The tag has no wiki summary.

learn more… | top users | synonyms

6
votes
1answer
150 views

Are direct solvers affect by the condition number of a matrix?

If I were to solve a relatively small problem, that is, a problem that can be handled by a direct method like LU, then does the condition number of the linear operator affect the accuracy of the ...
1
vote
1answer
104 views

Direct or iterative solver for ill-conditioned problems

I have to solve an ill-conditioned sparse matrix. Once I read that iterative solver are the better tool for such problems. Is that true? If yes, why?
3
votes
1answer
163 views

Designing a preconditioner for a very Ill-conditionned matrix

I am a physicist with limited numerical methods knowledge and I am trying to speed up the inversion of a very ill-conditioned problem ($rcond>10^{30}$). The same sparse square matrix is used ...
4
votes
2answers
155 views

Condition number from incomplete Cholesky factorization

I'm having difficulties patching together from what I read about obtaining the condition number of a real, symmetric, positive definite sparse matrix. In my code, I found that there is incomplete ...
2
votes
2answers
135 views

Does length unit in FEM affect numerical condition?

I try to solve a system of coupled PDEs using FEM. Unfortunately, the originating matrix has very poor condition. After days of double checking and thinking, I suspect the following reason: Given a ...
3
votes
1answer
103 views

Condition number of an algorithm

I am stuck with a problem about finding the condition number of an algorithm. I tried to find an example, but i couldn't. Can anybody help me, please? Given is $f(x)=\ln(x)$. We have the algorithm ...
12
votes
2answers
248 views

Is there any way to do “double preconditioning”

Question: Suppose that you have two different (factored) preconditioners for a symmetric positive definite matrix $A$: $$A \approx B^TB$$ and $$A \approx C^TC,$$ where the inverses of the factors $B, ...
8
votes
4answers
182 views

Are there any quad-double arithmetic sparse matrix package?

I am working on some ill-conditioned large sparse linear system of equations. I want to use double-double arithmetic or quad-double arithmetic to solve them. I know that there is a package named MPACK ...
3
votes
2answers
57 views

Parameter Fitting: Need measure of data 'support' for a parameter solution

I am estimating parameters on a dataset that would, for the most part, result in a weakly constrained solution. The dataset however also contains a few more data points that make the solution ...
11
votes
3answers
456 views

Is variable scaling essential when solving some PDE problems numerically?

In semiconductor simulation, it is common that the equations are scaled so they have normalised values. For example, in extreme cases electron density in semiconductors can vary over 18 order of ...
7
votes
1answer
75 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
135 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
3answers
2k 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 ...
8
votes
1answer
2k 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 ...
4
votes
2answers
153 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
421 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 ...
5
votes
2answers
148 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. ...
7
votes
3answers
2k 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
280 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 ...
11
votes
3answers
936 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
233 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
186 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 $c$?
9
votes
4answers
568 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 ...
20
votes
2answers
2k 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 ...
7
votes
4answers
1k 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 ...