如何用mathematica得到方程组的实数根?

mathematica

方程如下: NSolve[{x^2 + y^3 == 1, x^4 + y^4 == 2}, {x, y}] 得到的结果如下 {{x -> -0.428653 + 1.42567 I, y -> -0.552329 - 1.34942 I}, {x -> -0.428653 - 1.42567 I, y -> -0.552329 + 1.34942 I}, {x -> 1.15363, y -> -0.691637}, {x -> -1.20097 + 0.0500217 I, y -> 0.324147 - 0.698088 I}, {x -> -1.20097 - 0.0500217 I, y -> 0.324147 + 0.698088 I}, {x -> 0. + 0.71621 I, y -> 1.148}, {x -> 0. - 0.71621 I, y -> 1.148}, {x -> 1.20097 - 0.0500217 I, y -> 0.324147 - 0.698088 I}, {x -> 1.20097 + 0.0500217 I, y -> 0.324147 + 0.698088 I}, {x -> 0.428653 + 1.42567 I, y -> -0.552329 + 1.34942 I}, {x -> 0.428653 - 1.42567 I, y -> -0.552329 - 1.34942 I}, {x -> -1.15363, y -> -0.691637}} 但是对我来说有用的就是x和y都是实数的解,其余的复数解都没有用, 怎么办呢?

mathematica

Clear["Global`*"];(*清除所有变量*) eq1 = NSolve[{x^2 + y^3 == 1, x^4 + y^4 == 2}, {x, y}] (*求解方程*) eq2={x,y}/.eq1 (*只保留数据,去掉变量符号*) eq3 = Im[eq2] (*取得虚部*) Pick[eq2,eq3,0] (*只提取虚部等于零的数据*)

1. Clear["Global`*"];(*清除所有变量*)
2. eq1 = NSolve[{x^2 + y^3 == 1, x^4 + y^4 == 2}, {x, y}] (*求解方程*)
3. eq2={x,y}/.eq1 (*只保留数据,去掉变量符号*)
4. eq3 = Im[eq2] (*取得虚部*)
5. Pick[eq2,eq3,0] (*只提取虚部等于零的数据*)

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但是我得到的结果是 {{}, {}, {1.15363, -0.691637}, {}, {}, {1.148}, {1.148}, {}, {}, {}, \ {}, {-1.15363, -0.691637}} 这个结果总是与我想得到的要差那么些

mathematica

晕，我自己得到结果了，虽然是费了点小脑筋！ Select[eq2, Im[#] == {0, 0} &]

mathematica

1. Clear["Global`*"];(*清除所有变量*)
2. eq1=NSolve[{x^2+y^3==1,x^4+y^4==2},{x,y},10];(*求解方程,并得到10位数字*)
3. eq2={x,y}/.eq1;(*只保留数据,去掉变量符号*)
4. answer=Select[eq2,Im[#]=={0,0}&](*根据虚部都等于零提取得到最后的结果*)

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这下算是完美了，不过总有些自问自答的味道！

mathematica

再给一个复杂的 Clear["Global`*"];(*清除所有变量*) eq1=NSolve[{x^2+c^2==4^2,y^2+c^2==5^2, 2/x==b/c,2/y==a/c, a+b==c},{a,b,c,x,y},10] (*求解方程,并得到10位有效数字*) eq2={a,b,c,x,y}/.eq1 (*只保留数据,去掉变量符号*) eq3=Select[eq2,Im[#]=={0,0,0,0,0}&](*根根据虚部都等于零的结果*) (*提取结果都大于零的结果*) eq4=Select[eq3,(#[[1]]>0&&#[[2]]>0&&#[[3]]>0&&#[[3]]>0&&#[[4]]>0&&#[[5]]>0)&]

