# Tagged Questions

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

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### How should c be chosen to ensure rapid convergence of $x_{ n+1}= x_ n+c(f( x_ n))$ to $\alpha$?

Consider the rootfinding problem $f(x)=0$ with root $α$, with $f´(x)≠0$. Convert it to the fixed-point problem $x=x+cf(x)≡g(x)$ with $c$ a nonzero constant. How should c be chosen to ensure rapid ...
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### Numerical Analysis Gauss-Lobatto

I am trying to find the expression of the weights and nodes for the Gauss-Lobatto quadratures with 4 nodes. I am guessing this is a sum of weights? Does anyone here have experience working with this ...
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### Gently push my non-Positive Definite matrix back into the set of Positive Definite matrices

I have a matrix $\eta$ that should be Positive Definite but it is not. Is there a numerical method to gently push my non-Positive Definite matrix back into the set of Positive Definite matrices? ...
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### A basic question about randomly generated matrix

I have read in many research papers related with iteration methods to find the generalized inverses. Where to show efficiency of the methods randomly generated matrices of higher order have been ...
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### Runge function error second factor

I'm currently learning about the Runge function. On Wikipedia, I read the following: Consider the function: $\dfrac{1}{1+25x^2}$ Runge found that if this function is interpolated at ...
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### linear interpolation error estimate for non-smooth function

Suppose I have two points $x_1,x_2$ between which I would like to have a linear interpolation $P_1$. I know the value of the function $f$ at $x_1,x_2$. The error at any point between the two will be ...
338 views

### Intervals for Newton's method

I have a function $$F(x)= \frac{x^3 - 14x^2 + 7x + 203}{(x-3)(8-x)}$$ I need to use Newton's Method to find the max interval such that a number of constraints are valid. • $3 < a < b < 8$, ...
289 views

### Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve. Sorry if my ...
499 views

### Approximating a function with a piecewise constant function

I have some distribution X of values (which I don't know exactly but I can sample many times). I also have a function $f : X \to Y$ which may be complicated. I want to approximate $f$ with a piecewise ...
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### Bounds on the size of Voronoi cells

I am working on an algorithm for which bounds on the size of voronoi cells will come in handy. Suppose that the domain $D$ is partitioned according to the Voronoi cells $D_1,\dots,D_n$ with Voronoi ...
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### Is it possible to design a strongly stable linear multistep method of order 7 which has stiff decay

I'm studying for a test and I'd like to know is it possible to design a strongly stable linear multistep method of order 7 which has stiff decay. I have no clue to verify the claim. Can anyone give me ...
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### LU Decomposition vs. QR Decomposition for similar problems

Suppose I want to solve the 2D Poisson equation with Neumann boundary conditions. The solution is non-unique up to an additive constant. I have previously asked a related question here for the 1D ...
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### Looking analytically if one formula is better than another one

I'm studying errors in functions on numerical methods. On my notes, I've written the Heron's Formula: Let $a\geq b\geq c$:$$A=\sqrt{p(p-a)(p-b)(p-c)}\ \ ,$$ where $$p=\frac{a+b+c}{2}.$$ This formula ...
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### Kink, Antikink, solving the wave equation numerically.

I've got a task of solving the wave equation with a potential $$u_{tt}-u_{xx}+V'(u)=0,$$ where $$V(u) = \frac{u^2(1-u)^2(1+2u)^2}{2}$$ on Python. I'm not exactly sure how to do this; my lecturer ...
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### Why use the logarithm of the relative error?

In my numerical analysis course, we had an assignment to use MATLAB to numerically solve the Poisson Equation $-\nabla\cdot\nabla u = 0$ in one dimension. We computed the numerical solution, plotted ...
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### Gaussian and binary probability random variables numerical reconstruction

in my probability class I was given this question dealing with MATLAB code the purpose of which is to create and re estimate the random variables Z1, Z2, which reads as follows: rng('default') ...
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### 'Stable' Ways To Invert A Matrix

So lets say that I need to invert a matrix that is generally dense and is poorly conditioned. What are some ways I can get an accurate inverse? Here are my candidates: SVD Inverse Inverse Via ...
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### Which error does one usually need to consider in numerical analysis?

