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

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Newtons Method to solve equations

The question is the following: Use Newton’s method to solve $$x^3_1+x_2=1,\\ x^3_2−x_1=−1 $$ Indicate your initial condition and how many steps it requires to reach the tolerate of error to be ...
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0answers
13 views

How would I add the following numbers?

I have got two floating point numbers and need to calculate them in the following way: -1.724444389 (and a ton more digits) * 2^(126) + -1.342222094535 (and a ton more digits) * 2^(-108) My question ...
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1answer
23 views

find the root with the bisection method

The instructions of the problem are: Use bisection method to find a root of the function $$ \sin x + x \cos x = 0 $$ Indicate your initial condition and how many steps it requires to reach the ...
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2answers
17 views

Fixed Point Iteration Method

Can anyone explain or prove the Fixed Point Iteration method? I know the conditions of fixed point existence. A fixed point is a point of a function ${f}$ on a continuous interval ${(a,b)}$ which ...
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0answers
5 views

Introduction to the boundary element method, with convergence analysis

I've looked through several textbooks on the BEM, but while they show how to set up the boundary integral formulation and how to discretize it, none give any indication of what the convergence ...
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1answer
20 views

Stopping criterion for approximating exponential series

Given $e^{x}=1+x+\frac{x^2}{2!}+\frac{x^{3}}{3!}+\cdots $. Summing in the natural order, what stopping criterion should you use? Can you rearrange the series or regroup the terms in any way to get ...
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0answers
9 views

The Roots of Jacobi Polynomials

How can i obtain the roots of Jacobi polynomials of order n>50 ? ( α<0, β<0 and $\alpha+\beta=-1$ )
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0answers
11 views

Find the equations that determine minimizing x for the following

I don't quite understand what the question is asking and how to approach it. I am given the following two equations: i. P=1/2 $x^T A^T AX- x^TA^Tb$ ii. E=$||Ax-b||^2$ I could use some pointers ...
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17 views

Numerical integration of function with derivatives of implicit variables

I have an independent (array) variable $r = {r_0, r_1, ..., r_N}$, and three functions (arrays) of that variables, $n(r) ={n_0, n_1, ..., n_N}$, $p(r)$, and $E(r)$. How can I calculate the function ...
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0answers
12 views

relative condition number being undefined?

if we have a function $f(x) = x^{1/4}$ I'm wondering if the relative condition number is defined at $x = 0$ - I think that from the definition it is still defined as i found the condition number to be ...
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0answers
20 views

Show that the number 2.46^(1/64) is known within less than one unit in the place of its fifth significant digit

This question (from Hildebrand's Introduction to Analysis) states: Show that the number $2.46^{\frac{1}{64}}$ is known within less than one unit in the place of its fifth significant digit if ...
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0answers
11 views

Is there an efficent way to solve large systems of purely quadratic equations?

I have the following system of quadratic equations $$ b_1 = \sum_{k=1}^R x_{i_1, k} \ y_{j_1, k} $$ $$ \vdots $$ $$ b_p = \sum_{k=1}^R x_{i_p, k} \ y_{j_p, k} $$ where $i_1, \ldots, i_p \in ...
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1answer
30 views

How to solve Inequality with factorials

Im reading a book in Numerial analysis and I have the following which I dont understand involving inequalities and factorials, What i have is the following: $$\frac{1}{(2n+1)!(2n+1)} \leq 5*10^{-9}$$ ...
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1answer
23 views

Min exponent range in normalized floating-point system

In a floating-point system with precision $t = 6$ decimal digits, let $x = 1.23456$ and $y = 1.23579$. (a) If the floating-point system is normalized, what is the minimum exponent range for which ...
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0answers
11 views

solving/approximating the transcendental inequality $c \le αx + β(b^x) + γx(b^x)$

I couldn't find a representation of $x$ using Lambert $W$ function and I doubt this is even possible. Assuming there is no clean solution and numerical methods must be used, is there a way to ...
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2answers
22 views

Numerical Approximation for 2D Curvature

I have a list of points (x, y) that are taken from an unknown 2D parametric curve $\vec{f}(t)$. These points are monotonically increasing in t (ie: they're a "connect the dots" version of ...
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0answers
14 views

two questions on numerical stability and conditional numbers

1) Let $\tilde{f}$ define an algorithm to evaluate $y = f(x)$, let $\tilde{y} - y$ be the forward error and $\Delta x = \tilde{x} - x$ by the backwards error. We have that: $$\|y - \tilde{y} \| = ...
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1answer
18 views

Gaps between successive floating point numbers

(all numbers discussed are in decimal) lets say we have a floating point data type that is like : m * 10 ^ e ...
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1answer
21 views

