Can anybody tell me why the tactic "field" does not work when I try to prove the following goal involving rationals?
nat_to_Q 3 * nat_to_Q n * nat_to_Q n + nat_to_Q 3 * nat_to_Q n +
nat_to_Q 3 * nat_to_Q n + nat_to_Q 3 + nat_to_Q 3 * nat_to_Q n +
nat_to_Q 3 + nat_to_Q n * nat_to_Q n * nat_to_Q n + nat_to_Q n * nat_to_Q n +
nat_to_Q n * nat_to_Q n * nat_to_Q 2 + nat_to_Q n * nat_to_Q 2 ==
nat_to_Q 3 * nat_to_Q n * nat_to_Q n * nat_to_Q n +
nat_to_Q 6 * nat_to_Q n * nat_to_Q n + nat_to_Q 11 * nat_to_Q n +
nat_to_Q 6
Note: n is nat and nat_to_Q is (Z.of_nat n # Pos.of_nat 1).
Thanks a lot,
Marcus.
Let's remove the coercions to make this easier to read:
3 * n * n + 3 * n +
3 * n + 3 * n +
3 + n * n * n + n * n +
n * n * 2 + n * 2 ==
3 * n * n * n +
6 * n * n + 11 * n + 6
Now we can see the issue: the goal does not hold. The 3rd order coefficient on the left-hand side is 1, but on the right-hand side it is 3.
Related
I have these equations:
syms pm pr teta s
A1 = -2 * b1 * pm + 2 * b2 * pr + b * teta + (1-t) * s + (1-p) * a + c * (b1 - b2);
A2 = 2 * b2 * pm + 2 * b1 * pr + (1-b) * teta + t * s + p * a + c * (b1 - b2);
A3 = b * pm + (1-b) * pr - n * teta - c;
A4 = (1-t) * pm + t * pr - k * s - c;
eqns = [A1,A2,A3,A4];
F=#(pm, pr, teta, s) [A1
A2
A3
A4];
x0 = [10, 10, 10, 10];
fsolve(F, x0)
How I can solve them?
(When I use fsolve, it shows this error: FSOLVE requires all values returned by functions to be of data type double)
Since you tagged Mathematica
A1 = -2*b1*pm + 2*b2*pr + b*teta + (1 - t)*s + (1 - p)*a + c*(b1 - b2);
A2 = 2*b2*pm + 2*b1*pr + (1 - b)*teta + t*s + p*a + c*(b1 - b2);
A3 = b*pm + (1 - b)*pr - n*teta - c;
A4 = (1 - t)*pm + t*pr - k*s - c;
FullSimplify[Solve[{A1 == 10, A2 == 10, A3 == 10, A4 == 10}, {pm, pr, teta, s}]]
pm -> ((k ((-1 + b)^2 + 2 b1 n) + n t^2) (b (10 + c) k +
n (10 + c + 10 k - a k - b1 c k + b2 c k +
a k p - (10 + c) t)) + ((-1 + b) (10 + c) k +
k n (-10 + b1 c - b2 c + a p) - (10 + c) n t) (b k - b^2 k +
n (2 b2 k + t - t^2)))/(k n (1 - 2 b1 k + b^2 (1 + 4 b2 k) +
2 (b1 - 2 (b1^2 + b2^2) k) n - 2 t -
4 (b1 + b2) n t + (1 + 4 b2 n) t^2 +
2 b (-1 + 2 b1 k - 2 b2 k + t)))
pr -> (c +
b^2 (10 + c - (-20 + a + 2 b1 c - 2 b2 c) k) + 2 b1^2 c k n -
b2 (20 + c - 2 (-10 + a) k + 2 b2 c k) n - a (1 + 2 b2 k) n p -
2 b1 k (10 + c + n (10 - a p)) + 10 (1 + n - 2 t) -
c (2 + b2 n) t +
n (-30 + a + 20 b2 + a p) t + (10 + c - (-20 + a) n +
2 b2 c n) t^2 - b1 n (c + 20 t + c t (-1 + 2 t)) +
b (k (-10 + a + 20 b1 - 20 b2 - a p) + 20 (-1 + t) +
c (-2 + 3 b1 k - 3 b2 k + 2 t)))/(1 - 2 b1 k +
b^2 (1 + 4 b2 k) + 2 (b1 - 2 (b1^2 + b2^2) k) n - 2 t -
4 (b1 + b2) n t + (1 + 4 b2 n) t^2 +
2 b (-1 + 2 b1 k - 2 b2 k + t))
teta -> (10 - 20 b2 - b2 c -
20 b2 k + 2 a b2 k + 40 b2^2 k + 2 b2^2 c k +
2 b1^2 (20 + 3 c) k - a p -
2 a b2 k p + (-30 + 3 b2 (20 + c) + a (1 + p)) t - (-20 + a +
2 b2 (20 + c)) t^2 +
