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

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4
votes
2answers
145 views

Interpolate 4 points by an increasing polynomial

I need to create a polynomial function that passes through the points $(0,0)$, $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$ where $$ x_1 < x_2 < x_3\quad \text{ and } \quad y_1 < y_2 < y_3 $$ ...
0
votes
0answers
51 views

When can definite integration be numerically computable?

under what condition,can the integration $$\int_{\Delta}f(x_1,x_2,\dots,x_n)dx_1dx_2\dots dx_n, \text{where } \Delta \text{ is integration domain defined by function},f(x_1,x_2,\dots,x_n) \text{ ...
3
votes
0answers
55 views

Iterative equation

I have an equation that I want to try and solve iteratively. I don't have any background in numerical analysis so unsure as to how to go about it. Any help would be greatly appreciated. My equation ...
2
votes
1answer
267 views

Implementation of Richardson extrapolation in mesh independence study

I am busy with a mesh independence study in computational fluid dynamics, where I am systematically refining my mesh and monitoring a certain parameter of interest with the goal that the value should ...
0
votes
1answer
69 views

Finding the value of y using Lagrange Formula

Let $p_2(x)$ be the interpolating polynomial for the data $(0 , 0) , (0.5 , y) , (1,3)$ from Lagrange formula. The coefficient of $x^2$ in $p_2(x)$ is $-2$ , Find the value of $y$ .
0
votes
1answer
62 views

How to write approximations of a sequence $x_n = {1/3^n}$

Write three approximations of the sequence ${x_n} = {1/ 3^n}$, using the following scheme - $P_0= 1, P_1 = 0.33332$ and $P_n = (6/5)P_{n-1} - (1/5)P_{n-2}$ for $n = 2, 3,\dots$ Further, make a ...
1
vote
1answer
38 views

Converting scalar ODE to coupled system

I'm currently battling the following problem: \begin{align} u^{(iv)} (x) &= f(x)\quad\text{on }(0,1)\\ u(0) = u'(0) &= 0\\ u''(1) = u'''(1) &= 0 \end{align} which is, as I've understood, a ...
1
vote
4answers
71 views

$\sin(1/x)$ is not uniformly continuous on $(0, \frac{\pi}{2}]$

Show that $$ f(x)=\sin\frac1x $$ is not uniformly continuous on $(0,\frac\pi2]$. It is looking easy to do this problem if it is asked for $(0, 1)$ but I am not getting for the given range of ...
2
votes
2answers
123 views

Derive the weak form for nonlinear problem.

Let the equation be $$\frac{d^2 y}{dx^2}=\frac{y}{1+y}$$ For finite element formulation how to get the weak form? The major problem being the nonlinear rhs.`
1
vote
0answers
151 views

Solution of equations involving determinant and matrix inverse

$x$ and $y$ are two scalar unknowns. The two equations are $$|\mathbf{I}+x\mathbf{h}_1\mathbf{h}'_1+y\mathbf{h}_2\mathbf{h}'_2|=R$$ and ...
2
votes
1answer
41 views

Numerical Evaluation of a Series at a Point

I have a numerical calculus challenge to resolve using a C++ algorithm or scilab. The problem is the following: $f(x)=\sum_{n=1}^{\infty }a_{n}x^{n}$ where $a_{n}=\sqrt{n^2+1}-n$ This function is ...
2
votes
1answer
425 views

Why is Newton's method faster than gradient descent?

Can you provide some intuition as to why Newton's method is faster than gradient descent? Often we are in a scenario where we want to minimize a function f(x) where x is a vector of parameters. To do ...
0
votes
1answer
41 views

Is the optimization problem right?

If we want optimize the following problem $$ \min_x \{a(x)+c(x)\} $$ and we have $$ a = \min_y b(y) $$ then, could we directly optimize the following problem? $$ \min_x \{b(x)+c(x)\} $$
1
vote
1answer
40 views

How do I perform taylor expansion of the following?

Taylor expansion about $(x,y)$ of $f(x + a,\; y + k\; f(x + b,\; y + c))$ I do not understand what happens to the second $f$ inside. The inspiration for this question is Runge-Kutta methods.
0
votes
1answer
50 views

Interpolation polynomial types

I was wondering if both the Maclaurin and Taylor series are two types of interpolation polynomials? I was under the impression that they were not because they only go though one point in an interval ...
2
votes
0answers
53 views

Numerical Analysis, divided differences

This is what I have to prove: $$f[x_0, x_1, \dots, x_n] = \frac{(-1)^n}{(x_0+a)(x_1+a) \dots (x_n+a)}$$ where $f(x) = \frac{1}{x+a}$ and $f[x_0, \dots, x_n]$ is the divided difference of $f$ in ...
0
votes
0answers
42 views

For any deg $n-1$ polynomial, $\sum_{i=0}^n q(x_i) \prod_{j \neq i, j=0} (x_i - x_j )^{-1} = 0 $

I came across this (probably) easy problem: Prove for any polynomial $q$, of degree $n-1$, that: $$\sum_{i=0}^n q(x_i) \prod_{\substack{j \neq i \\ j=0}} (x_i - x_j )^{-1} = 0 $$ Do I only need to ...
0
votes
1answer
56 views

What algorithm to solve this integral similar to a normal CDF numerically?

