# WORKSHEET#12
#
# SOLUTION OF SYSTEMS OF POLYNOMIAL EQUATIONS
#
#
> f:=x^4-2*x*y^2-y^3-2;g:=x*y^4+x^2*y^2-4;\
4 2 3
f := x - 2 x y - y - 2
4 2 2
g := x y + x y - 4
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> with(plots);
[animate, animate3d, conformal, contourplot, cylinderplot, densityplot,
display, display3d, fieldplot, fieldplot3d, gradplot, gradplot3d,
implicitplot, implicitplot3d, loglogplot, logplot, matrixplot, odeplot,
pointplot, polarplot, polygonplot, polygonplot3d, polyhedraplot, replot,
setoptions, setoptions3d, spacecurve, sparsematrixplot, sphereplot,
surfdata, textplot, textplot3d, tubeplot]
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> F:=implicitplot(f,x=-4..4,y=-4..4):
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> G:=implicitplot(g,x=-4..4,y=-4..4):
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> display(F,title=`x^4-2*x*y^2-y^3=2`);display(G,title=`x*y^4+x^2*y^2=4`);\
display([F,G],title=`x^4-2*x*y^2-y^3=2 and x*y^4+x^2*y^2=4`);
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#
** Maple V Graphics **
#
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** Maple V Graphics **
#
>
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** Maple V Graphics **
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>
# The last plot indicates that there are 3 solutions. In the plot window you can point and
# click to see what the coordinates are. Try it now. I got (x,y)=(1.562,0.9796),
# (1.417,-1.06) and (0.4386,-1.642). We can use the solve command to fine exact solutions.
#
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> S:=solve({f,g},{x,y});
S := {y = %1,
367 17 15 2129 14 163 13 7453 12 1293 11
x = ---- %1 + ---- %1 - ------ %1 - ---- %1 - ------ %1 - ----- %1
1838 1838 117632 7352 235264 29408
727 10 6233 8 11563 6 2615 7 12329 4 67 16
+ ---- %1 - ----- %1 + ----- %1 - ----- %1 + ----- %1 + ----- %1
7352 29408 58816 29408 14704 29408
139 5 659 2 6599 9 255 18 1139 3
+ ---- %1 - ---- %1 + ----- %1 - ------ %1 + ---- + 2/919 %1
7352 7352 58816 117632 1838
1395 17
- ------ %1
235264
}
19 16 15 14 13 12 11
%1 := RootOf(_Z + 2 _Z + 2 _Z + 7 _Z + 16 _Z + 4 _Z - 4 _Z
10 8 7 4
+ 48 _Z + 92 _Z + 32 _Z + 64 _Z - 256)
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# This means that y is one of the 19 roots of the polynomial in Z and x is the stated
# expression in terms of the value of y. Notice that %1 has been assigned a value by the use
# of the assignment symbol ":=", but x and y have not. Check this.
#
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> %1;x;y;
19 16 15 14 13 12 11 10
RootOf(_Z + 2 _Z + 2 _Z + 7 _Z + 16 _Z + 4 _Z - 4 _Z + 48 _Z
8 7 4
+ 92 _Z + 32 _Z + 64 _Z - 256)
x
y
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# The next command assigns values S.
#
#
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> assign(S);
#
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> x;
367 17 15 2129 14 163 13 7453 12 1293 11
---- %1 + ---- %1 - ------ %1 - ---- %1 - ------ %1 - ----- %1
1838 1838 117632 7352 235264 29408
727 10 6233 8 11563 6 2615 7 12329 4 67 16
+ ---- %1 - ----- %1 + ----- %1 - ----- %1 + ----- %1 + ----- %1
7352 29408 58816 29408 14704 29408
139 5 659 2 6599 9 255 18 1139 3
+ ---- %1 - ---- %1 + ----- %1 - ------ %1 + ---- + 2/919 %1
7352 7352 58816 117632 1838
1395 17
- ------ %1
235264
19 16 15 14 13 12 11
%1 := RootOf(_Z + 2 _Z + 2 _Z + 7 _Z + 16 _Z + 4 _Z - 4 _Z
10 8 7 4
+ 48 _Z + 92 _Z + 32 _Z + 64 _Z - 256)
--------------------------------------------------------------------------------
> y;
19 16 15 14 13 12 11 10
RootOf(_Z + 2 _Z + 2 _Z + 7 _Z + 16 _Z + 4 _Z - 4 _Z + 48 _Z
8 7 4
+ 92 _Z + 32 _Z + 64 _Z - 256)
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# This has the side effect of changing the definitions of the functions f and g.
