1 / 71

Brief Guide to msharpmath ( cemmsharp )

2013. 03. 12. Brief Guide to msharpmath ( cemmsharp ). Lecture by www.msharpmath.com. Prof. Charn -Jung Kim Seoul National University. Why msharpmath ?. Need a paradigm shift : grammar-based to human thinking. Find a root of. MATLAB grammar-based. M#Math human-thinking.

ali-gentry
Télécharger la présentation

Brief Guide to msharpmath ( cemmsharp )

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 2013. 03. 12 Brief Guide tomsharpmath (cemmsharp) Lecture by www.msharpmath.com Prof. Charn-Jung Kim Seoul National University

  2. Why msharpmath ? Need a paradigm shift : grammar-based to human thinking Find a root of MATLAB grammar-based M#Math human-thinking function f = myfun(x) f = x^2 – 3*x + 2 >> fzero( @myfun, 0 ) #> solve .x ( x^2 = 3*x – 2 );

  3. How to download msharpmath • HomePage : www.msharpmath.com Click here

  4. msharpmath Screen

  5. Mathematics Calculator #> 3 + 4 * 5 ans = 23 #> ans/1000 + ans*100 ans = 2300.023 cursor in msharpmath #>3 + 4 * 5 ans = 23 #>ans / 1000 + ans * 100 ans = 2300.023 default variable name for results Variable “ans” can be used in subsequent commands Variable “ans” is changed to a new value

  6. ■ Unit conversion x.f// Fahrenheit to other temperatures x.c// Celsius to other temperatures x.ft// feet in meter x.acre// acre in square meter and many more #> 77.f 25 [C] 77 F 536.67 R 298.15 K ans = 298.15 #> 1.ft * 1 ans = 0.3048 #> 1.acre * 1 ans = 4046.8564 expression-look-alike !!!

  7. ■ Root finding solve.x{a} ( f(x)=g(x) ) // a root of f(x)=g(x) near a solve.x.y {a,b} ( u(x,y)=0, v(x,y)=0 ) solve.x.y.z {a,b,c} ( u(x,y,z)=0, v(x,y,z)=0, w(x,y,z)=0 ) #> solve.x ( exp(x) = x+3 ) ans = 1.5052415 #> solve.x ( exp(x) = x+3 ).span(-10,10) ans = [ -2.94753 -5.66165e-008 ] [ 1.50524 -1.89955e-008 ] #> solve.a.b ( a+b = 3, a*b = 2 ) ans = [ 2 ] [ 1 ] expression-look-alike !!!

  8. ■ Integration int.x(a,b) ( f(x) ) int.x(a,b).y(u(x),v(x)) ( f(x,y) ) int.x(a,b).y(u(x),v(x)).z(p(x,y),q(x,y)) ( f(x,y,z) ) #> int.x(1,2).y(x,2*x+1) ( exp(-x-y)*sin(x+y) ) ans = -0.0089769673 expression-look-alike !!!

  9. ● data type : double(real numbers) double msharpmath (m#math) double(real number) complex poly (polynomial) matrix vertex signal image ... MATLAB matrix real numbersare treated by 1 x 1 matrix

  10. Operators for double (1) double |a| // absolute -a // unary minus !a // return 1 if a=0, otherwise return 0 #> a = -1.7 a = -1.7 #> |a| ans = 1.7 #> -a ans = 1.7 #> !a ans = 0 #> !0 ans = 1

  11. Operators for double (2) double a + b // addition a - b // subtraction a * b // multiplication a / b // right division a \ b // left division a ^ b // power a % b // remainder, equal to .mod(a,b) #> 2 + 3 ans = 5 #> 2 - 3 ans = -1 #> 2 * 3 ans = 6 #> 2 / 3 ans = 0.66666667 #> 2 \ 3 ans = 1.5 #> 2 ^ 3 ans = 8 #> 13 % 5 ans = 3

  12. Operators for double (3) double Compound operations a += b // a = a+b a -= b // a = a-b a *= b // a = a*b a /= b // a = a/b a ^= b // a = a^b a %= b // a = a%b #> a = 3 a = 3 #> a += 4 a = 7 #> a ^= 2 a = 49

