作者harry901 (forcing to A cup)
看板Math
标题Re: [代数] 解三元联立
时间Wed Nov 2 01:24:43 2011
※ 引述《autnb (募集补教师资ing)》之铭言：
: x^2 + y^2 = 16
: x^2 + z^2 = 4+xy
: z^2 + y^2 = 4+yz(3)^(1/2)
: 解x,y,z
solved by mathematica
{x -> -3.35587, z -> -0.207159, y -> -2.17672}, {x -> -1.23399,
z -> -2.67815, y -> -3.8049}, {x -> -0.728872 - 3.8631 I,
z -> -5.15904 - 1.4911 I,
y -> -5.53633 + 0.508587 I}, {x -> -0.728872 + 3.8631 I,
z -> -5.15904 + 1.4911 I,
y -> -5.53633 - 0.508587 I}, {x -> 0.728872 - 3.8631 I,
z -> 5.15904 - 1.4911 I,
y -> 5.53633 + 0.508587 I}, {x -> 0.728872 + 3.8631 I,
z -> 5.15904 + 1.4911 I, y -> 5.53633 - 0.508587 I}, {x -> 1.23399,
z -> 2.67815, y -> 3.8049}, {x -> 3.35587, z -> 0.207159,
y -> 2.17672}
他也有numerical closed form 如下..... 请笑纳 自己慢慢打开吧
{x -> (-4096 Sqrt[2 (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])] -
2048 Sqrt[6 (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])] -
2048 Sqrt[
2 (-1 + 4 Sqrt[3]) (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])])/
8192, z -> (
6144 Sqrt[2 (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])] -
9216 Sqrt[6 (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])] +
3072 Sqrt[
2 (-1 + 4 Sqrt[3]) (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])])/
24576, y -> -Sqrt[
2 (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])]}, {x -> (
4096 Sqrt[2 (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])] +
2048 Sqrt[6 (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])] +
2048 Sqrt[
2 (-1 + 4 Sqrt[3]) (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])])/
8192, z -> (-6144 Sqrt[
2 (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])] +
9216 Sqrt[6 (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])] -
3072 Sqrt[
2 (-1 + 4 Sqrt[3]) (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])])/
24576, y -> Sqrt[
2 (10 - 3 Sqrt[3] - Sqrt[-1 + 4 Sqrt[3]])]}, {x -> (
4096 Sqrt[2 (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])] +
2048 Sqrt[6 (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])] -
2048 Sqrt[
2 (-1 + 4 Sqrt[3]) (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])])/
8192, z -> (-6144 Sqrt[
2 (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])] +
9216 Sqrt[6 (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])] +
3072 Sqrt[
2 (-1 + 4 Sqrt[3]) (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])])/
24576, y -> Sqrt[
2 (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])]}, {x -> (-4096 Sqrt[
2 (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])] -
2048 Sqrt[6 (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])] +
2048 Sqrt[
2 (-1 + 4 Sqrt[3]) (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])])/
8192, z -> (
6144 Sqrt[2 (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])] -
9216 Sqrt[6 (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])] -
3072 Sqrt[
2 (-1 + 4 Sqrt[3]) (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])])/
24576, y -> -Sqrt[
2 (10 - 3 Sqrt[3] + Sqrt[-1 + 4 Sqrt[3]])]}, {x -> (-4096 Sqrt[
2 (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])] +
2048 Sqrt[6 (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])] -
2048 I Sqrt[
2 (1 + 4 Sqrt[3]) (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])])/
8192, z -> (-6144 Sqrt[
2 (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])] -
9216 Sqrt[6 (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])] -
3072 I Sqrt[
2 (1 + 4 Sqrt[3]) (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])])/
24576, y -> -Sqrt[
2 (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])]}, {x -> (
4096 Sqrt[2 (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])] -
2048 Sqrt[6 (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])] +
2048 I Sqrt[
2 (1 + 4 Sqrt[3]) (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])])/
8192, z -> (
6144 Sqrt[2 (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])] +
9216 Sqrt[6 (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])] +
3072 I Sqrt[
2 (1 + 4 Sqrt[3]) (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])])/
24576, y -> Sqrt[
2 (10 + 3 Sqrt[3] - I Sqrt[1 + 4 Sqrt[3]])]}, {x -> (
4096 Sqrt[2 (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])] -
2048 Sqrt[6 (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])] -
2048 I Sqrt[
2 (1 + 4 Sqrt[3]) (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])])/
8192, z -> (
6144 Sqrt[2 (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])] +
9216 Sqrt[6 (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])] -
3072 I Sqrt[
2 (1 + 4 Sqrt[3]) (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])])/
24576, y -> Sqrt[
2 (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])]}, {x -> (-4096 Sqrt[
2 (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])] +
2048 Sqrt[6 (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])] +
2048 I Sqrt[
2 (1 + 4 Sqrt[3]) (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])])/
8192, z -> (-6144 Sqrt[
2 (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])] -
9216 Sqrt[6 (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])] +
3072 I Sqrt[
2 (1 + 4 Sqrt[3]) (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])])/
24576, y -> -Sqrt[2 (10 + 3 Sqrt[3] + I Sqrt[1 + 4 Sqrt[3]])]}
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