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

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48 views

How to choose a numeric approach for derivates

I would like to find the derivate from some combined logistic and exponential functions that all describe the same data numerically. $$f'(t)=\frac{f(t+h)-f(t)}{h}$$ seems not the best choise for ...
2
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1answer
41 views

Question about matrix discretisation numerical methods

Tomorrow I have an exam about Numerical Methods, and I came up with the following question. Let $$-\frac{d}{dr} \left ( \frac{1}{r} \frac{dy}{dr} \right ) = 1 $$with $r\in [1,2], y(1) = 1 \mbox{ and ...
2
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2answers
102 views

Recommendations for website/journal/magazine in applied mathematics

Which website/journal/magazine would you recommend to keep up with advances in applied mathematics? More specifically my interest are: multivariate/spatial interpolation numerical methods ...
2
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4answers
110 views

Chaotic iterative example needed

I'm using a very simple numerical method to find solutions to an equation. Start with an equation $\operatorname{f}(x)=0$ that you need to solve. Rearrange to give $x=\operatorname{g}(x)$ and then use ...
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2answers
50 views

Interpolation in 2-D co-ordinate system

Suppose there is a function f defined on (x, y) such that x, y $\in$ (-$\infty$, $\infty$), but function is not known. Let n data points are given f($X_0$, $Y_0$) = $Z_0$ f($X_1$, $Y_1$) = $Z_1$ ...
2
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0answers
48 views

Finding $k$ unknowns given the sum of their first $k$ powers

Motivation: The motivation for this question came from a Computer Science problem of finding duplicates in a list in constant time and constant space. If the list of numbers was $i_1, i_2, \ldots, ...
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1answer
2k views

estimating the error of $\sin(x) = x$ with Taylor's Theorem

I want to calculate the numerical error in approximating $\sin(x)=x$ with Taylor's Theorem. Furthermore, what values of $x$ will this approximation be correct to within $7$ decimal places? Here is ...
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1answer
30 views

how can i find best value of t in this equation?

I need to evaluate $$\tan^{-1} (x) - x = O(x^t)$$ as $x$ approaches $0$, in order to find the best value of $t$. Big O notation is described here. I tried: $$\lim_{x \to 0} \frac{\tan^{-1}(x) - ...
15
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3answers
996 views

Why does Monte-Carlo integration work better than naive numerical integration in high dimensions?

Can anyone explain simply why Monte-Carlo works better than naive Riemann integration in high dimensions? I do not understand how chosing randomly the points on which you evaluate the function can ...
2
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1answer
46 views

Point on an algebraic surface closest to another one

Given an algebraic surface $F(x,y,z)=0$, $F\in\mathbb{R}[x,y,z]$, and a point $P_0=(x_0,y_0,z_0)\in\mathbb{R}^3$, is there a possibility to (algorithmically) determine a point on $F(x,y,z)=0$ that is ...
1
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1answer
179 views

Consistency Trapezoidal rule

I want to prove that the consistency order of the trapezoidal rule is actually second order, that means that the error of the actual solution $x(t_{k+1})$ where we can restrict ourselves to the ...
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1answer
2k views

Minimum surface Curvature Interpolation Method

In this paper about Interpolation Methods, I am trying to learn Minimum curvature method. I have not done partial differential equations before; hence I am finding it tough to penetrate through this ...
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1answer
220 views

The function defined by $f(x)=\sin\pi x$ has zeros at every integer. Show that when $-1<a<0$ and $2<b<3$ , the bisection method converges to

The function defined by $f(x)=\sin\pi x$ has zeros at every integer. Show that when $-1<a<0$ and $2<b<3$ , the bisection method converges to (a) $0$, if $a+b<2$ (b) $2$, if $a+b>2$ ...
0
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1answer
649 views

Heun's method second order

I want to proof for the differential equation $x'=\lambda x$ that the error done by Heun's method in each step is $O(h^3)$. We have $x_{k+1}=x_{k}+\frac{h\lambda}{2}(x_{k+1}+x_{k})$ as the given ...
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3answers
13k views

What is difference between Finite Different Method, Finite Element Method and Finite Volume Method for PDE?

Can you help me explain the basic difference between FDM, FEM and FVM? What is the best and why? Advantage and disadvantage of them?
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0answers
160 views

closed-form solution for 1/tanh(x) - 1/x that can be evaluated at/near x=0?

I'm looking to evaluate $\frac{1}{\tanh x}-\frac{1}{x}$ over a range that includes x=0. Is there an alternate form that is both exact, and numerically stable at/near x=0? For now I'm using the Taylor ...
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1answer
112 views

Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires that $n>1000$.

