Matrix Theory -Vector Norms School of Mathematical Sciences Teaching Group Textbook: llse C.F.Ipsen,Numerical Matrix Analysis:Linear Systems and Least Squares.SIAM, 2009. Reference books: Fuzhen Zhang.Matrix Theory-Basic Results and Techniques,Second Edition. Springer,2011. Roger A.Horn and Charles A.Johnson:Matrix Analysis.Cambridge University Press,1985. Gene H.Golub and Charles F.Van Loan:Matrix Computations,Third Edition. Johns Hopkins Press,1996. Nicholas J.Higham.Accuracy and Stability of Numerical Algorithms,Second Edition.SIAM,2002. Y.Saad.Iterative Methods for Sparse Linear Systems,Second Edition.SIAM, Philadelphia,2003. Matrix Theory Vector Norms Maintained by Yan-Fei Jing
Textbook: Ilse C. F. Ipsen, Numerical Matrix Analysis: Linear Systems and Least Squares. SIAM, 2009. Reference books: ▸ Fuzhen Zhang. Matrix Theory-Basic Results and Techniques, Second Edition. Springer, 2011. ▸ Roger A. Horn and Charles A. Johnson: Matrix Analysis. Cambridge University Press, 1985. ▸ Gene H. Golub and Charles F. Van Loan: Matrix Computations, Third Edition. Johns Hopkins Press, 1996. ▸ Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms, Second Edition. SIAM, 2002. ▸ Y. Saad. Iterative Methods for Sparse Linear Systems, Second Edition. SIAM, Philadelphia, 2003. Maintained by Yan-Fei Jing Matrix Theory ––Vector Norms School of Mathematical Sciences Teaching Group Matrix Theory Vector Norms
Introduction Outline Introduction Definition Two Important Inequalities Exercises Normwise Errors Comprehensive Problems Hold Inequality again 参老年这头 Matrix Theory Vector Norms -2/39
Introduction Outline Introduction Definition Two Important Inequalities Exercises Normwise Errors Comprehensive Problems H¨old Inequality again Matrix Theory Vector Norms - 2/39
Introduction Introduction In the context of linear system solution,the error in the solution constitutes a vector. r=b-A×X. 奇电有这头 Matrix Theory Vector Norms -3/39
Introduction Introduction In the context of linear system solution, the error in the solution constitutes a vector. r = b − A × x. ▸ If we do not want to pay attention to individual components of the error, perhaps because there are too many components, then we can combine all errors into a single number. ▸ This is akin to a grade point average which combines all grades into a single number. ▸ Mathematically, this “combining” is accomplished by norms. Matrix Theory Vector Norms - 3/39
Introduction Introduction In the context of linear system solution,the error in the solution constitutes a vector. r=b-A×X. If we do not want to pay attention to individual components of the error,perhaps because there are too many components, then we can combine all errors into a single number. 奇电有这头 Matrix Theory Vector Norms -3/39
Introduction Introduction In the context of linear system solution, the error in the solution constitutes a vector. r = b − A × x. ▸ If we do not want to pay attention to individual components of the error, perhaps because there are too many components, then we can combine all errors into a single number. ▸ This is akin to a grade point average which combines all grades into a single number. ▸ Mathematically, this “combining” is accomplished by norms. Matrix Theory Vector Norms - 3/39
Introduction Introduction In the context of linear system solution,the error in the solution constitutes a vector. r=b-A×X. If we do not want to pay attention to individual components of the error,perhaps because there are too many components, then we can combine all errors into a single number. This is akin to a grade point average which combines all grades into a single number. 命电有这女 Matrix Theory Vector Norms -3/39
Introduction Introduction In the context of linear system solution, the error in the solution constitutes a vector. r = b − A × x. ▸ If we do not want to pay attention to individual components of the error, perhaps because there are too many components, then we can combine all errors into a single number. ▸ This is akin to a grade point average which combines all grades into a single number. ▸ Mathematically, this “combining” is accomplished by norms. Matrix Theory Vector Norms - 3/39