The condition-number tag has no wiki summary.
7
votes
1answer
56 views
Preconditioning and effects on precision of solution of LSE
In my courses on numerical analysis, I have been taught that the main and principal motivation for preconditioning linear systems of equations is to increase the convergence rate of iterative solvers ...
3
votes
1answer
38 views
Closed form for singular values of 2D Laplacian?
Does anyone know where to find an analytic form for the singular values of the finite-difference approximation to the 2D Laplacian, expressed in matrix form for a square grid? This would be for the ...
5
votes
2answers
201 views
Solving a sparse and highly ill-conditioned system
I intend to solve Ax = b where A is complex, sparse, unsymmetric and highly ill-conditioned (condition number ~ 1E+20) square or rectangular matrix. I have been able to solve the system with ZGELSS in ...
6
votes
1answer
131 views
How to approximate the condition number of a large matrix?
How do I approximate the condition number of a large matrix $G$, if $G$ is a combination of Fourier transforms $F$ (non-uniform or uniform), finite differences $R$, and diagonal matrices $S$?
The ...
2
votes
2answers
124 views
Problem Condition and Algorithm Stability
Consider 2 mathematical problems:
$$
f_1(x) = a - x \\
f_2(x) = e^x -1
$$
The condition number for a function is defined as follows:
$$
k(f) = \left| x \cdot \frac{f'}{f} \right|
$$
Lets analyze ...
2
votes
1answer
76 views
Non-linear root finding when the Jacobian is almost singular
I'm trying to solve a system non linear-equations:
$$
\frac{\partial K(\mathbf{\lambda})}{\partial \lambda_i} - c_i = 0
$$
for $i = 1, \dots, 15$, using Newton's method:
$$
\lambda^{k + 1} = \lambda^k ...
4
votes
2answers
108 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. ...
5
votes
3answers
262 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 ...
2
votes
2answers
140 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 ...
7
votes
3answers
285 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 $x$. ...
2
votes
1answer
134 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 ...
5
votes
2answers
158 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 $c$?
8
votes
4answers
278 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 ...
7
votes
4answers
679 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 ...