When analyzing performance of a numerical method, I have considered and plotted $$\| x_n - x^* \|$$ where $x_n$ is the $n$-th iteration and $x^*$ is the true value (in our toy examples, it wasn't hard ...
17 views

### Method to Linearise PDE

I have a Monge-Ampere-type PDE I wish to solve using a finite difference method: $$(1-u_{xx})(1-u_{yy}) -u_{xy}^2 = f(x,y).$$ Is there generally a preferred method for linearising the system after ...
22 views

### A proof of the $QR$ algorithm when $A$ is symmetric and tridiagonal that doesn't involve shifting

Consider the following special case of the $QR$ algorithm: Let $A$ be symmetric and tridiagonal. Define the following sequence of matrices: $A_1 :=A$, and for $n \ge 1$ decompose $A_n=Q_n R_n$ ...
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### Proof of Runge's phenomenon for a concrete case

Let $f(x)=\frac{1}{1+25x^2}$ and range is $[-1,1]$. Given $n+1$ equidistant points $x_0 = -1,x_1,...,x_n = 1$ and their values $f(x_0),f(x_1),..,f(x_n)$, perform polynomial interpolation by the $n+1$ ...
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### Very Basic Numerical Methods Book for Freshman students

To cut a long story short; the nature of this degree (it's not a college degree) is such that numerical methods is treated shortly after Calc I (single-var) and linear algebra, but before multi-var ...
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### Iterative methods: What happens when the spectral radius of a matrix is exactly 1?

I know that an iterative method (I'm using Jacobi and Gauss-Seidel in this case) will converge iff the spectral radius (max absolute value of eigenvalues) of its iterative matrix is strictly less than ...
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### How to find values of x where $a_i x$ are nearly integers? $a_i \in \Bbb R$

I have a set $\{a_i \in \Bbb R | \ i <=7 \}$, and I'm looking for a way to find values of $x$ where given $\epsilon > 0$, $$\forall i \ \exists n_i \in \Bbb{Z} \ \ |a_i x - n_i| < \epsilon$$ ...
45 views

### make the Exponential interpolation vanish and show it has a unique solution

I have several questions concerning different parts of the question: a) Is it sufficient to show that ${1, e^x,...,e^{nx}}$ are linearly independent over the vector space of differentiable ...
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### Deriving a New Iteration Method by Solving a Quadratic Equation

My Question: Derive a new iteration method for solving $f(x)=0$ by solving the quadratic equation $$f(x_k)+f'(x_k)(x-x_k)+\frac{1}{2}f''(x_k)(x-x_k)^2=0$$ Complete your algorithm by specifying ...
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### Gauss-Green cubature in 2d

Hello friends of maths, I've given an arbitrary polygonal cross section (in cartesian coordinates $y$ and $z$). On this cross section, there acts an arbitrary stress-field $\sigma = f(y,z)$ as ...
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### Is there a faster algorithm than $O(n^2)$ for calculating “cofactors” $C_k = \prod\limits_{j\neq k}(c_k - c_j)$?

Is there a faster algorithm than $O(n^2)$ for calculating "cofactors" $C_k = \prod\limits_{j\neq k}(c_k - c_j)$ ? (presumably $O(n \log_2 n)$ if one exists) In other words, if I have factors ...
33 views

### Computing Cholesky Factorisation by Hand

It is a common exam problem to compute the Cholesky factorisation of a small (typically 4x4) matrix. I know that this can be done by first finding the matrix $U$ in the $LU$-decomposition (e.g. by the ...
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### Why does this algorithm converge?

Consider the following problem. Let $p_1, \dots, p_n \in (0,1)$ such that $\sum p_i = 1$. Let $m > 0$ such that $$q_i := p_i + m \frac{p_i \log(p_i)}{\sum p_k \log(p_k)} < 1$$ Suppose ...
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### Fitting nonlinear differential equations to correspond a predefined solution

When modeling temporal dynamics of a biological process I stumbled upon a set of differential equations having the following matrix form: ...
I want to know how to implement the (nonhomogeneous) initial boundary value problem for a heat equation; $$u_{xx}=u_t ~~~x\in (-1,1),~t\in(0,1)$$ $$u(0,x)=u_0(x)$$ $$u(t,-1)=f(t), ~u_x(t,1)=0$$ Many ...