Back Won_Y._Yang_Wenw... 2.7 Cholesky Factorization of a Symmetric Positive Definite Matrix: If a matrix A is symmetric

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Back Won Y Yang Wenw 2 7 Cholesky Factorization Of A Symmetric Positive Definite Matrix If A Matrix A Is Symmetric 1 (44.78 KiB) Viewed 26 times

Back Won_Y._Yang_Wenw... 2.7 Cholesky Factorization of a Symmetric Positive Definite Matrix: If a matrix A is symmetric and positive definite, we can find its LU decomposition such that the upper triangular matrix U is the transpose of the lower triangular matrix L. which is called Cholesky factorization. Consider the Cholesky factorization procedure for a 4 x 4 matrix M000 W12 22 0 0 0 M22 23 24 12 022 2024 a ay ay ay 3 #23 # 0 0 04 0 WILD MINIA M 23 +2223 #1234 +234 +6+4 +42322 ++ ₁₁+1₂422 MIME+#₂23+#²33 ²+₁+²+ (P2.7.1) Equating every row of the matrices on both sides yields 1/1/₁₁/₁1 (P2.7.2.1) - #₂ (₂2)/22.024 (054-₁412)/22 (P2.7.2.2) 33√33-us-ui. #34 = (043-2423-1413)/33 (P2.7.2.3) au-₁-₁- (P2.7.2.4) 112 SYSTEM OF LINEAR EQUATIONS which can be combined into two formulas as for k=1: N (P2.7.3a) Midi- Min/₁ form=k+1: N and k=1: N (P2.7.3b) (a) Make a MATLAB routine "cholesky()", which implements these for- mulas to perform Cholesky factorization. (b) Try your routine "cholesky()" for the following matrix and check if UU-A=0 (U: the upper triangular matrix). Compare the result with that obtained by using the MATLAB built-in routine "chol()". 12 4 77 2 13 23 38 A= (P2.7.4) 4 23 77 122 7 38 122 294 #1112