1. Clear["Global`*"];(*清除所有变量*)
2. eq1=NSolve[{x^2+c^2==4^2,y^2+c^2==5^2,
3. 2/x==b/c,2/y==a/c,
4. a+b==c},{a,b,c,x,y},10] (*求解方程,并得到10位有效数字*)
5. eq2={a,b,c,x,y}/.eq1 (*只保留数据,去掉变量符号*)
6. eq3=Select[eq2,Im[#]=={0,0,0,0,0}&](*根根据虚部都等于零的结果*)
7. (*提取结果都大于零的结果*)
8. eq4=Select[eq3,(#[[1]]>0&&#[[2]]>0&&#[[3]]>0&&#[[3]]>0&&#[[4]]>0&&#[[5]]>0)&]

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不知道谁能把eq3和eq4简化一下!

mathematica

其实五楼的方程组是根据几何关系得到的，不知道谁能看出来？ 我当年就参加数学建模，看到了一本教材，就是因为这个感受到了mathematica的强大！

mathematica

代码总是解析会出问题，老是要把代码解析成公式，郁闷！

mathematica

五楼的保留20位的求解结果！ {{0.8190941687975817515, 1.0758780371486812001, 1.8949722059462629516, 3.5226524578350266188, 4.6269947415888808194}}

mathematica

对了，谁能够把五楼的方程组的符号解求解出来，我说的是符号解，不是数值解，并且提取出都大于零的解！

mathematica

好庞大的求解结果！看来只有mathematica能办到这事情了！ {{1/72 (36 \[Sqrt](33/2 - 1/2 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/2 \[Sqrt](182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)])) + 256 \[Sqrt]((33/2 - 1/2 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/2 \[Sqrt](182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]))/(91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3))) - 4 \[Sqrt]((91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)) (33/2 - 1/2 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/2 \[Sqrt](182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]))) - 4 \[Sqrt]((182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]) (33/2 - 1/2 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/2 \[Sqrt](182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]))) + 1/2 \[Sqrt]((91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)) (182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]) (33/2 - 1/2 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/2 \[Sqrt](182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)])))), 1/72 (36 \[Sqrt](33/2 - 1/2 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/2 \[Sqrt](182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)])) - 256 \[Sqrt]((33/2 - 1/2 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/2 \[Sqrt](182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]))/(91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3))) + 4 \[Sqrt]((91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)) (33/2 - 1/2 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/2 \[Sqrt](182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]))) + 4 \[Sqrt]((182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]) (33/2 - 1/2 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/2 \[Sqrt](182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]))) - 1/2 \[Sqrt]((91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)) (182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]) (33/2 - 1/2 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/2 \[Sqrt](182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)])))), \[Sqrt](33/ 2 - 1/2 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/2 Sqrt[ 182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]]), 1/144 (144 + 2176/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 145/8 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] + 1/8 (8789553 - 1119744 Sqrt[19])^(1/3) Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] + 9/8 (3 (4019 + 512 Sqrt[19]))^(1/3) Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/8 (91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3))^(3/2) + 16 Sqrt[182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]] - 1/8 (8789553 - 1119744 Sqrt[19])^(1/3) Sqrt[ 182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]] - 9/8 (3 (4019 + 512 Sqrt[19]))^(1/3) Sqrt[ 182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]] - 1/8 (182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)])^(3/2) + 17/4 \[Sqrt]((91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)) (182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]))), 1/144 (144 + 128/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] + 145/8 Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 1/8 (8789553 - 1119744 Sqrt[19])^(1/3) Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] - 9/8 (3 (4019 + 512 Sqrt[19]))^(1/3) Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)] + 1/8 (91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3))^(3/2) - 16 Sqrt[182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]] + 1/8 (8789553 - 1119744 Sqrt[19])^(1/3) Sqrt[ 182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]] + 9/8 (3 (4019 + 512 Sqrt[19]))^(1/3) Sqrt[ 182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)]] + 1/8 (182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)])^(3/2) + 1/4 \[Sqrt]((91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)) (182 - 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) - 3 (3 (4019 + 512 Sqrt[19]))^(1/3) + 1024/Sqrt[ 91 + 1/3 (8789553 - 1119744 Sqrt[19])^(1/3) + 3 (3 (4019 + 512 Sqrt[19]))^(1/3)])))}}