# Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

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### Lanczos convergence for symmetric matrix with eigenvalues $1,2,\cdots,2,100$

Suppose that $A$ is symmetric with eigenvalues $1,2,\cdots,2,100 \in \mathbb{R}^{100x100}$ and $b \in \mathbb{R}^{100}$ obtained by normalizing a standard normal random vector. Show that 1 and 100 are ...
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### Derive Lanczos algorithm by imitating the derivation of Arnoldi iteration algorithm.

If A is Hermitian, then everything above simplifies (e.g., Hessenberg matrices turn into tridiagonal), and we get what is know in the literature separately as Lanczos iteration. My attempt:- ...
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1 vote
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### How to transform an integro-differential equation into weak form for FEM

I have the following 2D boundary-value problem which I would like to solve numerically using FEM software: a(x,y)\nabla^2 u(x,y) + \int\int K(x,y,x',y')u(x',y') dx' dy' = f(x,y), \end{...
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### Recommended iterative numeric solver for generalized eigenvalue problem?

I have a generalized eigenvalue problem of the form $Av=\lambda Bv$, with the following conditions: $A$ is symmetric, but not sure if it is positive-definite. $B$ is diagonal with positive entries $A$...
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### Insights for Outcome Function Involving Multiple Interdependent Variables

I am working on a model involving multiple interdependent variables and systems of equations, and I am trying to gain insights into the behavior and properties of a specific outcome function. Despite ...
115 views

### Trapezoidal Rule on Infinitely Differentiable Periodic Functions

If I understand it correctly, the Euler-Maclaurin summation formula states that for a periodic and infinitely differentiable function, the error of the trapezoidal rule of the numerical integration of ...
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### Solve the matrix equation $A = - B A B^T + C$ without matrix inversion or vectorization
Let $B$ and $C$ be two $n \times n$ matrices, which may or may not be nicely behaved. I would like to solve for the matrix $A$ in the following equation: $$A = -B A B^T + C$$ This equation is linear ...