Fixed Point Iteration $x = g(x)$ method for $y_1 = e ^{-x}$ and $y_2= \cos x$

The question reads as follows: Find the x and y coordinates of the intersection points by means of the $x = g(x)$ method. ( I believe they are referring to the Fixed Point Iteration method) The ...
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0answers
21 views

Gauss-Jordan elimination in the form of (A|I)

So Gauss-Jordan elimination can be performed through the form of $(A|I)$ where $I$ is the identity matrix. We carry out row elementary operations as usual until the matrix becomes the form $(I|B)$, ...
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0answers
35 views

Diffusion of a chemical species inside a Y-shaped tube

I'm trying to model diffusion of a chemical species X inside a Y-shaped tube, whose diameter (thickness) is constant everywhere. The diffusion constant of X is $D$ ($\mu$m$^2$/s), so the concentration ...
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4answers
71 views

How do I find the error of nth iteration in Newton's Raphson's method without knowing the exact root

In our calculus class, we were introduced to the numerical approximation of root by Newton Raphson method. The question was to calculate the root of a function up to nth decimal places. Assuming that ...
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18 views

numerical solution of drift diffusion equation

0 down vote favorite in this link (in semiconductor physics section) you can see four coupled equations. do you know that finite element method is more accurate for discretization and numerical ...
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3 views

How to know the rate of convergence of a majorization - minimization algorithm?

The basic idea of majorization-minimization (MM) principlein optimization is to convert a hard problem (for example, non-smooth) into a sequence of simpler ones (for example smooth). To minimize ...
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1answer
38 views

Reduce this third order ordinary differential equation to first order to use Runge Kutta

The ODE I'm working with is $$\dddot{x} + t^2\ddot{x} + 4x = 0$$ with $$x(0)=1, \dot{x}(0)=0, \ddot{x}=-1$$ I've written a very basic program in C++ to use the RK4 method to approximate a solution to ...
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0answers
31 views

Sparse Matrices and Tridiagonalization.

Assume that we are given a sparse matrix,let it be 90*90(1000*1000), would you say that a vector with lots of zeros(let it be 90*1(1*1000),and 65(500) zeros are there),is a smart option to initialize ...
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0answers
14 views

Stick breaking point (discretized ODE)

I cannot find nontrivial solutions to the following problem. Let $x\in[0,1]$ and $y(x)$ be the deflection of the stick. Then this is described by the diff.eq.: $$\alpha^{-1} P y(x)+y(x)''=0 $$ where ...
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2answers
229 views

Deriving formula for derivative

I have a formula in my book for differentiating numerically. $$f'(x_0)=\frac{1}{12h}[-25f(x_0)+48f(x_0+h)-36f(x_0+2h)+16f(x_0+3h)-3f(x_0+4h)]+\frac{4}{5}f^{(5)}(\xi)$$ I was wondering if anyone ...
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0answers
17 views

Convergence of quadrature formulas and interpolating polynomials

There is a theorem of Polya (1933), which says: 1) If a interpolatory quadrature formula converges for all continuous functions on [a, b] and quadrature weights are all positive, then the formula ...
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0answers
15 views

Finding 1st,2nd and 3rd derivative for funtion of 2 variable

$E=g(p,v)$ $\frac{dp}{dv}=F$ $\frac{dE}{dv}$=$g_pF+g_v$ $\begin{align}\frac{d^2E}{dv^2}&=(g_pF+g_v)_pF+(g_pF+g_v)_v \\ &=g_{pp}FF+g_pF_pF+g_{vp}F+g_{pv}F+F_vg_p+g_{vv} \end{align}$ ...
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0answers
25 views

What is a one-parameter Newton's method?

The Newton's method that I know is defined as follows: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ However, I've recently encountered a paper that talks about a one-parameter family of Newton's method ...
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0answers
14 views

Mean value theorem for sequences

This is a problem I am trying to solve. Given a sequence $x_n$ defined $x_{n+1}=F(x_n)$. Assume $\lim_{n \to \infty}x_n=x$ and $F'(x)=0$. Need to show that $$x_{n+2}-x_{n+1}=o(x_{n+1}-x_{n}).$$ ...
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0answers
27 views

Trapezoidal rule - Multivariable

If I wanted to integrate the function $f(x,y)$ over the region $[a,b]\times[c,d]$ with two segments, am I going about this the right way? $$I(f) = \int_a^b \int_c^d f(x,y)\ dy\,dx = \int_a^b g(x) \ ...
6
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3answers
170 views

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

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 ...
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0answers
93 views

Evaluating a product (and sum) of polynomials in MATLAB [closed]

