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# Introduction to Computer Algebra Systems

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This presentation gives a brief overview of the capabilities of software packages known as 'computer algebra systems'. The stress is on symbolic and graphic computations. Maple is used as a vehicle to illustrate the concepts.

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### Introduction to Computer Algebra Systems

1. 1. Computer Algebra Systems Dr. V. N. Krishnachandran Department of Computer Applications Vidya Academy of Science and Technology Thrissur – 680 501 1
2. 2. Introduction to Computer Algebra Systems Outline Introduction Numerical computations Symbolic computations Some popular CAS’s : Maple, Matlab, … Maple in action Maple syntax Algebra with Maple Calculus with Maple Differential equations with Maple Maple packages LinearAlgebra package inttrans package Graphics with Maple V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 2
3. 3. Introduction to Computer Algebra Systems Introduction V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 3
4. 4. Introduction to Computer Algebra Systems Computer Algebra Systems A CAS is a software package having capabilities for • numerical computations • symbolic computations • graphical computations V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 4
5. 5. Introduction to Computer Algebra Systems Introduction Numerical computations V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 5
6. 6. Introduction to Computer Algebra Systems Numerical computation Example 1 l Let T = 2π g Find T when g = 981, π = 3.14, l = 51.5 . Use logarithm tables or an electronic calculator to calculate this expression and get T = 1.439 . V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
7. 7. Introduction to Computer Algebra Systems Numerical computation Example 2 Evaluate the following integral using trapezoidal rule: 1 x 2 − ∫ 0 e 2 dx V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 7
8. 8. Introduction to Computer Algebra Systems Introduction Symbolic computations V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 8
9. 9. Introduction to Computer Algebra Systems Symbolic computation Example 3 Solve the quadratic equation: ( a − b ) x + (b − c ) x + (c − a ) = 0 2 Solution : c−a x = 1, a−b V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
10. 10. Introduction to Computer Algebra Systems Symbolic computation Example 4 Obtain the general solution of the differential equation: 2 d y dy 2 + p = ax + b dx dx (See next slide for solution) V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 10
11. 11. Introduction to Computer Algebra Systems Symbolic computation Example 4 : Solution Complementary Function = C1 + C2 e − px 1 ⎛ a x 2 + ⎛b − a ⎞x ⎞ Particular Integral = ⎜ ⎜ ⎟ ⎟ p⎝2 ⎝ p⎠ ⎠ y = C. F. + P.I. V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
12. 12. Introduction to Computer Algebra Systems Introduction Graphical computations V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 12
13. 13. Introduction to Computer Algebra Systems Graphical computation Example 5 Draw the curve: (ax ) + (by ) = a − b 2/3 2/3 ( 2 ) 2 1/ 3 for a = 5 , b = 3 (See next slide for solution) V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 13
14. 14. Introduction to Computer Algebra Systems Graphical computation V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 14
15. 15. Introduction to Computer Algebra Systems Graphical computation Example 6 Draw the curve (polar coordinates): r = 2 − 3 sin ( 3θ ) (See next slide for solution) V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 15
16. 16. Introduction to Computer Algebra Systems Graphical computation V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 16
17. 17. Introduction to Computer Algebra Systems Graphical computation Example 7 Plot the surface: x − y = z 2 2 (See next slide for solution) V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 17
18. 18. Introduction to Computer Algebra Systems Graphical computation V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 18
19. 19. Introduction to Computer Algebra Systems Introduction Some popular CAS’s V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 19
20. 20. Introduction to Computer Algebra Systems Some popular CAS’s V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 20
21. 21. Introduction to Computer Algebra Systems Some popular CAS’s V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 21
22. 22. Introduction to Computer Algebra Systems Some popular CAS’s V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 22
23. 23. Introduction to Computer Algebra Systems Some popular CAS’s Symbolic Math Toolbox in V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 23
24. 24. Introduction to Computer Algebra Systems Maple in action V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 24
25. 25. Introduction to Computer Algebra Systems Maple in action Maple syntax V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 25
26. 26. Introduction to Computer Algebra Systems Maple syntax Operation Symbol Example Addition + a+b Subtraction - a-b Multiplication * a*b Division / a/b Exponentiation ^ (**) a^b(a**b) V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 26
27. 27. Introduction to Computer Algebra Systems Maple syntax Math Maple Math Maple sin x sin(x) sin -1 x arcsin(x) cos x cos(x) cos -1 x arccos(x) tan x tan(x) tan -1 x arctan(x) sec x sec(x) sec -1 x arcsec(x) cosec x csc(x) cosec -1 x arccsc(x) cot x cot(x) cot -1 x arccot(x) V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 27
28. 28. Introduction to Computer Algebra Systems Maple syntax Math Maple log x log(x) |x | abs(x) e^x exp(x) √x sqrt(x) V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 28
