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Type less...

vv 13832
November 18 2020
1 0

I always prefer to change myself the coordinates. This way I avoid the two distinct conventions used by Maple for spherical.
As a bonus, I type less!

int(eval((x^2 + y^2)*z, [x=sin(u)*cos(v), y=sin(u)*sin(v), z=cos(u)]) * sin(u), u = 0 .. Pi/2, v = 0 .. 2*Pi);

        Pi/2

 

select...

vv 13832
November 17 2020
2 1 
if type(lhs(ode),`*`) and rhs(ode)=0 then 
  ODE:=select(has, lhs(ode), diff) = 0
fi;

If lhs has a factor depending on y, but has not diff, you may keep it, or solve it for y.

examples...

vv 13832
November 16 2020
2 3
> 

restart;
> 

x[n](t) = x0 + int( f(tau, x[n-1](tau)), tau=t0..t);

x[n](t) = x0+int(f(tau, x[n-1](tau)), tau = t0 .. t)

 (1)
> 

# example 1
t0:=0; x0:=1;
f:= (t, x) -> t*x;
x[0]:= t -> x0;

0

  

1

  

proc (t, x) options operator, arrow; x*t end proc

  

proc (t) options operator, arrow; x0 end proc

 (2)
> 

for n to 5 do
x[n] := unapply( x0 + int( f(tau, x[n-1](tau)), tau=t0..t),  t);
print(x[n]= x[n](t));
od:

x[1] = 1+(1/2)*t^2

  

x[2] = 1+(1/8)*t^4+(1/2)*t^2

  

x[3] = 1+(1/48)*t^6+(1/8)*t^4+(1/2)*t^2

  

x[4] = 1+(1/384)*t^8+(1/48)*t^6+(1/8)*t^4+(1/2)*t^2

  

x[5] = 1+(1/3840)*t^10+(1/384)*t^8+(1/48)*t^6+(1/8)*t^4+(1/2)*t^2

 (3)
> 

# exaplle 2 (x[n], f are vector valued)
> 

t0:=0; x0:=<1,2>;
f:= (t, v) -> <t*v[1], v[2]> ;
x[0]:= t -> x0;

t0 := 0

  

Vector[column](%id = 18446747125427096142)

  

proc (t, v) options operator, arrow; `<,>`(t*v[1], v[2]) end proc

  

proc (t) options operator, arrow; x0 end proc

 (4)
> 

for n to 5 do
x[n] := unapply( x0 + int~( f(tau, x[n-1](tau)), tau=t0..t),  t);
print(x[n]= x[n](t));
od:

x[1] = (Vector(2, {(1) = 1+(1/2)*t^2, (2) = 2+2*t}))

  

x[2] = (Vector(2, {(1) = 1+(1/8)*t^4+(1/2)*t^2, (2) = t^2+2*t+2}))

  

x[3] = (Vector(2, {(1) = 1+(1/48)*t^6+(1/8)*t^4+(1/2)*t^2, (2) = 2+(1/3)*t^3+t^2+2*t}))

  

x[4] = (Vector(2, {(1) = 1+(1/384)*t^8+(1/48)*t^6+(1/8)*t^4+(1/2)*t^2, (2) = 2+2*t+(1/12)*t^4+(1/3)*t^3+t^2}))

  

x[5] = Vector[column](%id = 18446747125427074462)

 (5)
> 

 

NULL
 

Download Picard_iteration-vv.mw

 

Equal without assumptions...

