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Computer Algebra Systems
Dr. V. N. KrishnachandranDepartment of Computer Applications
Vidya Academy of Science and TechnologyThrissur – 680 501
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Outline
IntroductionNumerical computations
Symbolic computations
Some popular CAS’s : Maple, Matlab, …
Maple in actionMaple syntax
Algebra with Maple
Calculus with Maple
Differential equations with Maple
Maple packages
LinearAlgebra package
inttrans package
Graphics with Maple
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Introduction
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
A CAS is a software package having capabilities for
• numerical computations
• symbolic computations
• graphical computations
Computer Algebra Systems
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Introduction
Numerical computations
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Numerical computation
glT π2=
g = 981, π = 3.14, l = 51.5 .
Let
Find T when
Example 1
Use logarithm tables or an electronic calculator to calculate this expression and get .T = 1.439
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Numerical computation
∫−
1
0
2
2
dxex
Evaluate the following integral using trapezoidal rule:
Example 2
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Introduction
Symbolic computations
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Symbolic computation
Solve the quadratic equation:
0)()()( 2 =−+−+− acxcbxba
Example 3
Solution :
baacx
−−
= ,1
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Symbolic computation
Obtain the general solution of the differential equation:
baxdxdyp
dxyd
+=+2
2
Example 4
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(See next slide for solution)
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Symbolic computation
Complementary Function = px21 eCC −+
Particular Integral = ⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ −+ x
pabx
2a
p1 2
y = C. F. + P.I.
Example 4 : Solution
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Introduction
Graphical computations
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Graphical computation
( ) ( ) ( )3 ,5for
3/1223/23/2
==−=+
bababyax
Draw the curve:Example 5
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(See next slide for solution)
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Graphical computation
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Graphical computation
)3(sin32 θ−=r
Draw the curve (polar coordinates):Example 6
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(See next slide for solution)
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Graphical computation
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Graphical computation
zyx =− 22
Plot the surface:Example 7
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(See next slide for solution)
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Graphical computation
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Introduction
Some popular CAS’s
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Some popular CAS’s
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Some popular CAS’s
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Some popular CAS’s
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Symbolic Math Toolbox in
Some popular CAS’s
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Maple in action
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Maple in action
Maple syntax
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Operation Symbol Example
Addition + a + b
Subtraction - a - b
Multiplication * a*b
Division / a/b
Exponentiation ^ (**) a^b(a**b)
Maple syntax
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
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)
Maple syntax
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Math Maple
log x log(x)
|x | abs(x)
e^x exp(x)
√x sqrt(x)
Maple syntax
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Maple syntax
Example 8Mathematical expression
⎟⎠⎞
⎜⎝⎛
++++ −
216ax
axx1cbxe logsin)(cos
Maple expression:exp(a*x)*(cos(b*x+c))^6 + arcsin( sqrt(1 + log(x/(x+a^2)) ) )
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Maple in action
Algebra with Maple
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Algebra
Maple input> F:=expand((x-2*x^2*y)^3);
Example 9Expand the following and assign the expression to F:
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( )322 yxx −
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Algebra
Example 9 (continued)Maple output:
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Example 10To solve the quadratic equation
Maple input> solve( (a-b)*x^2+(b-c)*x+(c-a)=0,x);
0acxcbxba 2 =−+−+− )()()(
Algebra
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Algebra
Example 10 (continued)Maple output:
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Algebra
Example 11Solve the cubic equation
Maple input> solve(2*x^3+3*x^2-x+5=0,x);
= + − + 2 x3 3 x2 x 5 0
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Algebra
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Algebra
Example 12To find the solutions as floatingpoint numbers:
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Maple input> evalf(%);
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Algebra
Example 12 (continued)Maple output:
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Maple in action
Calculus with Maple
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
∂∂y ( )f , ,x y z
Maple input:> diff( f(x,y,z) , y );
Differentiation
General format to evaluate derivatives:
To find
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Let us consider the function:
Maple input> f := x^2 * exp(-z) + (2*y^3 - x) * arctan(x/z);
= ( )f , ,x y z + x2 e( )−z
( ) − 2 y3 x ⎛⎝⎜⎜
⎞⎠⎟⎟arctan
xz
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Differentiation
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Example 13To obtain the derivative of f with respect to x
Differentiation
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Maple input> diff(f,x);
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Example 13 (continued)
Maple output:
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Differentiation
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Example 14
Maple input> diff(f, y, z) ;
∂ ∂∂ 2
y z ( )f , ,x y z
To find
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Differentiation
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Example 14 (continued)Maple output:
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Differentiation
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Maple input> taylor( x/((x+1)*(x-2)), x=1, 4);
x( ) + x 1 ( ) − x 2
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Differentiation
Example 15To obtain the taylor series
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Example 15 (continued)Maple output:
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Differentiation
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Integration
The general format for evaluating indefinite integrals: To find
Maple input> int(f(x),x);
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dxxf∫ )(
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Integration
Maple input> int(x^2 - sin(x), x);
( )dxxx∫ − )sin(2
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Example 16A simple example:
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Integration
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Example 16 (continued)Maple output:
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Integration
Maple input> int( (a*x+b)/sqrt(p*x^2+q*x+r) , x );
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dxrqxpx
bax∫
++
+2
Example 17A very complicated integral
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Integration
Maple input> int(x^x,x);
∫ dxx x
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Example 18Sometimes Maple may not be able to obtain an explicit expression for an integral.