Let $f(x)=(x-1)^{10}$. The root of the equation , $p=1$. The approximates of the root, $p_n=1+\frac{1}{n}$ Show that $|f(p_n)|<10^{-3}$ whenever $n>1$ but that $|p-p_n|<10^{-3}$ requires ...
0
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1answer
121 views

Difficulty to solve the exercise of Bisection method.

Find an approximation to $ {25}^{\frac{1}{3}}$ correct to within $10^{-4}$ using the Bisection algorithm. How to solve it? Where are the function and interval here?
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3answers
372 views

I am not understanding what has asked to compute of the following exercise.

let $f(x)=(x+2)(x+1)x(x-1)^3(x-2)$. To which zero of $f$ does the Bisection method converges when applied on the interval $[-3,2.5]$ Have i asked to find the root of $f(x)$ ?
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1answer
85 views

Determine the number of iteration to find solutions accurate to within $10^{-2}$ for $f(x)=x^3-7x^2+14x-6=0$ on $[a,b]=[1,3.2]$

i got the number of iteration,$n$, to achieve the accuracy, $\epsilon=10^{-2}$ is $n=5.5\approx 6$ But in answer script, $n=8$. My procedure is $ \frac{(b-a)}{2^n}<\epsilon$ ...
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1answer
107 views

Condition or Proof: Minimizer of one function is maximizing another function

I have two real functions $f(X),g(X)$ where the argument $X$ is a real matrix. The solution $X^*$ for the problem of minimizing $f$ is ending up maximizing $g$ as well. I am looking for a way to prove ...
2
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1answer
133 views

Correct answer of the following math related to Bisection Method.

Use the Bisection method to find $p_3$ for $$f(x)=\sqrt x-\cos(x)$$ on $[0,1]$ I have got the answer $p_3=0.875$ But in answer script , $p_3=0.625$ Which one is correct? let $[a,b]=[0,1]$ ...
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1answer
83 views

Can we conclude that this matrix is definite positive? [duplicate]

Let $A$ be a $n\text{-by-}m$ matrix. Suppose that columns of $A$ are linearly independent. Can we conclude that $A^TA$ is definite positive? Could you help me with proof? Thanks.
2
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1answer
988 views

What is convection-dominated pde problems?

Can you explain for me what is convection-dominated problems? Definition and examples if possible. Why don't we can apply standard discretization methods (finite difference, finite element, finite ...
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1answer
83 views

Function Projection: Orthogonal Polynomials

I am currently reading a paper called "Numerical Quadrature" by Timothy J. Giese (2008) which describes the numerical quadrature technique in detail. At one point (just before equation 19) it states ...
7
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0answers
213 views

What’s the best way to cut an apple?

Take the apple in one hand, and the knife in the other. In the first cut, the apple is divided in two pieces: a small one that drops into the plate and a big one that is still hold with the hand. This ...
2
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1answer
267 views

Simpsons rule & Lagrange?

What is the relation between Lagrange interpolation and Simpson's rule to integrate some function with some points $x_0,f(x_0)$; ... $x_n, f(x_n)$ ?
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1answer
42 views

Ruled surface by skinning

I have the following problem. I have one curve and the article I am analyzing says: we can generate a ruled surface by skinning the curve in a direction D. I don't know how I can obtain a ruled ...
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1answer
159 views

How to show all eigenvalues are positive?

Could you help me to show that the following matrix has all its eigenvalues positive? $$H= \begin{bmatrix} \sum_{k=1}^ng_1(x_k)^2 & \sum_{k=1}^ng_1(x_k)g_2(x_k) & \cdots & ...
3
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1answer
546 views

How to show that the Hessian matrix of $G$ is positive definite?

Let $\{g_i:X\subset\mathbb{R}\rightarrow\mathbb{R};\;i=1,...,m\}$ be a linerly independet set of real functions. Given $n$ points $(x_1,y_1),...,(x_n,y_n)\in X$, consider the following function ...
0
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1answer
111 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 & ...
2
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1answer
108 views

I would like a hint in order to prove that this matrix is positive definite

Let $a_{ij}$ be a real number for all $i,j\in\{1,...,n\}$. Consider the matrix below. $$B=\begin{bmatrix} \sum_{k=1}^n(a_{1k})^2 & \sum_{k=1}^na_{1k}a_{2k} & \cdots & ...
4
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1answer
113 views

$\delta$ Notation in linear algebra

In this equation below, what is $\delta_{l,q}$ denoting? Is $\delta$ a standard notation, or anything to do with all one's or the basis matrix etc? $$A_{ij}=\delta_{l,q}\left(\sum_{h=1}^n B_{l,h} + ...
1
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1answer
315 views

Do the false position method really need that there exists only one root inside $[a; b]$?

