Numerical Analysis 5. Matrix Factorization: LU and Cholesky скачать в хорошем качестве

Numerical Analysis 5. Matrix Factorization: LU and Cholesky

This video covers matrix factorization, focusing on LU factorization and Cholesky factorization. LU Factorization Definition: Factoring a non-singular square matrix A into A=LU, where L is a lower triangular matrix (with ones on the diagonal) and U is an upper triangular matrix [00:51]. Uniqueness: If it exists for a non-singular matrix, it is unique [01:40]. Process: It is closely related to Gaussian elimination; the matrix U is the result of the elimination, while L contains the factors used during the process [13:15]. Application: Used to solve linear systems Ax=b more efficiently by splitting it into two simpler triangular systems: Ly=b (forward substitution) and Ux=y (backward substitution) [20:40]. Cholesky Factorization Definition: A special case for symmetric, positive definite matrices where A=LL T , with L being a lower triangular matrix [23:26]. Existence: A real matrix L with positive diagonal elements exists if A is symmetric and positive definite [24:13]. Efficiency: It requires approximately n 3 /6 multiplications, which is about half the computational cost of LU factorization (n 3 /3) [35:25]. Key Computational Steps LU: Linked to Gaussian multipliers [04:01]. Cholesky: Calculated systematically term-by-term using the property that each element A ij is the dot product of rows from L and columns from L T [30:55].