# Questions tagged [condition-number]

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1answer
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### Matrix condition number and reordering

Does the condition number change when a matrix is reordered by e.g. Cuthill Mckee or some other method?
1answer
79 views

### CHOLMOD condition number estimate

The CHOLMOD library provides a CHOLMOD_rcond function that estimates the reciprocal condition number (in the one norm) of a symmetric positive definite matrix from ...
1answer
66 views

### Missing something fundamental about condition number estimation

In Higham's Accuracy and Stability of Numerical Algorithms, Chapter 15, algorithm 15.3 and 15.4: The topic is ostensibly condition number estimation, but these algorithms show how to compute $\gamma$ ...
1answer
69 views

### Assessing numerical error in solving a least squares problem

I have a linear system of the type $$Ax = b$$ I want to minimise $|b - Ax|^2$. I know there are different approaches to directly solve the system (Normal equation + Cholesky, QR decomposition, SVD ...
1answer
83 views

### Conjugate gradient - ill-conditioning and numerical tolerance

I would like to solve system $Ax=b$, where $A$ is SPD, but very ill-conditioned ($\text{cond}(A)>10^{11}$). I am interested in using UNpreconditioned version of the conjugate gradient method. Is ...
2answers
150 views

### Condition number of a matrix

I saw some definition that if $$M=\max_{x\not=0} \tfrac{\|Ax\|}{\|x\|} \quad\text{and}\quad m=\min_{x\not=0} \tfrac{\|Ax\|}{\|x\|} \,,$$ the condition number of $A=M/m$. In my opinion, If we have a ...
1answer
110 views

2answers
181 views

### Imposing invertibility on a Matrix

I have a symmetric positive semidefinite covariance matrix $A$, which is approximately computed as the output of a quadratic regression. I then need to invert $A$, but often it is close to singular. I'...
3answers
3k views

### Fastest algorithm to compute the condition number of a large matrix in Matlab/Octave

From the definition of condition number it seems that a matrix inversion is needed to compute it, I'm wondering if for a generic square matrix (or better if symmetric positive definite) is possible to ...
2answers
514 views

### Ill-conditioned Gram Matrix Assembly

I'm trying to find the best approximation to the function $e^x$ in the finite dimensional polynomial space $P_4$ with respect to the standard basis vectors $B=\{1,x,x^2,x^3,x^4\}$ with inner product ...
3answers
4k views

### What are the symptoms of ill-conditioning when using direct methods?

Suppose we have a linear system and we know nothing about its conditioning and have no preliminary information about the solution. We blindly apply Gaussian elimination and obtain some solution $x$. ...
1answer
474 views

### Polynomial Regression using Semidefinite Programming

I'm trying to design the frequency response function for a low-pass filter. I need the function to be polynomial and to fulfill the following constraints: the coefficients must sum to 1, the function ...
2answers
218 views

### Condition number of (A + cI) matrix

For given matrix $A \in R^{n\times n}$, identity matrix $I$ and constant $c > 0$ is this possible to express $cond(A + cI)$ knowing $cond(A)$ and $c$?
4answers
2k views

### Condition number of A'A and AA' formulations

It's shown (Yousef Saad, Iterative methods for sparse linear systems, p. 260) that $cond(A'A) \approx cond(A)^2$ Is this true for $AA'$ as well? In case $A$ is $N\times M$ with $N \ll M$, I observe ...
2answers
8k views

### Does a tiny determinant imply ill-conditioning of a matrix?

If I have a square invertible matrix and I take its determinant, and I find that $\det(A) \approx 0$, does this imply that the matrix is poorly conditioned? Is the converse also true? Does an ill-...
4answers
2k views

### precision vs matrix condition number

I have an application in which I am computing a quantity which is approximated by an average over $M$ points. In theory, the average converges to the correct quantity when $M$ is infinite. In practice,...