Questions tagged [inverse]
The inverse tag has no usage guidance.
1
vote
1answer
60 views
Computing the Inverse of a matrix, using the Cholesky decomposition
I have to compute $CA^{-1}B$ and $CA^{-1}x$, where $A,B,C$ are conformable matrices and $x$ is a vector.
I've read that the a very computationally stable way to compute these inverses is by computing ...
2
votes
1answer
80 views
Inverting small matrices: canned factorization versus explicit formula
I am interested in solving a large number of small linear systems of equations, $Ax=b$, with $A$ either $2\times2$ or $3\times3$. Assuming none of these systems are actually singular, is there ...
4
votes
1answer
145 views
Do I really need to invert this matrix
I need to calculate a matrix $A$ (at least some elements of it, see below) as defined by the following equation
$$ A=B(\mathbb{1}-B)^{-1} $$
where B is a square matrix of dimension $N$ and $\mathbb{...
6
votes
1answer
220 views
How to directly compute the inverse of an ill-conditioned dense matrix
I know that it is generally a bad idea to compute the inverse matrix directly. However, if it is necessary to compute the inverse of an ill-conditioned invertible dense matrix, then what can I try?
...
3
votes
1answer
187 views
What is the fastest method to invert millions of matrices?
My project involves large simulation and estimation. For each simulation I need to solve 600,000 systems of nonlinear equations. Currently I am using Newton's method to find the solutions. That ...
1
vote
0answers
53 views
How to fast estimate derivates for calculating quantiles
I would like to know if there exists a package or how one can fast calculate the quantiles of a function within python, where the inverse of the function for calculating the quantile depends on the ...
2
votes
2answers
166 views
Inverting big symmetric and singular matrices
In this post I found a very similar probem to the one I have, but not a satisfactory answer for my purposes.
I have a set of matrices $C_\ell$. They are exactly symmetric by construction. ...
0
votes
1answer
77 views
Sparse matrix inverse with reduced bandwidth
I have a sparse symmetric matrix of dimension 1393x1393 (8308 no zero elements), with bandwidth 1380. By Cuthill–McKee algorithm, I could achieve a new matrix with ...
1
vote
0answers
75 views
Problem in analyzing the program of Gauss Jordan Inverse problem
I had to code a program which calculates Inverse of a matrix by Gauss-Jordan Inverse method , I was trying to analyse the program and then code it myself.
the link
http://hullooo.blogspot.in/2011/...
15
votes
2answers
516 views
Practical example of why it is not good to invert a matrix
I am aware about that inverting a matrix to solve a linear system is not a good idea, since it is not as accurate and as efficient as directly solving the system or using LU, Cholesky or QR ...
1
vote
0answers
39 views
Numerically inverting an exponentially growing function (defined by Chebyshev polynomials)
Assume a function $M(t)$ strictly increasing, essentially growing exponentially, and asymptoptically growing at a known rate $\bar{g}$, i.e. $\lim_{t\to\infty}M'(t)/M(t) = \bar{g}$
In a set of awful ...
10
votes
1answer
217 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 ...
0
votes
1answer
150 views
Fast computation of square root inverse of matrix, matrix being determined from Ax=b form
I have an equation of the form $J^Te=f$, where $e$ and $f$ are known vectors and $J$ is an unknown matrix.
How can I efficiently compute $J^T(JJ^T)^{-1/2}e$ ?
My motivation to address this problem ...
1
vote
1answer
188 views
Calculate inverse of dense matrix with entries of very different magnitude
I need to calculate the inverse of a dense matrix, with some elements taking values as high as 1e9 and some around 1e2. What would be the best method to do it?
Note:
I am more concerned about the ...
11
votes
2answers
5k views
Complexity of matrix inversion in numpy
I am solving differential equations that require to invert dense square matrices. This matrix inversion consumes the most of my computation time, so I was wondering if I am using the fastest algorithm ...
5
votes
1answer
101 views
Finding the matrix inverse given a solver for the matrix equation $Ax=b$
So I'm given a solver that can solve for $x$ in the matrix equation $\underset{=}{A} \underline{x} = \underline{b}$ where $b$ can be anything we specify. (NB: A is an NxN matrix).
I now want to find ...
1
vote
2answers
107 views
Inverse of “diagonally not dominant matrix”
I want to frame a higher order Central difference scheme of about $20^{th}$ order for first derivative. I'm using $20^{th}$ order because I need one scheme with good modified wave number. To find the ...
4
votes
1answer
327 views
Obtaining column vectors of pseudo-inverse of a matrix
I need to compute the pseudo-inverse of a very large rectangular dense matrix without any special structure or properties. I run out of memory/computing power and have no access to a large parallel ...
2
votes
0answers
47 views
Inverted value is not consistent with expectation
We have a group of observations
$$y = f(x_1, x_2, x_3) \enspace .$$
We have also a forward model $y = f(x_1, x_2)$. The forward model does not include $x_3$ because $x_3$ might include dozens of ...
5
votes
0answers
309 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, ...
3
votes
3answers
474 views
Exact analytical matrix inversion of sparse 100x100 matrices in C++
I need to invert a matrix. Of course, I'm not the first person in this situation, and I know that there's a wealth of powerful libraries out there, of which I only know a couple.
That being said, ...
6
votes
1answer
493 views
Is there a faster method to compute the geometric series of a matrix?
I want to calculate the geometric series of a matrix $A$:
$$S=I+A+A^2+\dots+A^n$$
and then apply to a vector $v$, $Sv$.
I've done it in Matlab with a loop and I think it's quite efficient applying ...
1
vote
1answer
958 views
computing the inverse of a large block diagonal sparse matrix in r
I would like to compute the inverse of some large block diagonal sparse matrix. The number of rows and columns is somewhat over 50,000. The blocks are 12 by 12 and are sparse (27 non zero elements).
...
3
votes
0answers
54 views
Dominant contributions of a quadratic form
Let $\Sigma$ be a covariance matrix (e.g. symmetric positive definite). For arbitrary vectors $\epsilon$, I need to compute $\chi^2 \equiv \epsilon^\top\Sigma^{-1}\epsilon$, which I do using a ...
1
vote
0answers
101 views
Efficient way to do congruent transformation using matrix inverse?
I know a square self-adjoint matrix $S_{vv}$ and I want to find:
$S_{rr} = HS_{vv}H^{\dagger}$
where $\dagger$ denotes conjugate transpose.
I do not know $H$ but I do know $H^{-1}$.
What is the ...
0
votes
2answers
298 views
closed form approximation of matrix inverse with special properties
I'm trying to find some theory to help me explicitly express the inverse of a matrix (or a close approximation of the inverse). My matrix has the following properties:
invertible
positive definite
...
5
votes
2answers
406 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 ...
8
votes
0answers
306 views
Updating matrix diagonal with Woodbury matrix identity and maintaining numerical accuracy
I have a dense matrix A and its corresponding inverse $A^{-1}$. The Woodbury matrix identity states:
$$ (A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1} $$
I wish to perform small ...
