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
123 views

Accuracy from approximating $\zeta(2)$ with a partial sum

This is for an introductory numerical analysis class. The answer shouldn't be too complicated, but if you have one, feel free to post it. Figure out what $n$ should be such that $$\sum_{k=n+1}^\infty ...
3
votes
1answer
379 views

Explain the error term in Euler method

Task: I had to find out some estimates for M and L to make sure the proportional accucrazy is not above $10^{-4}$ in the Euler method with the problem below. I am trying to understand the page 672 on ...
2
votes
2answers
651 views

Numerical Approximation of the Continuous Fourier Transform

Given a function $F(k)$ in frequency space (sufficiently nice enough, eg. a Gaussian), I would like to compute its Fourier inverse \begin{equation}f(x) = ...
2
votes
1answer
936 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..? ...
2
votes
2answers
7k views

How to solve simultaneous equations using Newton-Raphson's method?

I understand how to find roots of a polynomial equation programmatically using Newton-Raphson method as explained here. How to find the values of $x$ and $y$ from the simultaneous equation given ...
1
vote
1answer
33 views

Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
1
vote
0answers
20 views

floating-point operations do not satisfy the well-known laws for arithmetic operations

Introduction to Numerical Analysis, Stoer, Chapter: Error Analysis, Page 8 if $|y|<\frac{eps}{\beta}|x|$ where $eps = 0.5\times 10^{1-t}$ then $$fl(x+y)=x+^*y=x$$ where $fl(x)=$ normalized ...
1
vote
2answers
56 views

$B_{i}^{n}(x)={{n}\choose{i}}x^i(1-x)^{n-i}$, prove that $B_{i}^{n}(cu)=\sum\limits_{j=0}^{n}B_{i}^{j}(c)B_{j}^{n}(u)$

$B_{i}^{n}(x)={{n}\choose{i}}x^i(1-x)^{n-i}$, prove that $B_{i}^{n}(cu)=\sum\limits_{j=0}^{n}B_{i}^{j}(c)B_{j}^{n}(u)$ I tried to to solve it from the right side: ...
1
vote
2answers
209 views

resources to study PDE from

I am an undergrad engineering student. I recently completed my second year, with that said, I have taken several calculus courses. Most recently I completed differential equations and multivariable ...
1
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2answers
2k views

Gaussian quadrature three-point

Derive the one and two-point Gaussian quadrature formulas for $$I=\int^1_0xf(x)dx\approx \sum_{j=1}^nw_jf(x_j)$$ with weight function $w(x)=x$. Which I know how to do and which I attached below (I ...
1
vote
1answer
242 views

LU decomposition of matrices

Although I know how the LU decomposition is done, given the following two matrices: $\begin{pmatrix} 0 & 2 & 3\\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ and $ ...
1
vote
1answer
961 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. ...
1
vote
2answers
866 views

How did people calculate numerical values of transcendental and trigonometric functions?

I know that back in the Stone Age, people used tables on this thing called paper to look up values for functions like $\sin$ and $\ln$. But how did the guys who wrote the tables calculate those ...
1
vote
1answer
122 views

Do I use Euler -method with Differentials correctly?

The book here (sorry not in English) on page 676: $$\begin{cases}y'=-y^{2} \\ y(0)=1\end{cases}$$ when $x_{k}=\frac{k}{5}$ which means $h=0.2$ i.e. $\Delta x=x_{k+1}-x_{k}=0.2:=h$. The task is ...
1
vote
1answer
359 views

Numerical Solutions of ordinary differential equations

Numerically solving $y'=f(x,y)$ with $k_1=f(x_n,y_n), k_2=f(x_n+c_2*h, y_n+c_2*h*k_1), y_{n+1}=y_n+h(b_1*k_1+b_2*k_2)$ what would the local truncation error be? Also, how would we perform a ...
0
votes
0answers
42 views

QR Algorithm with Shifts Question

Why must QR Algorithm with Shifts make no progress when applied to this n x n matrix? (attached as image). Also, if a matrix A is orthogonal in a QR factorization, will R be tridiagonal? How would ...
0
votes
3answers
214 views

Subtracting two dates

I'm developing a software and I need to subtract two dates and then get a date again. I've been trying to solve this problem for a while and I have found some additional problems. One of these ...
0
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0answers
37 views

Numerical Solvers to deal simultaneously with very different types of Oscillatory Behaviour

I am trying to solve these two related problems numerically: \begin{align} &f^{(\mbox{v})}(y) -(f^5 (y))'-\frac{1}{6}yf(y)=0\\ f'(0)=f'''(0)=0, &\quad f(y) \sim Cy^{(-1/7)}\exp(\gamma ...
0
votes
1answer
100 views

Could someone help me to prove that this symmetric matrix is definite positive?

