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

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Gerschgorin Theorem singularity proof

I know how to prove the Gerschgorin Theorem but how exactly would one show that there are no values of $\mu$ s.t. $\mu<0$ for which $A-\mu B$ is singular where $$ A= \begin{bmatrix} ...
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12 views

Gaussian Elimination theoretical question

You know how Gaussian Elimination can be broken up into a sequence of L-U premultiplications right? Suppose that there is a matrix $A=a_{i,j} : j=1,...,n$ is an $n × n$ real matrix such that ...
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7 views

How do you calculate deviation from optimal value?

The variables I have: The optimal Tour Cost My Tour cost - I executed this 30 times, so I can get the average? Standard deviation just tells you how spread out the values are, I dont think thats the ...
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1answer
15 views

Prove the trapezoidal rule approximation

$\displaystyle L^i(x)= f(x_{i-1}) + \frac{f(x_i) - f(x_{i-1})}{x_i - x_{i-1}} (x-x_{i-1})$ Show that this linear approximation gives the trapezoidal rule. I know the formula of the trapezoid rule, ...
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1answer
16 views

Find t which minimizes ||A(x+ty)-b||$^2_2$

Let, f(t) = ||A(x+ty)-b||$^2_2$ = $(A(x+ty)-b)^T(A(x+ty)-b))$ ... $$= x^TA^TAx + 2tx^TA^TAy+t^2y^TA^TAy-2b^TA(x+ty)$$ Then, letting $f'(t) = 0$, we have $$ t = \frac{(b^TAy)-x^TA^TAy}{y^TA^TAy}$$ ...
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2answers
30 views

Jacobi method for any b

Determine if Jacobi method converges for any b for, $$\begin{bmatrix} 2 & 2\\ 3 & 4 \end{bmatrix}$$ The solution goes on like this... D-(L+U) = $$\begin{bmatrix} 2 & 0\\ 0 & 4 ...
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1answer
20 views

Polynomial Interpolation question

I do not understand the part that I have underlined in green. I thought that if you fit a polynomial $P_n$to given data for say $n+1$ distinct points you got an approximation of $f(x)$, where there is ...
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25 views

Aitkens Extrapolation

The Aitken's extrapolation can be written as $$X^n = X_{n-2} + \dfrac{(X_{n-1}- X_{n-2})^2}{(X_{n-1}- X_{n-2})-(X_n- X_{n-1})}$$ Verify it? And $X^n$ can be viewed being defined recursively by ...
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1answer
31 views

One step Gauss Seidel method

Apply one step of the Gauss Seidel method to $A\textbf{x} = b$ with A = $\begin{bmatrix} 4 & 2 & 1 \\ 1 & 4 & 1 \\ 1 & 2 & 4 \end{bmatrix}$, b = $\begin{bmatrix} 4\\ ...
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27 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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1answer
16 views

Using Givens Rotation on a vector

Say we have a vector v=$[3\ 0\ 4]$. Find a 3x3 orthogonal matrix Q such that only the second component of Qv is nonzero and such that this component is also positive. Is Q unique? I tried ...
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16 views

Inner Product inequality problem using Cauchy Schwarz, or what other way?

Let $<p,q>$ be an inner product on n. If p and q are both of degree n, show that $<p,q>^2$ $\leq$ $<p,p>$ $<q,q>$. I tried multiplying the right side out but am getting ...
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1answer
39 views

Show that the iteration $x_{n+1} = x_n - 2\frac{f(x_n)}{f'(x_n)}$ converges quadratically to $x_*$ provided $x_0$ is sufficiently close to $x_*$

We have the following conditions for the above slightly-modified Newton's method iteration: $f$ is a real function of one real variable $f''$ is Lipschitz continuous $f(x_*) = f'(x_*) = 0$ I also ...
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0answers
10 views

Gauss-Laguerre quadrature and error estimation

While using Gauss-Laguerre quadrature of varying orders, n, to estimate the value of integral what happens to the error as the value of n increases. would it increase or decrease and why?
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1answer
24 views

Is there a smooth map from the square to the deltoid?

Is there a $C^\infty$ map between a unit square in $\mathbb R^2$ and a deltoid like this one The deltoid is obtained by varying the angles $\theta_1$, $\theta_2$ in the equations \begin{align} x_2 ...
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0answers
12 views

Jacobi and Gauss Seidel Iteration for solution of ODEs

I have used the Jacobi and Gauss-Seidel iteration schemes for solution of the following ODE: $$y^{''}(x)-5y^{'}(x)+10y(x)=10x $$ I will outline my method below: Discretion the equation by ...
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0answers
13 views

Odd and Even Weight functions in orthogonal polynomials proof

Suppose now that w is an even function, i.e. $w(-x)$ = $w(x)$ for all x in $[-1,1]$ and let $p_0$,..., $p_n$ be a family of orthogonal polynomials with respect to w. Prove by induction that $p_k$ is ...
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1answer
18 views

Numerical integration: Quadrature method which one to use?

Since “it depends” is the proper answer to a question about what quadrature method to use in evaluating an integral, what are the things that one should consider when making a choice.
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1answer
14 views

Least Squares Approximation for odd functions question

Suppose that $f \in C[-1,1]$ is an odd function on $[-1,1]$. Show that polynomial $p_n$ of least squares approximation for $f$ in the norm $|\cdot|$ is an odd function on $[-1,1]$.
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34 views

Least squares polynomial approximation $(f-p_n,q)=0$ proof. [on hold]

I know how to do the other way around but I am getting stuck with showing the following If $<f-p_n,q>=0$ then $p_n$ is a polynomial of best least squares approximation in a norm $|\cdot|$ for a ...
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1answer
50 views

Floating Point Number System

I really have no idea of how to do these questions - in fact I have no idea of how to do any question in the paper - but I have tried to figure out what's going on in the course called Computational ...
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0answers
15 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
27 views

finite difference scheme for nonlinear partial differential equations

I have the following second order partial differential equation (PDE) on $[0,T] \times \mathbb{R},~ T >0 $: \begin{equation} \left(1 + \frac{1}{(1 + b f)^2}\right) \frac{\partial f}{\partial t} ...
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0answers
12 views

Finite Difference Scheme for a PDE on non-rectilinear coordinates

Consider poisson's equation on the domain $0 \leq x \leq 1$ $0 \leq y \leq H(x)$. Change the coordinates to $\xi=x$, $\eta=y/H(x)$. Construct a FDS that gives a positive definite symmetric matrix. ...
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0answers
17 views

numerial solution to fredholm integral equation

Consider the integral equation: $$ y(x) =1+\int_0^cK(x,t)\,y(t)\,dt, $$ where $x\ge0$ and $$ K(x,t) = \frac{\partial}{\partial ...
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0answers
7 views

Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
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0answers
18 views

Are there other well known oscillator systems besides Van der Pol oscillator? [on hold]

Is there any collections of oscillator systems similar to "matrix market"?
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1answer
25 views

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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13 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
23 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
22 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
31 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
12 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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0answers
12 views

Romberg Integration and Oscillating Functions

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

Solve the following nonlinear equations by using Bisection, Newton-Raphson, Secant, and method of false positions accurate

Solve the following nonlinear equations by using Bisection, Newton-Raphson, Secant, and method of false positions accurate to within $$ϵ_s=10^-4 $$ and compare the results. $$x^3-2x^2-5=0\text{ } ...
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0answers
21 views

cubic B-splines interpolation algorithm [closed]

Does anyone know if there is any fitting toolbox available to perform cubic B-splines interpolation? Thanks
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23 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
19 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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2answers
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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1answer
102 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
28 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
25 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
43 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
35 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 ...