29. 29. Introduction to Computer Algebra Systems Maple syntax Example 8 Mathematical expression −1 ⎛ x ⎞ e ax cos (bx + c ) + sin 6 1 + log ⎜ 2⎟ ⎝x +a ⎠ Maple expression: exp(a*x)*(cos(b*x+c))^6 + arcsin( sqrt(1 + log(x/(x+a^2)) ) ) V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 29
30. 30. Introduction to Computer Algebra Systems Maple in action Algebra with Maple V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 30
31. 31. Introduction to Computer Algebra Systems Algebra Example 9 Expand the following and assign the expression to F: (x − 2 x y ) 2 3 Maple input > F:=expand((x-2*x^2*y)^3); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 31
32. 32. Introduction to Computer Algebra Systems Algebra Example 9 (continued) Maple output: V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 32
33. 33. Introduction to Computer Algebra Systems Algebra Example 10 To solve the quadratic equation ( a − b ) x + (b − c ) x + ( c − a ) = 0 2 Maple input > solve( (a-b)*x^2+(b-c)*x+(c-a)=0,x); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 33
34. 34. Introduction to Computer Algebra Systems Algebra Example 10 (continued) Maple output: V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 34
35. 35. Introduction to Computer Algebra Systems Algebra Example 11 Solve the cubic equation 2 x +3 x −x +5 =0 3 2 Maple input > solve(2*x^3+3*x^2-x+5=0,x); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 35
36. 36. Introduction to Computer Algebra Systems Algebra V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 36
37. 37. Introduction to Computer Algebra Systems Algebra Example 12 To find the solutions as floating point numbers: Maple input > evalf(%); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 37
38. 38. Introduction to Computer Algebra Systems Algebra Example 12 (continued) Maple output: V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 38
39. 39. Introduction to Computer Algebra Systems Maple in action Calculus with Maple V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 39
40. 40. Introduction to Computer Algebra Systems Differentiation General format to evaluate derivatives: To find ∂ f( x , y , z ) ∂y Maple input: > diff( f(x,y,z) , y ); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 40
41. 41. Introduction to Computer Algebra Systems Differentiation Let us consider the function: f( x, y, z ) = x e 2 ( −z ) ⎛x⎞ + ( 2 y − x ) arctan ⎜ ⎟ 3 ⎜z⎟ ⎝ ⎠ Maple input > f := x^2 * exp(-z) + (2*y^3 - x) * arctan(x/z); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 41
42. 42. Introduction to Computer Algebra Systems Differentiation Example 13 To obtain the derivative of f with respect to x Maple input > diff(f,x); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 42
43. 43. Introduction to Computer Algebra Systems Differentiation Example 13 (continued) Maple output: V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 43
44. 44. Introduction to Computer Algebra Systems Differentiation Example 14 To find ∂2 f( x , y , z ) ∂y ∂z Maple input > diff(f, y, z) ; V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 44
45. 45. Introduction to Computer Algebra Systems Differentiation Example 14 (continued) Maple output: V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 45
46. 46. Introduction to Computer Algebra Systems Differentiation Example 15 To obtain the taylor series x (x + 1 ) (x − 2 ) Maple input > taylor( x/((x+1)*(x-2)), x=1, 4); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 46
47. 47. Introduction to Computer Algebra Systems Differentiation Example 15 (continued) Maple output: V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 47
48. 48. Introduction to Computer Algebra Systems Integration The general format for evaluating indefinite integrals: To find ∫ f ( x) dx Maple input > int(f(x),x); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 48
49. 49. Introduction to Computer Algebra Systems Integration Example 16 A simple example: ∫ (x − sin( x) dx ) 2 Maple input > int(x^2 - sin(x), x); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 49
50. 50. Introduction to Computer Algebra Systems Integration Example 16 (continued) Maple output: V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 50
51. 51. Introduction to Computer Algebra Systems Integration Example 17 A very complicated integral ax + b ∫ px + qx + r 2 dx Maple input > int( (a*x+b)/sqrt(p*x^2+q*x+r) , x ); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 51
52. 52. Introduction to Computer Algebra Systems V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 52
53. 53. Introduction to Computer Algebra Systems Integration Example 18 Sometimes Maple may not be able to obtain an explicit expression for an integral. ∫ x x dx Maple input > int(x^x,x); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 53
54. 54. Introduction to Computer Algebra Systems Integration Example 18 (continued) Maple output: V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 54
55. 55. Introduction to Computer Algebra Systems Integration The general format for evaluating definite integrals: To find b ∫ f ( x)dx a Maple input: > int ( f(x) , x=a..b ); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 55
56. 56. Introduction to Computer Algebra Systems Integration Example 19 Evaluate: 1 x ∫ 1+ x2 0 dx Maple input > int(x/(1+x^2), x = 0 .. 1); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 56
57. 57. Introduction to Computer Algebra Systems Integration Example 20 Limits can contain π (Pi) and ∞ (infinity) π /2 ∫ x sin( nx ) dx π − /2 Maple input > int(x*sin(n*x), x=-Pi/2 .. Pi/2); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
58. 58. Introduction to Computer Algebra Systems Integration Example 20 (continued) Maple output V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 58
