Transcription of Cholesky 分解ノート
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Cholesky 分解ノート
Cholesky .. 2008 6 9 , 2013 8 25 .. Cholesky .. Sylvester . L . U . U . 1 . , . Cholesky LU . ( . Hermite Hermite ) . LU . LU Cholesky . Cholesky . A . A = LDU (L , D , U ). LDU . A = AT = U T DT LT = U T DLT. LDU . L = UT , U = LT.. A = LDLT . 1. D D1/2 L . (LD1/2 L ). A = LLT.. A . P AP T = LDLT (P , L , D ). ( [2], [3] ) A 0 .. A = LDLT (L , D ). ( Cholesky ) A . A = LLT (L ). ( Cholesky ) . Trefethen and Bau III [9] [7], [5] . 2 Cholesky .. A = (aij ) M (N ; R) L = ( ij ) M (N ; R) .. A = LLT. A Cholesky ( Cholesky factorization ) 1 . Cholesky (1). Cholesky LU . 1. decompose . factorization .. 2.. (LU ) N N . (1) A GL(N ; R) P GL(N ; R), L GL(N ; R), U GL(N ; R) . (1) P A = LU.. (2) a 0 A GL(N ; R) . L GL(N ; R), U GL(N ; R) . (2) A = LU.. (3) (1), (2) LU . LU = L DU (L , U , D ).
が成り立つ。これをA のCholesky 分解(Cholesky factorization) と呼ぶ1。 2.2 Cholesky 分解の存在証明(1) Cholesky 分解の存在を証明するため、LU 分解について復習しよう。 1「分解する」という動詞には\decompose" が使われることが多いが、「分解」という名詞には\factorization"
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