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toy problems

最後の簡約は重いので無効化します(笑).

(%i1) postsimp:off$
Evaluation took 0.0000 seconds (0.0000 elapsed) using 32 bytes.

放物線と直線との交わり.

(%i2) solvex(y=x^2 %and x+y=1,x,y);
Evaluation took 5.4600 seconds (11.3400 elapsed) using 315.180 MB.
(%o2) [[x = (-sqrt(5)/2)-1/2,y = sqrt(5)/2+3/2],
       [x = sqrt(5)/2-1/2,y = 3/2-sqrt(5)/2]]

領域と直線との交わり.

(%i3) solvex(y>x^2 %and x+y=1,y);
Evaluation took 2.1600 seconds (4.7800 elapsed) using 120.922 MB.
(%o3) [[(-sqrt(5)/2)-1/2 < x,x < sqrt(5)/2-1/2,y = 1-x]]

x について解くと.

(%i4) solvex(y>x^2 %and x+y=1,x);
Evaluation took 2.2100 seconds (4.9300 elapsed) using 122.540 MB.
(%o4) [[3/2-sqrt(5)/2 < y,x = 1-y,y < sqrt(5)/2+3/2]]

放物線と領域との交わり.

(%i5) solvex(y=x^2 %and x+y<1,y);
Evaluation took 2.4200 seconds (5.0200 elapsed) using 146.358 MB.
(%o5) [[(-sqrt(5)/2)-1/2 < x,x < sqrt(5)/2-1/2,y = x^2]]

x について解くと.

(%i6) solvex(y=x^2 %and x+y<1,x);
Evaluation took 6.7700 seconds (11.4300 elapsed) using 462.988 MB.
(%o6) [[0 <= y,x = -sqrt(y),y < 3/2],[0 <= y,x = sqrt(y),y < 3/2-sqrt(5)/2],
       [3/2-sqrt(5)/2 < y,x = -sqrt(y),y < sqrt(5)/2+3/2]]

結果が見難いので,x を含む式とその他とに分離.

(%i7) sepaxa(%,x);
Evaluation took 0.0000 seconds (0.0000 elapsed) using 8.836 KB.
(%o7) [[[x = -sqrt(y)],[0 <= y,y < 3/2]],
       [[x = sqrt(y)],[0 <= y,y < 3/2-sqrt(5)/2]],
       [[x = -sqrt(y)],[3/2-sqrt(5)/2 < y,y < sqrt(5)/2+3/2]]]

解の選択.

(%i8) solvex(3*x^2=2 %and 1<x);
Evaluation took 0.2100 seconds (0.5500 elapsed) using 9.020 MB.
(%o8) [[false]]

空のようなので,改めて.

(%i9) solvex(2*x^2=3 %and 1<x);
Evaluation took 2.9600 seconds (6.4500 elapsed) using 211.316 MB.
(%o9) [[x = sqrt(3)/sqrt(2)]]

不等式の連言.

(%i10) solvex(2*x^2<=3 %and 1<x);
Evaluation took 2.6700 seconds (3.7200 elapsed) using 174.558 MB.
(%o10) [[1 < x,x <= sqrt(3)/sqrt(2)]]

いきなり 3 変数ですが.

(%i11) solvex(a*x^2=b %and a<x);
Evaluation took 2.7200 seconds (7.5900 elapsed) using 215.724 MB.
(%o11) [[0 < x-a,b = a*x^2]]

変数の指定が必要です.

(%i12) solvex(a*x^2=b %and a<x,x);
Evaluation took 6.7700 seconds (19.4000 elapsed) using 487.579 MB.
(%o12) [[0 < a,a^3-b < 0,x = sqrt(b/a)],[a < x,a = 0,b = 0],
        [a < 0,a^3-b < 0,b <= 0,x = -sqrt(b/a)],[a < 0,b <= 0,x = sqrt(b/a)]]

見易く整理.

(%i13) sepaxa(%,x);
Evaluation took 0.0000 seconds (0.0000 elapsed) using 12.602 KB.
(%o13) [[[x = sqrt(b/a)],[0 < a,a^3-b < 0]],[[a < x],[a = 0,b = 0]],
        [[x = -sqrt(b/a)],[a < 0,a^3-b < 0,b <= 0]],
        [[x = sqrt(b/a)],[a < 0,b <= 0]]]

http://www.cybernet.co.jp/maple/documents/pdf/product/maple/maple17/maple17_new_feature31.pdf
の 2 つ目の例.

(%i14) solvex(a*x^2<b %and a<x,x);
Evaluation took 31.4200 seconds (47.6300 elapsed) using 2388.419 MB.
(%o14) [[0 < a,a < x,a < 0,b < 0],[0 < a,a < x,-sqrt(b/a) < x,x < sqrt(b/a)],
        [0 < b,a < x,a = 0],[0 < b,a < x,a < 0],[x < -sqrt(b/a),a < x,a < 0],
        [a < x,sqrt(b/a) < x,a < 0]]

やはり簡約は必要ですね.

(%i15) plineq(%,1);
Evaluation took 0.2200 seconds (0.7700 elapsed) using 10.936 MB.
(%o15) [[0 < a,a < x,-sqrt(b/a) < x,x < sqrt(b/a)],[0 < b,a < x,a = 0],
        [0 < b,a < x,a < 0],[a < x,sqrt(b/a) < x,a < 0],
        [x < -sqrt(b/a),a < x,a < 0],[false]]

見易く整理.

(%i16) sepaxa(bra(ora(%)),x);
Evaluation took 0.4100 seconds (0.4100 elapsed) using 30.975 MB.
(%o16) [[[a < x,-sqrt(b/a) < x,x < sqrt(b/a)],[0 < a]],
        [[a < x],[0 < b,a = 0]],[[a < x],[0 < b,a < 0]],
        [[a < x,sqrt(b/a) < x],[a < 0]],[[a < x,x < -sqrt(b/a)],[a < 0]]]