Maple 2021 Questions and Posts

These are Posts and Questions associated with the product, Maple 2021

restart

with(PDEtools)

undeclare(prime)

`There is no more prime differentiation variable; all derivatives will be displayed as indexed functions`

(1)

with(DEtools)

``

with(DifferentialAlgebra)

"with(Student[ODEs][Solve]): "

with(IntegrationTools)

with(inttrans)

with(PDEtools)

with(Physics)

with(PolynomialTools)

with(RootFinding)

with(SolveTools)

with(LinearAlgebra)

with(sumtools)

pde := I*(diff(psi(x, t), t))+alpha*(diff(psi(x, t), `$`(x, 2)))+(beta[3]*abs(psi(x, t))+beta[4]*abs(psi(x, t))^2)*psi(x, t)+gamma*(diff(abs(psi(x, t))^2, `$`(x, 2)))*lambda^2*psi(x, t)/abs(psi(x, t)) = 0

case1 := {k = k, lambda = sqrt(-1/(18*alpha*beta[4]+18*gamma*beta[4]))*beta[3], w = -(9*alpha*k^2*beta[4]+2*beta[3]^2)/(9*beta[4]), A[0] = -beta[3]/(3*beta[4]), A[1] = beta[3]/(3*beta[4]), B[1] = 0}

" psi(x,t):=U(xi)*exp(I*(-k*x+w*t+theta))"

proc (x, t) options operator, arrow, function_assign; Physics:-`*`(U(xi), exp(Physics:-`*`(I, -Physics:-`*`(k, x)+Physics:-`*`(w, t)+theta))) end proc

(2)

" U(xi):=-(beta[3] (cosh(xi)-sinh(xi)))/(3 beta[4] cosh(xi))"

proc (xi) options operator, arrow, function_assign; -Physics:-`*`(beta[3], cosh(xi)-sinh(xi), Physics:-`^`(Physics:-`*`(3, beta[4], cosh(xi)), -1)) end proc

(3)

convert(U(xi), trig)

-(1/3)*beta[3]*(cosh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*k+x))-sinh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*k+x)))/(beta[4]*cosh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*k+x)))

(4)

xi := sqrt(-1/(18*alpha*beta[4]+18*gamma*beta[4]))*beta[3]*(2*alpha*k+x)

(-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*k+x)

(5)

S := psi(x, t)

-(1/3)*beta[3]*(cosh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*k+x))-sinh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*k+x)))*exp(I*(-k*x+t*w+theta))/(beta[4]*cosh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*k+x)))

(6)

solution := subs(case1, S)

-(1/3)*beta[3]*(cosh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*k+x))-sinh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*k+x)))*exp(I*(-k*x-(1/9)*(9*alpha*k^2*beta[4]+2*beta[3]^2)*t/beta[4]+theta))/(beta[4]*cosh((-1/(18*alpha*beta[4]+18*gamma*beta[4]))^(1/2)*beta[3]*(2*alpha*k+x)))

(7)

pdetest(psi(x, t) = -beta[3]*(cosh(sqrt(-1/(18*alpha*beta[4]+18*gamma*beta[4]))*beta[3]*(2*alpha*k+x))-sinh(sqrt(-1/(18*alpha*beta[4]+18*gamma*beta[4]))*beta[3]*(2*alpha*k+x)))*exp(I*(-k*x-(9*alpha*k^2*beta[4]+2*beta[3]^2)*t/(9*beta[4])+theta))/(3*beta[4]*cosh(sqrt(-1/(18*alpha*beta[4]+18*gamma*beta[4]))*beta[3]*(2*alpha*k+x)))*exp(I*(-k*x-(9*alpha*k^2*beta[4]+2*beta[3]^2)*t/(9*beta[4])+theta)), pde)

Error, (in pdetest) unable to determine the indeterminate function

 
 