59. 59. Introduction to Computer Algebra Systems Maple in action Differential equations with Maple V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 59
60. 60. Introduction to Computer Algebra Systems Differential equations Example 21 First order equations dy x +y=x dx Maple input > dsolve( x*diff(y(x), x) + y(x) = x ); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 60
61. 61. Introduction to Computer Algebra Systems Differential equations Example 21 (continued) Maple output V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 61
62. 62. Introduction to Computer Algebra Systems Differential equations Example 22 Second order equations 2 d y 2 +a y = x 2 dx Maple input > dsolve( diff(y(x),x,x)+a^2*y(x)=x); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 62
63. 63. Introduction to Computer Algebra Systems Differential equations Example 22 (continued) Maple output V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 63
64. 64. Introduction to Computer Algebra Systems Maple in action Maple packages V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 64
65. 65. Introduction to Computer Algebra Systems Packages Some packages combinat combinatorial functions inttrans integral transforms LinearAlgebra Linear algebra networks graph networks numtheory number theory plots displaying graphs V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 65
66. 66. Introduction to Computer Algebra Systems Maple in action LinearAlgebra package V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 66
67. 67. Introduction to Computer Algebra Systems LinearAlgebra package This is a collection of functions for symbolic computations involving vectors and matrices. V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 67
68. 68. Introduction to Computer Algebra Systems LinearAlgebra package Load LinearAlgebra package > with(LinearAlgebra); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 68
69. 69. Introduction to Computer Algebra Systems LinearAlgebra package To define the matrix ⎡ 1 −3 4⎤ ⎢ ⎢ 2 ⎥ ⎢ 3 4⎥ ⎥ ⎢ ⎢−4 ⎥ ⎣ 0 5⎥ ⎦ Maple input >A:=Matrix([[1, -3, 4],[2, 3, 4],[-4, 0, 5]]); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 69
70. 70. Introduction to Computer Algebra Systems LinearAlgebra package Example 23 To find the inverse of A Maple input > MatrixInverse(A); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 70
71. 71. Introduction to Computer Algebra Systems LinearAlgebra package Example 24 To find the characteristic polynomial in terms of lambda Maple input> CharacteristicPolynomial(A,lambda); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 71
72. 72. Introduction to Computer Algebra Systems LinearAlgebra package Example 25 To find the eigen values of A Maple input > Eigenvalues(A); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 72
73. 73. Introduction to Computer Algebra Systems Maple in action inttrans package V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 73
74. 74. Introduction to Computer Algebra Systems inttrans package The inttrans package is a collection of functions designed to compute integral transforms like Laplace transforms and Fourier transforms. V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 74
75. 75. Introduction to Computer Algebra Systems inttrans package To load inttrans packge > with(inttrans); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 75
76. 76. Introduction to Computer Algebra Systems inttrans package Example 26 To find the Laplace transform of 2 −4 t t sin( 3 t ) e Maple input > laplace(t^2*sin(3*t)*exp(-4*t), t, s); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 76
77. 77. Introduction to Computer Algebra Systems inttrans package Example 27 To find the inverse Laplace transform of s (s 2 + s +1 ) 2 Maple input > invlaplace(s/((s^2+s+1)^2), s, t); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 77
78. 78. Introduction to Computer Algebra Systems Maple in action Graphics with Maple V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 78
79. 79. Introduction to Computer Algebra Systems Graphics Example 28 To plot the graph of the function x −x+5 3 Maple input > plot( x**3 – x + 5 , x = -2..2 ); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 79
80. 80. Introduction to Computer Algebra Systems V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 80
81. 81. Introduction to Computer Algebra Systems Graphics Example 29 To plot the surface given by the function f ( x , y ) = sin( xy ) Maple input > plot3d(sin(x*y), x=-Pi..Pi, y=-Pi..Pi); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 81
82. 82. Introduction to Computer Algebra Systems V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 82
83. 83. Introduction to Computer Algebra Systems Graphics:plots package To use the plots package > with(plots); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 83
84. 84. Introduction to Computer Algebra Systems Graphics:plots package Example 30 To plot the curve given by the equation x + y = 3 xy 3 3 Maple input > implicitplot(x^3 + y^3 = 3*x*y, x = -2..2, y = -2..2); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 84
85. 85. Introduction to Computer Algebra Systems V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 85
86. 86. Introduction to Computer Algebra Systems Graphics:plots package Example 31 To plot the surface given by the equation x + y + z + 1 = ( x + y + z + 1) 3 3 3 3 Maple input > implicitplot3d( x^3 + y^3 + z^3 +1 = (x+y+z+1)^3, x=-2..2, y=-2..2, z=- 2..2); V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 86
87. 87. Introduction to Computer Algebra Systems V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 87
88. 88. Introduction to Computer Algebra Systems THANK YOU … V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501 88