  13. ● data type : poly(polynomial) poly msharpmath (m#math) double (real number) complex poly (polynomial) matrix vertex signal image ... MATLAB matrix polynomialsare treated by functions such as polyval(p,x) polyder(p,x)

  14. ascending poly poly poly( a0, a1, a2, a3, … , an ) [ a0, a1, a2, a3, … , an ] .apoly p(x) = a0 + a1 x + a2 x^2 + a3 x^3 + … + an x^n #> poly( 1,2,3,4,5 ) ans = poly( 1 2 3 4 5 ) = 5x^4 +4x^3 +3x^2 +2x +1 #> [ 1,2,3,4,5 ].apoly ans = poly( 1 2 3 4 5 ) = 5x^4 +4x^3 +3x^2 +2x +1

  15. descending poly poly [ an, … , a3, a2, a1, a0 ] .dpoly .[ an, … , a3, a2, a1, a0 ] .[ a ] // constant polynomial .[ a, b ] // 1st-order polynomial, ax+b .[ a, b, c ] // 2nd-order polynomial, ax^2+bx+c .[ a, b, c, d ] // 3rd-order polynomial, ax^3+bx^2+cx+d Note : a leading dot for descending polynomial !!! #> .[ 1,2,3,4,5 ] ans = poly( 5 4 3 2 1 ) = x^4 +2x^3 +3x^2 +4x +5 #> .[ 1,3,2 ] y = x^2 + 3x + 2 roots [ -1 ] [ -2 ] vertex at ( -1.5, -0.25 ) y_minimum = -0.25 y-axis intersect ( 0, 2 ) symmetry at x = -1.5 directrix at y = -0.5 focus at ( -1.5, 0 ) #> .[] ans = poly( 0) = 0 #> p = q = r = .[] p = poly( 0) = 0 q = poly( 0) = 0 r = poly( 0) = 0 .[] is a zero polynomial

  16. Operations for poly (1) poly -p // unary minus p’ // dp(x)/dx, derivative p~ // int_0^x p(x) dx, integration p.’ // p(x)-p(x-1), finite difference p.~ // p(1)+p(2)+…+p(n), cumulative sum #> .[1,0,0,0]' ans = poly( 0 0 3 ) = 3x^2 #> .[1,0,0,0].' ans = poly( 1 -3 3 ) = 3x^2 -3x +1 #> .[1,0,0,0]~ ans = poly( 0 0 0 0 0.25 ) = 0.25x^4 #> .[1,0,0,0].~ ans = poly( 0 0 0.25 0.5 0.25 ) = 0.25x^4 +0.5x^3 +0.25x^2

  17. Operations for poly (2) poly p + q // addition p - q // subtraction p * q // multiplication p / q // quotient(right division) q \ p // quotient(left division) p % q // remainder P %% q // synthetic division #> p = .[3,4,5] p = poly( 5 4 3 ) = 3x^2 +4x +5 #> q = .[1,2] q = poly( 2 1 ) = x +2 #> p + q ans = poly( 7 5 3 ) = 3x^2 +5x +7 #> p - q ans = poly( 3 3 3 ) = 3x^2 +3x +3 #> p * q ans = poly( 10 13 10 3 ) = 3x^3 +10x^2 +13x +10 #> p / q ans = poly( -2 3 ) = 3x -2 #> p % q ans = poly( 9 ) = 9

  18. Operations for poly (3) poly p(x) // double p(z) // complex p(q) // polynomial p(A) // matrix (element-by-element) p[A] // matrix polynomial, a0 I+a1 A+a2 A^2 + … #> p = .[1,0,0] p = poly( 0 0 1 ) = x^2 #> p(3) ans = 9 #> p(3+1i) ans = 8 + 6! #> p( .[3,4] ) ans = poly( 16 24 9 ) = 9x^2 +24x +16

  19. Operations for poly (4) poly p(x) // double p(z) // complex p(q) // polynomial p(A) // matrix (element-by-element) p[A] // matrix polynomial, a0 I+a1 A+a2 A^2 + … #> p( [1,2;3,4] ) ans = [ 1 4 ] [ 9 16 ] #> p[ [1,2;3,4] ] ans = [ 7 10 ] [ 15 22 ]