b (-10 - 4 b1^2 c k + a p +
b1 (20 + 40 k - 2 a k + c (3 + 4 b2 k - 2 t)) + 20 t - a t +
b2 (20 + c - 2 a k + 4 a k p - 2 (20 + c) t)) +
b1 (2 k (-10 + a p) + 20 (-1 + t) + c (-3 + (5 - 2 t) t)))/(1 -
2 b1 k + b^2 (1 + 4 b2 k) + 2 (b1 - 2 (b1^2 + b2^2) k) n - 2 t -
4 (b1 + b2) n t + (1 + 4 b2 n) t^2 +
2 b (-1 + 2 b1 k - 2 b2 k + t))
s -> (20 b1 + b1 c - b2 c -
2 b^2 (-10 + b1 c + b2 (20 + c)) + 20 b1 n + 40 b1^2 n -
20 b2 n + 40 b2^2 n + 2 b1^2 c n + 4 b1 b2 c n +
2 b2^2 c n - (b1 - b2) (20 + c) t +
4 b1 (-10 + b1 c - b2 c) n t - 10 (-1 + 2 b2 + t) +
a (-1 - b^2 - 2 b1 n + p + 2 (b1 + b2) n p + t - p t +
2 n (b1 + b2 - 2 b2 p) t - b (-2 + p + t)) +
b (-(-10 + b2 (20 + c)) (-3 + 2 t) + b1 (-20 + c - 2 c t)))/(1 -
2 b1 k + b^2 (1 + 4 b2 k) + 2 (b1 - 2 (b1^2 + b2^2) k) n - 2 t -
4 (b1 + b2) n t + (1 + 4 b2 n) t^2 +
2 b (-1 + 2 b1 k - 2 b2 k + t))
I have some equations which include some parameters and variables.
Each variable defines and also exists in other equations. these are my equations:
syms Pof s Pon teta landa PIon PIof PISC
Pof = (s * y - teta + T * teta - landa + Pi * landa + alpha * p + Pon * b1 + w * b2)/(2 * b2);
s = -(y * (w - Pof))/q;
Pon = (q * s * y + 2 * w * y - 2 * w * y^2 + 2 * q * alpha + 2 * q * Pi * landa - q * teta + 3 * q * T * teta - q * landa + q * Pi * landa - q * alpha * p - 2 * y * Pof + 2 * y^2 * Pof + C * q * b1 + q * w * b2)/(2 * q * b1);
teta = (C - 3 * C * T - Pon + 3 * T * Pon)/2 * q;
landa = (-1/2) * (-1 + Pi) * (C - Pon);
Don = (1-p) * alpha - b1 * Pon + b2 * Pof - (1-y) * s + T * teta + landa * Pi;
Dof = p * alpha - b1 * Pof + b2 * Pon + y * s - (1-T) * teta - landa * (1-Pi);
PIon = (Pon-C) * Don - (1/2) * n * teta^2 - (1/2) * q * landa^2;
PIof = (Pof-w) * Dof - (1/2) * L * s^2;
PISC = PIon + PIof;
How I can solve these in order to get a numeric answer for each variable?
(I don't want parametric answers)
The equations as stated are
which can be arranged as a linear system A x = b as follows
which you solve in Matlab as x = A \ b
On further investigation, it seems A is singular since it's 10×8 in size and cannot be inverted. So a least-squares solution is needed where
x = inv(AT* A)* AT b
I want to create a 4 x 4 matrix with each entry representing f(x,y) where both x and y take values 0, 1, 2 and 3. So the first entry would be f(0,0), all the way to f(3,3).
The function f(x,y) is:
3 * cos(0*x + 0*y) + 2 * cos(0*x + 1*y) + 3 * cos(0*x + 2*y) + 8 * cos(0*x + 3*y)
+ 3 * cos(1*x + 0*y) + 25 * cos(1*x + 1*y) + 3 * cos(1*x + 2*y)
+ 8 * cos(1*x + 3*y)
+ 3 * cos(2*x + 0*y) + 25 * cos(2*x + 1*y) + 3 * cos(2*x + 2*y)
+ 8 * cos(2*x + 3*y)
+ 3 * cos(3*x + 0*y) + 25 * cos(3*x + 1*y) + 3 * cos(3*x + 2*y)
- 90 * cos(3*x + 3*y)
I haven't used Matlab much, and it's been a while. I have tried turning f(x,y) into a #f(x,y) function; using the .* operator; meshing x and y, etc. All of it without success...
