# interact with slider

HI

Ubuntu 18.04,Jupyter notebook 9.0

from copy import deepcopy

''' Newton’s method. '''
show(LatexExpr(r"f'(x_n) \, (x-x_n) + f(x_n)"))
show(LatexExpr(r"x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}"))
print("source :","https://en.wikipedia.org/wiki/Newton%27s_method")
rd=5 # rounding display
def N_f(x_0=0.5,f=x^2,delta=10^(-6),n=0,xL=[]) :
global rRg,rd
"""real root approximation"""
fd=f.derivative(x)
epsilon=abs(f(x_0))
xnPlus1=0
xL.append((n,round(x_0,rd)))
rRg=[]
if epsilon > delta and n<100 and fd(x_0)!=0:
n+=1
xnPlus1=x_0-f(x_0)/fd(x_0)
print('x_'+str(n)+' : ',round(x_0,rd),'\t x_'+str(n+1),' : ',round(xnPlus1,rd),
' epsilon : ',round(epsilon,rd),' delta : ',round(delta,rd))
#xL.append((n,xnPlus1))

return N_f(xnPlus1,f,delta,n,xL)
else:
return x_0,n+1,delta,xL

'''test of function N_f '''

#R.<x> = PolynomialRing(RR)
f=x^3 - 2*x^2 + x - 1

for r in f.roots():
#show(r)
if r[0].imag_part()==0 :
realRoot=r[0]
show("real root: ",r[0])
show("real root numerical value:\t",r[0].n())

G = plot(f, (x, -1/2, 2), thickness=2, color='green', title="f(x)")
Gp0=point((realRoot,0),color='red',size=40)
Gt0 = text(r"$%s$" %(latex(f)),(0.75,1),color='green')
Gt1 = text(r"$%s$" %(latex(round(realRoot,rd))),(realRoot*(1+1/8),0.1),color='red')
Gl0=line([(0,realRoot), (2*realRoot,realRoot)],color='blue')

f_Coeffs=f.coefficients(sparse=False)
try: deltag
except NameError: deltag = None
if deltag ==None :
deltag= 5*10^(-2)

try: x_0g
except NameError: x_0g = None
if x_0g ==None :
x_0g=(realRoot-1/2).n()

#print(' \t \t delta : ',round(delta,rd))
GG=G+Gp0+Gl0+Gt0+Gt1
nbColorMaxg=16
rainbowG=rainbow(nbColorMaxg, 'rgbtuple')

try: colorNg
except NameError: colorNg = None
if colorNg ==None :
colorNg= 0

try: newtonG
except NameError: newtonG = None
if newtonG ==None :
newtonG= Graphics()

try: rRg
except NameError: rRg = None
if rRg ==None :
rRg=[]

@interact
#def _(x_0=input_box(round(x_0g,rd), width=20, label="x_0=")):
def _( deltal=slider(0.0001,0.5,0.0001,0.3)):

global GG,newtonG,delta,x_0g,colorNg,f,nbColorMaxg,realRoot,rRg
deltag=deltal;
if rRg != [] :
rRg=[]
rRg=N_f(x_0=x_0g,f=f,delta=deltag)
newtonG= GG
if rRg[3] !=[] :
newtonG=list_plot(rRg[3],color=rainbowG[colorNg % (nbColorMaxg-1) ],size=40)
colorNg+=1

(GG+newtonG).show()
#show(rRg[3])

Q1) How to supress warning messages: SageMath/local/lib/python3.7/site-packages/sage/repl/ipython_kernel/__main__.py:85: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR (x = ..., y = …) See http://trac.sagemath.org/5930 for details.

Q2) why do the values of the convergence curves accumulate in rRg [3]?. At the initialization we read in rRg:

[1.81956500676553,
4,
0.300000000000000,
[(0, 1.25488), (1, 2.55835), (2, 2.05721), (3, 1.81957)]]

Then after having positioned the cursor on the far left we read in rRg:

[1.75489290022857,
6,
0.000100000000000000,
[(0, 1.25488),
(1, 2.55835),
(2, 2.05721),
(3, 1.81957),
(0, 1.25488),
(1, 2.55835),
(2, 2.05721),
(3, 1.81957),
(0, 1.25488),
(1, 2.55835),
(2, 2.05721),
(3, 1.81957),
(4, 1.75876),
(0, 1.25488),
(1, 2.55835),
(2, 2.05721),
(3, 1.81957),
(4, 1.75876),
(5, 1.75489)]]

despite I clear rRg:

if rRg != [] :
rRg=[]

do a list(rRg) in the next cell and run it

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1. How to supress warning messages? The problem is your definition of the function f (line 31 of your code). Instead of writing

f = x^3 - 2*x^2 + x - 1

you should write

f(x) = x^3 - 2*x^2 + x - 1

That's it! Notwithstanding the fact that this seems to be an unimportant addition to the code, it is actually a necessary, for Sage needs to know the independent variable(s) of a function in order to make some operations, like the derivative, etc.

2. Why do the values of the convergence curves accumulate in rRg[3]? I think you just discovered a bug. I don't know if this is Sage bug or a Jupyter bug, so I am posting a related question. It seems that the list xL is being kept in memory from execution to execution of the interact. For example, if you uncomment your last line of code and execute the interact, you will notice that every time you move the slider, the list rRg[3], which comes from xL in your first subroutine, gets bigger and bigger. It is storing the convergent sequences for every deltal you use (i.e., one sequence for every position of the slider you use).

So, what do to do? I was able to solve this problem by changing your line number 85 from

rRg=N_f(x_0=x_0g,f=f,delta=deltag)

to the almost similar

rRg=N_f(x_0=x_0g,f=f,delta=deltag,xL=[])

That way, you force xL to become empty on every movement of the slider, so the information from previous executions (the previous convergent sequence for the previous slider position) is forgotten, and you get only the information for the current position of the slider.

I hope this helps!

more

Thank you @dsejas for Q1 it is ok now with what you recommand ,f(x), no more warning. my Q2 was about why there are many sequences in rRg[3] ? it should have been only

[(0, 1.25488),
(1, 2.55835),
(2, 2.05721),
(3, 1.81957),
(4, 1.75876),
(5, 1.75489)]
( 2020-04-04 17:22:26 +0200 )edit

Hello, @ortollj! Sorry about the confusion; I didn't sleep much last night, so I didn't understand clearly your question. I am updating my answer to solve your Q2.

( 2020-04-04 19:55:19 +0200 )edit
1

Oh, by the way, during the experiments I run to find the problem, I was able to simplify part of your code. I hop you don't mind. If it's of any help to you, feel free to modify and/or adapt it to your needs. You can find it here. In particular, I recommend NOT to use global variables.

( 2020-04-04 20:16:59 +0200 )edit

thank you very much @dsejas finally I unnecessarily complicated things ;-(.

one debian pastezone , tabulations are bad.

( 2020-04-04 21:30:54 +0200 )edit

Thank you, @ortollj for uploading the code to SageCell. I didn't post it that way, because the permalink was too large for the allowed number of characters in comments.

( 2020-04-04 22:36:01 +0200 )edit