Let $a_{ij}\in\mathbb{R}$ for all $i,j\in\{1,...,n\}$ and $m\in\mathbb{N}$. Consider the matrix below. $$B=\begin{bmatrix} \sum_{k=1}^n(a_{1k})^2 & \sum_{k=1}^na_{1k}a_{2k} & \cdots & ...
0
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3answers
344 views

Book in Numerical analysis

What books are good an introductory course in Numerical analysis? I look for a book with many applications, especially in biology
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1answer
483 views

Fast methods to check linearity of differentials? Generalizing linearity?

The L1 Mat-1.1010 -course here has taught me the linearity conditions $f(a x)=a f(x)$ and $f(a+b)=f(a)+f(b)$. I want to generalize it, some quite irrelevant slow investigation here. It requires time ...
0
votes
1answer
10k views

Solving a system with Newton's method in matlab?

I have the following non-linear system to solve with Newton's method in matlab: x²+y²=2.12 y²-x²y=0.04 What is the linear equation system to be solved? Should I ...
10
votes
5answers
471 views

Can this function be rewritten to improve numerical stability?

I'm writing a program that needs to evaluate the function $$f(x) = \frac{1 - e^{-ux}}{u}$$ often with small values of $u$ (i.e. $u \ll x$). In the limit $u \to 0$ we have $f(x) = x$ using L'Hôpital's ...
8
votes
2answers
580 views

A function for which the Newton-Raphson method slowly converges?

I'm doing a MATLAB assignment in which you work out and implement a better version of Newton-Raphson using a second degree Taylor polynomial instead of a first degree one. I have the algorithm worked ...
7
votes
1answer
197 views

Applications of the “soft maximum”

There is a little triviality that has been referred to as the "soft maximum" over on John Cook's Blog that I find to be fun, at the very least. The idea is this: given a list of values, say ...
6
votes
1answer
72 views

Solution of $\exp(z)=z$ in $\Bbb{C}$.

I have posted a related question here. I thinkg this one is more interesting: What about the solution of $\exp(z)=z$ in $\Bbb{C}$? My try : $z \mapsto e^z - z$ is entire non-constant. Perhaps ...
6
votes
1answer
190 views

Numerical approximation for log of incomplete beta function

Is there any known numerical approach to directly compute the log of the incomplete beta function? I would like to be able to compute $$ \log\left( \int_0^u x^{a-1} (1-x)^{b-1} dx \right) $$ ...
6
votes
2answers
452 views

Convergence of fixed point iteration for polynomial equations

I'm looking for the solution $x$ of $$x^n+nx-n=0.$$ Thoughts: From graphing it for several $n$ it seems there is always a solution in the interval $[\tfrac{1}{2},1)$. For $n=1$, the ...
5
votes
2answers
112 views

“Constrained” numerical solutions of ODEs with conservation laws?

Hi know little about numerical methods and I was considering the following problem that possibly has standard solution in the literature. Suppose you have an ODE for wich we already know that it must ...
5
votes
3answers
654 views

Projection matrices

I have found these two apparently contradicting remarks about projection matrices: 1) A matrix $P$ is idempotent if $PP = P$. An idempotent matrix that is also Hermitian is called a projection ...
5
votes
1answer
650 views

Some approximations for $\arccos(1/(1+x))$

I was trying to calculate the maximum ground distance you can see on mountains, with your elvation given. After some simple geometry, I was able to come up with the following formula: Let $h$ be ...
5
votes
1answer
216 views

how to solve $aX+bX^2=e^{cX}$

I build a model for our problem, but i cannot get a result from my model. Could anyone give me some idea to solve this formula: $aX+bX^2=e^{cX}$ Thx in advance!
5
votes
5answers
2k views

Numerically Efficient Approximation of cos(s)

I have an application where I need to run $\cos(s)$ (and $\operatorname{sinc}(s) = \sin(s)/s$) a large number of times and is measured to be a bottleneck in my application. I don't need every last ...
5
votes
1answer
1k views

Efficient and Accurate Numerical Implementation of the Inverse Rodrigues Rotation Formula (Rotation Matrix -> Axis-Angle)

I want to implement the Inverse Rodrigues Rotation Formula (also known as Log map from SO(3) to so(3)), in double precision code (MATLAB is fine for the example) preferably as a 3-parameter vector ...
5
votes
1answer
1k views

How does Knuth's algorithm for calculating logarithm work?

