A(I,J)- L(I,K) L(J,K)T endfor endfor endfor Marco Lapegna – Calcolo Parallelo e Distribuito II 8 Uso di BLAS/LAPACK for K= 1, NB L(K,K

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Marco Lapegna – Calcolo Parallelo e Distribuito II

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CPU multicore

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Fattorizzazione di cholesky a blocchi

A₁₁ L₁₁ L^T₁₁

A₂₂ L₂₂ L^T₂₂

0 A₂₁ 0

A^T₂₁

L₂₁

L^T₂₁

A = L L^T

A₁₁= L₁₁L^T₁₁ A^T₂₁= L₁₁L^T₂₁

A₂₁= L₂₁L^T₁₁ A₂₂= L₂₁L^T₂₁+L₂₂L^T₂₂

3

A₁₁= L₁₁L^T₁₁ L₁₁= Chol(A₁₁) A^T₂₁= L₁₁L^T21

A₂₁= L₂₁L^T₁₁ L₂₁= A₂₁L^-T₁₁

A₂₂= L₂₁L^T₂₁+L₂₂L^T22 L₂₂L^T₂₂=A₂₂-L₂₁L^T₂₁

4

L₁₁ L^T₁₁

A₃₃

0

0 A₃₁

A^T₃₁

L₂₁

L^T₂₁

A = L L^T

A₂₂= L₂₂L^T₂₂ A^T₃₂= L₂₂L^T₃₂

A₃₂= L₃₂L^T₂₂ A₃₃= L₃₂L^T₃₂+L₃₃L^T₃₃ A₁₁

A₂₂ A₂₁

A₃₂ A^T₂₁

A^T₃₂

L₃₁ L₃₃ L₂₂

L₃₂

0

L^T₃₁ L^T₂₂L^T₃₂ L^T₃₃ 0 0

0

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5

A₂₂= L₂₂L^T₂₂ L₂₂= Chol(A₂₂) A^T₃₂= L₂₂L^T32

A₃₂= L₃₂L^T₂₂ L₃₂= A₃₂L^-T₂₂

A₃₃= L₃₂L^T₃₂+L₃₃L^T33 L₃₃L^T₃₃=A₃₃-L₃₂L^T₃₂

6

L₁₁ L^T₁₁

A₃₃

0

A₃₁ 0

A^T₃₁

L₂₁

L^T₂₁

A = L L^T

A₃₃= L₃₃L^T33 L₃₃= Chol(A₃₃) A₁₁

A₂₂ A₂₁

A₃₂ A^T₂₁

A^T₃₂

L₃₁ L₃₃ L₂₂

L₃₂

0

L^T₃₁ L^T₂₂L^T₃₂ L^T₃₃ 0 0

0

7

Fattorizzazione LL^T a blocchi

for K= 1, NB

L(K,K) = Chol( A(K,K) )

for I= K+1, NB L(I,K) = A(I,K)L(K,K)^-T endfor

for I= K+1, NB for J= K+1, …

A(I,J) = A(I,J)- L(I,K) L(J,K)^T endfor

endfor endfor

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Uso di BLAS/LAPACK

for K= 1, NB

L(K,K) = Chol( A(K,K) )

for I= K+1, NB L(I,K) = A(I,K)L(K,K)^-T endfor

for I= K+1, NB for J= K+1, …

A(I,J) = A(I,J)- L(I,K) L(J,K)^T endfor

endfor endfor

DPOTF2 DTRSM

DGEMM

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9

Esempio NB=4

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2

10 9 7 4

8 6 3

5 2 1

3 4

5 6 7 8 9 10

10

Esempio NB=4

5 6 7 8 9 10

16 15 13

14 12 11

11

12 13

14 15 16

11

Esempio NB=4

19 18 17

14 15 16

17

18 19

12

Esempio NB=4

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19

20