{numerical-linear-algebra} questions involving algorithms for linear algebra computations.

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22 views

### departure time/overlap algorithm [on hold]

i'm looking for "departure time/overlap algorithm" or any other idea.
Suppose you have n trains and each one has a performance profile(how much electricity they need at the current time while driving ...

**2**

votes

**0**answers

29 views

### QR-Decomposition of matrix valued function

Suppose I have a matrix valued function
$$
F:\mathbb{R}\rightarrow\mathbb{R}^{m\times n},\qquad F(x)=\tilde Q\tilde R+xu_1v_1^T+xu_2v_2^T
$$
where $\tilde Q\in\mathbb{R}^{m\times m}$ is orthogonal, ...

**3**

votes

**1**answer

41 views

### submatrix of a given size with maximum frobenius norm

Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...

**0**

votes

**1**answer

58 views

### Convergence for symmetric, positive semi-definite operator

Assume $u$ is a vector in the Euclidean space $\mathbb{R}^N$, $\|u\|=\sqrt{\langle u, u\rangle}$, where $\langle u, v\rangle = \sum_{i=1}^N u_i v_i$.
I have that $\|u^{k+1}-u\|\leq \|I - c ...

**0**

votes

**1**answer

84 views

### Perturbation of Cholesky decomposition for matrix inversion

I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$
where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...

**10**

votes

**2**answers

284 views

### Determinant and eigenvalues of a specific matrix

This came up in a conversation with an engineer friend of mine.
Let $c>0$ be a constant. Let $A_{ij}$ be an $n$ by $n$ matrix with entries
$$
A_{ij} = e^{-c(i-j)^2}.
$$
Is there a name for this ...

**1**

vote

**1**answer

53 views

### Augmenting orthonormal system into complete orthonormal system in a numerically stable way

Let us suppose we have a, say, 10 dimensional real space with 3 orthogonal unit vectors given. How do I complete this orthonormal system with 7 additional vectors into a complete ONS in a way that is ...

**1**

vote

**0**answers

42 views

### How to solve a divergent linear system using iterative methods?

I have a matrix A which is symmetric and non-diagonal dominant. I tried to use Jacobi/Gauss-Seidel/SOR to solve it but it diverges. Is there any mechanism to condition the matrix for convergence using ...

**2**

votes

**2**answers

133 views

### Probability for a random positive-semidefinite matrix to not be positive-definite?

If I take $A^TA$, where $A$ is a full-rank random matrix (let's say with Gaussian-distributed independent entries), can I expect it to be positive-definite? It will be positive semi-definite ...

**5**

votes

**1**answer

156 views

### For a set of matrices $S$, find $X$ such that the elements of $SX$ commute

Let $S := \{A_0, A_1, \dots, A_d\}$, where $A_k \in \mathbb{C}^{n \times n}$, be a set of (generally noncommuting) matrices. I am interested in finding a nonsingular $X \in \mathbb{C}^{n \times n}$ ...

**2**

votes

**1**answer

78 views

### Updating $LU$ decomposition after adding a sparse matrix

How many elements of $LU$ decomposition of a symmetric matrix change after adding a sparse symmetric matrix? Is it more efficient to recompute $LU$ decomposition after adding a sparse matrix comparing ...

**1**

vote

**2**answers

175 views

### Eigenvalue computation using inverse iteration

I have a positive definite matrix $A$. I need to compute the max eigen value of $A$ using inverse iteration. The problem is that there are duplicate maximum eigen values and so inverse iteration ...

**2**

votes

**0**answers

103 views

### What would be a better method for numerical diagonalization of a certain Vandermonde-like matrix?

For the fractional iteration of the $\exp()$-function Hellmuth Kneser had 1942 proposed an analytic solution valid on the real numbers; there is a technical implementation for Pari/GP of this method ...

**1**

vote

**0**answers

99 views

### The rationale of QR algorithm for computing eigenvectors

For a symmetric matrix $A$, the rationale for the success of applying QR to compute the spectral decomposition of $A = UDU^T$ is, for large $k$, the QR factorization of $A^k = Q_kR_k$ obeys,
...

**6**

votes

**1**answer

228 views

### efficient rank-two updates of an eigenvalue decomposition (or more genearlly SVD)

Let $A$ be a symmetric matrix with eigenvalue decomposition $UDU^T$. Golub, et al. and Bunch, et al. have shown that given such an $A$, the eigenvalue decomposition of $A+\rho xx^t$ may be computed ...

