Field of values error estimates for evaluating functions of matrices - PowerPoint PPT Presentation

Field of values error estimates for evaluating functions of matrices via the Arnoldi method Bernhard Beckermann http://math.univ-lille1.fr/ ∼ bbecker Laboratoire Paul Painlev´ e UMR 8524 (´ equipe ANO-EDP) UFR Math´ ematiques, Universit´ e de Lille 1 Conference Harrachov 2007 Joint work with Lothar Reichel, Kent State University Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 1

Outline • the problem: approaching g ( A ) b via Arnoldi’s method • here: error estimates in terms of field of values W ( A ) = { y ∗ Ay : � y � = 1 } • link with best polynomial approximation of g on W ( A ) ? • • • Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 2

Outline • the problem: approaching g ( A ) b via Arnoldi’s method • here: error estimates in terms of field of values W ( A ) = { y ∗ Ay : � y � = 1 } • link with best polynomial approximation of g on W ( A ) ? • Explicit bounds for exp( A ) b • Explicit bounds for A κ b , − 1 < κ < 0 , and other Markov functions • Explicit bounds for general powers and log( A ) b Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 2

The problem How to approximately compute g ( A ) b , where A ∈ C d large, sparse, non-symmetric...? � b � = 1 , Arnoldi decomposition H m ∈ C m upper Hessenberg, AV m = V m H m + f m e ∗ m , and V m ∈ C d × n , V ∗ m V m = I m , V m e 1 = b , V ∗ m f m = 0 . We have for each polynomial p of degree < m : p ( A ) b = p ( A ) V m e 1 = V m p ( H m ) e 1 . Approximation via Arnoldi: • compute Arnoldi decomposition V m , H m for ”small” m • compute exactly g ( H m ) • approach g ( A ) b by V m g ( H m ) e 1 . Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 3

Error estimate with Crouzeix For each polynomial p of degree < m � g ( A ) b − V m g ( H m ) e 1 � = � ( g − p )( A ) b − V m ( g − p )( H m ) e 1 � ≤ � ( g − p )( A ) � + � ( g − p )( H m ) � . THEOREM 1: Let E ⊂ C be some convex and compact set containing the field of values W ( A ) = { y ∗ Ay : y ∈ C d , � y � = 1 } , and let g be analytic on E , then � g ( A ) b − V m g ( H m ) e 1 � ≤ 24 η m ( g, E ) , η m ( g, E ) := deg p<m � g − p � L ∞ ( E ) . min Proof: Michel Crouzeix in 2006 showed that � f ( B ) � ≤ 12 � f � L ∞ ( W ( B )) . Also, H m = V ∗ m AV m = ⇒ W ( H m ) ⊂ E . � Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 4

Riemann maps and Faber maps Let E convex and compact as before, D the closed unit disc, then there exists unique conformal maps φ : E c �→ D c with φ ( ∞ ) = ∞ , φ ′ ( ∞ ) > 0 , ψ := φ − 1 . R = { z ∈ E c : | φ ( z ) | > R } . Level sets for R > 1 defined by complement: E c Bernstein Theorem: If g analytic in E R then R − m η m ( g, E ) = deg p<m � g − p � L ∞ ( E ) ≤ 2 min 1 − R − 1 � g � L ∞ ( E R ) . Faber map F : A ( D ) �→ A ( E ) defined by � ψ ′ ( w ) 1 F ( G )( z ) = ψ ( w ) − z G ( w ) dw. 2 πi | w | =1 1 g = F ( G ) = ⇒ �F − 1 � η m ( G, D ) ≤ η m ( g, E ) ≤ 2 η m ( G, D ) . Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 5

Faber polynomials and Faber series Faber polynomial: F n ( z ) = F ( w n )( z ) polynomial of degree n , F n polynomial part of φ n , ovari ’67 ( E convex): � F n − φ n � L ∞ ( E ) ≤ 1 . Pommerenke and K˝ Examples: E = a + D R : F n ( z ) = ( z − a R ) n . E = [ − 1 , 1] : F n ( z ) = 2 T n ( z ) Chebyshev polynomials of first type. Faber series: For g ∈ A ( E ) , j ≥ 0 � ∞ � 1 g ( ψ ( w )) g j w j , g j = dw = ⇒ g = F ( G ) , G ( w ) = w j +1 2 πi | w | =1 j =0 where the last sum, and g ( z ) = � ∞ j =0 g j F j ( z ) , are absolutely convergent in D , and E , respectively. Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 6

THEOREM 2: Let E ⊂ C be some convex and compact set containing the field of values W ( A ) and let g = F ( G ) be analytic on E , then � g ( A ) b − V m g ( H m ) e 1 � ≤ 4 η m ( G, D ) . Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 7

Idea of proof of Theorem 2 Inspired from Crouzeix, Delyon, Badea, BB 02-07, in particular the CRAS ’05 of BB: � F n ( A ) � ≤ 2 . It is sufficient to show � h ( A ) � ≤ 2 � H � L ∞ ( D ) , h = F ( H ) + H (0) . Here W ( A ) ⊂ Int ( E ) for simplicity. We have  �  F m ( A ) if m = 0 , 1 , 2 , ... , 1 F ( w m )( A ) = w m ψ ′ ( w )( ψ ( w ) − A ) − 1 dw =  2 πi 0 if m = − 1 , − 2 , ... . | w | =1 Hence   �   � wψ ′ ( w )( ψ ( w ) − A ) − 1 � � wψ ′ ( w )( ψ ( w ) − A ) − 1 � ∗ h ( A ) = 1 dw   H ( w ) + iw .     2 π � �� � | w | =1 positive definite Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 8

