From: sychen@envmsa.eas.asu.edu (Chen, Shen Yeh)
Subject: Re: HELP!!(Cholesky Decompsition for negative-definite matrix ???)
News-Software: VAX/VMS VNEWS 1.4-b1  
Organization: Arizona State University, Tempe, AZ
Date: Thu, 21 Jan 1993 06:57:00 GMT

In article <1jkhnrINNm76@rave.larc.nasa.gov>, makarand@gazelle.larc.nasa.gov (Makarand A. Kulkarni) writes...
>In article <20JAN199313260117@envmsa.eas.asu.edu>, sychen@envmsa.eas.asu.edu (Chen, Shen Yeh) writes:
>|> 
>|>    I need to solve a linear system equation. In fact, that's the stiffness
>|>  matrix [K] from finite element method. The system is like :
>|>  [K]*{U}={P}.  where [K](nxn)
>|> 
>|>    [K] IS ALWAYS SYMETRIC, but not necessarily positive-definite. If it is
>|>  singular, we think this system to be unstable. Because of the symetry, I
>|>  always use Cholesky Decompsition to reduce the size of [K]. However, now
>|>  I have some situation that [K] is negative-definite but I still have to
>|>  solve the equation. I have not try Gauss method yet, however.
>|> 
>|>    Does anyone kow :
>|>    1) Is there any method I can use to reduce [K], and still solve the system
>|>       when [K] is negative-definite ? (Note that [K] IS ALWAYS SYMETRIC)
>|>    2) If such method exists, what's the criteria for positive- or negative-
>|>       definite?
>|>    3) If no such method exits,( We can not take advantage of symetry to
>|>       reduce the size of [k], I mean.), does Gauss method work? What's
>|>       the criteria for question (2)?
>|> 
>|>  NOTE : (1) In such system, we do not do pivotting among the rows of [K],
>|>             if using Gauss method. If there is a zero on the diagonal,
>|>             we consider it as unstable.
>|>         (2)      [ ] ====> matrix      { } ====> column vector
>|> 
>|>    It will be greatly appreciated if anyone can give me some idea. If you
>|>  do not want to type all that stuff to me, just tell me the theorem used.
>|>  Of course, it will be better if you can tell me reference books. Any
>|>  assistance will be greatly appreciated.
>|> 
>|>    Thank you very very very much !!!!!!
>|> 
> 
> 
>I think I dont understand your problem. If K is the stiffness matrix from
>the finite element method, how can K be NEGATIVE DEFINITE? 
>The qudratic form of K represents some kind of internal energy (eg. strain energy in solid mechanics), so K is always positive semi-definite, right????  It seems you ought not to worry about a negative definite K.
> 
>I realize that I am not shedding any light on your problem, but would
>appreciate it if you could clarify this.
>In particular, let me know what physical problem gives rise to a negative definite K.
>Regards
> 
>Makarand
>makarand@puma.larc.nasa.gov
Hi,

  This is from Shrn-yeh Chen, for further discription of the problem.

  This is hard to explain here. In fact, if a structure is under 'unloading'
condition, [k] will be negative-definite. That is, in general condition,
an incremental load will cause the total strain energy increase-if the
structure is 'intended' to be stable. Other wise, there is condition call
'snapping through ' happening.


   For example, a loaded sturcture like the FIG1   will have the behavior
described    above:

 ###########################################################################
 ##   @                   ##   |      *  *                                ##
 ## A  \                  ##   |   *   |    *                      *      ##
 ##      \                ##   | *     |      *                   *       ##
 ##        \              ##  L|*      |        *  <DEFLECTION>  *        ##
 ##          \ <----------##  O|-------|------------------------*-----    ##
 ##          /    LOAD    ##  A|       |          *            *          ##
 ##        /              ##  D|       |            *         *           ##
 ##      /                ##   |       |              *     *             ##
 ## B  /                  ##   |       |                 *                ##
 ##   @                   ##           |C               D|                ##
 ########< FIG1 >##########################< FIG2 >#########################

   Please see FIG2.    [K] will be sigular on C & D
                       [K] SHOULD be negative-definite from C to D

  HOWEVER, SOMEONE TELL ME THAT,  IF [K] IS NEGATIVE, THEN  -[K] SHOULD BE
  POSITIVE!! SO THERE IS NO PROBLEM!!!!!!!
  IS THIS CORRECT ???   IF YES(OR NO), CAN ANYONE PROVE IT???

  OR MAYBE [K] IS NOT NEGATIVE FROM C TO D. I HAVE PROVE IT TO BE NEGATIVE
  , ALTHOUGH IT MAY BE WRONG.


  THANK ALL OF YOU AGAIN !!