I'm looking to solve this integral numerically. It is a bit similar to a normal CDF. z and tau are deterministic. What kind of algorithm may do the job, ideally to be coded in C/C++? I'm ...
1
vote
0answers
162 views

Backward Euler method with a cross-product.

I want to solve the following differential equation with the backward Euler method ...
1
vote
0answers
96 views

the Gauss-Jordan algorithm requires how many multiplications/divisions and add/subtractions

I am trying to show this following result. The Gauss-Jordan algorithm requires $\frac{n^3}{2}+n^2-\frac{n}{2}$ multiplications/divisions and requires $\frac{n^3}{2}-\frac{n}{2}$ ...
1
vote
1answer
29 views

two solutions in $2^{nd}$ order linear differential equations

Could you please explain why we need two solutions $y_1$ and $y_2$ (fundamental set of solutions) for determine the general solution $y=cy_1+c_2y_2$ for a $2^{nd}$ order linear differential equation ...
1
vote
1answer
128 views

As I can put the Neumann boundary conditions? the Crank Nicholson scheme, the Heat Equation

How i can put the Neumann BC in my code? I tested but I get error, because the arrays are not the same size $$U_t=U_{xx},\quad 0<x<1$$ $$u_x(0,t)=0$$ $$u_x(1,t)=0$$ $$u(x,0)=f(x)$$ I have my ...
1
vote
1answer
39 views

Expanding $(x+yi)^c$ to series

I need to evaluate a complex expression $f(x,y)=(x+yi)^c$, where $x,y,c\in\mathbb{R}$, in double-precision arithmetic on the GPU. It is done in a usual way, i.e., computing $\exp(c \log(x + yi))$. ...
7
votes
3answers
302 views

The Integral of Multiple Tangent Functions

I need help to find the numerical values to the precision at least $50$ digits (the closed forms if possible) for the following integrals \begin{equation} ...
3
votes
1answer
218 views

Root Finding for Functions with many maxima and minima

I wondering if anyone can provide advice on the best combination of algorithms to find the roots (or any one root) of a function which is "dense" in that it has many local maxima and minima for ...
2
votes
1answer
31 views

Value of the sum (numerical analysis)

Let $x_0, x_1, \dots, x_n$ are different real numbers and $\omega(x) = (x-x_0)(x-x_1)\dots(x-x_n)$. Then what is the value of the following sum: $$\sum_{k=0}^{n}\frac{\omega''(x_k)}{\omega'(x_k)}$$ ...
0
votes
1answer
49 views

Expression of the exponential integral $E_1$ using standard functions with real arguments

In standard numeric packages (for C++) the function $$E_1(z)=\int_1^\infty \frac{e^{-zt}}{t}dt$$ is only implemented for real arguments. For a specific calculation I need to be able to evaluate this ...
2
votes
1answer
88 views

derivation of GMRES question: why is my result for the approximate solution to $Ax=b$ always exact?

I am trying to see if I understand the GMRES method and it's result. But somewhere I get confused and I wonder if I am making a mistake. We start with a system $Ax=b$. We look for approximate ...
1
vote
0answers
112 views

Tangent vector for a curve defined by a discrete set of points

I have a curve defined by a discrete set of points (x,y). How can I approximate the tangent vector at a point for such a curve?
0
votes
1answer
59 views

Rank of the evaluation of a polynomial matrix

Given a polynomial matrix $A(t)$ of rank $r$, I would like to know at what complex evaluations of $t$ the rank decreases. Some research with google told me these values are sometimes called the zeros ...
1
vote
0answers
59 views

Studying numerical methods

Is there a book which I can self-study numerical methods needed in engineering and which proves results rigorously? I would like to learn engineering mathematics needed in every day life as well as ...
0
votes
0answers
60 views

V-shape of error function of numerical derivative vs. analytical derivative

I'm given the following function: $$f(x) = \frac{x^2}{\sin(x)}$$ and I'm supposed to derive the derivative numerically at the point $x=1$ with the following central difference quotient: ...
-2
votes
1answer
184 views

Newton-Raphson- proving a recurrence relation

$$\def\ut#1{\underline{\text{#1}}}\def\vec#1{\mathbf{#1}} \def \d{\mathrm{d}} \def \p{\partial } \def \[{\left[} \def \]{\right]} \def \({\left(} \def \){\right)} \def \n{\boldsymbol{ \nabla}} ...
3
votes
0answers
48 views

How to compute a slowly converging series to 10 decimals places of accuracy?

