MaplePrimes - answers and comments on Question, Write a function in Maple

Rewrite this as a linear system of equations

jaytreiman — Thu, 17 Jan 2013 01:20:53 Z

If the values of f(v_i) are known for enough vectors v_i and the forms of q and p are known, eg all terms are of degree <= 4 in the components of the v's, one can try solving the system equations q(v_i)*f(v_i)=p(v_i). Since this system is linear in the coefficients for p and q, there are a number of techniques available.

As an example, if the dimension of v is 2 and the degree of the polynomials is at most 2, letting v_i=(x_i,y_i) one has equations of the form

A1*x_i^2+B1*y_i^2+C1*y_i*x_i+D1*x_1+E1*y_1+F1=f(v_i)*(A2*x_i^2+B2*y_i^2+C2*y_i*x_i+D2*x_1+E2*y_1+F2)

with

p=A1*x_i^2+B1*y_i^2+C1*y_i*x_i+D1*x_1+E1*y_1+F1 and q=A2*x_i^2+B2*y_i^2+C2*y_i*x_i+D2*x_1+E2*y_1+F2. Here one only requires the independence of 12 equations for a solution to exist.

One way

Markiyan Hirnyk — Thu, 17 Jan 2013 01:36:40 Z

If I correctly understand your question, you want to recover the rational function in x1,x2,x3,x4 over integers by its values.
The first difficulty to overcome is to write this function. The simple-minded approach fails:
>eval((sum(sum(sum(a[i, j, k, n]*x1^i*x2^j*x3^k*x4^(3-i-j-k), i = 0 .. 3), j = 0 .. 3-i), k = 0 .. 3-i-j))/
(sum(sum(sum(b[i, j, k, n]*x1^i*x2^j*x3^k*x4^(3-i-j-k), i = 0 .. 3), j = 0 .. 3-i), k = 0 .. 3-i-j)),
[x1 = 1/2, x2 = -2/3, x3 = 61/7, x4 = -11/9]);

(sum(sum(a[0, j, k, n]*(-2/3)^j*(61/7)^k*(-11/9)^(3-j-k)+(1/2)*a[1, j, k, n]*(-2/3)^j*(61/7)^k*(-11/9)^(2-j-k)+
(1/4)*a[2, j, k, n]*(-2/3)^j*(61/7)^k*(-11/9)^(1-j-k)+(1/8)*a[3, j, k, n]*(-2/3)^j*(61/7)^k*(-11/9)^(-j-k),
j = 0 .. 3-i), k = 0 .. 3-i-j))/(sum(sum(b[0, j, k, n]*(-2/3)^j*(61/7)^k*(-11/9)^(3-j-k)+(1/2)*b[1, j, k, n]*
(-2/3)^j*(61/7)^k*(-11/9)^(2-j-k)+(1/4)*b[2, j, k, n]*(-2/3)^j*(61/7)^k*(-11/9)^(1-j-k)+(1/8)*b[3, j, k, n]*
(-2/3)^j*(61/7)^k*(-11/9)^(-j-k), j = 0 .. 3-i), k = 0 .. 3-i-j))

Because of this reason, we'll go the other way.
> with(RandomTools):
> P := Generate(polynom(posint, {x1, x2, x3, x4}, degree = 3)):

#A random polynomial of degree 3 over naturals is created
> L := [op(P)]: nops(L);It includes 35 terms

35
>p := convert(map(c -> a[degree(c, x1), degree(c, x2), degree(c, x3),
degree(c, x4)]*x1^degree(c, x1)*x2^degree(c, x2)*x3^degree(c, x3)*x4^degree(c, x4), L), `+`)
#We replace the coefficients by the general ones, obtaining the numerator
a[0, 3, 0, 0]*x2^3+a[0, 0, 3, 0]*x3^3+a[0, 0, 2, 0]*x3^2+a[0, 2, 0, 0]*x2^2+a[1, 0, 0, 0]*x1+a[0, 1, 0, 0]*x2+
a[0, 0, 1, 0]*x3+a[0, 0, 0, 1]*x4+a[2, 0, 0, 0]*x1^2+a[3, 0, 0, 0]*x1^3+a[0, 0, 0, 3]*x4^3+a[0, 0, 0, 2]*x4^2+
a[0, 2, 0, 1]*x2^2*x4+a[0, 0, 1, 2]*x3*x4^2+a[0, 0, 2, 1]*x3^2*x4+a[1, 2, 0, 0]*x1*x2^2+a[2, 1, 0, 0]*x1^2*x2+
a[1, 0, 2, 0]*x1*x3^2+a[2, 0, 1, 0]*x1^2*x3+a[1, 0, 0, 2]*x1*x4^2+a[2, 0, 0, 1]*x1^2*x4+a[0, 1, 2, 0]*x2*x3^2+
a[0, 2, 1, 0]*x2^2*x3+a[0, 1, 0, 2]*x2*x4^2+a[0, 0, 0, 0]+a[0, 0, 1, 1]*x3*x4+a[1, 1, 0, 0]*x1*x2+
a[1, 0, 1, 0]*x1*x3+a[1, 0, 0, 1]*x1*x4+a[0, 1, 1, 0]*x2*x3+a[0, 1, 0, 1]*x2*x4+
a[0, 1, 1, 1]*x2*x3*x4+a[1, 1, 1, 0]*x1*x2*x3+a[1, 1, 0, 1]*x1*x2*x4+a[1, 0, 1, 1]*x1*x3*x4

Now the denominator is created
q := convert(map(c -> b[degree(c, x1), degree(c, x2), degree(c, x3),
degree(c, x4)]*x1^degree(c, x1)*x2^degree(c, x2)*x3^degree(c, x3)*x4^degree(c, x4), L), `+`):
Assigning the variables, we obtain the equation

