Questions tagged [numerical-linear-algebra]
Questions on the various algorithms used in linear algebra computations (matrix computations).
3,588
questions
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numerical integration quadrature rule Radau formula
Consider a numerical integration rule of the form $\int_{0}^{1} f(t)dt \approx af(0)+bf(c)$
a)Find a,b,c such that this quadrature rule has highest order of precision.
b)The quadrature rule above has ...
- 359
0
votes
0
answers
50
views
Power method and numerical analysis
Let $A\in \mathbb{R}^{{n}*{n}}$ be a symmetric matrix with eigenvalues satifying $\lambda_{1}>\lambda_{2}\ge\ldots \ge\lambda_{n-1}>\lambda_{n}$ and corresponding eigenvectors $x_{1},x_{2},\...
- 359
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votes
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What is the "lifted system of linear equations"?
I am reading this blog post, where it says
Here’s another variant of the same idea. Suppose we want to solve the linear system of equation $(D + uv^\top)x = b$ where $D$ is a diagonal matrix. Then we ...
- 836
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Is this generalization of determinant for a higher-order tensor a standard object?
The determinant of an $n$ by $n$ matrix $a$ can be defined as
$$ \mathrm{det}(a)= \sum_{\sigma} \mathrm{sgn}(\sigma) a_{1,\sigma(1)} a_{2,\sigma(2)} \dots a_{n,\sigma(n)}$$
where $\sigma$ is a ...
- 31
2
votes
0
answers
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need a Jacobian of an orthonormal basis for the column space of a matrix $\mathbf{X}$
I require to find a linear decomposition of any tall matrix $\mathbf{X}$ $\in \mathbb{R}^{M \times N}$, where $M \geq N$, into some uniquely defined decomposition $\mathbf{X}$ $=$ $\mathbf{A}$ $\...
- 41
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0
answers
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+50
Calculation of special subsets in high-dimensional binary matrices
I need to solve a rather specific problem related to binary matrices. The task is to count the number of specific "combinations", where "combination" means the following:
this is ...
- 337
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votes
0
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Show integral of element-wise square of matrix exponential + I is invertible
Question:
Let $\mathbf{Q}$ be a matrix whose eigenvalues have positive real parts, and let $\mathbf{B}$ be the matrix with entries
\begin{equation}
B_{ij} = \int_0^\infty ((e^{-Qt})_{ij})^2 dt.
\end{...
1
vote
0
answers
51
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Gerschgorin Circle Theorem and ordering of eigenvalues
Suppose $D\in \mathbb{R}^{n*n}$ is diagonal and $E\in \mathbb{R^{n*n}}$ be any matrix. Use Gerschgorin circle theorem to show that if $||E||_{\infty}<min_{i\neq j}|\frac{d_{ii}-d_{jj}}{2}|$ then ...
- 359
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0
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Finding the least squares solution of a linear system based on a QR factorization
One method of finding the least squares solution of the following "augmented system"
$$
\left[
\begin{matrix}
I & A \\
A^T & O
\end{matrix}
\right]
\left[
\begin{matrix}
r \\
x
\end{...
- 1,261
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The minimal eigenvalue of a symmetric matrix
Let $A$ be a symmetric matrix, and I want to show that for: $\epsilon>\lambda_{min}(A)$:
$$A+\epsilon I>0$$
where $\lambda_{min}(A)$ is the minimum eigenvalue of $A$.
My reasoning is as follows:
...
- 4,253
2
votes
0
answers
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views
How exactly is Sherman-Morrison-Woodbury formula used in Kalman Filter
I have seen many places mentioned that Sherman-Morrison-Woodbury formula can be used in Kalman Filter to speed up the matrix inverse. Even the linked Wikipedia page mentioned that.
I am not exactly ...
- 1,634
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0
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Reformulate an algorithm as a sequence of standard matrix operations
Consider the following code snippet
...
- 1
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7-point North-East ILU decomposition
I am using a 7-point difference operator (actually a 5-point difference operator with $b, f = 0$) to discretize the 2D model anisotropic problem. $ L_h $ in stencil notation is given. $ L_h = \begin{...
- 276
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Symmetry preserving quadratic forms
Let $A$ and $S$ be conformable matrices with $S$ symmetric. One frequent annoyance is that the product:
$$
\tilde{S} = ASA^\intercal
$$
has antisymmetric components due to rounding. Is there a simple ...
0
votes
1
answer
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Issue in numerical integration of $\frac{1}{4\pi^{3/2}i}\int_Ce^{-\frac{z^3}{12}-tz}z^{-1/2}dz$
I am trying to numerically integrate the integral representation of $\operatorname{Ai}^2(x)$. The representation is
$$\operatorname{Ai}^2(t)=\frac{1}{4\pi^{3/2}i}\int_Ce^{-\frac{z^3}{12}-tz}z^{-1/2}dz....
- 2,971