Lecture 4: Matrices. Math 98, Spring Math 98, Spring 2018 Lecture 4: Matrices 1 / 20

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1 Lecture 4: Matrices Math 98, Spring 2018 Math 98, Spring 2018 Lecture 4: Matrices 1 / 20

2 Reminders Instructor: Eric Hallman Login:!cmfmath98 Password: Class Website: Assignment Submission: Homework 3: 1 Due Feb 8 by 11:59pm on bcourses 2 Collaboration welcome Math 98, Spring 2018 Lecture 4: Matrices 2 / 20

3 HW3 Pointers Use nargin to set default input values. Math 98, Spring 2018 Lecture 4: Matrices 3 / 20

4 Recursion How do we compute the factorial of a number? { 1 n == 0 n! = n (n 1)! n > 0 A for loop will do nicely. function nfac = myfactorial(n) nfac = 1; for i = 1:n nfac = nfac * i; end end Math 98, Spring 2018 Lecture 4: Matrices 4 / 20

5 Recursion How do we compute the factorial of a number? { 1 n == 0 n! = n (n 1)! n > 0 We can also take advantage of the recursive definition, and define our function recursively: function nfac = myfactorial(n) if n == 0 nfac = 1; else nfac = n*myfactorial(n-1); end end Math 98, Spring 2018 Lecture 4: Matrices 5 / 20

6 Recursion How do we sort a list of numbers? There are many ways, but quicksort offers a simple recursive implementation. 1 Pick an element x v to be the pivot element. 2 Divide the rest of the list in two: those smaller than x and those larger than x. 3 output = [quicksort(smaller), x, quicksort(larger)] A few questions we need to answer when working out the details: What are the base cases that we need to handle? What if some numbers are the same size as x? Math 98, Spring 2018 Lecture 4: Matrices 6 / 20

7 Exercise Define the Fibonacci numbers as 0 n == 0 f (n) = 1 n == 1 f (n 1) + f (n 2) n >= 2 Write a recursive function to compute f (n), then write a non-recursive function (for loop) to do the same. The non-recursive function should compute all numbers f (0), f (1),..., f (n). Math 98, Spring 2018 Lecture 4: Matrices 7 / 20

8 Recursion Math 98, Spring 2018 Lecture 4: Matrices 8 / 20

9 Recursion The problem: our recursive definition did lots of unnecessary computation by not using previously computed values. >> fiborec(4) Computing f(4) Computing f(2) Computing f(0) Computing f(1) Computing f(3) Computing f(1) Computing f(2) Computing f(0) Computing f(1) ans = 3 Math 98, Spring 2018 Lecture 4: Matrices 9 / 20

10 Polynomials A quick overview of the main polynomial-related functions in Matlab: polyval roots poly conv polyfit For Matlab purposes, the polynomial a N x N + a N 1 x N a 1 x + a 0 will be stored as the vector p = [a N, a N 1,..., a 1, a 0 ]. Math 98, Spring 2018 Lecture 4: Matrices 10 / 20

11 Polynomials polyval(p,x) computes p(x) roots(p) finds all N roots of the degree-n polynomial p. poly([r 1,..., r N ]) returns a polynomial vector p with roots (r 1,..., r N ). >> p = poly([0,1,2]) >> roots(p) >> polyval(p,3) 6 Example with the polynomial p = x(x 1)(x 2) = x 3 3x 2 + 2x. Math 98, Spring 2018 Lecture 4: Matrices 11 / 20

12 Polynomials conv(p,q) returns the product of two polynomials polyfit(xs,ys,n) fits a degree-n polynomial to data If there are N + 1 data points, the fit is exact If there are more, it s a least-squares fit >> p = [1,1]; q = [1,2]; conv(p,q) >> xs = [1,2]; ys = [3,6]; p = polyfit(xs,ys,1) Example 1: (x + 1)(x + 2) = x 2 + 3x + 2. Example 2: polyfit fits {(1, 3), (2, 6)} with p(x) = 3x. Math 98, Spring 2018 Lecture 4: Matrices 12 / 20

13 Exercise Use polyfit to fit a degree-2 polynomial to the function f (x) = x at the points x = {1, 4, 9}, then use the polynomial with polyval to approximate 3. Math 98, Spring 2018 Lecture 4: Matrices 13 / 20

14 MATLAB = Matrix Laboratory Matrices: like vectors, but with more dimensions? >> ones(2, 3) >> ones(2) Try also: zeros(m,n) rand(m,n) randn(m,n) randi(k,m,n) What happens if you enter only m and leave out n? Math 98, Spring 2018 Lecture 4: Matrices 14 / 20

15 Matrices Other commands to create special types of matrices: >> eye(2) >> v = [3,5]; diag(v) >> diag(v,-1) + 2*eye(3) Math 98, Spring 2018 Lecture 4: Matrices 15 / 20

16 Matrices As with vectors, we have lots of ways to access and manipulate the entries of matrices. Try out the following commands: >> A = reshape(1:12,3,4) >> size(a) % also size(a,1) and size(a,2) >> A >> fliplr(a) % also flipud(a) >> A(2,3) >> A(:,[2,4]) >> p = randperm(3); q = randperm(4); A(p,q) >> reshape(a,2,6) >> A(5,5) = 1 Math 98, Spring 2018 Lecture 4: Matrices 16 / 20

17 Solving Ax = b Two ways to solve Ax = b: >> x = inv(a)*b >> x = A\b Which one should we use? Math 98, Spring 2018 Lecture 4: Matrices 17 / 20

18 Solving Ax = b Math 98, Spring 2018 Lecture 4: Matrices 18 / 20

19 Factorizations [P, L, U] = lu(a) returns matrices such that P A = L U, where L and U are lower and upper triangular, and P is a pivoting matrix (in other words, this is the algorithm for Gaussian elimination). [Q, R] = qr(a) computes the QR factorization of A. If A is m n where m n, use [Q,R] = qr(a,0) to return an m n matrix Q rather than an m m matrix. [V, D] = eig(a) computes the eigenvalue decompositon of A, so that A V = V D. These functions have many variations depending on what extra inputs you give and what outputs you request... check the documentation for more information. Math 98, Spring 2018 Lecture 4: Matrices 19 / 20

20 Exercise Solve the system Ax = b, where A = and b = Do not code A by hand... instead, build it using the vector v = [1, 2, 3, 4, 5]. Math 98, Spring 2018 Lecture 4: Matrices 20 / 20