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

**4**

votes

**2**answers

86 views

### Norm of triangular truncation operator on rank deficient matrices

Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$
It is known that for arbitrary $A_{n\times n}$
$$\|T\circ ...

**4**

votes

**1**answer

104 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 ...

**1**

vote

**1**answer

139 views

### Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation:
$X=c \cdot AXA' - diag(c \cdot AXA')+ I$,
where
(1) $A \in R^{n \times n}$ is a given matrix whose element ...

**10**

votes

**0**answers

143 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 ...

**7**

votes

**0**answers

123 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

**0**answers

110 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

**0**answers

87 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.
...

**5**

votes

**0**answers

125 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
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...

**4**

votes

**0**answers

64 views

### Efficiently factorize a KKT system with block diagonal upper corner

I have a system resulting from a quadratic energy minimization with linear equality constraints enforced with Lagrange multipliers which has the form:
\begin{equation}
A =
\left[\begin{array}{c|c}
...

**4**

votes

**0**answers

104 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 ...

**3**

votes

**0**answers

112 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 $W^T$).
...

**2**

votes

**0**answers

120 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 ...

**2**

votes

**0**answers

113 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,$-a_{i,j}$, $2a_{i,j}$}. ...

**2**

votes

**0**answers

45 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 ...

**2**

votes

**0**answers

136 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 ...

**2**

votes

**0**answers

257 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 ...

**2**

votes

**0**answers

168 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 ...

**2**

votes

**0**answers

373 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 ...

**1**

vote

**0**answers

54 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 ...

**1**

vote

**0**answers

116 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,
...

**0**

votes

**0**answers

119 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 ...