**4**

votes

**0**answers

94 views

### Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D ...

**0**

votes

**1**answer

99 views

### QR alogrithm for eigenvalue problem [closed]

Considering pure QR algorithm (without shifts and preliminary tridiagonal reduction) are there sufficient conditions for algorithm to converge to quasi-diagonal form?
For the the following matrix
$$
...

**5**

votes

**0**answers

85 views

### Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e.
...

**3**

votes

**2**answers

615 views

### Sparse approximation of the inverse of a sparse matrix

Is it possible to approximate an inverse of a sparse matrix with a sparse matrix?
The problem comes up in numerical non-linear quasi-Newton optimization: given a sparse Hessian a good starting point ...

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vote

**2**answers

135 views

### Alternative to Choleski Decomposition for Correlation Matrix

Let $\Sigma$ be a correlation matrix, ie. symmetric. The Choleski decompositon gives upper triangular $A$ such that $A^TA = \Sigma$. Instead of upper triangularity, we are looking for $A$ that is not ...

**6**

votes

**1**answer

233 views

### The singular values of the Hilbert matrix

The $n\times n$ Hilbert matrix $H$ is defined as
$H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$
What is known about the singular values $\sigma_1\geq\ldots\geq \sigma_n$ of $H$?
For example, ...

**4**

votes

**1**answer

54 views

### Algorithm to quickly compute the individual inverses of a linear sequence of matrices

Fix $n \times n$ real symmetric positive definite matrices $A$ and $B$. Fix vectors $x$ and $y$ in $\mathbf{R}^n$. I want to compute the following bilinear products quickly: $\{x^T (A+mB)^{-1} ...

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votes

**0**answers

99 views

### Error bound on matrix vector multiplication

I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate.
Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i,j}$, $2a_{i,j}$}. ...

**3**

votes

**1**answer

66 views

### Conjugate gradient algorithm where first search direction is not equal to residual

In usual formulation of conjugate gradient algorithm initial search direction is taken to be the residual (so residual and search direction spans Krylov subspace). However, in cases where inexact ...

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votes

**0**answers

115 views

### Problems where Conjugate gradient works much better than GMRES

I am interested in cases where Conjugate gradient works much better than GMRES method.
In general, CG is preferable choice in many cases of SPD because it requires less storage and theoretical bound ...

**7**

votes

**1**answer

130 views

### Who first observed that Conjugate Gradient for Symmetric Positive Definite linear systems is a Krylov method?

Conjugate gradient was originally presented in the 50's before the modern understanding of Krylov subspaces (and the resulting iterative methods) was fully realized. As such, the method was derived ...

**10**

votes

**0**answers

132 views

### What are the difficulties in proving almost-everywhere stability of Gaussian elimination?

It is well known that Gaussian elimination without pivoting is numerically unstable, and in practice Gaussian elimination is done with row pivoting (partial pivoting). A theorem of Wilkinson states ...

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votes

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76 views

### standard practice for large dense truncated svd computations?

What are the standard methods of computing the rank-k truncated SVD of large dense matrices? My literature search yields results only for large sparse matrices.
I assume for k small that you use a ...

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vote

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43 views

### Most efficient algorithm for computing norm of the residual for the least squares problem in the rank deficient case

I have a large $m\times n$ data matrix $A$, $m>n$, and response $m$-vector $b$. I need to calculate $E = ||Ax-b||_2$ as quickly as possible, where $x$ is the least squares solution. I don't need ...

**4**

votes

**1**answer

298 views

### Norm of inverse confluent Vandermonde matrix

Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $l_1+l_2+\dots+l_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as
$$V=
\begin{bmatrix}
...

**2**

votes

**0**answers

132 views

### Checking for error in conjugate gradient algorithm

What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality of search direction ...

**0**

votes

**1**answer

123 views

### Nonlinear matrix equation 2

Solve the following nonlinear equations for $v$ and $w$
$Avv^TAw+Bvv^TBw=\lambda_1v+\lambda_2w$
$Aww^TAv+Bww^TBv=\lambda_1w+\lambda_2v$
$v^Tw=w^Tv=0$
$v^Tv=w^Tw=1$
where $\lambda_1, \lambda_2, ...

