NUMERICAL MATHEMATICS & COMPUTING 7th Edition

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1 NUMERICAL MATHEMATICS & COMPUTING 7th Edition Ward Cheney/David Kincaid c UT Austin Engage Learning: Thomson-Brooks/Cole wwwengagecom wwwmautexasedu/cna/nmc7 October 16, 2011 Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING wwwengagecom[10pt] 7th EditionOctober 16, wwwmautexasedu/ / 33

2 Numerical Mathematics & Computing, 7th Ed NUMERICAL MATHEMATICS & COMPUTING 7th Edition Ward Cheney/David Kincaid c UT Austin Engage Learning: Thomson-Brooks/Cole wwwengagecom wwwmautexasedu/cna/nmc7 October 16, 2011 c 2012 Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING wwwengagecom[10pt] 7th EditionOctober 16, wwwmautexasedu/ / 33

3 23 Tridiagonal and Banded Systems In many applications, extremely large linear systems that have a banded structure are encountered Banded matrices often occur in solving ordinary and partial differential equations It is advantageous to develop computer codes specifically designed for such linear systems, since they reduce the amount of storage used Of practical importance is the tridiagonal system Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING wwwengagecom[10pt] 7th EditionOctober 16, wwwmautexasedu/ / 33

4 d 1 c 1 a 1 d 2 c 2 a 2 d 3 c 3 a i 1 d i c i a n 2 d n 1 c n 1 a n 1 d n x 1 x 2 x 3 x i x n 1 x n = b 1 b 2 b 3 b i b n 1 Here, all the nonzero elements in the coefficient matrix must be on the main diagonal or on the two diagonals just above and below the main diagonal (usually called superdiagonal and subdiagonal, respectively): All elements not in the displayed diagonals are 0 s b n (1) Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING wwwengagecom[10pt] 7th EditionOctober 16, wwwmautexasedu/ / 33

5 A tridiagonal matrix is characterized by the condition a ij = 0 if i j 2 In general, a matrix is said to have a banded structure if there is an integer k (less than n) such that a ij = 0 whenever i j k The storage requirements for a banded matrix are less than those for a general matrix of the same size Thus, an n n diagonal matrix requires only n memory locations in the computer, and a tridiagonal matrix requires only 3n 2 Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING wwwengagecom[10pt] 7th EditionOctober 16, wwwmautexasedu/ / 33

6 This fact is important if banded matrices of very large order are being used For banded matrices, the Gaussian elimination algorithm can be made very efficient if it is known beforehand that pivoting is unnecessary This situation occurs often enough to justify special procedures Here, we develop a code for the tridiagonal system and give a listing for the pentadiagonal system (in which a ij = 0 if i j 3) Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING wwwengagecom[10pt] 7th EditionOctober 16, wwwmautexasedu/ / 33

7 Tridiagonal Systems The routine to be described now is called procedure Tri It is designed to solve a system of n linear equations in n unknowns, as shown in System (1) Both the forward elimination phase and the back substitution phase are incorporated in the procedure, and no pivoting is used Thus, naive Gaussian elimination is used Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING wwwengagecom[10pt] 7th EditionOctober 16, wwwmautexasedu/ / 33

8 Step 1: we subtract a 1 /d 1 times row 1 from row 2, thus creating a 0 in the a 1 position Only the entries d 2 and b 2 are altered Observe that c 2 is not altered Step 2: the process is repeated, using the new row 2 as the pivot row Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING wwwengagecom[10pt] 7th EditionOctober 16, wwwmautexasedu/ / 33

9 Here is how the d i s and b i s are altered in each step: d 2 d 2 (a 1 /d 1 ) c 1 b 2 b 2 (a 1 /d 1 ) b 1 In general, we obtain d i d i (a i 1 /d i 1 ) c i 1 b i b i (a i 1 /d i 1 ) b i 1 (2 i n) Ward Cheney/David Kincaid c (UT Austin[10pt] NUMERICAL Engage Learning: MATHEMATICS Thomson-Brooks/Cole & COMPUTING wwwengagecom[10pt] 7th EditionOctober 16, wwwmautexasedu/ / 33

10 Tridiagonal Systems At the end of the forward elimination phase, the form of the system is as follows: d 1 c 1 x 1 b 1 d 2 c 2 x 2 b 2 d 3 c 3 x 3 b 3 d i c i d n 1 c n 1 d n x i x n 1 x n = b i b n 1 b n Of course, the b i s and d i s are not as they were at the beginning of this process, but the c i s are / 33

11 The back substitution phase solves for x n, x n 1,, x 1 as follows: x n b n /d n x n 1 (b n 1 c n 1 x n ) /d n 1 Finally, we obtain x i (b i c i x i+1 ) /d i (i = n 1, n 2,, 1) / 33

