Hello, Berwin,
Thanks a lot for answering! I prefer the first method, which seems easier to carry out. So, what I need now is some method to solve the non-linear equation c'(M_11-lam*I)^(-1) (M_11-lam*I)^(-1)c=1 for lam, where c is a vector, M_11 is a symmetric matrix and I is the identity matrix. I searched the R archive somehow, but didn't find anything valuable. Is there any function in R available for this problem? I am grateful of any hints. When c=0, my problem reduces to the typical minimization of b'M_11b w.r.t b'b=1. The solution is the normalized eigenvector associated with the minimum eigenvalue of M_11, right? Thanks, Yingfu PS: There is a type error in the first condition for b: the '+' should write to '-'. -----Original Message----- From: Berwin A Turlach [mailto:[EMAIL PROTECTED] On Behalf Of Berwin A Turlach Sent: den 10 augusti 2006 03:57 To: Yingfu Xie Cc: Rolf Turner; r-help@stat.math.ethz.ch Subject: Re: [R] minimization a quadratic form with some coef fixed and someconstrained >>>>> "YX" == Yingfu Xie <[EMAIL PROTECTED]> writes: YX> Thanks for reply! But I think that solution is right without YX> the constrain b'b=1. With this constrain, the solution is not YX> so simple. :( But simple enough. :) Write down the Lagrange function for the problem. Say, 'lam' is the Lagrange parameter for enforcing the constraint b'b=1. Then, using Rolf's notation: RT> [...] Write M as RT> | M_11 c | RT> | c' m | Then the system of equations that b and the Lagrange parameter have to fulfill is: b = (M_11 + lam*I)^{-1} c (with I being the identity matrix) and lam = b' M_11 b - b'c You can either use the first equation and do a (grid) search for the value of 'lam' that gives you b'b=1 (could be negative!), or start with lam=0 and then alternate between the two equations until convergence. At least I think that this will solve your problem. :) Thinking a bit about the geometry of the problem, I actually believe that if c=0, you might have an identifiability problem, i.e. there are at least two solutions, or, depending on M_11, infinitely many. Hope this helps. Cheers, Berwin ========================== Full address ============================ Berwin A Turlach Tel.: +61 (8) 6488 3338 (secr) School of Mathematics and Statistics +61 (8) 6488 3383 (self) The University of Western Australia FAX : +61 (8) 6488 1028 35 Stirling Highway Crawley WA 6009 e-mail: [EMAIL PROTECTED] Australia http://www.maths.uwa.edu.au/~berwin ########################################### This message has been scanned by F-Secure Anti-Virus for Mic...{{dropped}} ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
