T we lfth Le s s on F orm a lis in g a LUP De om p os ition Formath School 2012 – p.1
Motiv a tion T ake a las s i algorithm Corme n LUP algorithm How diffi ult to formalis e ? T e s t our lin e ar alge bra ( matrix.v ) Formath School 2012 – p.2
Lin e a r e qua tion s x + 2 y + 3 z = 5 2 x − 4 y + 6 z = 18 3 x − 9 y − 3 z = 6 Formath School 2012 – p.3
Lin e a r e qua tion s 1 2 3 x 5 2 − 4 6 y = 18 3 − 9 − 3 z 6 1 2 3 1 0 0 1 2 3 = 2 − 4 6 2 − 8 0 0 1 0 3 − 9 − 3 3 − 15 − 12 0 0 1 Formath School 2012 – p.4
Lin e a r e qua tion s 1 0 0 1 2 3 x 5 = 2 − 8 0 0 1 0 y 18 , , , , 3 − 15 − 12 0 0 1 z 6 1 0 0 x 1 5 1 2 3 x x 1 = = 2 − 8 0 y 1 18 0 1 0 y y 1 3 − 15 − 12 z 1 6 0 0 1 z z 1 x 1 = 5 y 1 = − 1 z 1 = 2 z = 2 y = − 1 x = 1 Formath School 2012 – p.5
Lin e a r e qua tion s 1 2 3 1 0 0 1 2 3 2 − 4 6 = 2 − 8 0 0 1 0 3 − 9 − 3 3 − 15 − 12 3 0 1 A = L U P j P = 1 i Formath School 2012 – p.6
Algorithm Easy ase: re ursive onstru tion Add to A one line and a olumn of zeros: = P A L U a w a w 1 1 = P A L U Formath School 2012 – p.7
Algorithm Main ase ( a � = 0 ) S hur omp lement A ′ = A − 1 /a ( v ∗ w ) a w A v a w A ′ Formath School 2012 – p.8
Algorithm Main ase: re ursive onstru tion ( A ′ = A − 1 /a ( v ∗ w ) ) = A ′ P L U u = 1 /a ( P ∗ v ) a w a w 1 1 = A ′ P A L U v u Formath School 2012 – p.9
Algorithm Sp e ial ase Line p ermutation a w = P A L U a w = P ′ A L U Formath School 2012 – p.10
Algorithm 1. Put a n on � ze ro e le me n t at (0,0) if n e e de d by lin e p e rmutation 2. Pe rform the re urs iv e all on the S hur omp le me n t 3. Re omp os e the re s ult of the re urs iv e all Formath School 2012 – p.11
F orm a lis a tion Write the algorithm Write the s p e ifi ation Prov e that the s p e ifi ation is me t Formath School 2012 – p.12
Wha t is a m a trix? Inductive matrix ( m n : nat) R := Matrix of {ffun ’I_ m ∗ ’I_ n → R }. Notation "’M[ R ]_( m , n )" := (matrix m n R ). Notation "’M[ R ]_( n )" := ’M[ R ]_( n , n ). Definition A : ’M[ R ]_( m , n ) := \ matrix(i,j) E . A i j : R Formath School 2012 – p.13
G e ttin g blo ks Us in g ordin al : whe re . A ul A ur A dl A dr lshift n ( i : ’I_ m ) : ’I_( m + n ) rshift m ( i : ’I_ n ) : ’I_( m + n ) split ( i : ’I_( m + n )) : ’I_ m + ’I_ n Inductive A + B := inl of A | inr of B Formath School 2012 – p.14
Row blo k [ A l | A r ] Definition row_mx A l A r : ’M[ R ]_( m , n 1 + n 2 ) : = \ matrix_( i , j ) match split j with inl j 1 ⇒ A l i j 1 | inr j 2 ⇒ A r i j 2 end. Definition lsubmx ( A : ’M[ R ]_( m , n 1 + n 2 )) : = \ matrix_( i , j ) A i (lshift n 2 j ). Definition rsubmx ( A : ’M[ R ]_( m , n 1 + n 2 )) : = \ matrix_( i , j ) A i (rshift n 1 j ). Formath School 2012 – p.15