**3**

votes

**1**answer

422 views

### Nonlinear matrix equation

Solve the following nonlinear equations for $v$ and $w$
$Avv^TAw=\lambda_1v+\lambda_2w$
$Aww^TAv=\lambda_1w+\lambda_2v$
$v^Tw=w^Tv=0$
$v^Tv=w^Tw=1$
where $\lambda_1, \lambda_2, \lambda_3$ are ...

**2**

votes

**0**answers

216 views

### Quantifying the failure of the Cholesky factorization test for indefinite matrices

The Cholesky factorization is the classic test to check if a matrix is positive definite. In infinite precision it is also an exact test: A matrix has a Cholesky factorization iff it is positive ...

**7**

votes

**2**answers

916 views

### Finding the smallest eigenvalues of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $M$. $M$ is a Laplacian matrix, and it has the following structure: ...

**0**

votes

**1**answer

158 views

### Ease of calculation of norm

I have SPD matrix A and two vectors z and b.
Is there exist a norm where I can calculate $||A^{1/2}b-z||$ without having to calculate $A^{1/2}b$ explicitly ?

**5**

votes

**0**answers

122 views

### reference for perturbation of projection result

Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...

**0**

votes

**0**answers

116 views

### Finding peaks and determining noise

Hello ,
Im having one matrix which is product of two FFT transforms of one fits image ( astronomical image ). In that matrix you could find 3 peaks. One largest in center, and two around central ...

**2**

votes

**2**answers

136 views

### Inflate a simplex, change rows to make the rank n

I have a simplex, n + 1 points in $\mathbb{R}^n$,
which may have rank $r < n$.
Is there a cheap way of "inflating" it to rank $n$,
changing a few, all but $r$, of the points ?
The points are ...

**2**

votes

**1**answer

222 views

### A question for solutions of perturbed linear systems

Consider a linear system
$$Ax=b\qquad (*)$$
and a sequence of perturbed linear systems $$(A+\delta A_n)x=b+\delta b_n. \qquad (n)$$
Suppose that all the linear systems are consistent (i.e., ...

**2**

votes

**0**answers

106 views

### Computing the norm of the columns of an implicitly defined matrix

I have an $n \times n$ matrix $M = \Sigma W$ where $\Sigma$ is diagonal and $W$ orthogonal. $W$ is implicitly defined, i.e. I can only perform matrix-vector products (but I also have access to $W^T$).
...

**3**

votes

**2**answers

1k views

### Interesting relationships between Cholesky decomposition and diagonalization

Let $\Sigma$ be a hermitian positive definite matrix and $L$ be it's Cholesky decomposition so that $LL^\ast=\Sigma$. Furthermore, let's diagonalize $\Sigma$ as $\Sigma = P\Lambda P^\ast$. $\Lambda$ ...

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vote

**1**answer

1k views

### Low rank Matrix factorization

Hello,
I've a SPD matrix A; which needs to be factorized as ${A=SS^{T}}$. But, using Cholesky for this purpose is prohibitive in terms of computational cost. Moreover, matrix is Dense and has a slow ...

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votes

**0**answers

160 views

### Could SVD be used to optimize the partial inner-products?

Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with
$m-$dimensional coordinates in ...

**7**

votes

**2**answers

330 views

### polynomials with minimal $L_\infty$ norm on multiple disjoint intervals

It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint ...

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votes

**0**answers

327 views

### How many iterations are required for the Lanczos algorithm to converge?

I am trying to find the n smallest eigenvalues and eigenvectors of a NxN SPD matrix using Lanczos method. What is the number of iterations usually required? I mean, does it scale as $O(N)$ or ...

**7**

votes

**0**answers

109 views

### How do I find elements of an algebra which generate an algebra contained in a fixed subspace?

Suppose $V$ is a linear subspace of a finite dimensional $C^*$-algebra $A$. (Feel free to assume $A$ is a multi-matrix algebra over $\mathbb C$.)
I would like to find $x \in V$ such that $\mathbb C ...

**5**

votes

**2**answers

616 views

### Factorizing a block symmetric matrix

Let $X,Y\in\mathbb{R}^{n\times n}$ be symmetric matrices. You may assume that $X$ is positive semidefinite and $Y$ negative semidefinite, if needed, but not that they are invertible.
I would like to ...

**4**

votes

**2**answers

810 views

### Is there some algorithms for solving non-linear matrix equations?

Is there some algorithms for solving non-linear matrix equations on field $\mathbb{C}$?
Especially, solving polynomial nonlinear matrix equations.
For instance, let $X$ be some matrix satisfying
...