  20. Member functions poly p.solve// roots p.pow(k) // replace x by x^k, p[i*k]=p[i] p.shift(k) // replace x by x-k p.up(k) // multiply by x^k, p[i+k]=p[i] p.newton(u) // a0+a1(x-u1)+a2(x-u1)(x-u2)+… #> .[1,2,5].solve ans = [ -1 - i 2 ] [ -1 + i 2 ] #> p = .[1,0,0]; p.pow(3); p = x^2 ans = x^6 #> p.shift(2) ans = = x^2 -4x +4 #> p.up(3) ans = x^5

  21. exercise for poly poly #> .[3,-5,7]^3 - poly(0,5,4)*.[1,-6,15,24] + 32*.[1,-17]^3 ans = poly( -156873 26889 -837 -753 433 -139 27 ) = 27x^6 -139x^5 +433x^4 -753x^3 -837x^2 +26889x -156873

  22. ● data type : complex complex msharpmath (m#math) double (real number) complex poly (polynomial) matrix vertex signal image ... MATLAB matrix complexnumbers are treated by 1 x 1 matrix

  23. Creating complex numbers complex 1i, 1j, 1! // i or j works only after digits 0-9 x+y! // postfix ! for pure imaginary part (r+t!).cyl// r cos(t) + r sin(t) ! #> 1i ans = 0 + 1! #> 1j ans = 0 + 1! #> 1! ans = 0 + 1! #> 3+pi! ans = 3 + 3.14159! #> (3+pi!).cyl ans = -3 + 3.67394e-016! ‘i’ after digits ‘j’ after digits ‘!’ after both digits and variables In polar coordinate

  24. Elements of complex complex z = x+y! |z| // absolute value ~z // conjugate, x-y! z.x, z.real// real part, x z.y, z.imag// imaginary part, x z.r, z.abs // equal to |z|=sqrt(x*x+y*y) z.t, z.arg // phase part in radian, tan^-1(y/x) z.deg // phase part in degree #> z = 3+4i z = 3 + 4! #> |z| ans = 5 #> ~z ans = 3 - 4! #> z.y ans = 4 #> z.r ans = 5 #> z.t ans = 0.92729522 #> z.deg ans = 53.130102

  25. Be cautious with sqrt(-1) complex A function “sqrt” returns ‘double’ for ‘double’, ‘complex’ for ‘complex’ #> sqrt(-1) ans = -1.#IND #> sqrt(-1+0i) ans = 6.12323e-017 + 1!

  26. ● data type : matrix matrix msharpmath (m#math) double (real number) complex poly (polynomial) matrix vertex signal image ... MATLAB matrix Matrix Laboratory All the data are processed based on matrix

  27. Matrix by a pair of brackets [] matrix [] // 0 x 0 matrix [ a11,a12,…,a1n; a21,a22,… ; … ; a_m1,…, a_mn ] // separate rows by a semi-colon ; // separate elements by a comma , #> A = [ 1,2,3; 4,5; 6 ] A = [ 1 2 3 ] [ 4 5 0 ] [ 6 0 0 ] #> [ A,-A; 2*A ] ans = [ 1 2 3 -1 -2 -3 ] [ 4 5 0 -4 -5 -0 ] [ 6 0 0 -6 -0 -0 ] [ 2 4 6 0 0 0 ] [ 8 10 0 0 0 0 ] [ 12 0 0 0 0 0 ] All zeros are assigned for undeclared elements Matrices can be also elements of a matrix

  28. Matrix by a colon operator matrix a:b // [ a,a+1,a+2, … ] if a < b a:h:b // [ a,a+h,a+2h, … ] #> 1:5 ans = [ 1 2 3 4 5 ] #> 5:1 ans = [ 5 4 3 2 1 ] #> 1 : 0.2 : 2 ans = [ 1 1.2 1.4 1.6 1.8 2 ] // [ a,a-1,a-2, … ] if a > b

  29. Matrix by built-in functions matrix .I(m,n=m) or .eye(m,n=m) .ones(m,n=m) .zeros(m,n=m) #> .I(2,3) ans = [ 1 0 0 ] [ 0 1 0 ] #> .ones(4) ans = [ 1 1 1 1 ] [ 1 1 1 1 ] [ 1 1 1 1 ] [ 1 1 1 1 ] #> .zeros(A) ans = [ 0 0 0 ] [ 0 0 0 ] [ 0 0 0 ] equal to eye(2,3) equal to ones(4,4) Same size with A

