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

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Intermediate Forms Between Parabolic and Hyperbolic PDE (numerically)

Greetings MSE community, I have recently conducted some rudimentary experiments in matlab coding of PDE's. I have explicit and implicit numerical solutions to both the heat and the wave equation, for ...
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2answers
32 views

Numerical analysis - Showing fixed point exists

Let $g(x) = \frac{1}{2}(e^{−x})\cos x$. Prove that $g(x)$ converges to a fixed point. The answer provided by my lecturer is: $g(x)$ is continuous on $[0,1]$, and we can easily verify that $0 ≤ ...
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11 views

Pairing Two Point Clouds

So I have two point clouds $X$ and $Y$ each with $N$ points in the familiar $\mathbb{R}^3$ euclidian 3D space. I then have an inter-point distance $d(\vec x_i,\vec y_j)$ which is zero if $\vec x_i$ is ...
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18 views

How to solve an inverse of derivative ode

How can I solve $$(\phi'(y))^{-1}=y^{-c_1}+y^{-c_2},$$ where $c_1,c_2$ are constants and $(\cdot)^{-1}$ is inverse? Since I have inverse of derivative and it's nonlinear I think it has to be done ...
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25 views

How to solve a second order ODE numerically with two boundary conditions at different points?

I can only find Runge-Kutta method in textbook to solve the equation numerically with boundary conditions like y(0)=$\alpha$, y'(0)=$\beta$, but how can I solve it with a bounday condition like ...
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35 views

stuck with some parts of the proof about “ matrix is normal iff each of its eigenvectors is also an eigenvector of its transpose conjugate matrix”

When I read the book Iterative Methods for Sparse Linear Systems, Second Edition, I get stuck with the following proof. The yellow highlight parts are the positions I have trouble to understand. ...
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45 views

Backwards Euler Method for $dx/dt=x$ [closed]

Apply the backward Euler method to the linear equation $dx/dt = x$ and show that the method converges to the true solution $x(t) = e^t$ as $t$ tends to infinity. I've already applied the forward ...
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2answers
35 views

How to use binary search to find a function

I am reading somewhere that $$(\phi'(y))^{-1}=y^{-c_1}+y^{-c_2},$$ $c_1,c_2$ are some numbers, can be solved for $\phi$ using binary search. I am surprised because binary search binary search is used ...
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21 views

Quadratic interpolation.

Quadratic interpolation means the following: If we are given a table of values $y_i=f(x_i), 0\leq i\leq 2n$, then on each interval $[x_{2j},x_{2j+2}], 0\leq j\leq n-1$,the quadratic interpolating ...
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16 views

prove relative error of X+Y<= max( relative error of X , relative error of Y) known X,Y have the same sign

Let X,Y such that X * Y > 0 (which means that X,Y are both positive or negetive) Let A be the relative error of X ,and B the relative error of Y prove that the relative error of (X+Y) <= max (A,B) ...
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21 views

Bounding error of Padé approximation

I'm trying to understand how one would understand the error of a given Padé approximation for a function. For instance, the $[2,1]$ approximant for $\log(1+x)$ is $\frac{x(6+x)}{6+4x}$. Is there a ...
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1answer
23 views

Numerical integration/Quadrature

I want to find constants a, b, c and d that will produce a quadrature formula: $$\int_{-1}^{1} af(-1) + bf(1) +cf'(-1)+df'(1)$$ that has degree of precision 3. I'm not sure how to go about this. Is ...
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1answer
27 views

Finite difference numerical differentiation

I needed to find an O(h2) method to find f'''(x). Using Taylor expansions, I found: $$f'''(x)=\frac{f(x+2h)-2f(x+h)-2f(x-h)+f(x-2h))}{2h^3} + O(h^2)$$. However, I have also found that: ...
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18 views

Oder of Approximation and Numerical Convergence Order

I have a Poisson Equation Problem in hand which states: $-u^{''}(x)= f(x)$ where $f(x)= -e^{x-1}$ subject to Mixed Boundary conditions as $u(0)= 0$ and $u^{'}(1)=1$; Neumann B.C will be ...
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1answer
17 views

Show tha triadiagonal is M-matrix.