Colum n blo k A u A d Definition col_mx A u A d : ’M[ R ]_( m 1 + m 2 , n ) : = \ matrix_( i , j ) match split i with inl i 1 ⇒ A u i 1 j | inr i 2 ⇒ A d i 2 j end. Definition usubmx ( A : ’M[ R ]_( m 1 + m 2 , n )) : = \ matrix_( i , j ) A (lshift m 2 i ) j . Definition dsubmx ( A : ’M[ R ]_( m 1 + m 2 , n )) : = \ matrix_( i , j ) A (rshift m 1 i ) j . Formath School 2012 – p.16
Blo k A ul A ur A dl A dr Definition col_mx A ul A ur A dl A dr : = col_mx (row_mx A ul A ur ) (row_mx A dl A dr ). Definition ulsubmx A := lsubmx (usubmx A ). Definition ursubmx A := rsubmx (usubmx A ). Definition dlsubmx A := lsubmx (dsubmx A ). Definition drsubmx A := rsubmx (dsubmx A ). Formath School 2012 – p.17
Pe rm uta tion i 2 [A] i 1 Definition row_perm ( s : ’S_ m ) A : = \ matrix_( i , j ) A ( s i ) j . Definition perm_mx n s : ’M[ R ]_ n : = row_perm s 1%:M. Definition xrow i 1 i 2 A : = row_perm (tperm i 1 i 2 ) A . Definition tperm_mx n i 1 i 2 : ’M[ R ]_ n : = perm_mx (tperm i 1 i 2 ). Formath School 2012 – p.18
Pi kin g Let P a predi ate on a finite type T i [A] where [pick x | P x ] : option T If there is a default value: Inductive option ( A : Type) : = Some of A | None. where odflt a [pick x | P x ] : T Definition odflt ( A : Type) ( a : A ) ( o : option A ) := if o is Some v then v else a . Formath School 2012 – p.19
Algorithm 1. Put a n on � ze ro e le me n t at (0,0) if n e e de d by lin e p e rmutation 2. Pe rform the re urs iv e all on the S hur omp le me n t 3. Re omp os e the re s ult of the re urs iv e all Formath School 2012 – p.20
Algorithm Let M n : ’M[ F ]_ n . = Fixpoint cormen_lup n : M n .+1 -> M n .+1 * M n .+1 * M n .+1 : = if n is _.+1 then fun A ⇒ let k : odflt 0 [pick k | A k 0 != 0 ] in = let A 1 : ’M_(1 + _) : xrow 0 k A in = let P 1 : ’M_(1 + _) : tperm_mx 0 k in = let Schur := drsubmx A 1 - (( A k 0 )^-1 *: dlsubmx A 1 ) *m ursubmx A 1 in let: ( P 2 , L 2 , U 2 ) := cormen_lup Schur in let P : block_mx 1 0 0 P 2 * P 1 in = let L : block_mx 1 0 (( A k 0 )^-1 *: ( P 2 *m dlsubmx A 1 )) L 2 in = let U : block_mx (ulsubmx A 1 ) (ursubmx A 1 ) 0 U 2 in = ( P , L , U ) else fun A ⇒ ( 1 , 1 , A ) . Formath School 2012 – p.21
S p e ifi a tion We hav e a de omp os ition : T he firs t omp on e n t is a p e rmutation : Lemma cormen_lup_correct n ( A : ’M_ n .+1) : let: ( P , L , U ) : = cormen_lup A in P * A = L * U . Definition perm_mx n s : ’M[ R ]_ n : = row_perm s 1%:M. Definition is_perm_mx n ( A : ’M[ R ]_ n ) : = existsb s : ’S_ n , A == perm_mx s . Lemma cormen_lup_perm n ( A : ’M[ F ]_ n .+1) : let: ( P , _, _) : = cormen_lup A in is_perm_mx P . Formath School 2012 – p.22