>eval(p/q, [x1 = -1/3, x2 = -6/5, x3 = -76/23, x4 = 12/7])=f(-1/3,-6/5,-76/23,12/7);

((5472/805)*a[0, 1, 1, 1]-(152/115)*a[1, 1, 1, 0]+(24/35)*a[1, 1, 0, 1]+
(304/161)*a[1, 0, 1, 1]-(2736/575)*a[0, 2, 1, 0]-(864/245)*a[0, 1, 0, 2]-
(912/161)*a[0, 0, 1, 1]+(2/5)*a[1, 1, 0, 0]+(76/69)*a[1, 0, 1, 0]-(4/7)*a[1, 0, 0, 1]+
(456/115)*a[0, 1, 1, 0]-(72/35)*a[0, 1, 0, 1]-(76/23)*a[0, 0, 1, 0]+(12/7)*a[0, 0, 0, 1]+
(1/9)*a[2, 0, 0, 0]-(1/27)*a[3, 0, 0, 0]+(1728/343)*a[0, 0, 0, 3]+(144/49)*a[0, 0, 0, 2]+
(432/175)*a[0, 2, 0, 1]-(10944/1127)*a[0, 0, 1, 2]+(69312/3703)*a[0, 0, 2, 1]-
(12/25)*a[1, 2, 0, 0]-(2/15)*a[2, 1, 0, 0]-(5776/1587)*a[1, 0, 2, 0]-(76/207)*a[2, 0, 1, 0]-
(48/49)*a[1, 0, 0, 2]+(4/21)*a[2, 0, 0, 1]-(34656/2645)*a[0, 1, 2, 0]+a[0, 0, 0, 0]-
(216/125)*a[0, 3, 0, 0]-(438976/12167)*a[0, 0, 3, 0]+(5776/529)*a[0, 0, 2, 0]+
(36/25)*a[0, 2, 0, 0]-(1/3)*a[1, 0, 0, 0]-(6/5)*a[0, 1, 0, 0])/
(-(216/125)*b[0, 3, 0, 0]-(438976/12167)*b[0, 0, 3, 0]+(5776/529)*b[0, 0, 2, 0]+
(36/25)*b[0, 2, 0, 0]-(1/3)*b[1, 0, 0, 0]-(6/5)*b[0, 1, 0, 0]-(76/23)*b[0, 0, 1, 0]+
(12/7)*b[0, 0, 0, 1]+(1/9)*b[2, 0, 0, 0]-(1/27)*b[3, 0, 0, 0]+(1728/343)*b[0, 0, 0, 3]+
(144/49)*b[0, 0, 0, 2]+b[0, 0, 0, 0]+(432/175)*b[0, 2, 0, 1]-(10944/1127)*b[0, 0, 1, 2]+
(69312/3703)*b[0, 0, 2, 1]-(12/25)*b[1, 2, 0, 0]-(2/15)*b[2, 1, 0, 0]-(5776/1587)*b[1, 0, 2, 0]-
(76/207)*b[2, 0, 1, 0]-(48/49)*b[1, 0, 0, 2]+(4/21)*b[2, 0, 0, 1]-(34656/2645)*b[0, 1, 2, 0]-
(2736/575)*b[0, 2, 1, 0]-(864/245)*b[0, 1, 0, 2]-(912/161)*b[0, 0, 1, 1]+(2/5)*b[1, 1, 0, 0]+
(76/69)*b[1, 0, 1, 0]-(4/7)*b[1, 0, 0, 1]+(456/115)*b[0, 1, 1, 0]-(72/35)*b[0, 1, 0, 1]+
(5472/805)*b[0, 1, 1, 1]-(152/115)*b[1, 1, 1, 0]+(24/35)*b[1, 1, 0, 1]+(304/161)*b[1, 0, 1, 1])=
f(-1/3,-6/5,-76/23,12/7)

At last, we solve a system of 70 such equations.

one_way.mw

PS. Timing in screen16.01.13.docx

One more difficulty

Markiyan Hirnyk — Thu, 17 Jan 2013 09:18:44 Z

Let us consider one of the simplest examples.
> f := unapply((2*x+3*y)/(4*x-5*y), x, y);


(x, y) -> (2*x+3*y)/(4*x-5*y)

> g := unapply((a*x+b*y)/(c*x+d*y), x, y);


(x, y) -> (a *x + b *y)/(c*x+d*y)

> eq1 := f(2/3, -1/5) = g(2/3, -1/5);

> eq2 := f(-2/7, -4/3) = g(-2/7, -4/3);

> eq3 := f(-12/25, -44/9) = g(-12/25, -44/9);

> eq4 := f(-12/25, -44/9) = g(-12/25, -44/9);

> sol := solve({eq1, eq2, eq3, eq4});


             { a = a, b = 3/2* a, c = 2* a, d = - 5/2* a }

> eval(sol, a = lcm(1, 2, 1, 2));

                 {2 = 2, b = 3, c = 4, d = -5}

Please clarify "One more difficulty"

Carl Love — Fri, 18 Jan 2013 20:46:01 Z

@Markiyan Hirnyk

I don't understand the point you are trying to make in "One more difficulty". Why did you make eq3 and eq4 the same?

Also, it would useful to sparsify and block-diagonalize the system of linear equations (especially since an exact solution is required, rather than floating point) by choosing 0 as an evaluation point as much as possible.

It looks like an error

Markiyan Hirnyk — Fri, 18 Jan 2013 21:18:46 Z

@Carl Love when copying and pasting. Of course, it should be

See execution.mw

Solution unique upto a common multiple

Carl Love — Fri, 18 Jan 2013 23:34:32 Z

@Markiyan Hirnyk

Ah, I see. Of course the solution can only be unique upto a common multiple of numerator and denominator.