I'm looking at a Project Euler problem, where a harmonic series is modified such that it excludes terms where a digit appears three times consecutively in the denominator. So this series would exclude ...
1
vote
0answers
106 views

Expand $\int_{-1}^0 e^{a\cos{\theta}}J_0(b\sin{\theta})\,d\cos{\theta}$ in spherical harmonics.

I want to solve the integral (a probability density function) $$ g(\gamma)=\int_{-1}^0 e^{-f\cos{\theta}\cos{\gamma}}J_0(-if\sin{\theta}\sin{\gamma})\,d\cos{\theta} $$ numerically, everything is ...
0
votes
1answer
56 views

Numerically solve integral with a function as variable of integration

I want to use a function as variable of integration for example in evaluating the integral: $\int_0^1 e^{\cos x}f(\sin x)d\cos x$ in which $f(x)$ is an arbitrary function.
1
vote
1answer
39 views

estimation of condition number for column equilibration

I have trouble with the following problem: Let $A$ be an invertible square matrix. Let $D$ be the diagonal matrix with entries $d_j=\dfrac{||A||_1}{\sum_i |a_{i,j}|}$. Show that $||D||^{-1}_\infty ...
9
votes
3answers
2k views

Finding the all roots of a polynomial by using Newton-Raphson method.

Is there a general formulation for finding all roots of a polynomial, especially the complex ones, by using the Newton-Raphson Method?
1
vote
1answer
40 views

Solving trancendental with variable argument. $20 = ax\sin(ax)$

Approaching transcendental equations is in general new to me. My experience with numerical methods is limited, and this equation seems to require such a method. But there's a catch - it contains an ...
2
votes
0answers
46 views

Can the Lanczos algorithm converge very fast by taking a good initial guess?

Suppose I have the two lowest eigenvectors $v_1$, $v_2$ of a matrix $M$. If slightly change $M$ to $M'$. Can I use $v_1$ or $v_2$ as an initial guess for $M'$? If so, which one should be used, $v_1$ ...
2
votes
1answer
220 views

Spectral convergence of coefficients of a Fourier series

I have seen claims that if a smooth function $f(x)$ is represented by its Fourier series, $f(x)=\sum_{n=-\infty}^\infty a_ne^{i(nt)}$, then as $|n|\rightarrow\infty$, then $|a_n|\rightarrow 0$ ...
0
votes
1answer
43 views

Determining unknown coefficients of cubic splines

The problem : Find $c$ in the following cubic spline. $S \scriptstyle{1}$$(x)$ = $\large4 - \large\frac{11}{4}x + \large\frac{3}{4}x^3$, on $[0,1]$ $S \scriptstyle{2}$$(x)$ = $\large2 - ...
2
votes
2answers
113 views

Floating point arithmetic: $(x-2)^9$

This is taken from Trefethen and Bau, 13.3. Why is there a difference in accuracy between evaluating near 2 the expression $(x-2)^9$ and this expression: $$x^9 - 18x^8 + 144x^7 -672x^6 + 2016x^5 - ...
0
votes
0answers
74 views

Composite Trapezoidal Rule for $\int_0^{\pi} \sin x\, dx$

Use the Composite Trapezoidal rule to find the approximation to $\int_0^\pi \sin x\,dx$ with $m = 1, 2, 4, 8, 16.$ Progress The Comp-Trap rule states: $$\int_a^b f(x)\,dx\approx ...
1
vote
1answer
139 views

Is there any direct method for Lagrange multiplier based domain decomposed problem?

In elastic problem, we often solve K * u = f, where K is the stiffness matrix, f the external force vector and u the displacement vector. I'm trying decompose the mesh to domains, using Lagrange ...
0
votes
0answers
40 views

In interpolation, why does my choice of $x_0…x_n$ matter?

This is more of a theoretical question regarding my choice of x's for my interpolation. I'm wondering if someone can explain to me why when I choose different x's for approximating a value at a point, ...
1
vote
0answers
47 views

How can I solve this PDE?

$\dfrac{\partial \hat{Q}}{\partial t} - \dfrac{Am}{\rho} \dfrac{\partial ^3Q}{\partial t \partial z^2} = 0$ I really do not know which method could I use to solve it!
0
votes
2answers
49 views

Cubic convergence of itearative method

thank you for your time at first! It's my homework, so I don't expect answer with result, only some hint. With given iteration method $$x_{n+1} = \frac{x_n(x_n^2 + 3U)}{3x_n^2 + U} $$ show cubic ...
1
vote
0answers
50 views

Finding the inverse of an integral

I'm looking for a computational approach here, since I don't think there is a closed-form solution. I have the following: $$ s(x) = \rho + \int_{\rho}^{x} \sqrt{ 1 + (\alpha \cos t - k)^2 } \, dt $$ ...
1
vote
0answers
62 views

Numerical Stability

In my numerical analysis class we have been working on approximating functions with Maclaurin Series. I am sort of confused by the definition of what makes an algorithm numerically stable. I ...