NULL

can any one tell me how test my solution in other example even not be like that

Download pde-solve.mw

restart;
Proc := proc(t) local t4, l3, R, r, eq, sol; _EnvHorizontalName := 'x'; _EnvVerticalName := 'y'; t4 := thickness = 4; l3 := linestyle = dot; R := 9; r := 1/2*R; geometry:-point(OO, 0, 0); geometry:-circle(Cir, [OO, R]); geometry:-point(K, R*cos(t), R*sin(t)); geometry:-point(Omega, r*cos(t), r*sin(t)); geometry:-circle(cir, [Omega, r]); eq := geometry:-Equation(cir); geometry:-line(XXp, y = 0); geometry:-line(YYp, x = 0); geometry:-line(L1, y = x); geometry:-line(L2, y = -x); geometry:-projection(M1, K, XXp); geometry:-coordinates(M1); geometry:-point(K2, geometry:-coordinates(M1)[1] - 2*R, 0); geometry:-coordinates(K2); geometry:-segment(sT, K2, M1); geometry:-point(N1, 0, R*sin(t)); subs(y = x, eq); sol := solve(%, x); geometry:-point(N2, sol[2], sol[2]); subs(y = -x, eq); sol := solve(%, x); geometry:-point(N3, sol[2], -sol[2]); plots:-display(geometry:-draw([Cir(color = blue, t4), cir(color = grey, t4), sT(color = black, t4), XXp(color = black, l3), YYp(color = black, l3), L1(color = black, l3), L2(color = black, l3), N1(color = blue, symbol = solidcircle, symbolsize = 15), N2(color = blue, symbol = solidcircle, symbolsize = 15), N3(color = blue, symbol = solidcircle, symbolsize = 15), M1(color = blue, symbol = solidcircle, symbolsize = 15)]), axes = none, view = [-30 .. 10, -10 .. 10], size = [800, 800]); end proc;
plots:-animate(Proc, [t], t = 0 .. 2*Pi, frames = 200);
NULL;
I am trying to program  this drawing, how to improve this code ? Thank you.

restart;
with(plots);
with(geometry);
_EnvHorizontalName := 'x';
_EnvVerticalName := 'y';
xA := 5;
yA := 0;
point(A, xA, yA);
xB := 5;
yB := -7;
point(B, xB, yB);
midpoint(C, A, B);
segment(sg1, A, B);
xP := -12;
yP := 0;
point(P, xP, yP);
PerpenBisector(L, C, P);
line(YYp, y = yB);
line(XXp, y = 0);
intersection(M, L, YYp);
line(PM, [P, M]);
projection(H, C, PM);
triangle(CMP, [C, M, P]);
triangle(ABH, [A, B, H]);
distance(B, H);
circle(cir, [B, 7]);
display(textplot([[coordinates(A)[], "A"], [coordinates(B)[], "B "], [coordinates(C)[], "C"], [coordinates(M)[], "M"], [coordinates(H)[], "H"], [coordinates(P)[], "P"]], align = {"above", 'right'}),
draw([YYp(color = red), XXp(color = black), PM(color = green), L(color = green), sg1(color = black), cir(color = magenta), P(color = black, symbol = solidcircle, symbolsize = 10), M(color = black, symbol = solidcircle, symbolsize = 10), H(color = black, symbol = solidcircle, symbolsize = 10), A(color = blue, symbol = solidcircle, symbolsize = 10), B(color = blue, symbol = solidcircle, symbolsize = 10), CMP(color = blue, filled = true, transparency = 0.8), ABH(color = red, filled = true, transparency = 0.8), C(color = blue, symbol = solidcircle, symbolsize = 10)]),
axes = none, view = [-15 .. 14, -15 .. 3]);
I want to change this figure when xP varies from -12 to 12; Is it possible to use Explore or animate ? Thank you.

Sq := proc(n::integer)
local aS, oS, aC, oC, s, dr, pc, u;
aS := -i/n;
oS := sum(1/s, s = 1 .. n);
aC := 1/2*aS;
oC := oS - 1/2*1/((n + 1)*n);
point(S, aS, oS); point(C, aC, oC);
MakeSquare(K, [S, 'center' = C]);
u := (x, i) -> sum(exp(-x*k)/k, k = 1 .. i);
pc := plot(u(x, n), x = 0 .. 4, color = green);
dr := draw([K]);
display({dr, pc});
end ;Sq(1);
Error, (in geometry:-draw) non-numeric coordinate encountered, cannot determine plot view
How to correct this procedure ?

