HI I have a simultaneous equation in terms of a, b, c, d. when I try to use NSolve to solve the equation and get the vales of a, b, c, d in terms of p,q,r,s it is just giving empty bracket as an answer. Any one can help me to solve this equation.
NSolve[{a*b + a*c + b*d + c*d == p &&
a*b + b*c - b*d == q &&
a*b + b*c + a*d + c*d == r &&
a*b + a*c - a*d == s},
{a, b, c, d}]
Thanks in advance.
That means there is no solution. Also since there are nonnumeric parameters, you might as well use Solve because that’s what NSolve will have to do anyway.
Possibly solve the equations stepwise. For example
In[49]:= Solve[a*b + a*c + b*d + c*d == p, a]
Out[49]= {{a -> (-b d - c d + p)/(b + c)}}
In[52]:= a = (-b d - c d + p)/(b + c);
Then use this result in second equation for b, etc.
In[3]:= Reduce[{ab + ac + bd + cd == p && ab + bc - bd == q &&
ab + bc + ad + cd == r && ab + ac - ad == s}, {a, b, c, d},
Backsubstitution → True]
Out[3]= (p == -q + r + s && a != 0 &&
b == (a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])/(2 a) &&
a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] != 0 &&
c == (-3 a^4 + 4 a^2 q - 4 a^2 s - s^2 +
3 a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] -
Sqrt[2] a^2 \[Sqrt](-(1/(
a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (-a^4 +
2 a^2 q - 2 a^2 s - s^2 +
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s -
Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])) &&
d == -((a^4 + 2 a^2 s + s^2 -
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] -
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
Sqrt[2] a^2 \[Sqrt](-(1/(
a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (-a^4 +
2 a^2 q - 2 a^2 s - s^2 +
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 +
s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))) || (p == -q +
r + s && a != 0 &&
b == (a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])/(2 a) &&
a^2 + s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] != 0 &&
c == (-3 a^4 + 4 a^2 q - 4 a^2 s - s^2 +
3 a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
Sqrt[2] a^2 \[Sqrt](-(1/(
a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (-a^4 +
2 a^2 q - 2 a^2 s - s^2 +
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s -
Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])) &&
d == -((a^4 + 2 a^2 s + s^2 -
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] -
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] -
Sqrt[2] a^2 \[Sqrt](-(1/(
a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (-a^4 +
2 a^2 q - 2 a^2 s - s^2 +
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 +
s - Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))) || (p == -q +
r + s && a != 0 &&
b == (a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])/(2 a) &&
a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] != 0 &&
c == (-3 a^4 + 4 a^2 q - 4 a^2 s - s^2 -
3 a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] -
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] -
Sqrt[2] a^2 \[Sqrt]((1/(
a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (a^4 -
2 a^2 q + 2 a^2 s + s^2 +
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s +
Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])) &&
d == (-a^4 - 2 a^2 s - s^2 -
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] -
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] -
Sqrt[2] a^2 \[Sqrt]((1/(
a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (a^4 -
2 a^2 q + 2 a^2 s + s^2 +
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s +
Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2]))) || (p == -q + r + s &&
a != 0 &&
b == (a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])/(2 a) &&
a^2 + s + Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] != 0 &&
c == (-3 a^4 + 4 a^2 q - 4 a^2 s - s^2 -
3 a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] -
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
Sqrt[2] a^2 \[Sqrt]((1/(
a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (a^4 -
2 a^2 q + 2 a^2 s + s^2 +
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s +
Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])) &&
d == (-a^4 - 2 a^2 s - s^2 -
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] -
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
Sqrt[2] a^2 \[Sqrt]((1/(
a^4))(a^4 - 4 a^2 q + 4 a^2 r + 2 a^2 s + s^2) (a^4 -
2 a^2 q + 2 a^2 s + s^2 +
a^2 Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2] +
s Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2])))/(2 a (a^2 + s +
Sqrt[a^4 - 4 a^2 q + 2 a^2 s + s^2]))) || (s == 0 &&
p == -q + r && a == 0 && b != 0 &&
c == (-b^2 + q - Sqrt[b^4 - 2 b^2 q + q^2 + 4 b^2 r])/(2 b) &&
d == (-b^2 - q - Sqrt[b^4 - 2 b^2 q + q^2 + 4 b^2 r])/(
2 b)) || (s == 0 && p == -q + r && a == 0 && b != 0 &&
c == (-b^2 + q + Sqrt[b^4 - 2 b^2 q + q^2 + 4 b^2 r])/(2 b) &&
d == (-b^2 - q + Sqrt[b^4 - 2 b^2 q + q^2 + 4 b^2 r])/(
2 b)) || (q == 0 && p == r + s && b == 0 && a != 0 &&
c == (-a^2 + s - Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2])/(
2 a) && -a^2 + s - Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2] != 0 &&
d == -((a (a^2 + 2 r + s + Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2]))/(
a^2 - s + Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2]))) || (q == 0 &&
p == r + s && b == 0 && a != 0 &&
c == (-a^2 + s + Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2])/(
2 a) && -a^2 + s + Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2] != 0 &&
d == -((a (a^2 + 2 r + s - Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2]))/(
a^2 - s - Sqrt[a^4 + 4 a^2 r + 2 a^2 s + s^2]))) || (r == -s &&
q == 0 && p == 0 && b == 0 && c == 0 && a != 0 &&
d == -(s/a)) || (s == 0 && q == 0 && p == r && a == 0 && b == 0 &&
c != 0 && d == r/c) || (s == 0 && r == 0 && q == 0 && p == 0 &&
a == 0 && b == 0 && c == 0)
