Linear Systems and LU
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1 Linear Systems and LU CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Linear Systems and LU 1 / 48
2 Homework 1. Homework 0 due Monday CS 205A: Mathematical Methods Linear Systems and LU 2 / 48
3 Homework 1. Homework 0 due Monday 2. Homework 1: Some coding CS 205A: Mathematical Methods Linear Systems and LU 2 / 48
4 Homework 1. Homework 0 due Monday 2. Homework 1: Some coding Standard in numerical analysis is Fortran CS 205A: Mathematical Methods Linear Systems and LU 2 / 48
5 Code: Gaussian Elimination parameter (n = 4) dimension a(n,n) data (a(1,j),j=1,n) /6.0,-2.0,2.0,4.0/ data (a(2,j),j=1,n) /12.0,-8.0,6.0,10.0/ data (a(3,j),j=1,n) /3.0,-13.0,9.0,3.0/ data (a(4,j),j=1,n) /-6.0,4.0,1.0,-18.0/ do 4 k=1,n-1 do 3 i=k+1,n xmult = a(i,k)/a(k,k) a(i,k) = 0.0 do 2 j=k+1,n a(i,j) = a(i,j) - xmult*a(k,j) 2 continue 3 continue 4 continue c do 6 i=1,n do 5 j=1,n print 7,i,j,a(i,j) 5 continue 6 continue c 7 format (1x, a(,i2,,,i2, ) =,e13.6) stop end CS 205A: Mathematical Methods Linear Systems and LU 3 / 48
6 Course CS 205A: Mathematical Methods Linear Systems and LU 4 / 48
7 Course (LOL) Courtesy A. Debray CS 205A: Mathematical Methods Linear Systems and LU 4 / 48
8 Linear Systems A x = b A R m n x R n b R m CS 205A: Mathematical Methods Linear Systems and LU 5 / 48
9 ( Case 1: Solvable ) ( x y ) = ( 1 1 ) Completely Determined CS 205A: Mathematical Methods Linear Systems and LU 6 / 48
10 Case 2: No Solution ( ) ( x y ) = ( 1 1 Overdetermined ) CS 205A: Mathematical Methods Linear Systems and LU 7 / 48
11 Case 3: Infinitely Many Solutions ( ) ( x y ) = ( 1 1 Underdetermined ) CS 205A: Mathematical Methods Linear Systems and LU 8 / 48
12 No Other Cases Proposition If A x = b has two distinct solutions x 0 and x 1, it has infinitely many solutions. CS 205A: Mathematical Methods Linear Systems and LU 9 / 48
13 Common Misconception Solvability can depend on b! ( ) ( ) ( ) 1 0 x 1 = 1 0 y 0 CS 205A: Mathematical Methods Linear Systems and LU 10 / 48
14 Dependence on Shape Proposition Tall matrices admit unsolvable right hand sides. Proposition Wide matrices admit right hand sides with infinite numbers of solutions. CS 205A: Mathematical Methods Linear Systems and LU 11 / 48
15 For Now All matrices will be: Square Invertible CS 205A: Mathematical Methods Linear Systems and LU 12 / 48
16 Inverting Matrices Do not compute A 1 if you do not need it. Not the same as solving A x = b Can be slow and poorly conditioned CS 205A: Mathematical Methods Linear Systems and LU 13 / 48
17 Example y z = 1 3x y + z = 4 x + y 2z = Permute rows Row scaling Forward/back substitution CS 205A: Mathematical Methods Linear Systems and LU 14 / 48
18 Row Operations: Permutation σ : {1,..., m} {1,..., m} P σ e σ(1) e σ(2) e σ(m) CS 205A: Mathematical Methods Linear Systems and LU 15 / 48
19 Row Operations: Row Scaling S a a a a m CS 205A: Mathematical Methods Linear Systems and LU 16 / 48
20 Row Operations: Elimination Scale row k by constant c and add result to row l. E I + c e l e k CS 205A: Mathematical Methods Linear Systems and LU 17 / 48
21 Solving via Elimination Matrices Reverse order! CS 205A: Mathematical Methods Linear Systems and LU 18 / 48
22 Introducing Gaussian Elimination Big idea: General strategy to solve linear systems via row operations. CS 205A: Mathematical Methods Linear Systems and LU 19 / 48
23 Elimination Matrix Interpretation A x = b E 1 A x = E 1 b E 2 E 1 A x = E 2 E 1 b. E k E 2 E 1 A x = E k E 2 E 1 b I n n A 1 CS 205A: Mathematical Methods Linear Systems and LU 20 / 48
24 ( ) A b = Shape of Systems CS 205A: Mathematical Methods Linear Systems and LU 21 / 48
25 Pivot CS 205A: Mathematical Methods Linear Systems and LU 22 / 48
