Theorem 3.5 (Cholesky Method): Let be symmetric, positive definite. Then A A can be factored in the form A=LLT. Operation counts: To find and is . L g=L-1f To find is operations. U Total is operations. Parallel LU Decomposition There are 6 convenient ways of writing the factorization step of the in LU nn A decomposition (see textbook). The two most common ways are as follows: kij loop: by row (daxpy) loop: by column (daxpy) A kji A for for k = 1, n 1 k = 1, n 1 − − for for i = k + 1, n p = k + 1, n lik = aik /akk lpk = apk /akk
Theorem 3.5 (Cholesky Method): Let A be symmetric, positive definite. Then A can be factored in the form A=LLT. Operation counts: To find L and g=L-1f is . To find U is operations. Total is operations. Parallel LU Decomposition There are 6 convenient ways of writing the factorization step of the nn A in LU decomposition (see textbook). The two most common ways are as follows: kij loop: A by row (daxpy) kji loop: A by column (daxpy) for k = 1, n − 1 for k = 1, n − 1 for i = k + 1, n for p = k + 1, n lik = aik /akk lpk = apk /akk