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

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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
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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 ...
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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: ...
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2answers
206 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 ...
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2answers
1k 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 ...
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1answer
239 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 $ ...
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1answer
927 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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2answers
828 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 ...
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1answer
120 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 ...
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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 ...
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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 ...
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3answers
181 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 ...
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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 ...
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1answer
99 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 & ...
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3answers
313 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
475 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 ...
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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 ...
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5answers
456 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 ...
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2answers
569 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 ...
6
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1answer
69 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
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1answer
175 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
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1answer
194 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 ...
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2answers
110 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 ...
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3answers
646 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
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1answer
631 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
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1answer
214 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
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5answers
1k 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
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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
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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 ...
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3answers
915 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
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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
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1answer
412 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 ...
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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 ...
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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
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2answers
42 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
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1answer
119 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 ...
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2answers
291 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
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2answers
283 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
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1answer
205 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 ...
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1answer
300 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 ...
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2answers
1k 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
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1answer
476 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
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1answer
337 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)?
3
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1answer
728 views

With Euler's method for differential equations, is it possible to take the limit as $h \to 0$ and get an exact approximation?

I was recently watching a tutorial on Euler's method for approximating differential equations, and the whole time I was thinking "why can't you just take the limit of the step size $h$ as it goes to ...
3
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2answers
410 views

linear least squares minimizing distance from points to rays - is it possible?

I'm writing a tool whose purpose is to process data from a sensor that provides the true bearing to a target, and combine measurements taken at various times into an estimate of the target's position ...
3
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1answer
5k views

use duplex method to find square root of given number

I was reading this Wikipedia page about finding square root of given number. But I did't understand the section on the "Vedic duplex method". Not because of lack of English vocabulary, but main ...
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1answer
149 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
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0answers
93 views

How to generate a random matrix which have given singular values?

I know one method: generate a random matrix, apply SVD decomposition, modify singular values, and then multiply those matrices back together. However, I'm wondering how random this method is. Since ...
2
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2answers
160 views

Is there a mean value theorem for higher order differences?

The standard mean value theorem tells us $\frac{f(x+h)-f(x)}{h} = f'(c)$ for some $c$ between $x$ and $x+h$. Rewriting this, we may see it as $\frac 1h\Delta_h f(x) = f'(c)$. This makes me wonder if ...
2
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2answers
1k views

Fixed-Point iteration technique.

I have to find the root of $x-e^{-x}=0$ by using fixed-point iteration. when i rewrite the equation as $x=e^{-x}$ , the iterative process converges to $0.567$ after $12$ iteration. But when i ...