How to show that a tridiagonal matrix $A=(-1,2,-1)$ is an M-matrix, meaning that the entries of its inverse are nonnegative?
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8 views

Relative Error with Respect to Frobenius Norm

I'm look at this tiny book called "Deblurring Images: Matrices, Spectra, and Filtering" by Hansen, Nagy, O'Leary. This is a self study, but I believe my question is broad enough so that it can be of ...
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1answer
59 views

Richardson Extrapolation - problems understanding how it works

I'm doing homework, and I am stumped on the first problem. I'm given this: Apply the extrapolation process described in Example 1 to determine $N_3(h)$, an approximation to $f(x_0)$, for the ...
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1answer
25 views

Inviscid Burger's Equation

I have a question about the following burger's equation. $u_t + (\frac12u^2)_x = 0 $ with $u(x,0) = sin(x)$ on $[0,2\pi]$ and periodic boundary conditions. When I studied this equation numerically, ...
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25 views

maximum error of a near-minimax approximation

I'm trying to find the maximum error in the degree $n$ near-minimax approximation to $g(t)=\tan^{-1}(t)$, $0 \leq t \leq 1$ for $n=1,\ldots,6$. Does anyone know of a formula to find this? The book ...
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24 views

Calculating bounds interpolation and approximation

If I have a function $f(x)=e^x$ and nodes $x_0=a-h/\sqrt(3)$ and $x_1=a+h/\sqrt(3)$ to linearly interpolate $f(x)$ on the interval $[a-h,a+h]$ for some real numbers $a$ and $h$, $h>0$. How do I ...
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37 views

interpolation and approximation/error in polynomial interpolation

How would I go about constructing a table of values of $f(x)=\sqrt{x}$ for $1\leq x\leq 100$ with values of $f(x)$ given for $x=0,h,2h,\ldots$? I'm supposed to choose $h$ so that when linear ...
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1answer
18 views

Steffensen's method in Numerical Analysis

In some sources, Steffensen's method is the development of Newton's method to avoid computing the derivative, http://bit.do/Um6M http://cims.nyu.edu/~donev/Teaching/NMI-Fall2010/Homework4.pdf ...
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2answers
17 views

What does it mean by one matrix is **unitarily similar** to another?

I am reading a tutorial about the Lanczos method for eigen problem / SVD. It mentioned "Then the tridiagonal matrix $B^∗B$ is unitarily similar to $A^∗A$. " What does it mean? I can derive this: ...
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1answer
19 views

conjugate gradient method for semi definite case

Show for a symmetric, positiv semi definite matrix $A$, a vector $b\in Ran(A)$ and initial vector $x_0$: (1) All directions $d_0,d_1,...,d_m$ of the conjugate gradient method are in the range ...
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1answer
32 views

Having trouble grasping curve fitting

I have an exam coming up next week in my Applied Numerical Methods class. Our professor gave us a list of about 12 things that we need to be able to do for the exam, all of which are pretty ...
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29 views

How to find complex roots using Müller's Method?

I have $f(x) = 2x^5 - 2x^4 + 6x^3 - 6x^2 + 8x - 8$ and $x_0 = 0.4$, $x_1 = 0.6$, $x_2 = 0.5$ with a tolerance Es = 10^-4. I solved it using Müller's method in 4 ...
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1answer
26 views

How to find initial estimate of roots from graphs?

I have f1(x,y) = x^2 + 3y^2 - 1 = 0 and f2(x,y) = (x-2)^2 + (y-1)^2 - 4 = 0 I am suppossed to find the roots of these nonlinear ...
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1answer
53 views

Numerical integration of a 2D advection equation with spacially varying coefficients.

I am interested in a simple algorithm for solving this PDE for $f(x,y,t)$: $$f_t + A y f_x - B x f_y =0$$ With $A,B>0$. The initial condition is some arbitrary smooth function (probably gonna ...
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35 views

When can definite integration be numerically computable?

under what condition,can the integration $$\int_{\Delta}f(x_1,x_2,\dots,x_n)dx_1dx_2\dots dx_n, \text{where } \Delta \text{ is integration domain defined by function},f(x_1,x_2,\dots,x_n) \text{ ...
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13 views

Finding the number of elements in a set when the base, the mantissa and lower and upper bounds are gi

I'm new to numerical methods and I cannot think of a way to start solving the following problem. I know the mantissa, base and the boundaries are related when defining a number but I cannot really put ...
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45 views

Iterative equation

I have an equation that I want to try and solve iteratively. I don't have any background in numerical analysis so unsure as to how to go about it. Any help would be greatly appreciated. My equation ...
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22 views

Householder reflector sign error

I am studying Householder reflectors from Trefethen and Bau but am having trouble creating a simple example for it. I am given the equation for the vector v that the Householder reflector H is based ...
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1answer
31 views

Implementation of Richardson extrapolation in mesh independence study

I am busy with a mesh independence study in computational fluid dynamics, where I am systematically refining my mesh and monitoring a certain parameter of interest with the goal that the value should ...
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1answer
41 views

