(* Fouriertransformation *)
a={-1,2,3,0,0}
{-1, 2, 3, 0, 0}
b={1,1,1,0,0}
{1, 1, 1, 0, 0}
(* beliebige Matrix *)
f={ {1,0,0,0,0},
{1,1,1,1,1},
{1,2,4,8,16},
{1,-1,1,-1,1},
{1,3,9,27,81}}
{{1, 0, 0, 0, 0}, {1, 1, 1, 1, 1}, {1, 2, 4, 8, 16},
{1, -1, 1, -1, 1}, {1, 3, 9, 27, 81}}
f //MatrixForm
1 0 0 0 0 1 1 1 1 1 1 2 4 8 16 1 -1 1 -1 1 1 3 9 27 81
f.a (* Anwendung auf a *)
{-1, 4, 15, 0, 32}
f.b (* Anwendung auf b *)
{1, 3, 7, 1, 13}
(f.a)*(f.b) (* komponentenweise Multiplikation *)
{-1, 12, 105, 0, 416}
fInv=Inverse[f] (* Berechnung der Inversen von f *)
5 3 1 1 1
{{1, 0, 0, 0, 0}, {-(-), -, -(-), -(-), --},
6 2 2 4 12
5 1 1 11 1 5 1 1 1
{-(-), -, -, --, -(--)}, {-, -1, -, -(-), -(--)},
6 4 6 24 24 6 2 4 12
1 1 1 1 1
{-(-), -, -(-), --, --}}
6 4 6 24 24
fInv // MatrixForm
1 0 0 0 0 5 3 1 1 1 -(-) - -(-) -(-) -- 6 2 2 4 12 5 1 1 11 1 -(-) - - -- -(--) 6 4 6 24 24 5 1 1 1 - - -(-) -(--) 6 -1 2 4 12 1 1 1 1 1 -(-) - -(-) -- -- 6 4 6 24 24
Transpose[(f.a)*(f.b)]
{-1, 12, 105, 0, 416}
fInv.(Transpose[(f.a)*(f.b)]) (* Konvolution *)
{-1, 1, 4, 5, 3}
(* direkte Berechnung der Konvolution *)
Expand[(-1+2x+3x^2)*(1+x+x^2)]
2 3 4 -1 + x + 4 x + 5 x + 3 x
(* Division *)
p=x^8+2x^7-2x^6+x^5+3x^4-x^3+3x^2+x-1
2 3 4 5 6 7 8 -1 + x + 3 x - x + 3 x + x - 2 x + 2 x + x
r1=PolynomialRemainder[p,(x-0)(x-1)(x-2)(x-3),x]
2 3 -1 + 2641 x - 4063 x + 1430 x
r2=PolynomialRemainder[p,(x-4)(x-5)(x-6)(x-7),x]
2 3 -33479041 + 22087449 x - 4888375 x + 366174 x
r11=PolynomialRemainder[r1,(x-0)(x-1),x]
-1 + 8 x
r12=PolynomialRemainder[r1,(x-2)(x-3),x]
-18523 + 9496 x
r21=PolynomialRemainder[r2,(x-4)(x-5),x]
-1622861 + 428688 x
r22=PolynomialRemainder[r2,(x-6)(x-7),x]
-28098295 + 5042672 x
r111=PolynomialRemainder[r11,(x-0),x]
-1
r112=PolynomialRemainder[r11,(x-1),x]
7
r121=PolynomialRemainder[r12,(x-2),x]
469
r122=PolynomialRemainder[r12,(x-3),x]
9965
r211=PolynomialRemainder[r21,(x-4),x]
91891
r212=PolynomialRemainder[r21,(x-5),x]
520579
r221=PolynomialRemainder[r22,(x-6),x]
2157737
r222=PolynomialRemainder[r22,(x-7),x]
7200409
p /. {x->0}
-1
p /. {x->1}
7
p /. {x->2}
469
p /. {x->3}
9965
p /. {x->4}
91891
p /. {x->5}
520579
p /. {x->6}
2157737
p /. {x->7}
7200409
(* Primitive Einheitswurzeln *)
w[i_,n_]:=E^(I 2 Pi i / n)
w[1,8]
I/4 Pi E
Simplify[(x-w[0,8])(x-w[4,8])(x-w[2,8])(x-w[6,8])]
4 -1 + x
Simplify[(x-w[1,8])(x-w[5,8])(x-w[3,8])(x-w[7,8])]
4 1 + x
Simplify[(x-w[0,8])(x-w[4,8])]
2 -1 + x
Simplify[(x-w[2,8])(x-w[6,8])]
2 1 + x
Simplify[(x-w[1,8])(x-w[5,8])]
2 -I + x
Simplify[(x-w[3,8])(x-w[7,8])]
2 I + x
(* Bit Operationen *)
a0=0
0
a1=1
1
a2=2
2
a3=3
3
b0=Mod[a0+a2,5]
2
b1=Mod[a1+a3,5]
4
b2=Mod[a0+4a2,5]
3
b3=Mod[a1+4a3,5]
3
c0=Mod[b0+b1,5]
1
c1=Mod[b0+4*b1,5]
3
c2=Mod[b2+2*b3,5]
4
c3=Mod[b2+8*b3,5]
2
c0=Mod[b0+b1,5]
1
c1=Mod[b0+4*b1,5]
3
c2=Mod[b2+2*b3,5]
4
c3=Mod[b2+8*b3,5]
2
Converted by Mathematica (July 7, 2003)