WELCOME TO MATH 165. This is a short introduction to the course while getting acquainted with MAPLE!
This course will introduce you to the basic ideas in SYMBOLIC ALGEBRAIC COMPUTATION.
and of EXPERIMENTAL MATHEMATICS.
f:=-6*x^15*y^2+6*x^10*z^2-3*x^14*y^3+3*z^5*x^8+42*x^14*y^3*z-126*x^13*y^4*z\134
^2+210*x^12*y^5*z^3-210*x^11*y^6*z^4+126*x^10*y^7*z^5-42*x^9*y^8*z^6+6*x\134
^8*y^9*z^7-42*x^9*z^3*y+126*x^8*z^4*y^2-210*x^7*z^5*y^3+210*x^6*z^6*y^4-\134
126*x^5*z^7*y^5+42*x^4*z^8*y^6-6*x^3*z^9*y^7+21*x^13*y^4*z-63*x^12*y^5*z\134
^2+105*x^11*y^6*z^3-105*x^10*y^7*z^4+63*x^9*y^8*z^5-21*x^8*y^9*z^6+3*x^7\134
*y^10*z^7+3*x^9*y*z^2-21*x^8*y^2*z^3+63*x^7*y^3*z^4-105*x^6*y^4*z^5+105*\134
x^5*y^5*z^6-63*x^4*y^6*z^7+21*x^3*y^7*z^8-3*x^2*y^8*z^9-3*z^3*x^13*y^2+2\134
1*z^4*x^12*y^3-63*z^5*x^11*y^4+105*z^6*x^10*y^5-105*z^7*x^9*y^6+63*z^8*x\134
^8*y^7-21*z^9*x^7*y^8+3*z^10*x^6*y^9-21*z^6*x^7*y+63*z^7*x^6*y^2-105*z^8\134 JSFH
*x^5*y^3+105*z^9*x^4*y^4-63*z^10*x^3*y^5+21*z^11*x^2*y^6-3*x*z^12*y^7;
factor(f);
We can also factorize integers using MAPLE:
ifactor(1302387694);
Now we look at other symbolic computation. How to do Symbolic integration:
# <---- This symbol can be use to comment code.
It is important to know how to ask for HELP!!
JSFH help(int);
f:=(x^2-1+3*(1+x^2)^3)/(1+x^2)^3*x/(x^2+2);
int(f,x);
int( x/(x^3-1), x );
It can of course do NUMERIC integration too!!
r := int( sech(x)*exp(-x^2), x = -infinity..infinity );
evalf(r);
int(1/(x*sqrt((b*x+c*x^2)^3)),x);
factor(%);
simplify(%) assuming positive;
subs(b+c*x=y/x,%);
expand(%);
MAPLE also has graphic capabilities: Suppose we want to plot the absolute value of the Riemann Zeta function
on the critical line of those complex numbers of real part=1/2.
plot(abs(Zeta(1/2+y*I)),y=0..36, numpoints=1000);
with(plots);
polyhedraplot([0,0,0],polytype=dodecahedron,style=PATCH,
scaling=CONSTRAINED,orientation=[71,66]);
pi := evalf(Pi);
p := seq([cos(t*pi/50)*(10+4*sin(9*(t*pi/50))),
sin(t*pi/50)*(10+4*sin(9*(t*pi/50))),
4*cos(9*(t*pi/50))],t=0..200):
polyhedraplot([p],polyscale=.4,polytype=hexahedron,scaling=CONSTRAINED,
orientation=[76,40]);
MAPLE can do Linear Algebra! (but you have to load the right package)
with(linalg);
We can generate random matrices
A:=randmatrix(5,5);
MAPLE is able to do FOR or WHILE loops
det(A);
We can use MAPLE to run some experiments on determinants of matrices:
numex:=7; # number of iterations
for n from 1 to numex do
A:=Matrix(n,n,shape=symmetric); print("Constructing matrix"):
for i to n do
for j to i do
A[i,j]:=x^(igcd(i,j));
od;
od; print(A);
factor(det(A));
print("factorization of the determinant",%);
od:
We can also work with 3-D graphics to our advantange to see things more clearly!
hel1:=[4*cos(t),4*sin(t),2*t];
spacecurve(convert(hel1,list),t=0..5*Pi,scaling=constrained,color=black,axes=framed, orientation=[40,77]);
tubeplot(convert(hel1,list), t=0..5*Pi,scaling=constrained,axes=framed, orientation=[40,77], shading=XYZ,lightmodel=light1,style=patchnogrid);
In this course we will on finding out whether a system of polynomial equations have solutions or not!
The main tool will be Grobner bases. There is a MAPLE package for Grobner bases
with(grobner);
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[x^2-2*x*z+5,x*y^2+y*z^3,3*y^2-8*z^3];
Groebner[gbasis]([x^2-2*x*z+5,x*y^2+y*z^3,3*y^2-8*z^3],plex(y,x,z));