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# set15 - a39 a38 S e t 1 5 K r y l o v M e t h o d s – G M...

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Unformatted text preview: a39 a38 S e t 1 5 : K r y l o v M e t h o d s – G M R E S K y l e A . G a l l i v a n D e p a r t m e n t o f M a t h e m a t i c s F l o r i d a S t a t e U n i v e r s i t y N u m e r i c a l L i n e a r A l g e b r a 1 F a l l 2 1 1 a39 a38 G e n e r a l i z e d M i n i m a l R e s i d u a l – G M R E S C h o o s e t w o s p a c e s o f d i m e n s i o n 1 ≤ k ≤ n , K m a n d L m P e t r o v- G a l e r k i n m e t h o d s s o l v e A x = b b y r e q u i r i n g x m ∈ x + K m b − A x m ⊥ L m G M R E S t a k e s K m = K ( A , r ) = s p a n [ r , A r , · · · , A m − 1 r ] L m = A K m 2 a39 a38 G M R E S R e c a l l t h a t d − M z ⊥ R ( M ) i f w = z m i n i m i z e s bardbl d − M w bardbl 2 . T h e r e f o r e , x m ∈ x + K m b − A x m ⊥ L m ↓ bardbl b − A x bardbl 2 = bardbl b − A ( x + V m y ) bardbl 2 = bardbl r − A V m y bardbl 2 = bardbl β v 1 − V m + 1 ¯ H m y bardbl 2 = bardbl V m + 1 ( β e 1 − ¯ H m y ) bardbl 2 = bardbl ( β e 1 − ¯ H m y ) bardbl 2 s i n c e V m + 1 i s a n i s o m e t r y . 3 a39 a38 F u l l G M R E S x a r b i t r a r y ; r = b − A x S e t v 1 = r β − 1 w h e r e β = bardbl r bardbl 2 U s e A r n o l d i- M G S t o c o n s t r u c t V m + 1 – a n o r t h o g o n a l b a s i s f o r K m + 1 ( A , r ) ¯ H m – m + 1 × m h e s s e n b e r g m a t r i x v m + 1 = w m η − 1 m + 1 , m e n d A r n o l d i fi n d y m t h a t m i n i m i z e s bardbl ( β e 1 − ¯ H m y ) bardbl 2 x = x + V m y m 4 a39 a38 L e a s t S q u a r e s A t e a c h s t e p w e h a v e ¯ H j = H j e T j η j + 1 , j G i v e n s r o t a t i o n s c a n b e u s e d t o c r e a t e G j ¯ H j = R j T j w h e r e R j i s j × j a n d u p p e r t r i a n g u l a r . ¯ d j = G j e 1 β ∈ R j + 1 i s a l s o c o m p u t e d . 5 a39 a38 G M R E S bardbl β e 1 − ¯ H j y j bardbl 2 = bardbl G j ( β e 1 − ¯ H j y j ) bardbl 2 = bardbl ¯ d j − G j ¯ H j y j bardbl 2 R j T j y j = d j δ j + 1 y j = R − 1 j d j bardbl b − A x j bardbl 2 = | δ j + 1 | 6 a39...
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set15 - a39 a38 S e t 1 5 K r y l o v M e t h o d s – G M...

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