math696HW3.html

Jayne Overgard
Math 696
HW#3 problem 1a.

>    f:=exp(.1*x)*sin(x);
for n from 1 to 6 do

   p:=NULL;
   for n from 1 to 6 do
g:=convert(taylor(f,x=0,n),polynom);
p:=p,g;od;
plot([f,p],x=-5..5);
p:=p,g;
end do;

HW#3 problem 1b.

> restart;

m:=exp(.1*x)*sin(x);
n:=convert(taylor(m,x=a,6),polynom);

unassign(a);

>

HW#3 problem 2a.

>    f := proc( a, b )
    local primes,p,numprimes,i;
    primes:= {};

       numprimes := 0;
       for i from a to b do
         i:= nextprime(i);
         primes := primes union {i};
         numprimes := numprimes + 1
       end do;

print("Finding the twin prime pairs in the interval.");
       for i from 2 to numprimes-1 do
         if primes[i] - primes[i-1] = 2 then
         print(primes[i-1],primes[i])
         end if;
       end do;
end proc:

> print("Type in f(a,b) where a and b is an interval. End with ; and then enter");

> f(1,60);

>

HW#3 problem 2b.

>    g := proc( a, b )
    local primes,p,numprimes,i,numtwins;

       primes:= {};
       numprimes := 0;

       for i from a to b do
       i:= nextprime(i);
       primes := primes union {i};
       numprimes := numprimes + 1
       end do;

         numtwins := 0;
         for i from 2 to numprimes-1 do
           if primes[i] - primes[i-1] = 2 then
           print(primes[i-1],primes[i]);
           numtwins := numtwins + 1
         end if;
         end do;

print(`Number of twin pairs is: `); print(numtwins);
end proc:
print("Finding the twin prime pairs in the interval.");
print("Type in g(a,b) where a and b is an interval. End with ; and then enter");

>

> g(1,60);

>

> restart;

HW#3 problem 3

>    r := proc(c,d)
     local a, n, countprimes;

countprimes :=0;
if gcd(c,d) = 1 then
   for n from 1 to 100 do
          a := c + n*d;
             if type(a, prime) then
             countprimes := countprimes + 1;
             print (a)
             end if;
   end do;

             else print("Those two numbers are not relatively prime. Please try again.");
             end if;
           print("The number of primes is:"); print(countprimes);
end proc:

> print("Please enter r(c,d) substituting c,d with integers. End with ; then enter.");

> r(3,501);

> r(3,5);

>

> restart;

HW#3 problem 4.

>    with(LinearAlgebra):
h :=proc(n)
    local y,z,i;
    y := HilbertMatrix(n);
    printf("\nHilbert Matrix\n");
    print(y);
    printf("\nMatrix Inverse\n");
    print(MatrixInverse(y));
    printf("\nDeterminant\n");
    print(Determinant(y));
    printf("\nConditional Number\n");
    print(ConditionNumber(y));
    printf("\nPrinciple Submatrices\n");


      for i from 2 to n-1 do
        z := HilbertMatrix(i);
        printf("\nSubmatrix %dx%d\n",i,i);
        print(z);
        printf("\nComputed %dx%d determinant is:\n",i,i);
        print(Determinant(z));
      end do;

  end proc: