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

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2
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1answer
34 views

Evaluating normal distribution integral

How one can show that $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{2.92}e^{-x^2/2}dx=0.99825$$
2
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0answers
40 views

(newbie) spectral derivative

I have data that form a scalar field on a 2D grid, evenly spaced. The grid has a finite size. There is no particular periodicity patern in my data. I want to calculate the value of the gradient at ...
0
votes
0answers
48 views

Newton-Raphson method fails!

I am trying to solve an equation like $R(x) = 0$, using Newton-Raphson method. To obtain the $x$ increment in each iteration I solve $dx = -(A)^{-1}\cdot R$ where $A = dR/dx$. But the convergence ...
-1
votes
0answers
11 views

Interpolation and finite differences question. [closed]

$u_x$ is a function of $x$ for which fifth differences are constant and $u_1$ $+$$ u_7$ $=$ $–786$ $ ,$ $u_2$ $+$ $u_6$ $=$ $686$, $u_3$ $+$ $u_5$ $=$ $1088$. Find $u_4$. Answer is $570.9$
0
votes
2answers
37 views

How to obtain Lagrange interpolation formula from Vandermonde's determinant

Assume that we have An interval $[a,b]$ A function $f(x)$ that is continuous on $[a,b]$ $n+1$ distinct points $a \le x_0<x_1<x_2<\cdots<x_n \le b$ And $f(x_0),f(x_1),\ldots,f(x_n)$ Now ...
0
votes
2answers
23 views

Minimum of Maximum of Addition of two vectors/arrays

Suppose you have two arrays and you want to compute the maximum of the addition of the two arrays. Now you move the second array one field to the right. Now you can compute the maximum again of the ...
3
votes
2answers
51 views

Does Newton's Method converge for f(x)

Does Newton's Method converge for: $$f(x)=(x-5)^2e^{x-5}$$ So Newton's Method is: $$x_{n+1}=x_n−\frac{f(x_n)}{f′(x_n)}$$ $$x_{n}=x_{n-1}-\frac{x_{n-1}-5}{x_{n-1}-3}$$ Error: $$e_n = x_{n} - ...
0
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0answers
17 views

B-Spline Question

Show that the cubic B-Spline with integer knots can be written as $$ s(x) = \frac{1}{6}\left [ x^3 \; x^2 \; x \; 1\right ]\begin{bmatrix} -1 &3 & -3 & 1 \\ 12 &-29 & 12 & 0 ...
1
vote
1answer
37 views

Homework for Gauss Seidel method

Let A be a strictly diagonally dominant matrix. Suppose we use Gauss Seidel method to solve $Ax=b$, a sequence of vectors {$x_{0},x_{1},...,x_{k},...$} is obtained (where $x_{0}$ is the initial guess) ...
0
votes
2answers
46 views

Use Bonnet recursion formula to prove by induction

Use Bonnet recursion formula: $P_{n+1}(x) = \frac{2n+1}{n+1} x P_n(x) - \frac{n}{n+1} P_{n-1}(x)$ to prove by induction 1) $P_n(1) = 1$ for all $n$ 2) $P_n(-x) = (-1)^n P_n(x)$ for all $n$ an for ...
1
vote
2answers
41 views

Weird computation error when using fnInt (numerical integral) on TI-84 Plus

Today in Calculus class I was bored so I decided to try and approximate $\pi$ by evaluating $ \left( \displaystyle \int_{-a}^a e^{-x^2} dx \right)^2$ on my calculator for larger and larger values of ...
0
votes
0answers
10 views

Would like in understanding a specific part of the Additive Operator Splitting scheme.

Can anybody help me to understand why the discretized version of this equation: $ \partial_tu = \partial_x \left( g(|\partial_x u_\sigma|^2) \partial_xu \right) $ (1) is the following: $ ...
0
votes
0answers
18 views

Using a casio calculator for numerical methods

I am just starting numerical methods, and I find myself typing in an equation into my calculator to get an answer, then re-inputting the answer to that iteration back into the equation, repeat etc. I ...
0
votes
1answer
36 views

Numerical approximate a convergent series

Consider i have a series $\sum_{i=1}^{\infty} X_i $ which i know converges in $\mathbb{R}$,but don't know exactly where. I am trying numerically approximate to the convergence point but not sure when ...
0
votes
1answer
45 views

What's the difference?

In my work we fit a parabola to some data in order to determine three parameters. I recently talked to someone who pointed out that the ISO standard related to the fit equation had changed. The ...
0
votes
1answer
31 views

Finding Eigenvalues with Gershgorin-Discs

$A=\begin{pmatrix}-5 & 0 & 0 \\ 2 & 2 & 1 \\ 3 & -5 & 4\end{pmatrix}$ Find the Gerschgorin-discs, where the eigenvalues of $A$ lie. According to the formula if we ...
1
vote
3answers
84 views

How to choose the starting point in Newton's method?