How to exploit Theorem 2? ∞ ∞ � � g j w j . � g ( A ) b − V m g ( H m ) e 1 � ≤ 4 η m ( G, D ) , g ( z ) = g j F j ( z ) , G ( w ) = j =0 j =0 LEMMA 3:   | g m |   ∞ � ≤ η m ( G, D ) ≤ | g m + j | . | g m + g m +( m +1) + g m +2( m +1) + ... |    j =0 | g m − g m +( m +1) + g m +2( m +1) ∓ ... | Knitznerman ’91 gave a similar upper bound with additional powers of m + j Hochbruck & Lubich ’97 gave a more complicated bound, weaker up to factor 0 . 75 . Idea of proof: Upper bound partial sum. First lower bound 1 G ( w ) 1 G ( w ) − q ( w ) � � deg q < m : g m = w m +1 dw = dw. 2 πi 2 πi w m +1 | w | =1 | w | =1 For second lower bound compute η m ( G, { exp( 2 πij m +1 ) : j = 0 , 1 , ..., m } ) = modulus of leading coefficient of interpolating polynomial at these roots of unity. � Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 9

Application 1: the exponential function Consider E symmetric with respect to real axis (e.g., A ∈ R d × d ), and g ( z ) = exp( τz ) , τ > 0 . We write ψ ( w ) = cap ( E ) w + c + O (1 / [ w | ) | w |→∞ . LEMMA 4: the case of ”large” j : there exist explicit ”modest” constants K, K 1 , K 2 > 0 such that for j ≥ τ cap ( E ) � � � f j − e τc [ τ cap ( E )] j [ τ cap ( E )] j [ τ cap ( E )] j [ τ cap ( E )] j � ≤ K � � , K 1 ≤ η j ( G, D ) ≤ K 2 . √ j j ! j ! j ! j ! Idea of proof: convexity plus ”elementary” properties of ψ allows to show that | ψ ( Re it ) − ψ ( R ) − R ( e it − 1) | ≤ | t | = ⇒ cap ( E ) R | e τ ( ψ ( Re it ) − ψ ( R )) − e τ cap ( E ) R ( e it − 1) | ≤ τ cap ( E ) | t | = ⇒ R � π e τ cap ( E )( u − R ) 1 � ≤ 1 τ cap ( E ) | t | dt R j = π τ cap ( E ) � � � � e − τψ ( R ) f j − du � � u j +1 R j +1 2 πi π R 2 | u | = R 0 j now take R = τ cap ( E ) ≥ 1 . � Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 10

Can we do something for j < τ cap ( E ) ? Sometimes one may exploit the trivial bound e τψ ( R ) | f j | ≤ min R j R ≥ 1 with minimum attained at R = 1 if j < τψ ′ (1) (notice that ψ ′ (1) = 0 if E has an outer angle > π at ψ (1) ), and else at � R being unique solution of τRψ ′ ( R ) = j . Example (Hochbruck and Lubich): let E = [ − 4 ρ, 0] then ψ ( w ) = cap ( E )( w + 1 Rψ ′ ( R ) = cap ( E )( R − 1 w − 2) , R ) , and thus for all 0 ≤ j ≤ τ cap ( E ) � � � � j 2 m 2 | f m | ≤ exp − , η j ( G, D ) ≤ K 3 exp − . 7 τ cap ( E ) 7 τ cap ( E ) Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 11

Application 2: powers/Markov functions Consider E ⊃ W ( A ) symmetric with respect to real axis (e.g., A ∈ R d × d ), and � β dµ ( t ) g ( z ) = z − t , α < β < γ = min { Re( z ) : z ∈ E } , µ ≥ 0 . α � 0 | t | κ Example: z κ = sin( π | κ | ) z − t dt for κ ∈ ( − 1 , 0) . −∞ π � β α φ ( t ) − j − 1 φ ′ ( t ) dµ ( t ) with sign ( − 1) j . Here Faber coefficients g j = − Improved: � ∞ j =0 | g m + j ( m +1) | ≤ η m ( G, D ) ≤ � ∞ j =0 | g m +2 j | sharp up to m +1 2 . COROLLARY 5: For Markov function � β | φ ′ ( t ) | | φ ( γ ) | m = 4 � g � L ∞ ( E ) | φ ( t ) | m − 1 ≤ 4 | g ( γ ) | dµ ( t ) � g ( A ) b − V m g ( H m ) e 1 � ≤ 4 | φ ( γ ) | m . | φ ( t ) | 2 − 1 α Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 12

A special case of Markov: FOM Consider x m := V m H − 1 m e 1 ∈ span ( V m ) = span ( b, Ab, ..., A m − 1 b ) then b − ( AV m ) H − 1 m e 1 = b − ( V m H m + f m e ∗ m ) H − 1 b − Ax m = m e 1 − f m e ∗ m H − 1 m e 1 ⊥ span ( V m ) = span ( b, Ab, ..., A m − 1 b ) = i.e., x m is FOM iterate. From previous slide for 0 �∈ E ⊃ W ( A ) symmetric with respect to real axis with g ( x ) = 1 /x � A − 1 b − x m � ≤ 4 | φ (0) | − m dist (0 , E ) . Closely related known estimate � A − 1 b − x m � � A � deg q<m � A − 1 b − q m − 1 ( A ) b � . ≤ inf � A − 1 b � dist (0 , E ) Bernhard Beckermann, Univ. Lille 1 back Harrachov 2007, page 13