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MA/CS 375

MA/CS 375. Spring 2002 Lecture 19. Matrices and Systems. Upper Triangular Systems. Upper Triangular Systems. Volunteer: tell us how to solve this? ( hint: this is not difficult ). Example: Upper Triangular Systems. Upper Triangular Systems. 1) Solve the N th equation

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MA/CS 375

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  1. MA/CS 375 Spring 2002 Lecture 19 MA/CS 375 Fall 2002

  2. Matrices and Systems MA/CS 375 Fall 2002

  3. Upper Triangular Systems MA/CS 375 Fall 2002

  4. Upper Triangular Systems Volunteer: tell us how to solve this? (hint: this is not difficult) MA/CS 375 Fall 2002

  5. Example: Upper Triangular Systems MA/CS 375 Fall 2002

  6. Upper Triangular Systems 1) Solve the Nth equation 2) Use the value from step 1 to solve the (N-1)th equation 3) Use the values from steps 1 and 2 to solve the (N-2)th equation 4) keep going until you solve the whole system. MA/CS 375 Fall 2002

  7. Backwards Substitution MA/CS 375 Fall 2002

  8. Matlab Code for Backward Substitution MA/CS 375 Fall 2002

  9. Testing Backwards Substitution MA/CS 375 Fall 2002

  10. Finite Precision Effects • Notice that x(1) depends on the values of x(2),x(3),…,x(N) • Remembering that round off rears its ugly head when we deal with (small+large) numbers • Volunteer to design a U matrix and b vector, which do not give good answers when used in this algorithm. • (goal is to give you some intuition for cases which will break an algorithm) MA/CS 375 Fall 2002

  11. Team Exercise • Who can build a “reasonable”, non-singular, U matrix and b vector which has the worst answer for U\b • 10 Minutes only MA/CS 375 Fall 2002

  12. LU Factorization • By now you should be persuaded that it is not difficult to solve a problem of the form Lx=b or Ux=b using forward/backward elimination • Suppose we are given a matrix A and we are able to find a lower triangular matrix L and an upper triangular matrix U such that: A = LU • Then to solve Ax=b we can solve two simple problems: • Ly = b • Ux = y • (sanity check LUx = L(Ux) = Ly =b) MA/CS 375 Fall 2002

  13. Example LU Factorization • Suppose we are given: • Then we can write A = LU where: • Let’s check that:  MA/CS 375 Fall 2002

  14. LU Factorization (in words) • Construct a sequence of multiplier matrices (M1, M2, M3, M4,…., MN-1) such that: • (MN-1…. M4 M3 M2 M1)A is upper triangle. • A suitable candidate for M1 is the matrix ( 4 by 4 case): MA/CS 375 Fall 2002

  15. LU Factorization cont. A suitable candidate for M2 is the matrix ( 4 by 4 case): MA/CS 375 Fall 2002

  16. LU Factorization cont. A suitable candidate for M3 is the matrix ( 4 by 4 case): MA/CS 375 Fall 2002

  17. LU Factorization cont. • So in this case: U=(M3M2M1)A is upper triangle • Each of the M matrices is lower triangle • The inverse of M2 looks like: • Using the fact that the inverse of the each M matrix just requires us to negate the below diagonal terms • A = (M3M2M1)-1 U = (M1)-1 (M2)-1 (M3)-1 U • Define L = (M1)-1 (M2)-1 (M3)-1 and we are done since the product of lower matrices is a lower matrix  MA/CS 375 Fall 2002

  18. Matlab Routine ForLU Factorization MA/CS 375 Fall 2002

  19. Matlab Routine ForLU Factorization • Build L and U • A is modified in place • Total cost O(N3) ( we can get this by noticing the triply nested loop) MA/CS 375 Fall 2002

  20. Matlab Routine ForLU FactorizationAnd Solution of Ax=b • Build rhs • Solve Ly=b • Solve Ux=y MA/CS 375 Fall 2002

  21. Team Exercise • Code up the factorization • Test your code to make sure it gives thecorrect answer as Matlab • Make sure abs(LU – Aorig) is small • Make sure abs(x - Aorig\b) is small • Make sure the routine actually finished without the break. • Time the code for an N=400 case • Compare with the timings for [Lmat,Umat] = lu(Aorig); • ASK IF YOU DO NOT UNDERSTAND ANY PART OF THIS!! MA/CS 375 Fall 2002

  22. When Basic LU Does Not Work • Remember that we are required to compute 1/A(i,i) for all the successive diagonal terms • What happens when we get to the i’th diagonal term and this turns out to be 0 or very small ?? • At any point we can swap the i’th row with one lower down which does not have a zero (or small entry) in the i’th column. • This is known as partial pivoting. MA/CS 375 Fall 2002

  23. Partial Pivoting Remember before that we constructed a setof M matrices so that is upper triangle Now we construct a sequence of M matricesand P matrices so that:is upper triangle. (MN-1…. M4 M3 M2 M1)A (MN-1 PN-1 …. M4 P4 M3 P3 M2 P2 M1 P1)A MA/CS 375 Fall 2002

  24. LU with Partial Pivoting MA/CS 375 Fall 2002

  25. LU with Partial Pivoting • find pivot • swap rows • Do elimination MA/CS 375 Fall 2002

  26. LU with Partial Pivoting • build matrix A • call our LU routine • check that: • A(piv,:) = L*U • Looks good  MA/CS 375 Fall 2002

  27. Team Exercise • Make sure you all have a copy of the .m files on your laptops (courtesy of Coutsias) • Locate GEpiv • For N=100,200,400,600 • build a random NxN matrix A • time how long GEpiv takes to factorize A • end • Plot the cputime taken per test against N using loglog • On the same graph plot N^3 using loglog • Observation MA/CS 375 Fall 2002

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