# Tagged Questions

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### Stability of matrix equations in MATLAB

Is there a method to check for unconditional stability or positive-definiteness of large matrices in MATLAB? For example, I know that matrices with property M (positive main diagonal elements and ...
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### Using entropy functions for increasing numerical stability

Regarding the numerical stabilization of two-dimensional advection equation, \dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z} - \...
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### Integrating a dynamical system until an algebraic condition is satisfied

I have a model given by a system of differential equations $$\frac{dy}{dt}=f(y)$$ with $y=(y_1,y_2,y_3)$ and $f:\mathbb{R}^3 \to \mathbb{R}^3$. The system works as follows : integrate the ode's ...
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### How to choose a good distribution for visualizing phase changes in the nature of the roots of a quadratic equation

I'm not sure if this SE site is the best one for this question, so let me know where it should be moved to if you think it doesn't belong here. After learning about the quadratic formula, I'm ...
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I'm working on some problems that ultimately boil down into a simple assembly of an overdetermined system of equations, $Ax=b$, where $A$ is $m \times n$ for $m \gg n$. I'm leveraging Armadillo's C++ ...
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### Single Precision a x plus y (SAXPY) terminology

I've been reading books which refers to vector update operations of the form: y := y + ax, where y and x are vector variables and a is a scalar as SAXPY. I understand ax plus y part, but why "single ...
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### Why the greatest exponent that can be represented in single-precision floating-pointer numbers is 127 (and not 128)?

I was told by a class mate that the smallest exponent that we can represent by a single-precision floating-point number (which uses 8 bits for the exponent) is $-126$ and the greatest is $127$. I ...
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### t digits are used to represent the mantissa in floating-point system, but rounding unit is calculated for doubles with 53 bits

I'm reading the book "A First Course in Numerical Methods" by U. Ascher and C. Greif, and in the 2nd chapter it's written that ... we associate $x$ a floating point representation $fl(x)$ of the ...
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### Conjugate Gradient, initial direction set to initial residual

In the (iterative) Conjugate Gradient (CG) algorithm: https://en.wikipedia.org/wiki/Conjugate_gradient_method The initial search direction $p_{0}$ is set to the initial residual $r_{0}$. But I can't ...
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### Stability in Discretization of 1D Stationary Boltzmann equation

I want to discretize and numerically solve the following PDE: $$v(k)\dfrac{\partial f}{\partial x} + E(x)\dfrac{\partial f}{\partial k} = S\{f\}$$ using finite volume (box ...
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### Stability in discretization of a PDE

Suppose I want to numerically solve for $f(x,k)$ the one-dimensional Boltzmann equation for electrons in steady-state condition, given as: \left( \dfrac{\hbar k}{m} \right) \dfrac{\...
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### Symplectic integration of PDE

I consider ordinary wave equation $$u_{tt} - u_{xx} = 0$$ with initial conditions $$u(x, 0) = \exp (-2 x^2) \\ u_t(x, 0) = 0$$ To solve this problem I approximate $u_{xx}$ with 4-th order ...
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### Find eigenvalues and eigenvectors using lanczos method [duplicate]

Can anyone help me out for finding eigenvalues and eigen vectors using Lanczos method?
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### Known issues with eigenvalue numerics?

Are there any known issues (such as precision issues) with $\mathsf{MATLAB}$ eig and charpoly functions for large enough $\{-1,0,+1\}$ matrices? Even if I change $1$ or $2$ entries between matrices ...
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### Can I make this numerical integration continuously differentiable?

Suppose I have the discrete values $f(x_i)$ for every discrete value $x_i$ greater than some $\varepsilon$, and I want to numerically calculate the following integral: n = \int_\...
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### Does artifical dissipation term makes scheme inconsistent?

Central schemes like JST uses artificial dissipation for the stabilization. This modification is an artificial one. Does this additional term makes system inconsistent? Can we expect this term to be ...
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### Numerical integration using RKF7(8) - different results

For my thesis, I look in trajectories of vehicles through an atmosphere at very high velocities. I have a set of equations of motion, which I propagate using a Runge-Kutta-Fehlberg (RKF) 7(8) ...
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### Integral in log-log space

I'm working with functions which, in general, are much smoother and better behaved in log-log space --- so that's where I perform interpolation/extrapolation, etc, and that works very well. Is there ...
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### Unexpected results of MATLAB's ode45

Whilst working with MATLAB recently I encountered something odd that I cannot explain. I was using the ode45 solver to solve a system of two coupled second order ODEs. I wasn't convinced about the ...
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### Accuracy of numerical methods in finding eigenvalues

I have a problem with assesing the accuracy of my numerical calculation. I have a 2nd order ODE. It is an eigenvalue problem of the form: $$y'' + ay' + \lambda^2y = 0,$$ and the boundary condiations ...
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### Relevance of fixed-point and arbitrary precision computations

I see very few non-floating point computing libraries/packages around. Given the various inaccuracies of floating point representation, the question arises why there aren't at least some fields where ...
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### Imposing invertibility on a Matrix

I have a symmetric positive semidefinite covariance matrix $A$, which is approximately computed as the output of a quadratic regression. I then need to invert $A$, but often it is close to singular. I'...