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

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calculate an approximate value of integral

Calculate an approximate value of integral : $$\int_1^{3.4}\frac {2}{\sqrt{x}+x}$$ Take 8-subintervals $n=8$ by using trapezoidal rule How can I calculate this?
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9 views

Computing area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions

I need to compute the area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions. As I am a non mathematics guy, it will be great if someone helps me out with the ...
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0answers
12 views

Numerical methods and Matlab

I am solving parabolic partial differential equation using Matlab and Finite difference method. I am new to Matlab so I do not know how to write ICs/BCs in Matlab numerically. If some one help me out ...
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1answer
16 views

Find the first two iteration of the Jacobi method for the following linear system, using $x^{(0)} = 0$

$$3x_{1} - x_{2} + x_{3} = 1,$$ $$3x_{1} + 6x_{2} + 2x_{3} = 0,$$ $$3x_{1}+3x_{2}+7x_{3} = 4$$ So, from this I got T = \begin{bmatrix} 0 & \frac{-1}{3} & \frac{1}{3} ...
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2answers
25 views

fourier series analysis, show that for every integer n, using euler's formulas relating trigonometric and exponential functions

Show that for every integer $n$, $$\int_0^{\pi} \cos nt~\sin t~\mathrm{d}t = \begin{cases} \dfrac{2}{1-n^2} & \text{if } n \text{ is even} \\[10pt] 0 &\text{if } n \text{ is odd} ...
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1answer
9 views

equivalent condition for interpolation polynomial

Let be $(x_1,y_1),...,(x_n,y_n)\in \mathbb{R}^2 $, where $x_i\neq x_j$ if $i\neq j$. Let be $p$ a polynomial such that, $$det(\begin{pmatrix} p(x)& 1 & x & x^2 &\dots & x^n ...
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6 views

Romberg Integration and Oscillating Functions

Why do adaptive quadrature methods produce better approximations than Romberg integration for oscillating functions?
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1answer
61 views

Derive The Midpoint Rule: $\int_{x0}^{x1}f(x)dx=hf(x_0+\frac{h}{2})+\frac{h^3}{24}f^{2}(\mu)$

The Given Question is: ================================================================== Expand the function $f(x)$ in a $1^{st}$ degree Taylor series about $x_0 + \frac{h}{2}$ with ...
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17 views

cubic B-splines interpolation algorithm [on hold]

Does anyone know if there is any fitting toolbox available to perform cubic B-splines interpolation? Thanks
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17 views

Runge Kutta stability region for forward euler and explicit midpoint

The interval of absolute stability is the intersection of the region of absolute stability in the complex plane with the real axis.Show that Runge Kutta forward Euler and RK explicit midpoint have the ...
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1answer
15 views

Interpolation of Polynomial using Lagrange

$f(x) = x^3 + 2x^2 + x + 1$. Find a polynomial of degree $4$ that interpolates the values of $f$ at $x = -2, -1, 0, 1, 2$. I was trying to use the Langrange algorithm, but I think i'm doing it ...
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19 views

Interpolation of Polynomial

Let $f(x) = x^3 + 2x^2 + x + 1$. Find the polynomial of degree $2$ that interpolates the values of $f$ at $x = -1,0,1$. I was able to do the an initial part of this problem (not written), but I ...
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1answer
29 views

Trapezoid Rule - Number of Points

How many points should we use in the trapezoid rule in computing an approximate value of $\int_{0}^{1} e^{x^2} dx$ if the answer is to be within $10^{-6}$ of the correct value? I'm looking at the ...
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0answers
25 views

Interpolation and Taylor's Theorem

I just answered a question where I used the fact that a $(n+1)$-times (continuously) differentiable function $f$ interpolated by a $n$th degree polynomial $p(x)$ through the $n+1$ points $x_0,...,x_n$ ...
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1answer
12 views

Polynomial Interpolation - Bound on Error

Let the function $f(x) = \ln(x)$ be approximated by an interpoation polynomial of degree of 9 with 10 nodes uniformly distributed in the interval $[1,2]$. What bound can be placed on the error? I've ...
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1answer
26 views

Natural Cublic Spline Confusion

Find the natural cubic spline which interpolates the data points $(1,0),\; (2,1),\; (3,0), \; (4,1), \; (5,0) $. I know how to check if a piecewise function is a natural cubic spline, but I don't ...
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0answers
9 views

How to define Rate of Convergence and Order of Convergence?