Finding the value of y using Lagrange Formula

Let $p_2(x)$ be the interpolating polynomial for the data $(0 , 0) , (0.5 , y) , (1,3)$ from Lagrange formula. The coefficient of $x^2$ in $p_2(x)$ is $-2$ , Find the value of $y$ .
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46 views

How to write approximations of a sequence $x_n = {1/3^n}$

Write three approximations of the sequence ${x_n} = {1/ 3^n}$, using the following scheme - $P_0= 1, P_1 = 0.33332$ and $P_n = (6/5)P_{n-1} - (1/5)P_{n-2}$ for $n = 2, 3,\dots$ Further, make a ...
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41 views

9 digit prime number with all digits from 1 to 9 [closed]

What is the 9 digit prime number which has all digits from 1 to 9(smallest and biggest)?? exclude zero.
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23 views

Tridiagonal linear system-linear equation

I was asked to write the tridiagonal linear system and i am confused as to know how to form it by the given values. Question: write the tridiagonal linear system to be solved for approximating ...
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14 views

curve created by the maximum values of a function as we vary a variable

I am trying to evaluate the curve created by the the maximum of a function over all $x$ values. Let $\theta = \cos^{-1} (\frac{a \cdot b}{||a||||b||})$, where $a= (x-P_{1x}, y - P_{1y}), b=(x-P_{2x}, ...
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1answer
25 views

Converting scalar ODE to coupled system

I'm currently battling the following problem: \begin{align} u^{(iv)} (x) &= f(x)\quad\text{on }(0,1)\\ u(0) = u'(0) &= 0\\ u''(1) = u'''(1) &= 0 \end{align} which is, as I've understood, a ...
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3answers
46 views

$\sin(1/x)$ is not uniformly continuous on $(0, \frac{\pi}{2}]$

Show that $$ f(x)=\sin\frac1x $$ is not uniformly continuous on $(0,\frac\pi2]$. It is looking easy to do this problem if it is asked for $(0, 1)$ but I am not getting for the given range of ...
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31 views

Derive the weak form for nonlinear problem.

Let the equation be $$\frac{d^2 y}{dx^2}=\frac{y}{1+y}$$ For finite element formulation how to get the weak form? The major problem being the nonlinear rhs.`
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Accumulation Points of a Set A

Let A denote a finite set, Find the set of accumulation points of A ? Now I know that in discrete metric, $d(x,y) = 1$ if $x ≠ y$ and $d(x,y) = 0$ if $x = y$. So is this mean I have to consider two ...
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Solution of equations involving determinant and matrix inverse

$x$ and $y$ are two scalar unknowns. The two equations are $$|\mathbf{I}+x\mathbf{h}_1\mathbf{h}'_1+y\mathbf{h}_2\mathbf{h}'_2|=R$$ and ...
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Lobatto IIIA - IIIB pair

Can someone explain me how to set up an implementation of a partitioned Runge-Kutta pair, consisting of the 3-steps Lobatto IIIA and IIIB methods for general functions $f$ and $g$, i.e. for solving ...
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Degree of precision Newton Cotes

I have a homework question that states: find the degree of precision of the degree four Newton-Cotes Rule $$\int_{x0}^{x4} f(x)dx \approx 2h/45(7y0+32y1+12y2+32y3+7y4)$$ I have been reading the book ...
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34 views

Numerical Evaluation of a Series at a Point

I have a numerical calculus challenge to resolve using a C++ algorithm or scilab. The problem is the following: $f(x)=\sum_{n=1}^{\infty }a_{n}x^{n}$ where $a_{n}=\sqrt{n^2+1}-n$ This function is ...
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53 views

Why is Newton's method faster than gradient descent?

Can you provide some intuition as to why Newton's method is faster than gradient descent? Often we are in a scenario where we want to minimize a function f(x) where x is a vector of parameters. To do ...
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40 views

Is the optimization problem right?

If we want optimize the following problem $$ \min_x \{a(x)+c(x)\} $$ and we have $$ a = \min_y b(y) $$ then, could we directly optimize the following problem? $$ \min_x \{b(x)+c(x)\} $$
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25 views

How do I perform taylor expansion of the following?

Taylor expansion about $(x,y)$ of $f(x + a,\; y + k\; f(x + b,\; y + c))$ I do not understand what happens to the second $f$ inside. The inspiration for this question is Runge-Kutta methods.
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25 views

Interpolation polynomial types

I was wondering if both the Maclaurin and Taylor series are two types of interpolation polynomials? I was under the impression that they were not because they only go though one point in an interval ...