I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm: given $x\in[1,2)$, do the ...
4
votes
3answers
949 views

Estimating the Gamma function to high precision efficiently?

I know there are several approximations of the Gamma function that provide decent approximations of this function. I was wondering, how can I efficiently estimate specific values of the Gamma ...
4
votes
1answer
70 views

How to safely solve a pair of elementary equations in a floating point computing system?

I wrote a simple short computer program to solve a pair of equations of the format , $y = a1 * x + b1$ $y = a2 * x + b2$ . But , it outputs clearly wrong answers sometimes when $abs(a1)$ or ...
4
votes
1answer
415 views

Hermite Interpolation of $e^x$. Strange behaviour when increasing the number of derivatives at interpolating points.

I am trying to understand Hermite Interpolation. Here is my pedagogical example. I want to approximate $f(x)=e^x$ on the domain $[-1,1]$ using Hermite interpolation. I choose the Chebyshev zeros ...
4
votes
2answers
1k views

Fast and robust root of a cubic polynomial with constraints

I'm looking for a fast and robust method for finding a root of a cubic polynomial $x^3 + px^2 + qx + r$ To make the search more robust and faster, I'd like to leverage these properties: The ...
3
votes
0answers
55 views

estimations of solutions

Let $\Omega=\mathbb{R}^2_+=\{(x,y) \in \mathbb{R}^2; y > 0\}$, $f \in L^2(\Omega)$, $\lambda \in \mathbb{R}^*_+$, $A=(a_{ij})_{1\leq i,j \leq 2}$, $a_{ij} \in \mathbb{R}$ and there exist $\alpha ...
3
votes
2answers
44 views

Determine best possible Lipschitz constant

I'm slightly confused by a homework problem here...I've been given the function: $ f(u) = log(u) $ With the bounds: $ 2 \leq u \lt \infty $ Now I thought I understood what the Lipschitz Condition ...
3
votes
1answer
122 views

Accuracy of the Newton-Cotes formulas for polynomials of degree $n+1$ and even $n$

Let $f$ be a polynomial of degree $n+1$. The Newton-Cotes formula is given by $$\int_{-t}^tf(x)\text{ d}x\approx\sum_{k=0}^nf(x_k)\int_{-t}^t\omega_{n+1}(x)\text{ d}x \tag{*}$$ where ...
3
votes
1answer
84 views

Fermat-quotient of “order” 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known?

I'm rereading an older text on fermat-quotients (see wikipedia) from which I have now the Question for $$ b^{p-1} \equiv 1 \pmod{ p^m} \qquad \text{ with $p \in \mathbb P $, $1 \lt b \lt p$ and ...
3
votes
2answers
302 views

Explanation of Lagrange Interpolating Polynomial

Can anybody explain to me what Lagrange Interpolating Polynomial is with examples? I know the formula but it doesn't seem intuitive to me.
3
votes
2answers
290 views

Solving 3 simultaneous cubic equations

I have three equations of the form: $$i_1^3L_1+i_1K+V_1+(i_2+i_3+C)Z_n=0$$ $$i_2^3L_2+i_2K+V_2+(i_1+i_3+C)Z_n=0$$ $$i_3^3L_3+i_3K+V_3+(i_1+i_2+C)Z_n=0$$ where $L_1,L_2,L_3,K,V_1,V_2,V_3,C$ and $Z_n$ ...
3
votes
1answer
209 views

Understanding accuracy of Newton's Method

In a numerical analysis book I'm reading it says that using the Newton error formula we can find an expression for the number of correct digits in an approximation using Newton's Method. Here's the ...
3
votes
1answer
305 views

Understanding rate of convergence and order of convergence

I am trying to understand what is the difference between 'rate of convergence' and 'order of convergence'. Does anyone know an intuitive explanation of the difference between them? For example, say I ...
3
votes
2answers
2k views

Modern formula for calculating Riemann Zeta Function [duplicate]

Possible Duplicate: How to evaluate Riemann Zeta function I have an amateur interest in the Zeta Function. I have read Edward's book on the topic, which is perhaps a little dated. I would ...
3
votes
1answer
495 views

Unstable linear inverse problem: which “dampening” Tikhonov matrix should I use?

A linear inverse problem is given by: $\ \mathbf{d}=\mathbf{A}\mathbf{m}+\mathbf{e}$ where d: observed data, A: theory operator, m: unknown model and e: error. The Least Square Error (LSE) model ...
3
votes
1answer
341 views

Why Monte Carlo integration is not affected by curse of dimensionality?

What is the common sense explanation behind that fact that MC integration is free of "curse of dimensionality" in contrast to deterministic integration rules (e.g. trapezoidal rule)?