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Integration
Example 18 (continued)Maple output:
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
The general format for evaluating definite integrals: To find
Maple input:> int ( f(x) , x=a..b );
Integration
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∫b
a
dxxf )(
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Integration
Example 19Evaluate:
Maple input> int(x/(1+x^2), x = 0 .. 1);
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∫ +
1
021
dxx
x
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Integration
Example 20Limits can contain (Pi) and (infinity)
Maple input> int(x*sin(n*x), x=-Pi/2 .. Pi/2);
π ∞
∫−
2/
2/
)sin(π
π
dxnxx
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Integration
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Example 20 (continued)Maple output
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Maple in action
Differential equations with Maple
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Differential equations
Example 21First order equations
Maple input> dsolve( x*diff(y(x), x) + y(x) = x );
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xydxdyx =+
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Differential equations
Example 21 (continued)Maple output
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Differential equations
Example 22Second order equations
Maple input> dsolve( diff(y(x),x,x)+a^2*y(x)=x);
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xyadx
yd=+ 2
2
2
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Differential equations
Example 22 (continued)Maple output
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Maple in action
Maple packages
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Packages
Some packages
combinat combinatorial functions
inttrans integral transforms
LinearAlgebra Linear algebra
networks graph networks
numtheory number theory
plots displaying graphs 65
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Maple in action
LinearAlgebra package
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
LinearAlgebra package
This is a collection of functions for symbolic computations involving vectors and matrices.
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
LinearAlgebra package
Load LinearAlgebra package> with(LinearAlgebra);
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
LinearAlgebra package
To define the matrix
Maple input>A:=Matrix([[1, -3, 4],[2, 3, 4],[-4, 0, 5]]);
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
1 − 3 42 3 4
− 4 0 5
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
LinearAlgebra package
Example 23
To find the inverse of A
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Maple input> MatrixInverse(A);
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
LinearAlgebra package
Example 24To find the characteristic polynomial in terms of lambda
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Maple input> CharacteristicPolynomial(A,lambda);
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
LinearAlgebra package
Example 25To find the eigen values of A
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Maple input> Eigenvalues(A);
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Maple in action
inttrans package
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
inttrans package
The inttrans package is a collection of functions designed to compute integral transforms like Laplace transforms and Fourier transforms.
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
inttrans package
To load inttrans packge> with(inttrans);
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
inttrans package
Example 26To find the Laplace transform of
Maple input> laplace(t^2*sin(3*t)*exp(-4*t), t, s);
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tett 42 )3sin( −
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
inttrans package
Example 27To find the inverse Laplace transform of
Maple input> invlaplace(s/((s^2+s+1)^2), s, t);
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( )22 1++ sss
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Maple in action
Graphics with Maple
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Graphics
Example 28To plot the graph of the function
Maple input> plot( x**3 – x + 5 , x = -2..2 );
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53 +− xx
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Graphics
Example 29To plot the surface given by the function
Maple input> plot3d(sin(x*y), x=-Pi..Pi, y=-Pi..Pi);
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)sin(),( xyyxf =
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Graphics:plots package
To use the plots package> with(plots);
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Graphics:plots package
Example 30To plot the curve given by the equation
Maple input> implicitplot(x^3 + y^3 = 3*x*y, x = -2..2, y = -2..2);
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xyyx 333 =+
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
Graphics:plots package
Example 31To plot the surface given by the equation
Maple input> implicitplot3d( x^3 + y^3 + z^3 +1 = (x+y+z+1)^3, x=-2..2, y=-2..2, z=-2..2);
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3333 )1(1 +++=+++ zyxzyx
Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501
THANK YOU …
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Introduction to Computer Algebra Systems
V. N. Krishnachandran, Vidya Academy of Science and Technology, Thrissur - 680501