vv 13832
November 08 2020
1 1

> 

restart;
> 

aa := {{y = -2*X1*X2*alpha[1, 8]*alpha[2, 6]/(sqrt((X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X1*((-2*X1*X2*alpha[3, 6] - 2*X2*alpha[2, 2] + 2*X3)*alpha[2, 6] + X1*X2*alpha[2, 4]*(alpha[2, 8] + alpha[3, 9]))*alpha[1, 8] + X2^2*(alpha[2, 8] + alpha[3, 9])^2)*alpha[1, 8]^2) + alpha[1, 8]^2*alpha[2, 4]*X1^2 + X2*alpha[1, 8]*(alpha[2, 8] + alpha[3, 9])), z = (-sqrt((X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X1*((-2*X1*X2*alpha[3, 6] - 2*X2*alpha[2, 2] + 2*X3)*alpha[2, 6] + X1*X2*alpha[2, 4]*(alpha[2, 8] + alpha[3, 9]))*alpha[1, 8] + X2^2*(alpha[2, 8] + alpha[3, 9])^2)*alpha[1, 8]^2) - alpha[1, 8]^2*alpha[2, 4]*X1^2 + (-alpha[3, 9] - alpha[2, 8])*X2*alpha[1, 8])/(2*alpha[1, 8]^2*alpha[2, 6]*X1)}, {y = -2*X2*alpha[1, 8]*alpha[2, 6]*X1/(-sqrt((X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X1*((-2*X1*X2*alpha[3, 6] - 2*X2*alpha[2, 2] + 2*X3)*alpha[2, 6] + X1*X2*alpha[2, 4]*(alpha[2, 8] + alpha[3, 9]))*alpha[1, 8] + X2^2*(alpha[2, 8] + alpha[3, 9])^2)*alpha[1, 8]^2) + alpha[1, 8]^2*alpha[2, 4]*X1^2 + X2*alpha[1, 8]*(alpha[2, 8] + alpha[3, 9])), z = (sqrt((X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X1*((-2*X1*X2*alpha[3, 6] - 2*X2*alpha[2, 2] + 2*X3)*alpha[2, 6] + X1*X2*alpha[2, 4]*(alpha[2, 8] + alpha[3, 9]))*alpha[1, 8] + X2^2*(alpha[2, 8] + alpha[3, 9])^2)*alpha[1, 8]^2) - alpha[1, 8]^2*alpha[2, 4]*X1^2 + (-alpha[3, 9] - alpha[2, 8])*X2*alpha[1, 8])/(2*alpha[1, 8]^2*alpha[2, 6]*X1)}}:
> 

bb := {{y = -2*X2*alpha[2, 6]*X1/(-sqrt(X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X2*alpha[1, 8]*((alpha[2, 8] + alpha[3, 9])*alpha[2, 4] - 2*alpha[2, 6]*alpha[3, 6])*X1^2 - 4*alpha[1, 8]*alpha[2, 6]*(X2*alpha[2, 2] - X3)*X1 + X2^2*(alpha[2, 8] + alpha[3, 9])^2) + (alpha[2, 8] + alpha[3, 9])*X2 + alpha[1, 8]*alpha[2, 4]*X1^2), z = (sqrt(X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X2*alpha[1, 8]*((alpha[2, 8] + alpha[3, 9])*alpha[2, 4] - 2*alpha[2, 6]*alpha[3, 6])*X1^2 - 4*alpha[1, 8]*alpha[2, 6]*(X2*alpha[2, 2] - X3)*X1 + X2^2*(alpha[2, 8] + alpha[3, 9])^2) - alpha[1, 8]*alpha[2, 4]*X1^2 + (-alpha[3, 9] - alpha[2, 8])*X2)/(2*alpha[1, 8]*alpha[2, 6]*X1)}, {y = -2*X2*alpha[2, 6]*X1/(sqrt(X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X2*alpha[1, 8]*((alpha[2, 8] + alpha[3, 9])*alpha[2, 4] - 2*alpha[2, 6]*alpha[3, 6])*X1^2 - 4*alpha[1, 8]*alpha[2, 6]*(X2*alpha[2, 2] - X3)*X1 + X2^2*(alpha[2, 8] + alpha[3, 9])^2) + alpha[1, 8]*alpha[2, 4]*X1^2 + (alpha[2, 8] + alpha[3, 9])*X2), z = (-sqrt(X1^4*alpha[1, 8]^2*alpha[2, 4]^2 + 2*X2*alpha[1, 8]*((alpha[2, 8] + alpha[3, 9])*alpha[2, 4] - 2*alpha[2, 6]*alpha[3, 6])*X1^2 - 4*alpha[1, 8]*alpha[2, 6]*(X2*alpha[2, 2] - X3)*X1 + X2^2*(alpha[2, 8] + alpha[3, 9])^2) - alpha[1, 8]*alpha[2, 4]*X1^2 + (-alpha[3, 9] - alpha[2, 8])*X2)/(2*alpha[1, 8]*alpha[2, 6]*X1)}}:
> 

