Maxima: How to replace the variables to simplify the equation? - maxima

(%i1) r: sqrt(x^2+y^2+z^2);
(r) sqrt(z^2+y^2+x^2)
(%i2) dx: diff(r,x);
(dx) x/sqrt(z^2+y^2+x^2)
I just show a simple code because my code is long and complex.
I want to simplify dx and get the result is x/r not x/sqrt(z^2+y^2+x^2).
However, I can't find the useful command.
Could somebody help me to solve this problem?

In this specific case, you can use subst, although ratsubst is probably useful in a greater number of cases.
(%i1) linel:65;
(%o1) 65
(%i2) r: sqrt(x^2+y^2+z^2);
2 2 2
(%o2) sqrt(z + y + x )
(%i3) diff (r, x);
x
(%o3) ------------------
2 2 2
sqrt(z + y + x )
(%i5) subst (r = 'r, %o3);
x
(%o5) -
r
(%i6) ratsubst ('r, r, %o3);
x
(%o6) -
r
Note that the single quote mark prevents evaluation, so that 'r is the symbol r instead of the value of r (namely sqrt(x^2 + y^2 + z^2)).

Related

Why is Maxima failing to give a solution?

I have a function in Maxima I am differentiating then attempting to find the value at which this is zero. When I use solve(), however, I am not given a solution. Why is this, and how can I work around it?
(%i1) f(x):=(-5*(x^4+5*x^3-3*x))/(x^2+1);
(%o1) f(x):=((-5)*(x^4+5*x^3+(-3)*x))/(x^2+1)
(%i2) df(x):=''(diff(f(x), x));
(%o2) df(x):=(10*x*(x^4+5*x^3-3*x))/(x^2+1)^2-(5*(4*x^3+15*x^2-3))/(x^2+1)
(%i3) solve(df(x), x);
(%o3) [0=2*x^5+5*x^4+4*x^3+18*x^2-3]
The function solve is not too strong; there are many problems it can't solve. A stronger version is under development. In the meantime, try the add-on package to_poly_solve. Here's what I get:
(%i1) df(x) := (10*x*(x^4+5*x^3-3*x))/(x^2+1)^2-(5*(4*x^3+15*x^2-3))/(x^2+1) $
(%i2) load (to_poly_solve) $
(%i3) to_poly_solve (df(x), x);
(%o3) %union([x = - 2.872468527640942], [x = - 0.4194144025323134],
[x = 0.3836388367122223], [x = 0.2041221431132173 - 1.789901606296292 %i],
[x = 1.789901606296292 %i + 0.2041221431132173])
Something which is maybe a little surprising is that to_poly_solve has returned a numerical solution instead of exact or symbolic. Tracing allroots shows that to_poly_solve has constructed a quintic equation and punted it to allroots. Since the general quintic doesn't have a solution in terms of radicals, and even in special cases it's probably very messy, maybe it's most useful to have a numerical solution anyway.
Try plot2d(df(x), [x, -3, 1]) to visualize the real roots returned above.
You can try to find a numerical solution. I don't know why solve does not try this. Either you take the ouput of aolveor you do hte folölowing:
(%i1) f(x):=(-5*(x^4+5*x^3-3*x))/(x^2+1);
4 3
(- 5) (x + 5 x + (- 3) x)
(%o1) f(x) := ---------------------------
2
x + 1
(%i2) df(x):=''(diff(f(x), x));
4 3 3 2
10 x (x + 5 x - 3 x) 5 (4 x + 15 x - 3)
(%o2) df(x) := ---------------------- - --------------------
2 2 2
(x + 1) x + 1
Bring it to a common denominator and extract the numerator:
(%i3) xthru(df(x));
4 3 2 3 2
10 x (x + 5 x - 3 x) - 5 (x + 1) (4 x + 15 x - 3)
(%o3) ------------------------------------------------------
2 2
(x + 1)
(%i4) num(%);
4 3 2 3 2
(%o4) 10 x (x + 5 x - 3 x) - 5 (x + 1) (4 x + 15 x - 3)
use allsrootsto find the roots of a polynomial numerically
(%i5) allroots(%);
(%o5) [x = 0.3836388391066617, x = - 0.4194143906217701,
x = 1.789901606296292 %i + 0.2041221431132174,
x = 0.2041221431132174 - 1.789901606296292 %i, x = - 2.872468734711326]
skip the complex solutions
(%i6) sublist(%,lambda([t],imagpart(rhs(t))=0))
;
(%o6) [x = 0.3836388391066617, x = - 0.4194143906217701,
x = - 2.872468734711326]