How to choose the starting point in Newton's method ? If $p(x)=x^3-11x^2+32x-22$ We only learnt that the algorithm $x_{n+1}:=x_n-\frac{f(x_n)}{f'(x_n)}$ converges only in some ...
1
vote
1answer
15 views

How can I apply Newton's method with boundaries?

I am trying to use Newton's method to minimize the distance between a line segment and a bezier curve. The distance function $f(x, t)$ that I'm minimizing is only defined for $x_1 \le x \le x_2$ and ...
0
votes
1answer
20 views

Approximation Theory: Iterative Methods

Can someone explain to me the general idea of what's going on? I don't get how the functions are being formed, nor do I know why we redefine $h(x)$ to ensure convergence after we defined $g$ ...
4
votes
2answers
61 views

Newton's Method for Roots of Polynomials

The standard way to use Newton's Method for finding a root of a polynomial $p(x)$ is to use the iteration formula $$x_{n+1}=x_n-{p(x)\over p'(x)}$$ I recently thought of a new way of finding the ...
0
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0answers
19 views

How to differentiate Lagrange Basis Polynomial?

How to differentiate Lagrange Basis Polynomial ? I don't know, if the term is correct, but the question is: If $x_0,...,x_n\in\mathbb R$ are pairwise distinct ...
0
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0answers
14 views

cubic B-spline interpolation function

I read that the B-spline basis functions are the follows: $B_0(x)=(1-x)^3/6$ $B_1(x)=(3x^3-6x^2+4)/6$ $B_2(x)=(-3x^3+3x^2+3x+1)/6$ $B_3(x)=x^3/6$ The cubic b-spline interpolation function it ...
0
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0answers
50 views

convergence for symmetric, positive semi-definite operator

Assume $u$ is a vector in the Euclidean space $\mathbb{R}^N$, $||u||=\sqrt{\langle u, u\rangle}$, where $\langle u, v\rangle = \sum_{i=1}^N u_i v_i$. I have that $||u^{k+1}-u||\leq ||I - c ...
0
votes
3answers
50 views

Improved Euler method for second order ODE

I am trying to solve the simple harmonic oscillator problem with various Euler methods. Having managed to solve it with simple and modified Euler methods now I am trying to solve it with the improved ...
1
vote
3answers
79 views

How do I solve an equation involving $e^t$ and $t$ on one side? $18=0.5e^t-0.5t-0.5$

The equation is: $$18=0.5e^t-0.5t-0.5$$ How do I solve for $t$? The answer manual gives $3.706$, but it gives no explanation on how to it got there. This is from a recommended dynamics problem, ...
0
votes
1answer
26 views

Reference request: nonlinear systems, optimization, ode/pde

Could someone suggest me one or more good books on the following topics: Nonlinear systems: fixed point and Newton's method Optimization: steepest descent and Newton's-quasi newton methods ODE ...
2
votes
2answers
54 views

dual of $H^1_0$: $H^{-1}$ or $H_0^1$?

I have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...
2
votes
1answer
55 views

Bisection Method Question, Multiple Roots

I understand how to do the bisection method and how to do it with a point of intersection. My question is should this not actually have multiple points of intersection? and if you're not given any ...
0
votes
0answers
15 views

Hermite interpolation with interior points

I am trying to solve the following problem: Given the conditions on a curve c(u) of degree 4 at the points -1, 0, 1 as: c(-1) = 4; c'(-1) = 4; c(0) = 6; c(1) = -4; c'(1) = -6; find the generalized ...
2
votes
0answers
40 views

Standard symmetric tridiagonal matrix Eigenvalue decomposition algorithm?

Hi I am trying to generate an arbitrary Gauss quadrature rule by using the Golub-Welsh algorithm (here). I need to code this on C++ for my personal project. This algorithm involves the eigenvalue ...
0
votes
0answers
15 views

Backward Difference Formula: Solving for First Derivative with a Limited Set of Knowns

I am trying to solve for $f''(x)$ by using only the following in the set ${ f(x_0), f((x_0-h), f(x_0 +h)}$. I realize that I am suppose to use Taylor's Theorem. This should help with the cancellation, ...
1
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0answers
9 views

By what factor do winning chances increase based on total value?

Say I am entering 24/7 in endless sweepstakes, contests, giveaways, drawings, etc. Assuming each one I enter has no less than 1 in 1,000 chances, but no more than 1 in 1 million (and I enter at least ...
1
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0answers
27 views

Numerical method for indefinite integral

let be the indefinite integral $$ F(x)= \int_{0}^{x} g(t)dt $$ the integral depends on the parameter 'x' i can use a linear transformation to turn this integral into $$ F(x)= ...
2
votes
3answers
87 views

Numerical Solution of $\frac{x}{1-e^{-x}} -5 = 0$

I am working on a problem at the moment which cuts down to the following question: How do I get a numerical solution for: $$\frac{x}{1-e^{-x}} -5 = 0?$$ I've been thinking about using Newton's ...
2
votes
1answer
22 views

How many bits of difference in a relative error?

