tailieunhanh - numerical mathematics and scientific computation volume 1 Episode 5

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Error Propagation and Condition Numbers 135 Definition . An algorithm is stable if the computed solution w satisfies where Cl a not too large constant u is the unit roundoff and K is the condition number of the problem. By the definition of the condition number K it follows that backward stability implies forward stability but the converse is not true. Sometimes it is necessary to weaken the definition of stability. Often an algorithm can be considered stable if it produces accurate solutions for well-conditioned problems. Such an algorithm can be called weakly stable. Weak stability may be sufficient for giving confidence in an algorithm. Example . Higham 24 The outer product of two vectors x y G R is A xyT flij where ữij XiPj. In floating point arithmetic we compute A fl xyT flij where ãij Xi yffl ỏij ij u and so A xy1 ỵ. A u xyT . This is a satisfactory result for many purposes but the computation is not backward stable. The computed A is not in general a rank one matrix and thus it is not possible to find perturbations and Ay so that Ẵ x Ax x Ay T. In the method of normal equations for computing the solution of a linear least squares problem one first forms the matrix ATA. This product matrix can be expressed in outer form as m V . yy ữioĩ i i where áị is the ith row of A . AT ill 2 0-m By the result above it follows that this computation is not backward stable . it is not true that fl AT A A E T A E for some small error matrix E. In order to avoid loss of significant information double precision need to be used. Backward stability is easier to prove when there is a sufficiently large set of input data compared to the number of output data. This makes it harder to show backward stability when the input data is structured rather than general. goodbreak Example . A Toeplitz matrix T is a matrix whose entries are constant along every diagonal T i to tl . tn 1 t-1 to tn 2 T t-n l t-n 2 to G R and is defined by the