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

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1answer
232 views

finding derivative at intermediate point of known data set

I have a function $y = f(x)$, $ x \in [0,1] $ and $ y \in [0,1]$ Set of values $(x_i,y_i)$ are known for n points. I need to find derivative at point $x_{\zeta}$ such that $y(x_{\zeta}) = 0.5$ Now ...
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1answer
87 views

Solving Linear Equation with Floating point Arithmetic

Given the matrix A = $\begin{pmatrix} 0.005 & 1 \\ 1 & 1 \\ \end{pmatrix}$ and the vector b = $\begin{pmatrix} 0.5 \\1 \end{pmatrix} $ we have to solve for x in Ax = b in three ...
3
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1answer
120 views

Numerical Integration of $\int_0^1 \log(x) dx$

Do you guys have any idea how to handle something like $\int_0^1 log(x) dx$ numerically in Matlab (I'm only interested in the real part btw)? I have used Quad, Quadl, "integral" etc all of them ...
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1answer
175 views

Optimize material in a can

A can in the shape of a right circular cylinder is to be constructed to contain 1000 cm$^3$. The circular top and bottom of the can must have a radius of 0.25 cm more than the radius of the can so ...
4
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2answers
96 views

Minimizing a functional

I am trying to follow this paper. In it they define a functional $$J(f) = \sum_{x \in \Omega} \psi (f(x) - u(x)) + \beta \sum_{x \in N_x} \phi(f(x) - f(y)), $$ where, for my purposes, $f$ and $u$ ...
2
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1answer
109 views

Approximate a function with polynomial degree $n$ instead of $n+1$

If I have a $n+1$ degree polynomial $f(x)$ that approximate the function $g(x)$ on the interval $[-1,1]$. After that, if I lose $g(x)$, just know $f(x)$, how to find a $n$ degree polynomial that is ...
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1answer
469 views

Rate of convergence of Gauss-Seidel iteration method.

Help me: $2x-y=7\\ -x+2y-z=1\\ -y-2z=1$ Show Gauss-Seidel iteration scheme converges and find the rate of convergence. My Attempt: The iteration matrix for this system of equation is $$ H = ...
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1answer
403 views

Stratified Monte Carlo

Consider the integral $I=\int_{0}^{1}e^{-x}dx$. Now consider the stratifed Monte Carlo estimate $\hat{I^{s}}$, that has $N_{st}=8$ strata. What is the variance of $\hat{I^{s}}$? What is the percent ...
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2answers
325 views

Solving a nonlinear recurrence relation

All of that being said, while trying to come up with a computational algorithm to solve a particular nonlinear PDE, using the method of finite differences, I ran into a nonlinear recursion. The ...
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1answer
381 views

Weak Formulations and Lax Milgram:

I have a question on how to put a PDE into weak form, and more importantly, how to properly choose the space of test functions. I know that for an elliptic problem, we want to start with a problem ...
2
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2answers
284 views

Evaluate a differential equation at a given point

I have a second order differential equation $x^2\cdot y''(x)+x\cdot y'(x)+(x^2-10)=0$ with the initial condition $y(100)=1$ and $y'(100)=0$. I want to evaluate $y(x)$ when $x=103$ with 8 digit ...
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1answer
64 views

Books for mathematics used in computer games.

I'm looking for a good book (idiot proof) for learning all the magic behind computing matrices, quaternions, euler angles, orientation in 3d space and more... Book needs to have examples and ...
2
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0answers
179 views

all eigenvalues of a large sparse symmetric matrix

my question is similar to how to diagonalize a large sparse symmetric matrix, to get the eigenvalues and eigenvectors however i wish to be more concrete and ask if one can, on a standard PC (e.g. a ...
1
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1answer
183 views

Method of Undetermined Coefficients

I am trying to solve a problem using method of undetermined coefficients to derive a second order scheme for ux using three points, ...
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1answer
66 views

showing a function has exactly one root using Numerical Analysis Methods

Im trying out the Banach Caccioppoli Contradiction Principle but having a few problems.. f(x) = exp(x/2) - 25x^2 How would i show that this function f has exactly one root x^ in (-Infinity,0) ? Some ...
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1answer
12k views

Newton Raphson Iteration method in Matlab

I am currently trying to learn how to use ${\tt Matlab}$ properly but having a bit of trouble, I want to write a code that, say i had a differentiable function, will take as input $f$, $f'$, $x_0$ ...
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0answers
181 views

Gradient Descent: Optimal fixed step size for a quadratic objective?

