CS 100Lecture 231 CS100J Lecture 23 n Previous Lecture –MatLab and its implementation, continued. n This Lecture –MatLab demonstration

CS 100Lecture 232 MatLab The notation j:k gives a row matrix consisting of the integers from j through k. The notation j:k gives a row matrix consisting of the integers from j through k. >> 1:8 ans = The notation j:b:k gives a row matrix consisting of the integers from j through k in steps of b. The notation j:b:k gives a row matrix consisting of the integers from j through k in steps of b. >> 1:2:10 >> 1:2:10 ans = To get a vector of n linearly spaced points between lo and hi, use linspace(lo,hi,n) To get a vector of n linearly spaced points between lo and hi, use linspace(lo,hi,n) >> linspace(1,3,5) ans = n To transpose a matrix m, write m’ >> (1:3)’ ans = 1 2 3

CS 100Lecture 233 MatLab Variables n MatLab has variables and assignment: >> evens = 2.*(1:8) evens = n To suppress output, terminate line with a semicolon (not to be confused with the vertical concatenation operator) >> odds = evens - 1; >> n To see the value of variable v, just enter expression v: >> odds odds =

CS 100Lecture 234 MatLab Idiom n MatLab has for-loops, conditional statements, subscripting, etc., like Java. But the preferred programming idiom uses uniform operations on matrices and avoids explicit loops, if possible. n Essentially every operation and built in function takes a matrix as an argument.

CS 100Lecture 235 sum, prod, cumsum, cumprod Function sum adds row elements, and the function prod multiplies row elements: Function sum adds row elements, and the function prod multiplies row elements: >> sum(1:1000) ans = >> prod(1:5) ans = Function cumsum computes a row of the partial sums and cumprod computes a row of the partial products: Function cumsum computes a row of the partial sums and cumprod computes a row of the partial products: >> cumprod(1:7) ans = >> cumsum(odds) ans =

CS 100Lecture 236 Compute Pi by Euler Formula n Leonard Euler ( ) derived the following infinite sum expansion:  2 / 6 =  1/j 2 (for j from 1 to  ) >> pi = sqrt( 6.* cumsum(1./ (1:10).^ 2)); >> plot(pi) n To define a function, select New/m-file and type definition : function e = euler(n) function e = euler(n) % Return a vector of approximations to pi. % Return a vector of approximations to pi. e = sqrt( 6.* cumsum(1./ (1:n).^ 2)); e = sqrt( 6.* cumsum(1./ (1:n).^ 2)); n Select SaveAs and save to a file with the same name as the function. n To invoke: >> pi = euler(100); >> plot(pi)

CS 100Lecture 237 Help System n Use on-line help system >> help function >> help function... description of how to define functions description of how to define functions... >> help euler >> help euler Return a vector of approximations to pi. Return a vector of approximations to pi.

CS 100Lecture 238 Compute Pi by Wallis Formula n John Wallis ( ) derived the following infinite sum expansion: (2 * 2) * (4 * 4) * (6 * 6) *... (2 * 2) * (4 * 4) * (6 * 6) *...  / 2 = (1 * 3) * (3 * 5) * (5 * 7) *... (1 * 3) * (3 * 5) * (5 * 7) *... Terms in Numerator Terms in Numerator evens.^ 2 evens.^ 2 n Terms in Denominator odds odds odds odds *3 3*5 5*7 7*9 9* *3 3*5 5*7 7*9 9*11... i.e., odds.* (odds + 2) i.e., odds.* (odds + 2) Quotient Quotient prod( (evens.^ 2)./ (odds.* (odds+2)) ) prod( (evens.^ 2)./ (odds.* (odds+2)) ) Successive approximations to Pi Successive approximations to Pi pi = 2.* cumprod( (evens.^2)./ (odds.* (odds+2)) ) pi = 2.* cumprod( (evens.^2)./ (odds.* (odds+2)) )

CS 100Lecture 239 Wallis Function Function Definition Function Definition function w = wallis(n) % compute successive approximations to pi. evens = 2.* (1:n); odds = evens - 1; odds2 = odds.* (odds + 2); w = 2.* cumprod( (evens.^ 2)./ odds2 ); Contrasting Wallis and Euler approximations Contrasting Wallis and Euler approximations >> plot(1:100, euler(100), 1:100, wallis(100))

CS 100Lecture 2310 Compute Pi by Throwing Darts n Throw random darts at a circle of radius 1 inscribed in a 2-by-2 square. n The fraction hitting the circle should be the ratio of the area of the circle to the area of the square: f =  / 4 f =  / 4 n This is called a Monte Carlo method x y 1 1

CS 100Lecture 2311 Darts The expression rand(h,w) computes an h-by-w matrix of random numbers between 0 and 1. The expression rand(h,w) computes an h-by-w matrix of random numbers between 0 and 1. >> x = rand(1,10); >> y = rand(1,10); Let d2 be the distance squared from the center of the circle. Let d2 be the distance squared from the center of the circle. >> d2 = (x.^ 2) + (y.^ 2); Let in be a row of 0’s and 1’s signifying whether the dart is in (1) or not in (0) the circle. Note 1 is used for true and 0 for false. Let in be a row of 0’s and 1’s signifying whether the dart is in (1) or not in (0) the circle. Note 1 is used for true and 0 for false. >> in = d2 > in = d2 <= 1; Let hits(i) be the number of darts in circle in i tries Let hits(i) be the number of darts in circle in i tries >> hits = cumsum(in); Let f(i) be franction of darts in circle in i tries Let f(i) be franction of darts in circle in i tries >> f = hits./ (1:10); Let pi be successive approximations to pi Let pi be successive approximations to pi >> pi = 4.* f;

CS 100Lecture 2312 Compute Pi by Throwing Needles In 1777, Comte de Buffon published this method for computing  : In 1777, Comte de Buffon published this method for computing  : N needles of length 1 are thrown at random positions and random angles on a plate ruled by parallel lines distance 1 apart. The probability that a needle intersects one of the ruled lines is 2/ . Thus, as N approaches infinity, the fraction of needles intersecting a ruled line approaches 2/ .

CS 100Lecture 2313 Subscripting n Subscripts start at 1, not zero. >> a = [1 2 3; 4 5 6; 7 8 9] ans = >> a(2,2) ans = 5 n A range of indices can be specified: >> a(1:2, 2:3) ans = n A colon indicates all subscripts in range: >> a(:, 2:3) ans =

CS 100Lecture 2314 Control Structures: Conditionals if expression list-of-statements list-of-statementsend if expression list-of-statements list-of-statementselse end if expression list-of-statements list-of-statements elseif expression list-of-statements list-of-statements elseif expression list-of-statements list-of-statementselse end

CS 100Lecture 2315 Control Structures: Loops while expression list-of-statementsend for variable = lo:hi list-of-statementsend for variable = lo:by:hi list-of-statementsend