
[Sponsors] 
June 15, 2001, 04:26 
Conditional number

#1 
Guest
Posts: n/a

I have a silly question: how to calculate the conditional number of a matrix? I should know it myself, but I graduated a few years ago and I have completly forgotten some basic facts.
Pawel 

June 15, 2001, 05:19 
Re: Conditional number

#2 
Guest
Posts: n/a

condition number of square matrix A = A inv(A)
where . is some matrix norm and inv(A) is the matrix inverse of A. If you use one particular norm, which I dont recall at the moment, the condition number takes a simple form, cond(A) = max eigenvalue / min eigenvalue 

June 15, 2001, 11:48 
Re: Conditional number

#3 
Guest
Posts: n/a

I think the norm you are refering to is the L2 norm. Condition numbers for matrices resulting from discretization of standard operators like the Laplacian can be easily estimated. Estimation of condition numbers for more general matrices requires numerical algorithms.


June 17, 2001, 14:07 
Re: Conditional number

#4 
Guest
Posts: n/a

that's the one I'm familiar with. It's essentially the stiffness if the system.


June 18, 2001, 06:02 
Re: Conditional number

#5 
Guest
Posts: n/a

The norm of a matrice is the product  A  *  A^1  where  ..  is a norm such as:
1)  A _1 = max(j) sum_i (a_ij) 2)  A _2 = sqrt( rho(A^t A ) ) where rho(A^t A ) is the spectral radius of A^t*A 3)  A _inf = max(i) sum_j( a_ij ) or when the matrix is symetric the condition number is given by: max(  lambda_i ) / min (  lambda_i  ) where lambda are the eigenvalues of A. 

June 18, 2001, 09:30 
Re: Conditional number

#6 
Guest
Posts: n/a

If I have Ax=b, when will I have to find the conditional number? and what is the physical meaning of conditional number and the spectral radius? Thank you very much.
Atit Koonsrisuk 

June 18, 2001, 17:01 
Re: Conditional number

#7 
Guest
Posts: n/a

In real life, You seldom (never) have to compute the condition number. For some systems, such as the discretization of a poisson type equation, the condition number can directly computed (or estimated) from known expressions.
The condition number estimates the effect of small perturbations on the B vector or the coefficients of the matrix A will have the solution. When the condition number is high, the small perturbations will cause big change on the solution. (this is the case for poisson type equations). The spectral radius is the lagest eigenvalue of the matrix. 

Thread Tools  
Display Modes  


Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Mesh Refinement  Luiz Eduardo Bittencourt Sampaio (Sampaio)  OpenFOAM Mesh Utilities  41  January 17, 2013 03:43 
DecomposePar unequal number of shared faces  maka  OpenFOAM PreProcessing  6  August 12, 2010 09:01 
BlockMeshmergePatchPairs  hjasak  OpenFOAM Native Meshers: blockMesh  11  August 15, 2008 07:36 
Unaligned accesses on IA64  andre  OpenFOAM  5  June 23, 2008 10:37 
Trimmed cell and embedded refinement mesh conversion issues  michele  OpenFOAM Other Meshers: ICEM, Star, Ansys, Pointwise, GridPro, Ansa, ...  2  July 15, 2005 04:15 