes
4answers
294 views

A numerical evaluation of $\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x^{(n)} dx$

I would like to obtain a numerical evaluation of the series $$S=\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x(x+1)\cdots(x+n-1)\: dx$$ to five significant digits. I've used ...
9
votes
2answers
2k views

Big-O Interpretation

I have trouble understanding what the "Big O" notation, or asymptotic notation means. For instance, if you have $\sin(x)=x+O(x^3)$, what does this mean? Can anyone describe it in a simple way? I tried ...
24
votes
1answer
521 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
12
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7answers
4k views

How to derive a function to approximate $\sqrt{3}$?

The problem is to used fixed-point iteration method to find an approximation to $\sqrt{3}$. The equation from the book is $f(x) = \dfrac{1}{2}\left(x + \dfrac{3}{x}\right)$. It make senses to me ...
11
votes
2answers
781 views

Relation of Brownian Motion to Helmholtz Equation

one can obtain solutions to the Laplace equation $$\Delta\psi(x) = 0$$ or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, ...
8
votes
2answers
2k views

Numerical method for finding the square-root.

I found a picture of Evan O'Dorney's winning project that gained him first place in the Intel Science talent search. He proposed a numerical method to find the square root, that gained him $100,000 ...
5
votes
1answer
352 views

Conjugate Gradient Method and Sparse Systems

What is it about conjugate gradient that makes it useful for attacking sparse linear systems. Why would steepest descent be significantly worse? Please keep in mind that I am still trying to fully ...
6
votes
3answers
4k views

rewriting to avoid catastrophic cancellation

why is rewriting $x^2 -y^2$ as $(x+y)(x-y)$ a way to avoid catastrophic cancellation? We are still doing $(x-y)$. Is it because the last operation in the second form is a multiplication?
3
votes
3answers
1k views

Numerical Analysis References

Could anyone suggest any good (perhaps online ref papers) reference material on numerical analysis focusing on determining accuracy/estimated errors, rates/orders of convergence especially when ...
7
votes
1answer
467 views

Eigenvalues of a tridiagonal trigonometric matrix

Let $D$ be the diagonal matrix w/alternating in sign diagonal entries: $$D_{kk}=(-1)^{k+1}\tan(\frac{k\pi}{2n+1}),$$ where $k=1,2,\dots n\in N$, and let $B$ be the $n$ by $n$ square $(0,1)$-matrix ...