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65 views

The fast, and The Backward-Stable (left) $3\times 3$ matrix inverse

I need to compute a lot of $3\times3$ matrix inverses (for Newton iteration polar decomposition), with very small number of degenerate cases ($<0.1\%$). Explicit inverse (via matrix minors divided ...
88 views

FLOPS of a linear system

I have two questions that I want to ask. Consider the following system: $$BM = A$$ i) $B$ is a $n$ by $n$ tridiagonal matrix and $A$ is a diagonal matrix. What is the leading order computation cost ...
64 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula $$(A+uv^T)^{-1} = A^{-1} - \frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u}$$ results in small errors in relation to the standard matrix inverse operation after each application, ...
88 views

solve linear system of equation of a large sparse symetric positive definite matrix

I want to invert large matrices ($10^4x10^4$ to $10^6x10^6$) but sparce (less than 100 non-zero entries per line) on clusters with 16 to 48 processors per node. I'm looking for an efficient method to ...
166 views

$AX=B$: How to solve for $X$ if elements of matrix A are matrices

Objective: I am trying to solve for $C$ in 2D space (x,y) and time from following PDE. $$\text{PDE: }\frac{\partial C}{\partial t} + \nabla\left(v.C - D\nabla{C} \right)= \alpha.C$$ Method: I ...
72 views

Solving large system of equations, is linear programming best option?

I have a problem where I am trying to solve many systems of equations, that have very few variables per equation, but a lot of equations. For example potentially 10 variables max in a single ...
269 views

MATLAB: code for restarted gmres

I have a question about Matlab and restarted gmres. I would like to use gmres.m provided here. This code seems to be a popular for the scientific computation ...
95 views

Avoid arithmetic overflow in matrix multiplication

I am solving the following matrix equation for $\mathbf{x}$: $$(J^{\mathbf{T}}J)\mathbf{x}=J^{\mathbf{T}}\mathbf{r}$$ $J$ is $m\times n$ matrix $\mathbf{x}$ is vector of size $n$ $\mathbf{r}$ is ...
85 views

Solve eigenvalue problem using finite differences without vectorization

I am interested in solving the problem $-A u = \lambda u$ on a finite differences grid on a square. In my case, the operator $A$ is of the type $-\Delta + \mu I$, where $\mu$ is there to impose some ...
175 views

Least-squares for a diagonal matrix

This is a follow-up to a different question I asked with more detail. For $v\in\mathbb{R}^n$, denote $D_v\in\mathbb{R}^n$ as the diagonal matrix with elements in $v$. Given a "tall" matrix ...
245 views

SVD and HITS Algorithm Power Iterations

As we know, computing the authority (or hub) score of HITS ranking method, means to use the following matrix equation: $$\textbf{a}^{k}=A^T A\textbf{a}^{(k-1)}$$ and apply the power iteration ...
146 views

How to find out if it is possible to contruct a binary matrix with given row and column sums

How to find out if it is possible to contruct a binary matrix with given row and column sums. Input : The first row of input contains two numbers 1≤m,n≤1000, the number of rows and columns of the ...
1k views

Best PARALLEL numerical solver of first order differential equation

I have a system of 256 differential equations that I want to solve numerically. The system represents the Liouville equation, which is a first order, linear differential equation with complex numbers. ...
248 views

PETSc: Blocking matrices using MatCreateSeqBAIJ and MatSetValuesBlocked

I am a little confused with PETSc's documentation for MatSetValuesBlocked. The code below works fine for matrices when I choose small block sizes, but I get errors ...
681 views

What is the best solver for solving a large sparse indefinite system

What's the best solver that can solve a large sparse but indefinite matrix?
69 views

On solution of a class of discrete-time Lyapunov equation for systems with multiplicaitve noise

Let's consider the following equation $$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$ where $p$ is an positive integer and $C$ is a known positive semidefinite matrix. If we augment $F=[F_{1}...F_{p}]$ ...