All of the literature I'm reading immediately skips to the idea of adjusting the step size as you iterate (as far as I can tell) to maximize the rate of convergence. In the context of neural network ...
4
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1answer
192 views

The spectral norm of projection matrices

Given any $n \times n$ non-symmetric projection matrix $P$, i.e., $P^2 = P$ but $P^T \ne P$, is the spectral norm of $P$ bounded by a constant which is independent of the dimension $n$?
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2answers
104 views

Evaluate $\cos(x)$ at a particular point

I am trying to evaluate $\cos(x)$ at the point $x=3$ with $7$ decimal places to be correct. There is no requirement to be the most efficient but only evaluate at this point. Currently, I am thinking ...
3
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2answers
89 views

Discrete approximations of $\nabla^2{\bf v}$

I am writing a Navier Stokes solver. The vector field is represented as a grid with integer coordinates I am looking at other people's computer code. I don't entirely understand the vector calculus, ...
0
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1answer
275 views

Finding the slope for Euler's method of evaulating differential equations

The change in the belocity of a body falling at a relatively slow speed over a short distance is given by $\frac{\mathrm{d}v}{\mathrm{d}t}= g - kv$, where $g$ is the acceleration due to gravity and ...
2
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1answer
367 views

Derivation of weak form for variational problem

My question is about understanding the derivation of the weak form of a variational problem (to be used for the solution via the finite element method). The problem is as follows (it is an image ...
2
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1answer
31 views

skinned surface

In the representation of a skinned surface using $B$-Spline, I have $K+1$ given curves of degree $p$ on a common partition $U$ and I want to construct the surface $S(u,v)$ with these curves as ...
4
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1answer
382 views

Matlab numerical triple integral involving quotient of two small quantities

Math people: I am trying to use Matlab's "triplequad" to numerically evaluate the triple integral $$\int_0^1 \int_0^{r_1} \int_{r_1-r_0}^{r_1+r_0} \left[\frac{Ov(r_0,r_1,t)^2}{(\frac{4}{3}\pi ...
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1answer
182 views

Determining slope of line relative to a maximum

In the following scientific report (Seismic Q estimation), a mathematical procedure of linear curve-fitting is described in words. The authors state: The stratigraphic effects are minimized by ...
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1answer
2k views

Rate of convergence of modified Newton's method for multiple roots

I've got a problem with a modified Newton's method. We've got a function $f \in C^{(k+1)}$ and $r$ which is it's multiple root of multiciplity $k$. Also $f^{(k)}(r) \neq 0$ and $f'(x) \neq 0 $ in the ...
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1answer
181 views

Cubic interpolating spline

Consider calculating a cubic interpolating spline with the additional boundary conditions $s''(x_0)=0$ and $s''(x_n)=0$. Show that $$\int_{x_0}^{x_n}[s''(x)]^2dx \leq ...
2
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3answers
47 views

Matrix decomposition again

If some matrix (M×N) can be expressed as product of (M×1) and (1×N) vectors: what is proper term for such kind of decomposition? how to tell if such kind of decomposition exists for given matrix? ...
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1answer
1k views

Writing a program using the trapezoidal rule

Write a program to evaluate $I=\int_a^bf(x)dx$ using the trapezoidal rule with $n$ subdivisions, calling the result $I_n$. Use the program to calculate the following integrals with ...
2
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2answers
325 views

Hermite problem formula

Consider the Hermite problem $$p^{(r)}(x_i)=y_i^{(r)},\ i=1,2 ;\ r=0,1,2$$ with $p(x)$ a polynomial of degree $\leq 5$. a - Give a Lagrange type of formula for $p(x)$. b - Give a Newton ...
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2answers
755 views

Trapezoid rule error analysis

How can I prove that the max error of the trapezoid rule for the integral $\int_{a}^{b}{f(x)\, \mathrm{d}x} $ is: $$\Delta=-\frac{1}{12n^2}f''(c)(b-a)^3 \text{for } c \in (a,b) \ ?$$ I know that to ...
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2answers
106 views

Numerical Analysis Taylor Method

using the taylor method perform two steps when $y' = -2t - y$ when $y(0) = -1$ and $h = 0.1$ what is $\frac{df}{dt},\quad \frac{d^2f}{dt^2}$ ... I found $\frac{df}{dt} = -2 - y'$ and I do not think ...
0
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1answer
45 views

Least Squares Approx.