Substitute variable in Maxima

newbie Maxima question
I have a transfer function in Maxima
E1 : y = K_i*s/(s^2 + w^2);
I'd like to have the closed-form of the equation affter applying the bilinear transform
E2 : s = (2/Ts*(z-1)/(z+1));
I would like to get the transfer function for z, by substituing s by equation E2. How should I proceed?
Regards
Note that subst can apply one or more substitutions stated as equations. In this case, try subst(E2, E1).
That will probably create a messy result -- you can simplify it somewhat by applying ratsimp to the result.
Here's what I get from that.
(%i2) E1 : y = K_i*s/(s^2 + w^2);
K_i s
(%o2) y = -------
2 2
w + s
(%i3) E2 : s = (2/Ts*(z-1)/(z+1));
2 (z - 1)
(%o3) s = ----------
Ts (z + 1)
(%i4) subst (E2, E1);
2 K_i (z - 1)
(%o4) y = ------------------------------
2
4 (z - 1) 2
Ts (z + 1) (------------ + w )
2 2
Ts (z + 1)
(%i5) ratsimp (%);
2
2 K_i Ts z - 2 K_i Ts
(%o5) y = -----------------------------------------------
2 2 2 2 2 2 2
(Ts w + 4) z + (2 Ts w - 8) z + Ts w + 4

Maxima: How to factor a expression in an expected form

I have an expression:
(b+2*ab+a+1)/c
I want to use Maxima to factor the equation treating (b+1) as a factor.
i.e. I want the expression in the following form:
[(b+1)(1+a)+ab]/c
Any help would be appreciated.
Well, my advice is first isolate the numerator, then get the quotient and remainder after dividing by b + 1, then put the pieces back together.
(%i1) display2d : false $
(%i2) expr : (b + 2*a*b + a + 1)/c $
(%i3) num (expr);
(%o3) 2*a*b+b+a+1
(%i4) divide (num (expr), b + 1);
(%o4) [2*a+1,-a]
(%i5) first(%o4) * (b + 1) + second(%o4);
(%o5) (2*a+1)*(b+1)-a
(%i6) (first(%o4) * (b + 1) + second(%o4)) / denom (expr);
(%o6) ((2*a+1)*(b+1)-a)/c
(%i7) is (equal (%o6, expr));
(%o7) true
Note that divide returns two values; first is the quotient and second is the remainder.

Maxima - differentiating a piecewise function

Suppose you have a function defined by intervals, such as
f(x):=block(if x<0 then x^2 else x^3);
When we differentiate it with
diff(f(x),x);
we get
d/dx (if x<0 then x^2 else x^3)
whereas I'd like to get
(if x<0 then 2*x else 3*x^2)
Is there a way to obtain such result?
This may help in a simple case:
(%i1) f(x):= charfun(x<0)*x^2 + charfun(x>=0)*x^3$
(%i2) gradef(charfun(y), 0)$
(%i3) diff(f(x),x);
2
(%o3) 2 x charfun(x < 0) + 3 x charfun(x >= 0)
charfun, gradef
You can try also Pw.mac package from Richard Hennessy.
Here's a different approach using a simplification rule for "if" expressions. The unsolved part here is to detect discontinuities and generate delta functions for those locations. If you want to ignore those, you can define FOO to return 0. Note that I didn't attempt to implement the function discontinuities; that part is unsolved here. I can give it a try if there is interest.
(%i1) display2d : false $
(%i2) matchdeclare ([aa, bb, cc], all, xx, symbolp) $
(%i3) 'diff (if aa then bb else cc, xx) $
(%i4) tellsimpafter (''%, apply ("if", [aa, diff (bb, xx), true, diff (cc, xx)]) + FOO (aa, bb, cc, xx)) $
(%i5) FOO (a, b, c, x) := 'lsum ((ev (c, x = d) - ev (b, x = d)) * delta (d, x), d, discontinuities (a, x)) $
(%i6) diff (if x > 0 then x^2 else x^3, x);
(%o6) (if x > 0 then 2*x else 3*x^2)+'lsum((d^3-d^2)*delta(d,x),d,
discontinuities(x > 0,x))
Building on slitinov's answer I wrote this quite naive implementation for functions with more than two "pieces":
gradef(charfun(dummy),0)$
/* piecewise function definition */
itv: [[x<0],[x>=0,x<1], [x>=1]]; /* intervals */
fi: [ 1, x^2+1, 2*x ]; /* local functions */
/* creation of global function f and its derivative df */
f:0;
for i: 1 thru 3 do f:f+charfun(apply("and",itv[i]))*fi[i];
df:diff(f,x);
/* display of local functions and derivatives */
for i: 1 thru 3 do (
apply(assume,itv[i]),
newline(),
print(itv[i]),
print("f = ",ev(f)),
print("df = ",ev(df)),
apply(forget,itv[i])
);
plot2d([f,df],[x,-2,3],[y,-1,5],[style,[lines,4,3],[lines,2,2]]);

symbolically replace expressions in maxima

I'm having trouble finding out how to do this:
x=a+b
y=c+d
z=x*y
I would like the output to be
z=ac+ad+bc+bd
not
z=xy
Like this?
(%i1) x: a+b;
(%o1) b + a
(%i2) y: c+d;
(%o2) d + c
(%i3) z: x*y;
(%o3) (b + a) (d + c)
(%i4) z: expand (z);
(%o4) b d + a d + b c + a c
(%i5)
Assignment in maxima is done by :, not = (which is used for checking for equality)
Actually, to get the output he's requesting without assigning a lot of variables,
you can just do this:
(%i1) z = x*y, x = a+b, y = c+d, expand;
(%o1) z = b d + a d + b c + a c
This is an old question, but the canonical solution in my opinion is the subst() function

Resources