aa:=simplify(rationalize(aa)):
bb:=simplify(rationalize(bb)):
> 

indets(aa);

{X1, X2, X3, y, z, alpha[1, 8], alpha[2, 2], alpha[2, 4], alpha[2, 6], alpha[2, 8], alpha[3, 6], alpha[3, 9], ((X1^4*alpha[1, 8]^2*alpha[2, 4]^2+2*X1*((-2*X1*X2*alpha[3, 6]-2*X2*alpha[2, 2]+2*X3)*alpha[2, 6]+X1*X2*alpha[2, 4]*(alpha[2, 8]+alpha[3, 9]))*alpha[1, 8]+X2^2*(alpha[2, 8]+alpha[3, 9])^2)*alpha[1, 8]^2)^(1/2)}

 (1)
> 

indets(bb);

{X1, X2, X3, y, z, alpha[1, 8], alpha[2, 2], alpha[2, 4], alpha[2, 6], alpha[2, 8], alpha[3, 6], alpha[3, 9], (X1^4*alpha[1, 8]^2*alpha[2, 4]^2+2*X2*alpha[1, 8]*(-2*alpha[3, 6]*alpha[2, 6]+alpha[2, 4]*(alpha[2, 8]+alpha[3, 9]))*X1^2-4*alpha[1, 8]*alpha[2, 6]*(X2*alpha[2, 2]-X3)*X1+X2^2*(alpha[2, 8]+alpha[3, 9])^2)^(1/2)}

 (2)
> 

ra:=indets(aa,radical)[]:
rb:=indets(bb,radical)[]:
> 

simplify(ra^2/rb^2);

alpha[1, 8]^2

 (3)
> 

AA:=eval(aa, [ra=R*s*t, alpha[1,8]=t]):  # So, s = +1 or -1
> 

BB:=eval(bb, [rb=R, alpha[1,8]=t]):
> 

simplify( eval(y, AA[1])-eval(y,BB[1]) ), simplify( eval(z, AA[1])-eval(z,BB[1]) );
simplify( eval(y, AA[2])-eval(y,BB[2]) ), simplify( eval(z, AA[2])-eval(z,BB[2]) );

(1/2)*X2*R*(s-1)/(t*((X1*alpha[3, 6]+alpha[2, 2])*X2-X3)), -(1/2)*R*(s-1)/(t*alpha[2, 6]*X1)

  

-(1/2)*X2*R*(s-1)/(t*((X1*alpha[3, 6]+alpha[2, 2])*X2-X3)), (1/2)*R*(s-1)/(t*alpha[2, 6]*X1)

 (4)
> 

simplify( eval(y, AA[1])-eval(y,BB[2]) ), simplify( eval(z, AA[1])-eval(z,BB[2]) );
simplify( eval(y, AA[2])-eval(y,BB[1]) ), simplify( eval(z, AA[2])-eval(z,BB[1]) );

(1/2)*X2*R*(s+1)/(t*((X1*alpha[3, 6]+alpha[2, 2])*X2-X3)), -(1/2)*R*(s+1)/(t*alpha[2, 6]*X1)

  

-(1/2)*X2*R*(s+1)/(t*((X1*alpha[3, 6]+alpha[2, 2])*X2-X3)), (1/2)*R*(s+1)/(t*alpha[2, 6]*X1)

 (5)
> 

# ==> aa = bb even for complex parameters
> 

 

 

StringTools...

vv 13832
November 03 2020
1 1 
restart;
ic:=[y(0)=0,D(y)(0)=3];
s:=Physics:-Latex(ic, output=string);
s1:=StringTools:-SubstituteAll(s, "D\\left(y \\right)", "y'");   # or,
s2:=StringTools:-SubstituteAll(s, "D\\left(y \\right)", "y^{\\prime}");

 

Polya...

vv 13832
October 21 2020
2 0

Try to prove (without Maple) the nice formulae due to G. Polya.