26 Row Scaling 1 CS 205A: Mathematical Methods Linear Systems and LU 23 / 48
27 Forward Substitution CS 205A: Mathematical Methods Linear Systems and LU 24 / 48
28 Forward Substitution CS 205A: Mathematical Methods Linear Systems and LU 25 / 48
29 Upper Triangular Form CS 205A: Mathematical Methods Linear Systems and LU 26 / 48
30 Back Substitution CS 205A: Mathematical Methods Linear Systems and LU 27 / 48
31 Back Substitution CS 205A: Mathematical Methods Linear Systems and LU 28 / 48
32 Back Substitution CS 205A: Mathematical Methods Linear Systems and LU 29 / 48
33 Steps of Gaussian Elimination 1. Forward substitution: For each row i = 1, 2,..., m Scale row to get pivot 1 For each j > i, subtract multiple of row i from row j to zero out pivot column 2. Backward substitution: For each row i = m, m 1,..., 1 For each j < i, zero out rest of column CS 205A: Mathematical Methods Linear Systems and LU 30 / 48
34 Total Running Time O(n 3 ) CS 205A: Mathematical Methods Linear Systems and LU 31 / 48
35 Problem ( 0 1 ) A = 1 0 CS 205A: Mathematical Methods Linear Systems and LU 32 / 48
36 Even Worse ( ε 1 ) A = 1 0 CS 205A: Mathematical Methods Linear Systems and LU 33 / 48
37 Pivoting Pivoting Permuting rows and/or columns to avoid dividing by small numbers or zero. Partial pivoting Full pivoting CS 205A: Mathematical Methods Linear Systems and LU 34 / 48
38 Reasonable Use Case A x 1 = b 1 A x 2 = b 2. Can we restructure A to make this more efficient? Does each solve take O(n 3 ) time? CS 205A: Mathematical Methods Linear Systems and LU 35 / 48
39 Observation Steps of Gaussian elimination depend only on structure of A. Avoid repeating identical arithmetic on A? CS 205A: Mathematical Methods Linear Systems and LU 36 / 48
40 Another Clue: Upper Triangular Systems CS 205A: Mathematical Methods Linear Systems and LU 37 / 48
41 After Back Substitution No need to subtract the 0 s explicitly! O(n) time CS 205A: Mathematical Methods Linear Systems and LU 38 / 48
42 Next Pivot: Same Observation Observation Triangular systems can be solved in O(n 2 ) time. CS 205A: Mathematical Methods Linear Systems and LU 39 / 48
43 Upper Triangular Part of A A x = b. M k M 1 A x = M k M 1 b Define: U M k M 1 A CS 205A: Mathematical Methods Linear Systems and LU 40 / 48
44 Lower Triangular Part U = M k M 1 A A = (M1 1 M 1 k )U LU CS 205A: Mathematical Methods Linear Systems and LU 41 / 48
45 Why Is L Triangular? S diag(a 1, a 2,...) E I + c e l e k Proposition The product of triangular matrices is triangular. CS 205A: Mathematical Methods Linear Systems and LU 42 / 48
46 Solving Systems Using LU A x = b LU x = b 1. Solve L y = b using forward substitution. 2. Solve U x = y using backward substitution. O(n 2 ) (given LU factorization) CS 205A: Mathematical Methods Linear Systems and LU 43 / 48
47 LU: Compact Storage U U U U L U U U L L U U L L L U Assumption: Diagonal elements of L are 1. Warning: Do not multiply this matrix! CS 205A: Mathematical Methods Linear Systems and LU 44 / 48
48 Computing LU Factorization Small modification of forward substitution step to keep track of L. 1 1 See textbook for pseudocode. CS 205A: Mathematical Methods Linear Systems and LU 45 / 48
49 Question Does every A admit a factorization A = LU? CS 205A: Mathematical Methods Linear Systems and LU 46 / 48
50 Recall: Pivoting Pivoting Permuting rows and/or columns to avoid dividing by small numbers or zero. Partial pivoting Full pivoting CS 205A: Mathematical Methods Linear Systems and LU 47 / 48
51 Pivoting by Swapping Columns (E k E 1 ) elimination A (P 1 P l ) (Pl P1 ) x permutations inv. permutations = (E k E 1 ) b A = LUP Next CS 205A: Mathematical Methods Linear Systems and LU 48 / 48
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