  30. Operations for matrices matrix + // addition - // subtraction * // multiplication / // right division, A/B = A*B^-1 \ // left division, A\B = A^-1*B ^ // power, A^3 = A*A*A ‘ // complex conjugate transpose #> A = [ 1,2; 3,4 ]; b = [ 5; 6 ] A = [ 1 2 ] [ 3 4 ] b = [ 5 ] [ 6 ] #> A \ b ans = [ -4 ] [ 4.5 ]

  31. Operations with scalars matrix Scalar is expanded as a matrix #> A = [ 1,2,3; 4,5,6 ] A = [ 1 2 3 ] [ 4 5 6 ] #> A + 5 ans = [ 6 7 8 ] [ 9 10 11 ] #> A ^ 5 runtime error#90700: matrix not square #> A = 5 A = [ 5 5 5 ] [ 5 5 5 ] in MATLAB, A = 5 results in a scalar (1 x 1 matrix)

  32. Element-by-element operations matrix .* // element-by-element multiplication ./ // element-by-element right division .\ // element-by-element left division .^ // element-by-element power .‘ // pure transpose #> A = [ 1,2; 3,4 ]; B = [ 5,6; 7,8 ] A = [ 1 2 ] [ 3 4 ] B = [ 5 6 ] [ 7 8 ] #> A .* B ans = [ 5 12 ] [ 21 32 ] #> A ./ B ans = [ 0.2 0.333333 ] [ 0.428571 0.5 ] #> A .\ B ans = [ 5 3 ] [ 2.33333 2 ] #> B .^ A ans = [ 5 36 ] [ 343 4096 ]

  33. Dot and Kronecker Products matrix A ** B // dot product, .dot(A,B) A ^^ B // Kronecker product, .kron(A,B) #> A = [ 1,10,-100; -10,1;1000 ]; B = [ 1,2,3; 4,5,6; 7,8,9 ] A = [ 1 10 -100 ] [ -10 1 0 ] [ 1000 0 0 ] B = [ 1 2 3 ] [ 4 5 6 ] [ 7 8 9 ] #> A ^^ B ans = [ 1 2 3 10 20 30 -100 -200 -300 ] [ 4 5 6 40 50 60 -400 -500 -600 ] [ 7 8 9 70 80 90 -700 -800 -900 ] [ -10 -20 -30 1 2 3 0 0 0 ] [ -40 -50 -60 4 5 6 0 0 0 ] [ -70 -80 -90 7 8 9 0 0 0 ] [ 1000 2000 3000 0 0 0 0 0 0 ] [ 4000 5000 6000 0 0 0 0 0 0 ] [ 7000 8000 9000 0 0 0 0 0 0 ] #> B ** B ans = 285

  34. Concatenations matrix A | B // horizontal concatenation, .horzcat(A,B,C,…) A _ B // vertical concatenation, .vertcat(A,B,C,…) A \_ B // diagonal concatenation, .diagcat(A,B,C,…) #> A = [ 1,2,3 ]; B = [ -4,5 ] A = [ 1 2 3 ] B = [ -4 5 ] #> A | B ans = [ 1 2 3 -4 5 ] #> A _ B ans = [ 1 2 3 ] [ -4 5 0 ] #> A \_ B ans = [ 1 2 3 0 0 ] [ 0 0 0 -4 5 ]

  35. Set operations matrix A ++ B // set union A -- B // set difference A /\ B // set intersection #> A = [ 2,3,4,3,4,6 ]; B = [ 1,2,3,2,3,8,8,9 ] A = [ 2 3 4 3 4 6 ] B = [ 1 2 3 2 3 8 8 9 ] #> A ++ B ans = [ 2 3 4 6 1 8 9 ] #> A -- B ans = [ 4 6 ] #> B -- A ans = [ 1 8 9 ] #> A /\ B ans = [ 2 3 ]

  36. Compound assignments matrix A += B // A = A + B A -= B // A = A - B A *= B // A = A * B A /= B // A = A / B = A(B^-1) A |= B // A = A | B, horizontal concatenation A _= B // A = A _ B, vertical concatenation A\_= B // A = A\_ B, diagonal concatenation A .*= B // A = A .* B A ./= B // A = A ./ B A .^= B // A = A .^ B A ++= B // A = A ++ B, set union A --= B // A = A –- B, set difference A /\= B // A = A /\ B, set intersection

