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

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0
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1answer
40 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
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1answer
29 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.
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1answer
27 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 ...
1
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0answers
22 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
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1answer
25 views

Benchmarks for numerical methods [closed]

In computer science it is common to benchmark algorithms, programming languages, hardware, etc. For example, the Computer Language Benchmark Game benchmarks similar algorithms in various languages. ...
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0answers
27 views

Find the order of accuracy

Let $\displaystyle\cases{ y'(t)=3y-2t=f(t,y) & \cr y(0)=\frac29 }$ for $t\in[0,2]$ and the method $\large y_{n+1}=y_n+h_nf(t_n,y_n)+\frac{h_n^2}{2}f'(t_n,y_n)$ find the accuracy of the ...
0
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0answers
36 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
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1answer
26 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
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0answers
50 views

Backward Euler method with a cross-product.

I want to solve the following differential equation with the backward Euler method ...
1
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0answers
16 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
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1answer
26 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
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1answer
42 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
35 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 $\exp(c \log(x + yi))$. ...
6
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3answers
264 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
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1answer
58 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
23 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: $$\sum_{k=0}^{n}\frac{\omega''(x_k)}{\omega'(x_k)}$$ ...
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0answers
16 views

Error of linear Interpolation with intermediate points obtained from an explicit RKM

For the initial value problem $y'(t)=f(t,y(t))$ $(f\in C^\infty(\mathbb{R^2}))$ with $t\in [a,b]$ and $y(a)=y_0$ let $u_k, k=0,...,n$ be the approximation of $y(t_k)$ obtained from an explicit ...
0
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1answer
32 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
41 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 ...
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0answers
47 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
39 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
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0answers
26 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 ...
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0answers
17 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: ...
0
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1answer
26 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
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0answers
32 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 ...
0
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0answers
12 views

SOR and Conjugate Gradient Method

Let $H_n=[H_{ij}]\in\mathbb{R}^{n\times n}$ be Hilbert Matrix, define $h_{ij}=\frac{1}{i+j-1}$ and $x=\left(1\quad 1\quad\cdots\quad 1\right)\in\mathbb{R}^{n\times n}$ such that $b_n=H_nx$. Use SOR ...
1
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0answers
49 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
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0answers
34 views

Trying to solve this system with Gauss-Seidel

I'm trying to solve this system: $$ \begin{cases} {-x}+5y+3z=2\\ 7x+4y+2z=7\\ 3x-y+5z=5 \end{cases} $$ I have to use Gauss-Seidel, but no matter how I try the system does not converge. So my question ...
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0answers
17 views

Finite element solution

I need to obtain the solution of the following finite element formulation: "given $A_h^{n+1}$ and $\hat{Q}_h^{n+1}$, find $\tilde{Q}_h^{n+1} \in V_h^0$ such that: $\Bigg( ...
0
votes
1answer
30 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
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1answer
18 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
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3answers
461 views
+100

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
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1answer
29 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
23 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, $v_1$ ...
1
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1answer
45 views

Which numerical method is more accurate than Range-Kutta. [closed]

I need a numerical method is more accurate than Range_Kutta to solve the differential equation especially for third-order and more.
2
votes
1answer
41 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
25 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
99 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
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0answers
26 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
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0answers
34 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 ...
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0answers
24 views

Trapezoidal Method

Use Taylor Expansion to show that the implicit Trapezoidal Method $Y_{k+1} = Y_k+ ∆t/2 (f(t_{k+1}, Y_{k+1})+f(t_k, Y_k))$ has a local truncation error of order $∆t^2$. My understanding: The ...
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0answers
28 views

Does it really matter that we are using the Taylor polynomial and remainder?

Assuming that the quadrature rule $I_n$ integrates all polynomials of degree less than or equal to N exactly: $I_n(p)$=$I(p)$ for all p $\epsilon$ $P_N$. Using this it could be proved that for any ...
0
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0answers
31 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, ...
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0answers
17 views

How to solve this matrix equation with Hadamard product?

I am having trouble in solving $X$ in the following equation: $AX+B\otimes X=C$ where the first product is the usual matrix product and the second is a element-wise multiplication (aka Hadamard ...
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0answers
43 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!
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0answers
21 views

Determine for what scalar a function has multiple intersections

I am unfamiliar with numerical analysis, and would like some help figuring out how to find when two functions intersect on multiple points. In particular, I would like to determine for what $\lambda ...
0
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2answers
32 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
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0answers
30 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
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0answers
17 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 ...
0
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2answers
31 views

Numerical problem

The value of 1001 to the power 3?. Any trick for quick answer?