When the limit approaches from left, the result must be zero. What cause the wrong? Will introducing an extra parameter M affect the result in W1?

W := -1/2+(1/2)*tanh(3.6*(tan(Pi*(r-2.1)/(2*7.9))-.6^2/tan(Pi*(r-2.1)/(2*7.9)))/Pi)

-1/2+(1/2)*tanh(1.145915590*tan(.1988349781*r-.4175534540)-.4125296124/tan(.1988349781*r-.4175534540))

(1)

limit(W, r = 10, left)

-1.000000000

(2)

W1 := -1/2+(1/2)*tanh(3.6*(tan(Pi*(r-2.1*M)/((2*7.9)*M))-.6^2/tan(Pi*(r-2.1*M)/((2*7.9)*M)))/Pi)

-1/2+(1/2)*tanh(1.145915590*tan(.1988349781*(r-2.1*M)/M)-.4125296124/tan(.1988349781*(r-2.1*M)/M))

(3)

limit(W1, r = 10*M, left)

-1.000000000

(4)

NULL

Download limit.mwlimit.mw

eqn := B(n) = -sum(binomial(n + 1, k)*B(k), k = 0 .. n - 1)/(n + 1);
init := B(0) = 1, B(1) = -1/2;
sol := rsolve({eqn, init}, B(n));
Why doesn’t this give me any solution ? Thank you.

How to show that the angle QF2P remains constant when M moves on the ellipse ? Perhaps with Explore ?
restart;
with(plots);
with(geometry);
_EnvHorizontalName := 'x';
_EnvVerticalName := 'y';
x0 := 100;
y0 := 40;
a := 7;
b := 5;
c := sqrt(a^2 - b^2);
ellipse(el, x^2/a^2 + y^2/b^2 - 1);
point(F1, -c, 0);
point(F2, c, 0);
eq := simplify((a^2 - x0^2)*(y - y0)^2 + (b^2 - y0^2)*(x - x0)^2 + 2*x0*y0*(x - x0)*(y - y0)) = 0;
sol := solve({eq}, {y});
line(tang1, op(sol[1]));
line(tang2, op(sol[2]));
sol2 := op(solve({op(sol[1]), x^2/a^2 + y^2/b^2 - 1 = 0}, {x, y}));
xM2 := rhs(sol2[1]);
yM2 := rhs(sol2[2]);
point(A, xM2, yM2);
sol3 := op(solve({op(sol[2]), x^2/a^2 + y^2/b^2 - 1 = 0}, {x, y}));
xM3 := rhs(sol3[1]);
yM3 := rhs(sol3[2]);
point(B, xM3, yM3);
line(Pol, [A, B]);
simplify(Equation(Pol));
isolate(%, y);
xM := 4;
sol := solve({subs(x = xM, x^2/a^2 + y^2/b^2 - 1 = 0)}, {y});
yM := rhs(op(sol[1]));
point(M, xM, yM);
line(Tang, x*xM/a^2 + y*yM/b^2 - 1 = 0);
intersection(P, tang1, Tang);
intersection(Q, tang2, Tang);
line(PF2, [P, F2]);
line(QF2, [Q, F2]);
alpha := FindAngle(PF2, QF2);
arctan(alpha);
evalf(%);
display(textplot([[-c, 0, "F1"], [c, 0, "F2"], [coordinates(B)[], "B"], [coordinates(A)[], "A "], [coordinates(M)[], "M "], [coordinates(P)[], "P "], [coordinates(Q)[], "Q "]], align = {"above", 'right'}), draw([el(color = red), A(color = black, symbol = solidcircle, symbolsize = 16), PF2(color = brown), QF2(color = brown), B(color = black, symbol = solidcircle, symbolsize = 16), M(color = black, symbol = solidcircle, symbolsize = 16), P(color = black, symbol = solidcircle, symbolsize = 16), tang1(color = green), tang2(color = green), Tang(color = green), F1(color = blue, symbol = solidcircle, symbolsize = 16), Q(color = blue, symbol = solidcircle, symbolsize = 16), F2(color = red, symbol = solidcircle, symbolsize = 16)]), axes = none, view = [-7 .. 15, -7 .. 12]);