S p e ifi a tion T he s e on d omp on e n t is a lowe r trian gular matrix with on e of the diagon al T he third omp on e n t is an up p p e r trian gular matrix Lemma cormen_lup_lower n ( A : ’M[F]_ n .+1) ( i j : ’I_ n .+1) : let: (_, L , _) : = cormen_lup A in i <= j → L i j = ( i == j )%:R. Lemma cormen_lup_upper n ( A : ’M[F]_ n .+1) ( i j : ’I_ n .+1) : let: (_, _, U ) : = cormen_lup A in j < i → U i j = 0 . Formath School 2012 – p.23
Prov in g I Lemma cormen_lup_correct n ( A : ’M[ F ]_ n .+1) : ( P , L , U ) : = cormen_lup A in P * A = L * U . let: Lemma mulmx_block ( m 1 m 2 n 1 n 2 p 1 p 2 : nat) ( A ul : ’M[ R ]_( m 1 , n 1 )) ( A ur : ’M[ R ]_( m 1 , n 2 )) ( A dl : ’M[ R ]_( m 2 , n 1 )) ( A dr : ’M[ R ]_( m 2 , n 2 )) ( B ul : ’M_( n 1 , p 1 )) ( B ur : ’M_( n 1 , p 2 )) ( B dl : ’M_( n 2 , p 1 )) ( B dr : ’M_( n 2 , p 2 )), block_mx A ul A ur A dl A dr *m block_mx B ul B ur B dl B dr = block_mx ( A ul *m B ul + A ur *m B dl ) ( A ul *m B ur + A ur *m B dr ) ( A dl *m B ul + A dr *m B dl ) ( A dl *m B ur + A dr *m B dr ) Lemma mxE F : matrix_of_fun F =2 F . Lemma mx11_scalar ( A : ’M[ R ]_1) : A = ( A 0 0)%:M. Formath School 2012 – p.24
Prov in g I Lemma cormen_lup_correct n ( A : ’M[ F ]_ n .+1) : ( P , L , U ) : = cormen_lup A in P * A = L * U . let: Proof. elim: n ⇒ [| n IHn ] /= in A *; first by rewrite !mul1r. set k : = odflt _ _; set A 1 : ’M_(1 + _) : = xrow _ _ _. = _ - _; move/(_ A ′ ): IHn ; case: cormen_lup ⇒ [[ P ′ L ′ U ′ ]] /= IHn . set A ′ : rewrite -mulrA -!mulmxE -xrowE -/ A 1 /= -[ n .+2]/(1 + n .+1)%N -{1}(submxK A 1 ). rewrite !mulmx_block !mul0mx !mulmx0 !add0r !addr0 !mul1mx -{ L ′ U ′ }[ L ′ *m _] IHn . rewrite -scalemxAl !scalemxAr -!mulmxA addrC -mulrDr { A ′ } subrK. congr (block_mx _ _ (_ *m _) _). rewrite [_ *: _]mx11_scalar !mxE lshift0 tpermL {}/ A 1 {}/ k . case: pickP ⇒ /= [ k nzAk0 | no k ]; first by rewrite mulVf ?mulmx1. rewrite (_ : dlsubmx _ = 0 ) ?mul0mx //; apply/colP ⇒ i . by rewrite !mxE lshift0 (elimNf eqP ( no k _)). Qed. Formath School 2012 – p.25
Prov in g II Lemma cormen_lup_perm n ( A : ’M[ F ]_ n .+1) : let: ( P , _, _) : = cormen_lup A in is_perm_mx P . Lemma perm_mx_is_perm ( n : nat) ( s : ’S_ n ) : is_perm_mx (perm_mx s ). Lemma is_perm_mxMr ( n : nat) ( A B : ’M[ R ]_ n ) : is_perm_mx B → is_perm_mx ( A *m B ) = is_perm_mx A . Definition lift0_mx A : ’M[ R ]_(1 + n) := block_mx 1 0 0 A . Lemma lift0_mx_is_perm ( n : nat) ( s : ’S_ n ) : is_perm_mx (lift0_mx (perm_mx s )). Formath School 2012 – p.26
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