12 Tridiagonal Systems In procedure Tri for a tridiagonal system, we use single-dimensioned arrays (a i ), (d i ), and (c i ) for the diagonals in the coefficient matrix and array (b i ) for the right-hand side, and store the solution in array (x i ) Notice that the original data in arrays (d i ) and (b i ) have been changed / 33

13 Pseudocode Tri procedure Tri(n, (a i ), (d i ), (c i ), (b i ), (x i )) integer i, n; real xmult real array (a i ) 1:n, (d i ) 1:n, (c i ) 1:n, (b i ) 1:n, (x i ) 1:n for i = 2 to n xmult a i 1 /d i 1 d i d i (xmult)c i 1 b i b i (xmult)b i 1 end for x n b n /d n for i = n 1 to 1 step 1 do x i (b i c i x i+1 )/d i end for end procedure Tri / 33

14 Tridiagonal Systems A symmetric tridiagonal system arises in the cubic spline development A general symmetric tridiagonal system has the form d 1 c 1 x 1 c 1 d 2 c 2 x 2 c 2 d 3 c 3 x 3 c i 1 d i c i c n 2 d n 1 c n 1 c n 1 d n x i x n 1 x n = b 1 b 2 b 3 b i b n 1 b n (2) / 33

15 One could overwrite the right-hand side vector b with the solution vector x as well Thus, a symmetric linear system can be solved with a procedure call of the form call Tri(n, (c i ), (d i ), (c i ), (b i ), (b i )) This reduces the number of linear arrays from five to three / 33

16 Strictly Diagonal Dominance Since procedure Tri does not involve pivoting, it is natural to ask whether it is likely to fail Simple examples can be given to illustrate failure because of attempted division by zero even though the coefficient matrix in System (1) is nonsingular On the other hand, it is not easy to give the weakest possible conditions on this matrix to guarantee the success of the algorithm We content ourselves with one property that is easily checked and commonly encountered If the tridiagonal coefficient matrix is diagonally dominant, then procedure Tri does not encounter zero divisors / 33

17 Strictly Diagonal Dominance Definition A general matrix A = (a ij ) n n is strictly diagonally dominant if a ii > n j=1 j i a ij (1 i n) In the case of the tridiagonal system of System (1), strict diagonal dominance means simply that (with a 0 = a n = 0) d i > a i 1 + c i (1 i n) / 33

18 Let us verify that the forward elimination phase in procedure Tri preserves strictly diagonal dominance The new coefficient matrix produced by Gaussian elimination has 0 elements where the a i s originally stood, and new diagonal elements are determined recursively by { d 1 = d 1 d i = d i ( a i 1 / d i 1 ) ci 1 (2 i n) where d i denotes a new diagonal element The c i elements are unaltered Now we assume that d i > a i 1 + c i, and we want to be sure that d i > c i / 33

19 Obviously, this is true for i = 1 because d 1 = d 1 If it is true for index i 1 (that is, d i 1 > c i 1 ), then it is true for index i because d i = d i ( a i 1 / d i 1 ) ci 1 d i a i 1 c i 1 / d i 1 > a i 1 + c i a i 1 = c i While the number of long operations in Gaussian elimination on full matrices is O(n 3 ), it is only O(n) for tridiagonal matrices Also, the scaled pivoting strategy is not needed on strictly diagonally dominant tridiagonal systems / 33

20 Pentadiagonal Systems The principles illustrated by procedure Tri can be applied to matrices that have wider bands of nonzero elements procedure Penta is given here to solve the five-diagonal system: d 1 c 1 f 1 a 1 d 2 c 2 f 2 e 1 a 2 d 3 c 3 f 3 e 2 a 3 d 4 c 4 f 4 e i 2 a i 1 d i c i f i e n 4 a n 3 d n 2 c n 2 f n 2 e n 3 a n 2 d n 1 c n 1 e n 2 a n 1 d n x 1 x 2 x 3 x 4 x i x n 2 x n 1 x n = b 1 b 2 b 3 b 4 b i b n 2 b n 1 b n / 33

21 Pentadiagonal Systems (cont) In the pseudocode, the solution vector is placed in array (x i ) Also, one should not use this routine if n 4 (Why?) procedure Penta(n, (e i ), (a i ), (d i ), (c i ), (f i ), (b i ), (x i )) integer i, n; real r, s, xmult real array (e i ) 1:n, (a i ) 1:n, (d i ) 1:n, (c i ) 1:n, (f i ) 1:n, (b i ) 1:n, (x i ) 1:n r a 1 s a 2 t e / 33

