OVERVIEW. Basics Generalized Schur Form Shifts / Generalized Shifts Partial Generalized Schur Form Jacobi-Davidson QZ Algorithm

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2 OVERVIEW Basics Generalized Schur Form Shifts / Generalized Shifts Partial Generalized Schur Form Jacobi-Davidson QZ Algorithm

3 BASICS Generalized Eigenvalue Problem Ax lbx We call (, x) Left eigenpair = or ( A- lb) x = 0 l a (right) eigenpair and ( AB, ) a (matrix) pencil ya= lyb Ay= lby Pencil ( AB, ) regular if det( A lb) - is not identically zero (for all l) é1 0ù Example singular/degenerate pencil: A = 0 0 and B êë úû det A- lb = 0 for any l. Pencil is singular as ( ) é1 0ù = 0 0 êë úû Regular pencil has finite number of eigenvalues characteristic equation is polynomial of degree m n zeros. For singular B possible that char. polynomial is nonzero constant.. If polynomial not identically zero, then at most m

4 BASICS Regular pencil has finite number of eigenvalues p l det A lb Characteristic polynomial ( ) ( ) = - is poly of degree m n If polynomial not identically zero, then at most m zeros For singular B possible that characteristic polynomial is nonzero constant no eigenvalues If l ¹ 0eigenvalue of ( AB, ) then m = l -1 eigenvalue of ( BA, ) If B singular and Bx = 0( x ¹ 0), then ( 0, x ) eigenpair of ( BA, ) Hence natural to consider (,x ) eigenpair of ( AB, ) Moreover, if polynomial degenerate (not of full degree) then some eigenvalues must be infinite (and B must be singular)

5 GENERALIZED SCHUR FORM Generalized Schur form for regular ( AB, ) Unitary UVexist, such that S U AV, T U BV = = both uppertriangular s t l = Easy for triangular problem note better to thin of, than l Generalized eigenvalues: det( S - lt) = 0 l = and ( TS) Eigenvalues of (, ) ST are eigenvalues of (, ) AB : t, ( A lb) x ( A lbvv ) -1 x U ( A lbvv ) ( -1 x) - = 0 - = 0 - = 0 Eigenvalue l unchanged and eigenvector transformed to -1 Same for left eigenvector y U y Prefer unitary UV, for numerical stability V -1 x s

6 GENERALIZED SCHUR FORM ( U A- B V) = ( U) ( V) ( A- lb) Note that ( l ) det det det det Nonsingular UVdo, not change the zeros, so eigenvalues and multiplicities are preserved. For unitary UV, determinants are unit scalars p = A - B is characteristic polynomial ( ) ( l) det AB, ( l ) Algebraic multiplicity of eigenvalue is its multiplicity as root of char. pol. If m degree of p ( AB, )( l ), then (, ) AB has infinite eigenvalue of mult. n- m

7 SHIFTING THE PROBLEM If ( AB, ) regular and B nonsingular: -1 Ax = lbx B Ax = lx Useful both for analysis and algorithms. For singular B (and efficient algorithms) we may want to shift the problem Tae l m = m = l-mwl 1 + wl ( ) - If ( B + wa) is well-conditioned we can wor with ( w ) 1 Ax = lbx Ax - mwax = mbx Ax = m B + wa x In practice, near singular B is just as problematic as singular B If ( AB, ) ( l, x) then ( w ) æ AB, A l ö +, x è ç 1 + wl ø B + A Ax = mx

8 MORE GENERAL SHIFT We can shift more generally. æw w ö If A and ç w w nonsingular, and çè ø CD, = wa+ wbwa, + w B let ( ) ( ) Then (,,x) ab is eigenpair of ( AB, ) +, +, eigenpair of ( CD, ) if ( w a w b w a w b x) Can be used to mae methods that require inverse (in some way) of one of the matrices better behaved. Can be used to base analysis of generalized problem on analysis of standard problem.

9 PARTIAL GENERALIZED SCHUR FORM For large problems we are typically interested only in selected eigenvalues and corresponding right (and/or left) eigenspace: partial generalized Schur form. Consider Ax = lbx = a bbx bax - abx = 0 Partial generalized Schur form: Find Q, Z R, R Î upper triangular such that A B Let ( A a = R ) and ( B R ) i i n Î with orthonormal cols and AQ = R and BQ A b = be diagonal coefficients If ( a, b, ) is generalized eigenpair of ( A B, ) i i y (,, Qy) a b is generalized eigenpair of ( AB, ) i i Note that solving ( A B R a b R ) y 0 i i R R, then = Z R. B - = is simple (upper triangular)

10 JACOBI DAVIDSON QZ Assume we have approximate space V for Q and W for Z, we use (so-called) u Îrange V u = Vy such that Petrov-Galerin approximation: Find ( ) hau -zbu ^W hw AVy - zw AVy = 0 So, given VW, we solve a small generalized eigenvalue problem (LAPACK) for (, ) WAVWBV (typically using, dense, standard QZ algorithm) Find unitary S, S such that Now L R ( ) = z L R A i ( ) = z L R B i S W AV S T S W BV S T (diag) (diag) VS approximates Q and WS approximates Z R L range W = range n AV + m BV We can tae W s.t. ( ) ( ) 0 0

11 JACOBI-DAVIDSON QZ Improve by solving JD correction equation for t æ pp ö ç ( )( ) ç I - A B I uu t r pp h - z - =- çè ø where r = ha- zb u ( ) p = n Au + m Bu 0 0 ^ u for pencil ha- zb Solve correction equation (very) approximately: t (exact solve yields quadratic convergence near a solution) Extend spaces: ( ) v = t - VV t / t - VV t (accurately) m+ 1 m m m m w = n Av + m Bv 0 m 0 m 2 and w ( w W W w ) / ( w W W w ) = = then p = WS e R 1 R L 1 u V S VS e Note if ( ) 1 m 1 m m m m + = - - 2

12 DEFLATION AND RESTART A We have AQ = Z R and B BQ = Z R Extend é A R aù 1 AQ é qù éz zù - ê = ë -1 úû êë -1 úû 0 a ê ë ú û and New generalized Schur pair ( ab,,q): é B R bù 1 BQ é qù éz zù - ê = ë -1 úû êë -1 úû 0 b ê ë ú û Q q = 0 and ( - 1 )( )( I -Z Z ba-ab I - Q Q ) q = where a = Z Aq and b = Z Bq Hence ( ab,,q) is eigenpair of (( I -Z Z ) A( I -Q Q ),( I -Z Z ) B( I - Q Q )) Restarting by using Generalize Schur Form and taing desired part

13 GOOD READING

Synopsis of Numerical Linear Algebra

Synopsis of Numerical Linear Algebra Eric de Sturler Department of Mathematics, Virginia Tech sturler@vt.edu http://www.math.vt.edu/people/sturler Iterative Methods for Linear Systems: Basics to Research