It seems like there exists a bug when taking the following limit in Maple (I tried Maple 2021):

If I run this command:
> evalf(limit(CylinderU(0,CylinderU(0,x)),x=0));
1.2722774800
the result is 1.2722774800, which seems to be incorrect.
evalf(limit(CylinderU(0,CylinderU(0,x)),x=0));
evalf(limit(CylinderU(0,CylinderU(0,x)),x=0))

However, when I run this command:
> evalf(CylinderU(0,limit(CylinderU(0,x),x=0)));
0.5456799403
the result is 0.5456799403, which seems to be correct.
evalf(CylinderU(0,limit(CylinderU(0,x),x=0)))
evalf(CylinderU(0,limit(CylinderU(0,x),x=0)))

Finally, when I run this command:
> evalf(CylinderU(0,CylinderU(0,0)));
0.5456799403
the result is 0.5456799403 which is also correct.
evalf(CylinderU(0,CylinderU(0,0)))
evalf(CylinderU(0,CylinderU(0,0)))

My expectation is that all three commands must return the same result, thus I consider this a bug.
I also run the following command in WolframAlpha
> limit ParabolicCylinderU(0,ParabolicCylinderU(0,x)) as x->0.0
and obtained the correct result 0.54568, confirming that in Maple this is evaluated incorrectly.

Would appreciate if anybody can confirm that this is a bug.

How such Maple bug should be reported?

restart;
with(combinat);
F := unapply(rsolve({F(1) = 1, F(2) = 1, F(n + 1) = F(n) + F(n - 1)}, F(n)), n);
combine(expand(F(n + 1)*F(n + 2) - F(n)*F(n + 3)));
F := n -> fibonacci(n);
combine(expand(F(n + 1)^2 - F(n)*F(n + 3) + (-1)^n));
is(F(n + 1)^2 = F(n)*F(n + 2) + (-1)^n);#should be true
G := n -> arctan(1/F(n));
is(G(4) = G(5) + G(6));
is(G(4) = G(5) + G(6));
is(G(2*n) = G(2*n + 1) + G(2*n + 2));#should be true
How to establish these formulas ? Thank you.

How to correct this code ? Thank you.

restart;
with(plots);
with(geometry);
_EnvHorizontalName := 'x';
_EnvVerticalName := 'y';
x0 := 10;
y0 := 9;
a := 7;
b := 5;
c := sqrt(a^2 - b^2);
ellipse(el, x^2/a^2 + y^2/b^2 - 1);
point(F1, -c, 0);
point(F2, c, 0);
point(P, x0, y0);
eq := simplify((a^2 - x0^2)*(y - y0)^2 + (b^2 - y0^2)*(x - x0)^2 + 2*x0*y0*(x - x0)*(y - y0)) = 0;
sol := solve({eq}, {y});
line(tang1, op(sol[1]));
slope(tang1);
line(tang2, op(sol[2]));
slope(tang2);
sol2 := op(solve({op(sol[1]), x^2/a^2 + y^2/b^2 - 1 = 0}, {x, y}));
xM2 = rhs(sol2[1]);
yM2 = rhs(sol2[2]);
point(M2, xM2, yM2);
sol3 := op(solve({op(sol[2]), x^2/a^2 + y^2/b^2 - 1 = 0}, {x, y}));
xM3 = rhs(sol3[1]);
yM3 = rhs(sol3[2]);
point(M3, xM3, yM3);
assume(xM2 - xM3 <> 0);
line(Pol, [M2, M3]);
coordinates(M2);
display(textplot([[-c, 0, "F1"], [c, 0, "F2"], [coordinates(P)[], "P"]], align = {"above", 'right'}), draw([el(color = red), P(color = black, symbol = solidcircle, symbolsize = 16), tang1(color = green), tang2(color = green), Pol(color = red, thickness = 3), F1(color = blue, symbol = solidcircle, symbolsize = 16), F2(color = red, symbol = solidcircle, symbolsize = 16)]), axes = none);
Warning, data could not be converted to float Matrix