22 Pentadiagonal Systems (cont) for i = 2 to n 1 xmult r/d i 1 d i d i (xmult)c i 1 c i c i (xmult)f i 1 b i b i (xmult)b i 1 xmult t/d i 1 r s (xmult)c i 1 d i+1 d i+1 (xmult)f i 1 b i+1 b i+1 (xmult)b i 1 s a i+1 t e i end for / 33

23 xmult r/d n 1 d n d n (xmult)c n 1 x n (b n (xmult)b n 1 )/d n x n 1 (b n 1 c n 1 x n )/d n 1 for i = n 2 to 1 step 1 x i (b i f i x i+2 c i x i+1 )/d i end for end procedure Penta / 33

24 Pentadiagonal Systems (cont) To be able to solve symmetric pentadiagonal systems with the same code and with a minimum of storage, we have used variables r, s, and t to store temporarily some information rather than overwriting into arrays This allows us to solve a symmetric pentadiagonal system with a procedure call of the form call Penta(n, (f i ), (c i ), (d i ), (c i ), (f i ), (b i ), (b i )) This reduces the number of linear arrays from seven to four Of course, the original data in some of these arrays are corrupted The computed solution is stored in the (b i ) array Here, we assume that all linear arrays are padded with zeros to length n in order not to exceed the array dimensions in the pseudocode / 33

25 Block Pentadiagonal Systems Many mathematical problems involve matrices with block structures In many cases, there are advantages in exploiting the block structure in the numerical solution This is particularly true in solving partial differential equations numerically We can consider a pentadiagonal system as a block tridiagonal system / 33

26 D 1 C 1 A 1 D 2 C 2 A 2 D 3 C 3 A i 1 D i C i A n 2 D n 1 C n 1 A n 1 D n X 1 X 2 X 3 X i X n 1 X n = B 1 B 2 B 3 B i B n 1 B n where [ d2i 1 c D i = 2i 1 a 2i 1 d 2i ] [ e2i 1 c, A i = 2i 1 0 e 2i ] [ f2i 1 0, C i = c 2i 1 f 2i ] Here, we assume that n is even, say n = 2m If n is not even, then the system can be padded with an extra equation x n+1 = 1 so that the number of rows is even / 33

27 Block Pentadiagonal Systems The algorithm for this block tridiagonal system is similar to the one for tridiagonal systems Hence, we have the forward elimination phase { Di D i A i 1 D 1 i 1 C i 1 B i B i A i 1 D 1 i 1 B i 1 (2 i m) The back substitution phase is { Xn D 1 n B n X i D 1 i (B i C i X i+1 ) (m 1 i 1) Here, D 1 i = 1 where = d 2i d 2i 1 a 2i 1 c 2i 1 [ d 2i a 2i 1 c 2i 1 d 2i 1 ] / 33

28 Code for solving a pentadiagonal system using this block procedure is left as an exercise The results from the block pentadiagonal code are the same as those from the procedure Penta, except for roundoff error Also, this procedure can be used for symmetric pentadiagonal systems (in which the subdiagonals are the same as the superdiagonals) / 33

29 Later, we discuss two-dimensional elliptic partial differential equations For example, the Laplace equation is defined on the unit square A 3 3 mesh of points are placed over the unit square region, and they are ordered in the natural ordering (left-to-right and up) Figure: Mesh points in natural order / 33

30 In the Laplace equation, second partial derivatives are approximated by second-order centered finite difference formulas This results in an 9 9 system of linear equations having a sparse coefficient matrix with this nonzero pattern: A = / 33

31 Here, nonzero entries in the matrix are indicated by the symbol, and zero entries are a blank This matrix is block tridiagonal, and each nonzero block is either tridiagonal or diagonal Other orderings of the mesh points result in sparse matrices with different patterns / 33

32 Summary 23 For banded systems, such as tridiagonal, pentadiagonal, and others, it is usual to develop special algorithms for implementing Gaussian elimination, since partial pivoting is not needed in many applications The forward elimination procedure for a tridiagonal linear system A = tridiagonal[(a i ), (d i ), (c i )] is d i d i (a i 1 /d i 1 ) c i 1 b i b i (a i 1 /d i 1 ) b i 1 (2 i n) The back substitution procedure is x i (b i c i x i+1 ) /d i (i = n 1, n 2,, 1) / 33

33 A strictly diagonally dominant matrix A = (a ij ) n n is one in which the magnitude of the diagonal entry is larger than the sum of the magnitudes of the off-diagonal entries in the same row and this is true for all rows, namely, a ii > n j=1 j i a ij (1 i n) For strictly diagonally dominant tridiagonal coefficient matrices, partial pivoting is not necessary because zero divisors are not encountered The forward elimination and back substitution procedures for a pentadiagonal linear system A = pentadiagonal [(e i ), (a i ), (d i ), (c i ), (f i )] is similar to that for a tridiagonal system / 33