Not sure, what you've tried exactly, but using meshgrid is the correct idea.
% Function defintion (abbreviated)
f = #(x, y) 3 * cos(0*x + 0*y) + 2 * cos(0*x + 1*y) + 3 * cos(0*x + 2*y)
% Set up x and y values.
x = 0:3
y = 0:3
% Generate grid.
[X, Y] = meshgrid(x, y);
% Rseult matrix.
res = f(X, Y)
Generated output:
f =
#(x, y) 3 * cos (0 * x + 0 * y) + 2 * cos (0 * x + 1 * y) + 3 * cos (0 * x + 2 * y)
x =
0 1 2 3
y =
0 1 2 3
res =
8.00000 8.00000 8.00000 8.00000
2.83216 2.83216 2.83216 2.83216
0.20678 0.20678 0.20678 0.20678
3.90053 3.90053 3.90053 3.90053
Here is the code:
syms G1 G2 G3 M1 M2 M3 P1 P2 P3 D1 D2 D3
S = solve(0 == 0.9 * (20 - G1) - 0.012 * D3 * G1, ...
0 == 0.9 * (20 - G2) - 0.012 * D1 * G2, ...
0 == 0.9 * (20 - G3) - 0.012 * D2 * G3, ...
0 == -0.0033 * M1 + 0.002 * G1, ...
0 == -0.0033 * M2 + 0.002 * G2, ...
0 == -0.0033 * M3 + 0.002 * G3, ...
0 == 0.1 * M1 - 0.0033 * P1 + 2 * 0.5 * D1 - 2 * 0.025 * (P1 ^ 2), ...
0 == 0.1 * M2 - 0.0033 * P2 + 2 * 0.5 * D2 - 2 * 0.025 * (P2 ^ 2), ...
0 == 0.1 * M3 - 0.0033 * P3 + 2 * 0.5 * D3 - 2 * 0.025 * (P3 ^ 2), ...
0 == -0.5 * D1 + 0.025 * (P1 ^ 2) + 0.9 * (20 - G2) - 0.012 * D1 * G2, ...
0 == -0.5 * D2 + 0.025 * (P2 ^ 2) + 0.9 * (20 - G3) - 0.012 * D2 * G3, ...
0 == -0.5 * D3 + 0.025 * (P3 ^ 2) + 0.9 * (20 - G1) - 0.012 * D3 * G1)
The problem is that I need real solution, but I have got answers with roots. How can I get real answers?
Assuming by "real solution" you mean a numeric value instead of the exact root-of-polynomial form, there are two options depending on how you intend on using the results:
double: this will evaluate the RootOf expressions, assuming there are no free parameters, and return a numeric output of class double. While the output is now limited to double-precision, if your goal is to use the roots in computation and care about performance, this is the fastest option.
vpa: this will evaluate the RootOf expressions, assuming there are no free parameters, and return a Symbolic output of class sym. While the output is now able to be approximated to varying degrees of accuracy at evaluation by calling digits beforehand, there may be a large computational overhead in subsequent calculations due to its Symbolic nature.
I have a matrix, let's say 5x5 looking like this:
0 0 0 1 0
0 0 0 4/5 1/5
3/5 1/5 1/5 0 0
1/5 2/5 1/5 1/5 0
1/10 1/10 2/5 1/5 1/5
I need it to solve it like a system of linear equations looking like this (I can transpose it myself, but then multiplying it with the symbolic variables gets me into troubles):
0 * a + 0 * b + 3/5 * c + 1/5 * d + 1/10 + e = a
0 * a + 0 * b + 1/5 * c + 2/5 * d + 1/10 + e = b
0 * a + 0 * b + 1/5 * c + 1/5 * d + 2/5 + e = c
1 * a + 4/5 * b + 0 * c + 1/5 * d + 1/5 + e = d
0 * a + 1/5 * b + 0 * c + 0 * d + 1/5 + e = e
a + b + c + d + e = 1
I can easily solve this in wxMaxima, but I have to manually write all the values there, which is increasingly tedious with bigger matrices.
Is there a way to get the results after some steps using matlab operator \ for solving system of linear equations?
You can solve the equation set no?
>>[A-eye(5);ones(1,5)]\[0,0,0,0,0,1]'
ans =
0.1729
0.2061
0.1345
0.4350
0.0515
>> sum(ans)
ans =
1.0000
And a symbolic solution:
M=sym(A);
v=sym('[a;b;c;d;e]');
sol=solve(M*v==v,sum(v)==1);
returns solutions in the form sol.a, sol.b, ...