Đang chuẩn bị liên kết để tải về tài liệu:
Burden - Numerical Analysis 5e (PWS, 1993) Episode 2 Part 5

Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) episode 2 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 6.2 Pivoting Strategies 343 Gaussian Elimination with Scaled-Column Pivoting The only steps in this algorithm that differ from those of Algorithm 6.2 are Step 1 For i 1 . n sew max 1 sjsn if Sj 0 then OUTPUT no unique solution exists STOP. set AWW i. Step 2 For i 1 . . n 1 do Steps 3-6. Elimination process. Step 3 Let p be the smallest integer with i p n and a NROW p z _ a NROW j z s NROW p SS s NROW jy The first additional computations required for scaled column pivoting result from the determination of the scale factors n 1 comparisons for each of the n rows for a total of n n 1 comparisons. To determine the correct first interchange n divisions are performed and n 1 comparisons are made. The first interchange determination then adds a total of n n 1 n - 1 comparisons and n divisions. Since the scaling factors are computed only once the second step requires n 2 comparisons and n 1 divisions. Proceeding in a similar manner the scaled column pivoting procedure adds a total of 6.7 n n 1 2 1 q n n Í comparisons k 2 2 and y k 1 - 1 divisions Ố 2 to the Gaussian elimination procedure. The time required to perform a comparison is about the same as an addition subtraction. Since the total time to perform the basic Gaussian elimination procedure is ơ n3 3 multiplications divisions and ớ n3 3 additions subtractions scaled-column pivoting does not add significantly to the computational time requứed to solve a system for large values of n. To emphasize the importance of choosing the scale factors only once consider the amount of additional computation that would be required if the procedure were modified so that new scale factors were determined each time a row interchange decision was to be made. In this case the term n n 1 in Eq. 6.7 would be replaced by ẳ k k - 1 ịn n2 - 1 . Jt 2 344 CHAPTER 6 Direct Methods for Solving Linear Systems As a consequence this modified scaled-column pivoting technique would add O n3 3 comparisons in addition to the ra n l 2 1 divisions. If a