There are lots of mathematical explainations available for Rate of Convergence and Order of Convergence but they all include too much of mathematical notations. Rate of convergence is the speed at ...
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1answer
26 views

Numerical Integration Confusion

Derive a numerical integration formula (i.e. determine A,B,C and $\alpha$) of the form $$\int_{-1}^{1}|x|f(x) \approx Af(-\alpha)+Bf(0) + Cf(\alpha) $$ that is exact for polynomials of degree $\leq ...
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1answer
24 views

division by sum of exponentials of large negative numbers

I need to evaluate the following numerically: $$ f = \frac{\exp(a)}{\exp(a)+\exp(b)+\exp(c) + \exp(d)} $$ $a,b,c$ and $d$ are large negative numbers, they are smaller than -1000. Numerically ...
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0answers
28 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
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1answer
38 views

Solving a problem using Householder's method

For the following points on a plane: $(-1,1),(0,0),(1,1),(1,-1)$, we look for a polynomial $p(x)=a+bx$ such that: $$ \sum_{i=1}^4{(p(x_i)-y_i)^2} = min $$ How do I formulate this as problem as a ...
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1answer
13 views

Multistep method Numerical Analysis : IVP

Given an explicit multistep method , we need to find the constants of the terms such as f(xi) , f(xi-1)..... and method order ..etc. A general approach is to write the corresponding Taylor series and ...
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1answer
24 views

How can I solve an ODE when $F(x_0)=F'(x_0)=0$ is given at an unknown point $x=x_0$ using bvp5c?

I'm attempting to solve the following ODE using MATLAB bvp5c. I've used bvp5c for other typical multipoint boundary value problems but I have no idea how to deal with ODEs with conditions given at an ...
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15 views

Is fourth order Runge-Kutta method validity

I wonder whether the fourth-order Runge-Kutta method is suitable for a second-order linear ODE with dissipative terms modelling free fall of an object through a viscous medium under the act of ...
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8 views

Can multidimensional eigenfunction problems be solved to arbitrary precision in constant memory usage?

Suppose we have a differential operator like a quantum mechanical Hamiltonian: $$\hat H=-\nabla^2+U$$ with zero Dirichlet boundary conditions. In one dimension its eigenvalues can be easily found ...
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22 views

Heuristics for sequence convergence

Having a finite sequence of double precision floating point numbers (obtained using the fixed point iteration of a function), is there any algorithm which can be used to determine that this sequence ...
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0answers
25 views

how to prove this curious identity with the Chebyshev polinomials

we defined the Tm like this (where Tm are the Chebyshev polinomials) Then I showed this: And now I have no idea how to proove this: I also have to make the remark that I also proved that the ...
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0answers
17 views

Numerical methods to compute multivariate integrals

I would like to compute an multivariate integral as: $$\iiint f(x_1, x_2, x_3) \ g(x_1) \ g(x_2) \ g(x_3) \ dx_1 \ dx_2 \ dx_3$$ It is indeed an expectation a function ($f$) of independent marginals ...
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2answers
27 views

Absolute value of complex number in Numerical Recipes

In numerical recipes in C, absolute value of complex number $a+ib$ is implemented as $b*\sqrt{1+\left(\frac{a}{b}\right)^2}$ if $|b|$ is greater than $|a|$ and $a*\sqrt{1+\left(\frac{b}{a}\right)^2}$ ...
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23 views

Approximating the function $ f(x) = \frac{1}{1+a^2x^2} \text{with } a=4 \text{ in the interval }[-1,1]$ with Legendre Polynomials

Given: $$ f(x) = \frac{1}{1+a^2x^2} \text{with } a=4 \text{ in the interval }[-1,1]$$ Approximate the function $f(x)$ in the least squares sense using legendre polynomials up to order 2. The ...
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2answers
65 views

Evaluate the integral $I=\int_0^{1.6}\frac {1}{1+x^4} dx$

$I=\int_0^{1.6}\frac {1}{1+x^4} dx$ by using generalized trapozoidal rule $n=8$ the final answer don't equal the correct answer .I need the final answer and how can i solve it ? The answer to your ...
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1answer
22 views

Numerical verification of the ternary Goldbach conjecture

In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this ...
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1answer
34 views

Evaluating normal distribution integral

How one can show that $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{2.92}e^{-x^2/2}dx=0.99825$$
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38 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 ...
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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 ...
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10 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 $570.9$
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32 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 ...
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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 ...
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2answers
47 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} - ...
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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 ...
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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) ...
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2answers
44 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 ...
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2answers
40 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 ...
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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: $ ...
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0answers
16 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 ...
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1answer
35 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 ...
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1answer
44 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 ...
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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 ...
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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 ...
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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 ...