  37. Access to elements (1) matrix A(i,j), A(k) // real element A{i,j), A{k} // imaginary element A[i,j], A[k] // complex element #> A = [ 1, 3-4! ; -2+5i, 6; 7,-8i ] A = [ 1 3 - i 4 ] [ -2 + i 5 6 ] [ 7 -0 - i 8 ] #> A(2,1); A{2,1}; A[2,1]; ans = -2 ans = 5 ans = -2 + 5! This is a unique feature of msharpmath compared with MATLAB and similar interactive languages

  38. Access to elements (2) matrix A.row(i), A(i,*) // i-th row vector (1 x n matrix) A.col(j), A(*,j) // j-th column vector (m x 1 matrix) A.diag(k) // diagonal vector (k=0 where i=j) #> A = [ 2,5,8; 3,6,9; 4,7,10 ] A = [ 2 5 8 ] [ 3 6 9 ] [ 4 7 10 ] #> A.col(3) ans = [ 8 ] [ 9 ] [ 10 ] #> A.diag(0) -= 20 A = [ -18 5 8 ] [ 3 -14 9 ] [ 4 7 -10 ] #> A.row(2) *= 100 A = [ -18 5 8 ] [ 300 -1400 900 ] [ 4 7 -10 ]

  39. Access to elements (3) matrix A.(matrix) // entry of elelements, A.entry(matrix) #> A = [ 10,40,70,100; 20,50,80,110; 30,60,90,120 ] A = [ 10 40 70 100 ] [ 20 50 80 110 ] [ 30 60 90 120 ] #> A.( [2,6,10] ) ans = [ 20 60 100 ] #> A.( [2,6,10] ) = 0 A = [ 10 40 70 0 ] [ 0 50 80 110 ] [ 30 0 90 120 ] 2nd, 6th, and 10th elements

  40. Access to elements (4) matrix A..(rows)(columns) // submatrix, A.sub(rows)(columns) #> x = 1:6 x = [ 1 2 3 4 5 6 ] #> A = [ x+10; x+20; x+30; x+40; x+50; x+60 ] A = [ 11 12 13 14 15 16 ] [ 2122 23 2425 26 ] [ 31 32 33 34 35 36 ] [ 41 42 43 44 45 46 ] [ 51 52 53 54 55 56 ] [ 61 62 63 64 65 66 ] #> A ..( 2,1,[4,3],6) ( 2,[5,4],1 ) ans = [ 22252421] [ 12 15 14 11 ] [ 42 45 44 41 ] [ 32 35 34 31 ] [ 62 65 64 61 ]

  41. Deleting elements matrix A.row(i) = [] // A = A.rowdel(i) A.col(j) = [] // A = A.coldel(i) A..(rows)(columns) = [] // A = A.subdel(rows)(columns) #> A = [ 11,12,13,14; 21,22,23,24; 31,32,33,34 ] A = [ 11 12 13 14 ] [ 21 22 23 24 ] [ 31 32 33 34 ] #> A.row(2) = [] A = [ 11 12 13 14 ] [ 31 32 33 34 ] #> A.col(3) = [] A = [ 11 12 14 ] [ 31 32 34 ]

  42. Extending matrices (1) matrix A.resize(m,n) // new m x n dimension, while copying A.ext(m,n,s) // extend m rows, n columns and assign s #> A = [ 1,2,3; 4,5,6 ] A = [ 1 2 3 ] [ 4 5 6 ] #> A.resize(4,6) ans = [ 1 2 3 0 0 0 ] [ 4 5 6 0 0 0 ] [ 0 0 0 0 0 0 ] [ 0 0 0 0 0 0 ] #> A.ext(3,2, -1) ans = [ 1 2 3 -1 -1 ] [ 4 5 6 -1 -1 ] [ -1 -1 -1 -1 -1 ] [ -1 -1 -1 -1 -1 ] [ -1 -1 -1 -1 -1 ] #> A.resize(4,2) ans = [ 1 2 ] [ 4 5 ] [ 0 0 ] [ 0 0 ]