I'm new to MATLAB (and programming in general) and there's something I've been having a lot of trouble with. I want to evaluate the Lebesgue function with MATLAB. The function is as follows: $ L(x)= ...
2
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0answers
31 views

Definite integral of a hypergeometric function of an imaginary argument

How would one deal with such an integral? $$\int_0^\infty\frac{e^{-n r}}{r}{}_1F_1(i/k+1;2;2i kr) \, \mathrm{d} r$$ Here $F$ is the confluent hypergeometric function, $n\in\mathbb{N}$ and $k>0$ ...
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0answers
29 views

What is the best method to solve the ill-conditioned non-linear systems? [closed]

What is the best method to solve the ill-conditioned non-linear systems? for example: $$ x^2 − 2x + 3y = − 1 \\ 2x^2 - 3.9999x + 6.0001y = - 1.9999 $$
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1answer
34 views

Simpson's 3/8 Rule

When deriving Simpson's 1/3 Rule, I used a second order polynomial $P(x) = Ax^2 + Bx + C$, and integrated over the region $[-h,h]$ Integrating gave me: $ \ \dfrac{h}{3}(2Ah^2 +6C)$ I evaluated ...
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3answers
136 views

How can I solve this equation $x^{x^{x^{x^{.^{.^{.}}}}}}-a=0$

I always use the Newton-Raphson Method if I want to find the roots of any equation as follow $$x_{1}=x_{0}-\frac{y_{0}}{y'_{0}}$$ But I don't know how to use this method if the equation takes the ...
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1answer
29 views

What is the relation between analytical Fourier transform and DFT?

First of all let me state that I searched for this topic before asking. My question is as follows we have the Analytical Fourier Transform represented with an ...
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2answers
29 views

Evaluating differential entropies with Matlab: NaN issue

With Matlab I am trying to evaluate differential entropies. These are integrals like $$\int_\mathbb{R} p(x) \log (p(x)) \mathrm{d}x$$ where $p(x)$ is a probability density function. My $p(x)$ is ...
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1answer
14 views

Derivation of $f(x)=\prod_{m=0}^{n}(x-x_m)^{m+1}\tan(x), x_m=m\pi, M>0$

I have the following function: $$f(x)=\prod_{m=0}^{n}(x-x_m)^{m+1}\tan(x), x_m=m\pi, M>0$$ I would like to calulate the numeric root of: $n\pi, n\ge0.$ In order to do that, I want to use ...
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0answers
28 views

Calculate Derivative while Runge Kutta

I am thinking about writing a C++ code to solve an ODE using Runge Kutta method. As you know, RK method calculates the state space vector $X'$ in a few mid-points and uses these mid-points for ...
0
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1answer
10 views

Using divide difference formula find the value of $f\left[x_{0},x_{1},x_{2},…,x_{10}\right]$

Consider the polynomial $f(x)=x^{10}+x-1$ , $x\in \mathbb R$ & let $x_{k}=k$ for $k=0,1,2,...,10$. Then the value of the divide difference $f\left[x_{0},x_{1},x_{2},...,x_{10}\right]=$ (a) $-1$ ...
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2answers
22 views

Give an estimate for the error.

Use the first three nonzero terms of Taylor’s formula for $\sin x$ to find an approximate value for the integral $\int_0^1 \frac{\sin x}{x}$ and give an estimate for the error.(It is understood that ...
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0answers
27 views

Fourier series question - represent $x$ as a series of $\cos$

I was asked to represent $f(x)=x$ in $(0,\pi)$ as a sum of $\cos$ functions, using fourier series. I couldn't solve it on my own, but here is what the teacher did, and I don't fully understand why ...
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0answers
11 views

Is there a formula of coefficients of Newton-Cotes Method in numerical intergation?

We know the coefficients of Newton-Cotes method in numerical integration are: 2-points $ 0.5$ , $0.5$ 3-points $ 1/6$, $2/3$, $ 1/6$ 4-points $1/8$, $3/8$, ...
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0answers
26 views

Lipschitz Constant (Burden and Faires Exercise)

There's an exercise in Burden & Faires Numerical Analysis book, Section 5.1 #2a, where they appear to want the reader to verify that a Lipschitz constant exists for the following ODE: ...
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1answer
20 views

polynomial approximation - basic chebyshev question

I was asked to find the best linear approximation to $f(x)=x^2$ in $x \in [0,1]$ using chebyshev polynomials, meaning, using the known property that $2^{1-n}T_n(x)$ is the best approximation to $0$ at ...
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0answers
19 views

How to improve stability of numerical solutions to partial differential equations

This is a quite general question, but I am working with a system of partial differential equations in two variables. There is one time direction $t$ and one spatial direction $z$ and the numerical ...