If I was using a least squares approximation of the form $y = A_1 + A_2\sin(wx) + A_3\cos(wx)$, would you be minimising the function $\sum_{i=0}^n (y_i - (A_1 + A_2\sin(wx) + A_3\cos(wx))^2$ ? I've ...
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1answer
223 views

Derivative of B-Spline curve

Using the notation in the book by Piegl, when I have to compute the derivative of a B-Spline curve, I write $\sum_{i=0}^n N_{i,p}P_i$ on the knot vector ...
4
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2answers
62 views

Numerical Methods for ODEs Precision

I have come across the following statement : Higher order (Ode stepper) does not always mean high accuracy (from Numerical Recipies, third edition). Why so ? Thank you in advance.
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0answers
195 views

Reference for Finite Difference Schemes

Is there any place that I can find a list of different PDEs and common finite difference schemes used for each? I have seen tables of finite difference coefficients such as the one here ...
2
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1answer
1k views

Polynomial root finding

I have an univariate polynomial of some degree - how do I numerically find all of its real roots? I never thought I would ask this question - everyone knows how to find polynomial roots, right..? ...
9
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3answers
462 views

$w^4+(w')^2 = g(t)$

I have a question about a first order non-linear differential equation. I have tried many method to solve this problem but not successful yet. Here is my question; $$w^4 + (w')^2 = g(t)$$ $$w' = ...
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3answers
564 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 ...
2
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1answer
196 views

Integration - Quadrature

I need to integrate the following between x = $-1$ and $1$: $$ f(x) =x (\cos(x)/\sin(x))$$ As far as I know it is not possible to obtain exact integration to this integral. I was trying to solve ...
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1answer
1k views

Three step Adams-Moulton functional iteration

I'm given an IVP: $$y' = e^y, \;\;\;\; 0 \le t \le 0.20, \;\;\;\;\; y(0) = 1$$ The solution is: $$y(t) = 1 - \ln(1 - et)$$ Applying the three-step Adams-Moulton method to the problem is equivalent ...
2
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3answers
746 views

How to find the period of a periodic function?

I'm working on some numerical analysis problem, and I'm studying functions that "seem" to be periodic. Now what I would like to do, is to determine their period. Only, the methods I actually use are ...
2
votes
1answer
345 views

FFT Algorithm for an interpolating polynomial

I'm trying to use the Fast Fourier transform algorithm to determine the trigonometric interpolating polynomial of degree $16$ for $f(x) = x^2\cos(x)$ on $[-\pi,\pi]$ I see a computer result in my ...
2
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1answer
197 views

Numerical Analysis using MATLAB. Find the condition number $\mu$

Find the condition number $\mu = |A||A^{-1}|$ for the Hilbert Matrix $A$ using the uniform form. $A = \left( \begin{array}{cccc} 1 &\frac{1}{2} & \frac{1}{3} &\frac{1}{4} \\ \frac{1}{2} ...
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1answer
1k views

Derivative of $f(x,y)$ with respect to another function of two variables $k(x,y)$

Suppose that we have a function $f(x,y)$ of two variables: $$f(x,y) = g(x) + h(y) + 5(x-y) = x^2 + y^2 + 5(x-y)$$ where $g(x) = x^2$ and $h(y) = y^2$ are also functions of $x$ and $y$, respectively. ...
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0answers
74 views

Looking for a binomial system solver

I am interested in solving binomial systems of the form $$ \begin{cases} a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} + b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\ ...
3
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1answer
125 views

A strange quantum potential: $V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$

So I have a strange quantum potential I have been playing with: $$V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$$ where $\mu$ is the Möbius function. This is what ...
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2answers
1k views

Numerically integrating a function of 3 variables with respect to 2 variables in Matlab

Math people: I have Googled this question, tried some suggestions, and none of them worked. I browsed the Similar Questions here and didn't find what I wanted. I apologize if this is a duplicate. ...
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1answer
151 views

Iteration convergence.

How can I solve this problem? Let $$x(n+1)=-\frac{\exp(x(n)/2)}{5}$$ be a given sequence. Prove using the Banach contraction principle that this sequence converges to some fixed point $X$ with ...
4
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2answers
7k views

Numerical solution to x = tan (x)

I needed to find, using the bisection method, the first positive value that satisfy $x = \tan(x)$. So I went to Scilab, I wrote the bisection method and I got $1.5707903$. But after some reasoning I ...