  43. Extending matrices (2) matrix A.insrow(i,B) // insert row A.inscol(j,B) // insert column A.ins(i,j,B) // insert matrix #> A = (1:9)._3 A = [ 1 4 7 ] [ 2 5 8 ] [ 3 6 9 ] #> A.insrow(2, [11,12,13,14] ) ans = [ 1 4 7 0 ] [ 11 12 13 14 ] [ 2 5 8 0 ] [ 3 6 9 0 ] #> A.inscol(3, [11,12]) ans = [ 1 4 11 12 7 ] [ 2 5 0 0 8 ] [ 3 6 0 0 9 ]

  44. Member functions matrix A.m// m of an m x n matrix A.n// n of an m x n matrix A.mn // mn of an m x n matrix A.det // determinant A.rank// rank of a matrix A.inv // inverse of a matrix, A^(-1) A.eig // eigenvalues of a matrix and many more #> A = [ 1,2; 4,3 ] A = [ 1 2 ] [ 4 3 ] #> A.det ans = -5 #> A.inv ans = [ -0.6 0.4 ] [ 0.8 -0.2 ] #> A = [ 11,12,13; 21,22,23 ] A = [ 11 12 13 ] [ 21 22 23 ] #> A.m; A.n; A.mn; ans = 2 ans = 3 ans = 6

  45. ■ Tuples (a1,a2,a3,…,an) // n-tuple #> (a,b) = (1,2) a = 1 b = 2 #> (b,a) = (a,b) b = 1 a = 2 Swap values by tuples

  46. ■ Logical (true or false), conditional 0 // false only for the zero Non-zero // true for all non-zeros a && b // and, .and(a,b) a || b // or, .or(a,b) a !& b // exclusive and, .xand(a,b) a !| b // exclusive or, .xor(a,b) #> [ 0 && 0, 0 && 1, 1 && 0, 1 && 1 ] ans = [ 0 0 0 1 ] #> [ 0 || 0, 0 || 1, 1 || 0, 1 || 1 ] ans = [ 0 1 1 1 ] #> [ 0 !& 0, 0 !& 1, 1 !& 0, 1 !& 1 ] ans = [ 1 0 0 1 ] #> [ 0 !| 0, 0 !| 1, 1 !| 0, 1 !| 1 ] ans = [ 0 1 1 0 ]

  47. ■ relational a > b // greater than a < b // less than a >= b // greater than or equal to a <= b // less than or equal to a == b // equal to a != b // not equal to #> 3 > 2 ans = 1 #> 3 < 2 ans = 0 #> 3 >= 2 ans = 1 #> 3 <= 2 ans = 0 #> 3 == 2 ans = 0 #> 3 != 2 ans = 1

  48. ■ relational operators for matrices only A .mn. B // true if A and B are of same dimension A .eq. b // true if all a_ij == b_ij A .ne. b // true if all a_ij != b_ij A .gt. B // true if all a_ij > b_ij A .lt. B // true if all a_ij < b_ij A .ge. B // true if all a_ij >= b_ij A .le. B // true if all a_ij <= b_ij #> A = [ 1,2,3 ]; B = [ 4,5,6 ]; A = [ 1 2 3 ] B = [ 4 5 6 ] #> A .mn. B ans = 1 #> A .mn. B' ans = 0

  49. ■ Array Data can be declared as an array. Access to its elements is done by [] double x[n] is equal to x[0],x[1],..., x[n-1] Be careful !!! Index starts at 0 #> double x[5] #> x[0] = 3 ans = 3 #> x[1] = 5 ans = 5 #> x x = double [5] [0] = 3 [1] = 5 [2] = 2.0694473e+161 [3] = 1.8875371e+219 [4] = 3.7699368e-317 Note that x[5] causes an error since it tries to refer 6-th element (not existing)

  50. ■ ternary operation, a ? b : c a ? b : c // b if a is true, otherwise c #> 3 ? 10 : 20; ans = 10 #> 0 ? 10 : 20; ans = 20 #> (a,b) = (2,3); .max(a,b); .min(a,b); a = 2 b = 3 ans = 3 ans = 2 #> a > b ? a : b b = 3 #> a < b ? a : b a = 2

More Related