--------------------------------------------------------------------------------
> f;
367 17 15 2129 14 163 13 7453 12 1293 11
(---- %1 + ---- %1 - ------ %1 - ---- %1 - ------ %1 - ----- %1
1838 1838 117632 7352 235264 29408
727 10 6233 8 11563 6 2615 7 12329 4 67 16
+ ---- %1 - ----- %1 + ----- %1 - ----- %1 + ----- %1 + ----- %1
7352 29408 58816 29408 14704 29408
139 5 659 2 6599 9 255 18 1139 3
+ ---- %1 - ---- %1 + ----- %1 - ------ %1 + ---- + 2/919 %1
7352 7352 58816 117632 1838
1395 17 367 17 15 2129 14 163 13
- ------ %1 )^4 - 2 (---- %1 + ---- %1 - ------ %1 - ---- %1
235264 1838 1838 117632 7352
7453 12 1293 11 727 10 6233 8 11563 6 2615 7
- ------ %1 - ----- %1 + ---- %1 - ----- %1 + ----- %1 - ----- %1
235264 29408 7352 29408 58816 29408
12329 4 67 16 139 5 659 2 6599 9 255 18
+ ----- %1 + ----- %1 + ---- %1 - ---- %1 + ----- %1 - ------ %1
14704 29408 7352 7352 58816 117632
1139 3 1395 17 2 3
+ ---- + 2/919 %1 - ------ %1 ) %1 - %1 - 2
1838 235264
19 16 15 14 13 12 11
%1 := RootOf(_Z + 2 _Z + 2 _Z + 7 _Z + 16 _Z + 4 _Z - 4 _Z
10 8 7 4
+ 48 _Z + 92 _Z + 32 _Z + 64 _Z - 256)
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# The values previously assigned to x and y are used in computing f(x,y). If we want to use
# the functions again we'll have to convert x and y back into variables.
--------------------------------------------------------------------------------
> X:=x:Y:=y:x:='x':y:='y':
--------------------------------------------------------------------------------
# The assigned values of x and y are saved in X and Y. Then x and y are changed back into
# variables. To find explicit numerical values for the intersection points we can do the
# folloeing.
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> allvalues(Y);
-1.647432204, - 1.337971368 - .8833243765 I, - 1.337971368 + .8833243765 I,
-1.057852433, - .8832686595 - .7210045074 I,
- .8832686595 + .7210045074 I, - .2558788231 - 1.256358136 I,
- .2558788231 + 1.256358136 I, - .07576148742 - 1.477425325 I,
- .07576148742 + 1.477425325 I, .05303535957 - 1.209109819 I,
.05303535957 + 1.209109819 I, .8442838121 - .8587454679 I,
.8442838121 + .8587454679 I, .9961067967, 1.242260568 - .9836072175 I,
1.242260568 + .9836072175 I, 1.267889518 - .8320938192 I,
1.267889518 + .8320938192 I
--------------------------------------------------------------------------------
# Notice that we get 19 values, which we expect since we are solving a degree 19
# polynomial. Only 3 of the roots are real; this agrees with our expectation that there are
# only 3 real intersection points. Let's give a name to the degree 19 polynomial.
--------------------------------------------------------------------------------
> p:=op(Y);
19 16 15 14 13 12 11 10
p := _Z + 2 _Z + 2 _Z + 7 _Z + 16 _Z + 4 _Z - 4 _Z + 48 _Z
8 7 4
+ 92 _Z + 32 _Z + 64 _Z - 256
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> Yreal:=fsolve(p,_Z);
Yreal := -1.647432204, -1.057852433, .9961067967
#
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# To find the corresponding x values we substitute into the expression for X.
--------------------------------------------------------------------------------
> for k from 1 to 3 do Xreal[k]:=subs(%1=Yreal[k],X) od;
Xreal[1] := .46378423
Xreal[2] := 1.412154629
Xreal[3] := 1.572087242
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# This gives us our 3 real solutions. We should verify that these are solutions by
# substituting back into the original polynomials.
--------------------------------------------------------------------------------
> for k from 1 to 3 do print(Xreal[k],Yreal[k],subs(x=Xreal[k],y=Yreal[k],f),subs(x=Xreal[k],y=Yreal[k],g)) od;
-7 -6
.46378423, -1.647432204, .68*10 , -.131*10
-7 -8
1.412154629, -1.057852433, .13*10 , .8*10
-7 -8
1.572087242, .9961067967, -.102*10 , -.4*10