Why this code does not work. Thank you.
restart;
with(plots);
with(geometry);
_EnvHorizontalName := 'x';
_EnvVerticalName := 'y';
x0 := 10;
y0 := 9;
a := 7;
b := 5;
c := sqrt(a^2 - b^2);
ellipse(el, x^2/a^2 + y^2/b^2 - 1);
point(F1, -c, 0);
point(F2, c, 0);
point(P, x0, y0);
eq := simplify((a^2 - x0^2)*(y - y0)^2 + (b^2 - y0^2)*(x - x0)^2 + 2*x0*y0*(x - x0)*(y - y0)) = 0;
eq := (a^2 - x0^2)*m^2 + 2*x0*y0*m + b^2 - y0^2 = 0;
sol := solve(%, m);
m1 := sol[1];
m2 := sol[2];
(y - y0 - m1) + (-x + x0);
(y - y0 - m2) + (-x + x0);
line(tang1, (y - y0 - m1) + (-x + x0));
line(tang2, (y - y0 - m2) + (-x + x0));
display*[textplot*([[-c, 0, "F1"], [c, 0, "F2"], [coordinates(P)[], "P"]], align = {"above", 'right'}), draw*([el(color = red), P(color = black, symbol = solidcircle, symbolsize = 16), tang1(color = green), tang2(color = green), F1(color = blue, symbol = solidcircle, symbolsize = 16), F2(color = red, symbol = solidcircle, symbolsize = 16)], axes = none)];
 

sol := y = -3283/4253 - (3283*x)/4253, How can I determine the value of the coefficient of x?
How can I take the value of the coefficient of x? Thank you.

restart;
with(geometry);
with(plots);
_EnvHorizomtalName = 'x';
_EnvVerticalName = 'y';
_local(D);
line(delta, y = 1/3*x - 2, [x, y]);
line(deltap, y = (-1)/4*x + 1, [x, y]);
line(D, y = 3*x - 5, [x, y]);
point(S, 3, 0);
omega := Pi/3;
intersection(P, delta, D);
intersection(Pp, deltap, D);
projection(H, S, delta);
projection(K, S, deltap);
projection(M, S, D);
circle(c1, [H, K, M], 'centername' = O1);
display*[textplot*([[coordinates(S)[], "S"], [coordinates(P)[], "P"], [coordinates(H)[], "H"], [coordinates(K)[], "K"], [coordinates(M)[], "M"]], font = [times, bold, 16], align = [above, right]), draw*([delta(color = blue), deltap(color = blue), D(color = red), c1(color = black), S(color = black, symbol = solidcircle, symbolsize = 16), P(color = black, symbol = solidcircle, symbolsize = 16)], scaling = constrained, axes = none, view = [-15 .. 15, -15 .. 15])];
               [                /[             [9  -13     ]  
plots:-display [plots:-textplot |[[3, 0, "S"], [-, ---, "P"], 
               [                \[             [8   8      ]  

  [33  -9     ]  [52  4      ]  [9  2     ]]  
  [--, --, "H"], [--, --, "K"], [-, -, "M"]], 
  [10  10     ]  [17  17     ]  [5  5     ]]  

                                                  \              
  font = [times, bold, 16], align = [above, right]|, geometry:-d\
                                                  /              

  raw ([delta(color = blue), deltap(color = blue), 

  D(color = red), c1(color = black), 

  S(color = black, symbol = solidcircle, symbolsize = 16), 

  P(color = black, symbol = solidcircle, symbolsize = 16)], 

  scaling = constrained, axes = none, view = [-15 .. 15, -15 .. 15])];
  display*[textplot*([[3, 0, "S"], [9/8, -13/8, "P"], [33/10, -9/10, "H"], [52/17, 4/17, "K"], [9/5, 2/5, "M"]], font = [times, bold, 16], align = [above, right]), draw*([delta(color = blue), deltap(color = blue), D(color = red), c1(color = black), S(color = black, symbol = solidcircle, symbolsize = 16), P(color = black, symbol = solidcircle, symbolsize = 16)], scaling = constrained, axes = none, view = [-15 .. 15, -15 .. 15])]
Why the program will not run; Thank tou.

