From dmday@sandia.gov Mon May 7 09:56:33 2007 Date: Mon, 07 May 2007 09:55:52 -0600 From: David Day To: Evangelos A. Coutsias Subject: Rayleigh quotient iteration Hi Vageli, The short answer is B.N.Parlett, Symmetric eigenvalue problem (SEP) The important point about RQI is that it is only defined for a symmetric matrix. The Rayleigh quotient is rho(A,x) = x'Ax/x'x. RQI is 1.rho_k = rho(A, x_k) 2.Solve (A-rho_k) y_k+1 = x_k ... unless A-rho_k is singular ... 3.normalize, x_k+1 = y_k+1 / norm(y_k+1) 4.stop if norm(y_k+1) is big enough An excuse to open my file cabinet is the Ancient History of RQI: cubic convergence S.H. Crandall, Proc. Royal Soc. 207, 416-423, 1951 rate constant for cubic convergence A. M. Ostrowski Arch. Rational Mech. Anal. 1, 233-241, 1958 Theorem 4.9.1 of SEP. Let (x_k) be the Rayleigh sequence generated by any unit vector xo. As k-> infinity, 1. (rho_k) converges and either 2. (rho_k, x_k) -> (lambda,z) cubically, where Az=z lambda, or 3. x_2k -> x+ and x_2k+1 -> x-, where x+ and x_ are the bisectors of a pair of eigenvectors whose eigenvalues have mean rho = lim_k rho_k. People today still fuss about the relationship between nonlinear solvers, like Newton's method, and RQI. In the 1970's, the connection between RQI and QR iteration was something that people cared about. For nonsymmetric matrices, RQI is not defined, and the obvious extension is called the Raleigh quotient shift. Changing from Rayleigh to Raleigh is subtle. With the Raleigh quotient shift, there are period doubling bifurcations. The reference is a paper by S. Batterson and J. Smilie in Math Comp. circa 1988. Now you are an expert, David Day Evangelos A. Coutsias wrote: > > Hi David, could you suggest any reference(s) on the convergence of RQI? > I was not aware how quirkily it seems to behave. It has no trouble > converging very fast to something, but I don't quite understand to what. > I suspect there may be a way to rescale a matrix to encourage it to > converge to a specific eigenvalue, but I can't quite figure out how close > the eigenvector estimate needs to be before I can hope to steer it to a > specific place. Any suggestions would be greatly appreciated, > best regards, > Vageli > > ******************************************************************** > * Evangelos Coutsias (Professor, Applied Mathematics) * > * University of New Mexico, Albuquerque, NM 87131 USA * > * Department of Mathematics and Statistics * > * Tel.:+1-505-2773310; Fax.:+1-505-2775505; Sec.:+1-505-2774613 * > * http://math.unm.edu/~vageli/ (home tel.: +1-505-2427610 ) * > ******************************************************************** > > > From dmday@sandia.gov Tue May 8 10:56:51 2007 Date: Tue, 08 May 2007 10:56:30 -0600 From: David Day To: Evangelos A. Coutsias Subject: Re: Rayleigh quotient iteration Dear Vageli: Methods of inverse - iteration type, like Rayleigh quotient iteration, tend to converge to the eigenvalues nearest to zero. Methods like the power method tend to converge to the eigenvalues farthest from zero. For the largest eigenvalue, RQI requires a very good initial guess. The power method is slow yet reliable. You might think a little about the initial guess. The standard way to speed up the power method is to use the Lanczos algorithm to reduce T_n to T_m with m< David, thanks! I did observe, and was quite puzzled by, the oscillation > of thm. 4.9.1 and that prompted the question. Hopefully I can pick up > Parlett's book from Louis (not in our library). I greatly appreciate the > crash course. I was looking (still am) for the absolutely fastest, > practical way to find the largest eigenpair of a symmetric (tridiagonal) > matrix. > Vageli > > ******************************************************************** > * Evangelos Coutsias (Professor, Applied Mathematics) * > * University of New Mexico, Albuquerque, NM 87131 USA * > * Department of Mathematics and Statistics * > * Tel.:+1-505-2773310; Fax.:+1-505-2775505; Sec.:+1-505-2774613 * > * http://math.unm.edu/~vageli/ (home tel.: +1-505-2427610 ) * > ******************************************************************** > > > On Mon, 7 May 2007, David Day wrote: > > >> Hi Vageli, >> The short answer is B.N.Parlett, Symmetric eigenvalue problem (SEP) >> The important point about RQI is that it is only defined for a symmetric >> matrix. >> >> The Rayleigh quotient is rho(A,x) = x'Ax/x'x. RQI is >> 1.rho_k = rho(A, x_k) >> 2.Solve (A-rho_k) y_k+1 = x_k ... unless A-rho_k is singular ... >> 3.normalize, x_k+1 = y_k+1 / norm(y_k+1) >> 4.stop if norm(y_k+1) is big enough >> >> An excuse to open my file cabinet is the Ancient History of RQI: >> cubic convergence S.H. Crandall, Proc. Royal Soc. 207, 416-423, 1951 >> rate constant for cubic convergence A. M. Ostrowski Arch. Rational Mech. >> Anal. 1, 233-241, 1958 >> >> Theorem 4.9.1 of SEP. Let (x_k) be the Rayleigh sequence generated by >> any unit vector xo. As k-> infinity, >> 1. (rho_k) converges and either >> 2. (rho_k, x_k) -> (lambda,z) cubically, where Az=z lambda, or >> 3. x_2k -> x+ and x_2k+1 -> x-, where x+ and x_ are the bisectors of >> a pair of >> eigenvectors whose eigenvalues have mean rho = lim_k rho_k. >> >> People today still fuss about the relationship between nonlinear >> solvers, like Newton's method, and RQI. >> In the 1970's, the connection between RQI and QR iteration was something >> that people cared about. >> >> For nonsymmetric matrices, RQI is not defined, and the obvious extension >> is called the Raleigh quotient shift. >> Changing from Rayleigh to Raleigh is subtle. With the Raleigh quotient >> shift, there are period doubling bifurcations. >> The reference is a paper by S. Batterson and J. Smilie in Math Comp. >> circa 1988. >> >> Now you are an expert, >> David Day >> >> >> >> >> Evangelos A. Coutsias wrote: >> >>> Hi David, could you suggest any reference(s) on the convergence of RQI? >>> I was not aware how quirkily it seems to behave. It has no trouble >>> converging very fast to something, but I don't quite understand to what. >>> I suspect there may be a way to rescale a matrix to encourage it to >>> converge to a specific eigenvalue, but I can't quite figure out how close >>> the eigenvector estimate needs to be before I can hope to steer it to a >>> specific place. Any suggestions would be greatly appreciated, >>> best regards, >>> Vageli >>> >>> ******************************************************************** >>> * Evangelos Coutsias (Professor, Applied Mathematics) * >>> * University of New Mexico, Albuquerque, NM 87131 USA * >>> * Department of Mathematics and Statistics * >>> * Tel.:+1-505-2773310; Fax.:+1-505-2775505; Sec.:+1-505-2774613 * >>> * http://math.unm.edu/~vageli/ (home tel.: +1-505-2427610 ) * >>> ******************************************************************** >>> >>> >>> >>> >> >> > > >