> 

Int(x^(-x), x=0..1) = Sum(n^(-n), n=1..infinity);
Int(x^(x), x=0..1) = Sum((-1)^(n-1)*(n)^(-n), n=1..infinity);

Int(x^(-x), x = 0 .. 1) = Sum(n^(-n), n = 1 .. infinity)

  

Int(x^x, x = 0 .. 1) = Sum((-1)^(n-1)*n^(-n), n = 1 .. infinity)

 (1)
> 

evalf([%%,%])

[1.291285997 = 1.291285997, .7834305107 = .7834305107]

 (2)

 

Unstable...

vv 13832
October 20 2020
1 2

JordanForm does non accept numeric matrices containing floats, because the Jordan form is very unstable numericallly (and SingularValues is recommended for numeric tasks).

(Rank is also unstable, but it accepts floats; probably it is considered "less unstable").

Of course it's possible to cheat e.g. JordanForm(A+'sin(0)')  or convert to rationals.

 

syntax...

vv 13832
October 18 2020
1 0

@Stretto You want to use your own syntax! Note that Maple already has the operators `--` and `++`.

a:=10: b:=100:
a--+b;  [a,b];
                             a := 9
                              110
                            [9, 100]
 

oversight...

vv 13832
October 18 2020
1 0

--  IsZero works for lists too by design
-- HasZero returns false if the argument is not a rtable (and not identical 0) by design
-- AllNonZero returns true if the argument is a list because it returns  not HasZero. Really oversight. 

A simple (but not very efficien...

vv 13832
October 15 2020
1 3

A simple (but not very efficient) way.

> 

with(GraphTheory):
n := 4: degrees:=[1,2,2,3]:
L:=NonIsomorphicGraphs(n, output=graphs, outputform=adjacency):
map(proc(u) local g:=Graph(u),d:=sort(Degree(g)); `if`(d=degrees,g,NULL) end, [L]);

[GRAPHLN(undirected, unweighted, [1, 2, 3, 4], Array(1..4, {(1) = {4}, (2) = {3, 4}, (3) = {2, 4}, (4) = {1, 2, 3}}), `GRAPHLN/table/8`, 0)]

 (1)
> 

print~(DrawGraph~(%))[];

 

 

{x}...

vv 13832
October 14 2020
0 4 
eq:=x^3 - 3*x^2 + 3*x - 1=0;
op~(solve(eq,[x]))[];

 

Answer...

vv 13832
October 10 2020
2 1

If you want the book result (i.e. a submatrix of A) then call LinearAlgebra:-Basis with the list of the column vectors of A.

parametric...

vv 13832
October 10 2020
2 0 
eq:=(2*x+a^2-3*a)=a*(x-1):
solve(eq,x, parametric);

           

Shorter solution...

vv 13832
October 09 2020
2 1 
IsElementary:=proc(A::Matrix(square))
  local Z:=rtable_elems(A-1), ij;
  if   nops(Z)=1 then true 
  elif nops(Z)<>4 then false
  elif nops((ij:=lhs~(Z)))=2 and Z = {(ij[])=1, (ij[2],ij[1])=1, (ij[1]$2)=-1, (ij[2]$2)=-1} then true else false fi
end;

 

+ Pi...

vv 13832
October 09 2020
2 2

It's too simple to find a bug here. You must add +Pi

f:=-Pi/2 - arctan(25*x) - Pi + 2*Pi/3:
solve(f,x);
eval(f, x= 1/(25*sqrt(3)) );
solve(f+Pi, x); 
evalf(%) = fsolve(f+Pi,x);

 

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