I would like to know if there is a formula or any other way to find out how many bits of difference between two values given the relative error. For instance: $$\epsilon_{\text{rel}} = \frac{V - ...
0
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0answers
25 views

Function plotting

I have a function $f(x)=\binom{N}{K} \ln(1-F(x)), x \geq 0$, where $F(x)$ is a cumulative distribution function. Then, $\ln(1-F(x))$ is negative for various values of $x$ as $F(x) \geq 0$. Also, ...
0
votes
0answers
29 views

How to use derivatives at points with interpolation

I am given given $n$ points with $x$ and $y$ values. I am also given the derivatives at each of these points. How can I use the derivatives to calculate or to improve my interpolation? I've been ...
3
votes
1answer
52 views

Computational Maths

I'm trying to revise for a test and these 2 questions I just don't really understand what I'm meant to do, any pointers would be good. Any help I'd be very grateful for.
1
vote
1answer
34 views

how to solve these two quadratic equations

Can someone help me find the solution for these two quadratic equations ? $ 2(z^2) \ - \ 3.023bz \ + \ 0.115(b^2) \ + \ 2.0814b \ + \ 0.142z \ - \ 0.5856 \ = \ 0 $ $ 6.0828(z^2) \ + \ 2.0414bz \ + \ ...
3
votes
1answer
51 views

How to solve 5 simultaneous equations having 3 linear and 2 non linear equations

How do i solve the below equations? $$ x+y+z-1=0 $$ $$ 2a+2b+4z-2.0272=0$$ $$ x+2y+b-1.5778=0$$ $$ 1.0115xb-ya=0$$ $$ 1.0207a^2-z(x+y+z+a+b)=0$$
0
votes
0answers
30 views

Superlinearly convergent

A sequence $\{p_n\}$ is said to be superlinearly convergent to $p$ if $$\lim_{n\to \infty}{\frac{|p_{n+1}-p|}{|p_n-p|}}=0$$ a. Show that if $p_n\to p$ of order $\alpha$ for $\alpha>1$, then ...
1
vote
2answers
37 views

Proof that Lipschitz condition guarantees well posedness of initial value problems

In the proof of the theorem which states that the Lipschitz condition guarantees well posed-ness of an initial value problem $y'=f(x,y)$, $y(x_0)=y_0$, I came across this Let the perturbed problem be ...
0
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0answers
17 views

IVP and Newton-Cotes

I need to derive a RK method for solving the following IVP and would appreciate some help: y'(t) = f(t,y); y(0) = y0 by using the 3-points open Newton-Cotes formula, and get the LTE as well. How ...
0
votes
1answer
26 views

How to evaluate a condition number for a function of several variables.

I'm trying to get the condition number of a function $f(a,b,c)$ to see if it is stable. It is multivariate. I am reading the information here ...
0
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0answers
21 views

Diagonal Pivoting Algorithm

Commonly in LU factorization, partial pivoting is used. I know there is another pivoting which is diagonal pivoting. However, on the internet very few resources discussing diagonal pivoting (Only ...
0
votes
1answer
37 views

Checking convergence of an iteration

How to check if an algorithm converges in an interval, for example $x_{k+1}:=\frac{1}{11}(1-\cos(x_k))$ does it converge for any startpoint $x_0\in (-\pi/2,\pi/2)\setminus\{0\}$ ? (as hint: ...
0
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0answers
25 views

What is convergence degree of False Position method

What is the degree of convergence in False Position (Regula Falsi) method? Somewhere I heard that it equals: $$P=\frac{1+\sqrt5}{2}$$ Can anyone please prove it or prove it is not correct? Thanks.
0
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0answers
30 views

Solve Poisson's equation

I want to solve Poissons equation $$ C=\nabla^2 v $$ where $C$ is a constant and v my variable. I want to solve over some 2D domain D with the boundary condition that v is zero on the edge. How does ...
0
votes
1answer
37 views

A Simple Algorithm for Imposing Semi-definite Constraints

What is the simplest algorithm to implement, to impose semi-definite constraints? $\min_{X\succeq 0} f(X) $, where $X$ is an $n \times n$ symmetric matrix, and $f$ is a general smooth convex ...
0
votes
1answer
38 views

Find the rate of convergence?

Given is the iteration $x_{k+1}=\frac{1}{11}(1-\cos(x_{k}))$ with $x_{0}\in (-\frac{\pi }{2},\frac{\pi }{2})$ without $0$. Check if the sequence converges to $x^{*}=0$ and find its convergence rate. ...