I'm studying the False Position Method for finding zeroes of real functions and in the book I'm reading the author says that it is required that only one root of $f$ is contained inside the initially ...
2
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2answers
259 views

How to determine the starting values for linear multistep methods?

I am so confused with how to determine the starting values for linear multistep methods. I have searched the wiki page for linear multistep method. And it says that for ...
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0answers
61 views

Geodesic Interpolation of a Vector

I have two vectors given and I want to estimate another vector by using geodesic interpolation, how can I do this? Thanks
2
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3answers
453 views

The Weierstrass Approximation Theorem Vs The Runge's Phenomenon

I am learning about different interpolation methods in my internship. Today as I was looking this article on Wikipedia to learn about the Runge's Phenomenon exhibited by Polynomial Interpolation. I ...
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2answers
2k views

Rate of convergence with matlab

I am supposed to determine the order of convergence of Heun's method just by evaluating $ y'(t)=\lambda y(t)$ for several $\lambda$, several step sizes and several number of grid points. I already ...
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0answers
145 views

Effects of numerical integration stepsize on impulse inputs (e.g., delta function)

Some models of neurons treat synaptic input (from other neurons) as a single impulse, such as the Dirac delta. But doesn't this make the magnitude of that impulse a function of numerical integration ...
3
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0answers
56 views

Converting a linear program into standard form

In especially, I have a question about the demand that if I have $ Ax \leq b$, then I can convert this into $A'x'=b$ for some new $A'$ and $x'$. I have given the system of equations: $20x_1+30x_2 ...
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1answer
431 views

numerically evaluate a continued fraction

I am looking at a continued fraction of the form $$ F_n = \cfrac{1}{1+\cfrac{p_1}{1+\cfrac{p_2}{1+\cfrac{p_3}{1+\ldots}}}} $$ where $p_n$ is a function I know. For simplicity I just take it to $p_n=n$ ...
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1answer
382 views

How to prove that the following iteration process converges?

I have the following iteration process: $$ p_{n+1} = \frac{{p_{n}}^3 + 3 a p_{n} }{3 {p_{n}}^2 + a } , $$ where $a > 0$. Q1: How to prove that this iteration process converges for every number ...
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0answers
195 views

Best method of interpolation?

I am learning different interpolation methods, and their pros and cons. Which interpolation method do you think is the best for practical use? If you can give me links to research papers about various ...
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0answers
186 views

How to solve the projectile problem with numerical method in matlab

i wanna ask how to solve the projectile problem using matlab? could you give me the source code in matlab? the equations is x"=-(1+0.1*x)^2 , with x(0)=0, x'(0)=1. thanks before.
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0answers
56 views

Help setting up non-linear parabolic BVP for Newton's method for Non-linear Systems

I am trying to apply Newton's method for non-linear systems to this equation: $$\frac{\partial u}{\partial t}=\frac{\partial ^{2} u}{\partial x^{2}}+(1-u^{2})u+f(x,t) , x \in [-1,1], t>0$$ ...
2
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1answer
298 views

One Sided Approximation for Mixed Derivatives

Consider the function u(x,y,z) I am trying to approximate the partial derivative at point (i,j,k) by one sided finite difference method. Now using one sided 2nd order finite difference approxmation ...
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106 views

Bilinearly Blended Coons Patch

I have four quintic Bézier curves and I want to create the Coons Surface that interpolates them. How can I obtain the 25 control points of the surface? Thanks.
2
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2answers
822 views

Error estimation for spline interpolation

Can you please indicate a reference for the proof of the fact that the error when interpolating a $C^4$ function by a cubic spline is bounded by $Ch^4\sup_{[x_{i},x_{i+1}]} |f''''(x)|$?
0
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0answers
105 views

How to solve $a(\exp(b/x)-1)=c(\exp(b/(x-d))-1)$ for $x$?

How to solve $a(\exp\left(\frac{b}{x}\right)-1)=c(\exp\left(\frac{b}{x-d}\right)-1)$ for $x$? where $a,b,c,d$ can be treated as constants. This problem is a simplified version of dividing the ...
3
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4answers
49 views

Can we conclude that this summation is positive?

The summation is: $$S_n=n\sum_{i=0}^n x_i^2- \left ( \sum_{i=0}^nx_i\right)^2$$ where $n>1$ and $x_1,x_2,\ldots,x_n\in \mathbb{R}$. I'm trying to prove that if $x_i\neq x_j$ for $i\neq j$ then ...