Download - Introduction to Computer Algebra Systems

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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A CAS is a software package having capabilities for

• numerical computations

• symbolic computations

• graphical computations

Computer Algebra Systems

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Introduction

Numerical computations

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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



Numerical computation

∫−

1

0

2

2

dxex

Evaluate the following integral using trapezoidal rule:

Example 2

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Introduction

Symbolic computations

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Symbolic computation

Solve the quadratic equation:

0)()()( 2 =−+−+− acxcbxba

Example 3

Solution :

baacx

−−

= ,1




Obtain the general solution of the differential equation:

baxdxdyp

dxyd

+=+2

2

Example 4

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(See next slide for solution)




Complementary Function = px21 eCC −+

Particular Integral = ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −+ x

pabx

2a

p1 2

y = C. F. + P.I.

Example 4 : Solution



Introduction

Graphical computations

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Graphical computation

( ) ( ) ( )3 ,5for

3/1223/23/2

==−=+

bababyax

Draw the curve:Example 5

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)3(sin32 θ−=r

Draw the curve (polar coordinates):Example 6

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zyx =− 22

Plot the surface:Example 7

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Introduction

Some popular CAS’s

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Symbolic Math Toolbox in


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Maple in action

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Maple in action

Maple syntax

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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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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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Math Maple

log x log(x)

|x | abs(x)

e^x exp(x)

√x sqrt(x)

Maple syntax

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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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Maple in action

Algebra with Maple

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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 −



Algebra

Example 9 (continued)Maple output:

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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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Algebra


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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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Algebra

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Algebra

Example 12To find the solutions as floatingpoint numbers:

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Maple input> evalf(%);



Algebra


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Maple in action

Calculus with Maple

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∂∂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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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



Example 13To obtain the derivative of f with respect to x

Differentiation

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Maple input> diff(f,x);



Example 13 (continued)

Maple output:

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Differentiation



Example 14

Maple input> diff(f, y, z) ;

∂ ∂∂ 2

y z ( )f , ,x y z

To find

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Differentiation




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Differentiation



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




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Differentiation



Integration

The general format for evaluating indefinite integrals: To find

Maple input> int(f(x),x);

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dxxf∫ )(



Integration

Maple input> int(x^2 - sin(x), x);

( )dxxx∫ − )sin(2

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Example 16A simple example:



Integration

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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



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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.



Integration


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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 )(



Integration

Example 19Evaluate:

Maple input> int(x/(1+x^2), x = 0 .. 1);

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∫ +

1

021

dxx

x



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



Integration

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Example 20 (continued)Maple output



Maple in action

Differential equations with Maple

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Differential equations

Example 21First order equations

Maple input> dsolve( x*diff(y(x), x) + y(x) = x );

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xydxdyx =+





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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





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Maple in action

Maple packages

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Packages

Some packages

combinat combinatorial functions

inttrans integral transforms

LinearAlgebra Linear algebra

networks graph networks

numtheory number theory

plots displaying graphs 65



Maple in action


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This is a collection of functions for symbolic computations involving vectors and matrices.

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Load LinearAlgebra package> with(LinearAlgebra);

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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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Example 23

To find the inverse of A

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Maple input> MatrixInverse(A);




Example 24To find the characteristic polynomial in terms of lambda

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Maple input> CharacteristicPolynomial(A,lambda);




Example 25To find the eigen values of A

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Maple input> Eigenvalues(A);



Maple in action

inttrans package

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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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inttrans package

To load inttrans packge> with(inttrans);

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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( −



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



Maple in action

Graphics with Maple

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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



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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 =



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Graphics:plots package

To use the plots package> with(plots);

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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 =+



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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



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THANK YOU …

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