restart;
with(geometry);
with(plots);
_EnvHorizontalName := 'x';
_EnvVerticalName := 'y';
cb := color = blue;
t3 := thickness = 3;
l3 := linestyle = 3;
xA := 3;
yA := 0;
xB := -3;
yB := 0;
c := 6;
point(A, xA, yA);
point(B, xB, yB);
R := 5;
alpha := arctan(3/4);
evalf(%*180/Pi);
                          36.86989765

xH := (xA + xB)/2;
yH := (yA + yB)/2;
point(H, xH, yH);
xO1 := 0;
yO1 := 4;
point(O1, xO1, yO1);
xO2 := 0;
yO2 := -4;
point(O2, xO2, yO2);
segment(sAB, A, B);
segment(sHO, H, O1);
segment(sAO, A, O1);
segment(sBO, B, O1);
                              sAB

alpha1 := arctan((yO1 - yA)/(xO1 - xA));
beta := Pi + arctan((yO1 - yB)/(xO1 - xB));
AR1 := plottools[arc]([xO1, yO1], R, alpha1 .. beta, l3);
AR2 := plottools[arc]([xO2, yO2], R, -beta .. -alpha1, l3);
N := 80;
dt := (beta - alpha1)/(N - 1);
dr := draw({O1, O2, sAB, sHO, sAO(cb), sBO(cb)});
tex := textplot([[xA, yA - 0.5, "A"], [xB, yB - 0.5, "B"], [xO1, yO1 + 0.5, "O1"], [xO2, yO2 - 0.5, "O2"], [xH, yH - 0.5, "H"]]);
M1 := seq(plottools[disk]([R*cos(dt*t + alpha1) + xO1, R*sin(dt*t + alpha1) + yO1], 0.2, color = orange), t = 0 .. N);
M2 := seq(plottools[disk]([R*cos(dt*t + Pi + alpha1) + xO2, R*sin(dt*t + Pi + alpha1) + yO2], 0.2, color = orange), t = 0 .. N);
P1 := seq(plottools[polygon]([[R*cos(dt*t + alpha1) + xO1, R*sin(dt*t + alpha1) + yO1], [xA, yA], [xB, yB]], color = aquamarine, linestyle = dash), t = 0 .. N);
P2 := seq(plottools[polygon]([[R*cos(dt*t + Pi + alpha1) + xO1, R*sin(dt*t + Pi + alpha1) + yO1], [xA, yA], [xB, yB]], color = aquamarine, linestyle = dash), t = 0 .. N);
for t to N do
    E[t] := display(dr, tex, AR1, AR2, M1[t], P1[t]);
    F[t] := display(dr, tex, AR1, AR2, M2[t], P2[t]);
end do;
display([seq(E[t], t = 1 .. N), seq(F[t], t = 1 .. N)], insequence = true, axes = none, scaling = constrained, view = [-10 .. 10, -10 .. 10]);

NULL;
angle that does not turn on the lower arc. Thank you.

trouverTripletsDecroissants := proc(tripletInitial) local m, n, a, b, c, triplets, dernierTriplet; triplets := []; dernierTriplet := tripletInitial; m := dernierTriplet[3]; do m := m - 1; for n to m - 1 do if igcd(m, n) = 1 and (m - n) mod 2 = 1 then a := m^2 - n^2; b := 2*m*n; c := m^2 + n^2; if a^2 + b^2 = c^2 and a < dernierTriplet[1] and b < dernierTriplet[2] and c < dernierTriplet[3] then dernierTriplet := [a, b, c]; triplets := [op(triplets), dernierTriplet]; break; end if; end if; end do; break; if 0 < nops(triplets); end do; return triplets; end proc;
tripletInitial := [275, 252, 373];
tripletsDecroissants := trouverTripletsDecroissants(tripletInitial);
print(tripletsDecroissants);
               tripletInitial := [275, 252, 373]

           tripletsDecroissants := [[273, 136, 305]]

                       [[273, 136, 305]]

;
trouverTripletsDecroissants(275, 252, 373);
Error, (in trouverTripletsDecroissants) final value in for loop must be numeric or character
How to correct this error. Thank you.

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