Brief Introduction to Maple

Maple is a symbol manipulator.

Exact arithmetic:

> 1/29+1/31;

> arcsin(1/2);

> exp(I*Pi);

Can give numerical answers to arbitrary precision:

> evalf( arcsin(1/2), 10);

> evalf( arcsin(1/2), 40);

> evalf( Pi, 1000);

Note default of evalf is 10 digits:

> evalf(Pi);

Algebraic expressions:

> restart; Also discuss saving work

> a:=x^2*exp(x)*cos(x);

> diff(a,x);

> int(a,x);

> I:=int(a,x=-Pi..Pi);

Error, illegal use of an object as a name

> I1:=int(a,x=-Pi..Pi);

> simplify(I1);

> factor(I1);

> plot(a,x=0..Pi/2);

Note last thing is NOT a symbolic procedure!

Using Maple to solve equations:

> restart;

> a:= x^2-3*x+1=0 ;

> solve(a,x);

> solve( {x+s*y=t,2*x-y=s*t} , {x,y} );

${y = -t*(-2+s)/(2*s+1), x = t*(s^2+1)/(2*s+1)}$

> solve( {x+s*y=t,2*x-y=s*t} , {t,y} );

${y = -x*(-2+s)/(s^2+1), t = x*(2*s+1)/(s^2+1)}$

> a:= x=(1+x^2)*cos(x) ;

> solve(a,x);

> fsolve(a,x);

This is numeric....but matlab does not really have it

> plot( (1+x^2)*cos(x)-x , x = -3..3);

> fsolve(a,x=-3..-1);

Some linear algebra

> restart;

> with(linalg);

Warning, the protected names norm and trace have been redefined and unprotected

> M1:=matrix(3,3,[1,3,4,t,1,6,-3,t+4,1] );

> det(M1);

> solve(det(M1)=0,t);

> M2:=matrix(3,3,[1,3,4,-5,1,6,-3,-1,1] );

> det(M2);

> colspace(M2);

> eigenvals(M2);

> eigenvects(M2);

$[3/2+3/2*I*15^(1/2), 1, {vector([-5/8-3/8*I*15^(1/2), 11/8-3/8*I*15^(1/2), 1])}], [3/2-3/2*I*15^(1/2), 1, {vector([-5/8+3/8*I*15^(1/2), 11/8+3/8*I*15^(1/2), 1])}], [0, 1, {vector([1, -13/7, 8/7])}]$

> eigenvals(M1); this is going to involve solving a cubic....maple is bad at this.

There is now a package to do strictly numerical linear algebra on maple....but I use matlab

Programming in maple:

> x:=1/2; for i from 1 to 10 do x:=5/2*x*(1-x) end do;

> x:=1/2; for i from 1 to 20 do x:=evalf(5/2*x*(1-x)) end do;

This of course would better be done in matlab

More generally can write functions known as procedures

Graphics

> restart;

> with(plots);

Warning, the name changecoords has been redefined

Note: plotting is de facto numeric...better done in matlab but maple has nice procedures (and I know it)

> animate( sin(x*t), x=-10..10, t=0.5..1);

> implicitplot3d( x^2-x+2*y+3*z=1, x=-5..5, y=-5..5, z=-10..5, axes=boxed, style=patchnogrid);

>

> animate3d(cos(t*x)*sin(t*y),x=-Pi..Pi, y=-Pi..Pi,t=1..2);

>