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Why solve function can't solve system which can be solved manually

asked 2023-08-07 04:21:59 +0200

avi9526 gravatar image

updated 2023-08-08 22:43:07 +0200

slelievre gravatar image

This returns an empty list:

reset()
var('i_B10 i_B6 i_B11 i_C10 i_C6 i_C11 beta i_in i_out i_E10 i_E6 i_E11')
solve([
  i_C10 == i_B10 * beta,
  i_E10 == i_C10 + i_B10,

  i_C6 == i_B6 * beta,
  i_E6 == i_B6 + i_C6,

  i_C11 == i_B11 * beta,
  i_E11 == i_B11 + i_C11,

  i_E10 == i_B11 + i_E6,

  i_E11 + i_E6 == i_in + i_out,

  i_B11 == i_B6,
  i_in == i_B10 + i_C11,
  i_out == i_C10

], i_out)

Output:

[]

But it can be solved manually

reset()
var('i_B10 i_B6 i_B11 i_C10 i_C6 i_C11 beta i_in i_out i_E10 i_E6 i_E11')

i_C10 = i_B10 * beta
i_E10 = i_C10 + i_B10
i_C6 = i_B6 * beta
i_E6 = i_B6 + i_C6
i_C11 = i_B11 * beta
i_E11 = i_B11 + i_C11
i_B10 = (i_E10 == i_B11 + i_E6).solve(i_B10)[0].right()
i_B10 = i_B10(i_B11=i_B6)
i_C10 = i_C10(i_B10=i_B10)
i_B6 = solve(i_in == i_B10 + i_C11(i_B11=i_B6), i_B6)[0].right()
i_out = i_C10(i_B6=i_B6)

i_out = i_out.simplify_rational()
i_out

Output:

(beta^2 + 2*beta)*i_in/(beta^2 + 2*beta + 2)

Why solve doesn't work? Do I need to set some assumptions or etc? Or is because i_out depends only on one variable? How to make "solve" function work in this case?

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2 Answers

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answered 2023-08-21 04:02:00 +0200

avi9526 gravatar image

This system of equations is based on electronic circuit. Seems 'solve' works fine if asked to find all unknown values: ('beta' and 'i_in' is excluded as they are known)

reset()
var('i_B10 i_B6 i_B11 i_C10 i_C6 i_C11 beta i_in i_out i_E10 i_E6 i_E11')
i=solve([
  i_C10 == i_B10 * beta,
  i_E10 == i_C10 + i_B10,

  i_C6 == i_B6 * beta,
  i_E6 == i_B6 + i_C6,

  i_C11 == i_B11 * beta,
  i_E11 == i_B11 + i_C11,

  i_E10 == i_B11 + i_E6,

  i_E11 + i_E6 == i_in + i_out,

  i_B11 == i_B6,
  i_in == i_B10 + i_C11,
  i_out == i_C10

], [i_B10, i_B6, i_B11, i_C10, i_C6, i_C11, i_out, i_E10, i_E6, i_E11])

i[0][6].simplify_rational()

output

i_out == (beta^2 + 2*beta)*i_in/(beta^2 + 2*beta + 2)
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answered 2023-08-09 10:45:55 +0200

Emmanuel Charpentier gravatar image

updated 2023-08-21 08:08:50 +0200

Your program rebinds the Python variables initially bound to the objects representing your mathematical variables. Therefore, it does not solve the *same *system of equations.

This can be illustrated by tracing the state of your variables. Your initial setup, with a slight modification :

reset()
# Collect variable names
Vars = var('i_B10 i_B6 i_B11 i_C10 i_C6 i_C11 beta i_in i_out i_E10 i_E6 i_E11')
# Collect equations
Sys = [i_C10 == i_B10 * beta,
       i_E10 == i_C10 + i_B10,
       i_C6 == i_B6 * beta,
       i_E6 == i_B6 + i_C6,
       i_C11 == i_B11 * beta,
       i_E11 == i_B11 + i_C11,
       i_E10 == i_B11 + i_E6,
       i_E11 + i_E6 == i_in + i_out,
       i_B11 == i_B6,
       i_in == i_B10 + i_C11,
       i_out == i_C10]
# *Text* of the program
Txt="""i_C10 = i_B10 * beta
i_E10 = i_C10 + i_B10
i_C6 = i_B6 * beta
i_E6 = i_B6 + i_C6
i_C11 = i_B11 * beta
i_E11 = i_B11 + i_C11
i_B10 = (i_E10 == i_B11 + i_E6).solve(i_B10)[0].right()
i_B10 = i_B10(i_B11=i_B6)
i_C10 = i_C10(i_B10=i_B10)
i_B6 = solve(i_in == i_B10 + i_C11(i_B11=i_B6), i_B6)[0].right()
i_out = i_C10(i_B6=i_B6)"""

We create a copy by renaming variables (and modify program text accordingly).

# New (= renamed) variables
uVars = var([repr(u).replace("i_","u_") for u in Vars])
# use them in program text (and split it in lines)
uTxt = Txt.replace("i_","u_").split("\n")

Now, we can execute the program step by step, and trace identifier's values after each step :

State = []
for t in uTxt:
    exec(t)
    State.append([eval(repr(u)) for u in uVars])
# Effective history of identifiers :
view(table(State, header_row=uVars, header_column=["After"]+[u.replace("_","\\_") for u in uTxt]))

The result cannot currently be displayed in this site, whose typesetting is questionable, but is visible in Sagecell.

HTH,

EDIT : avi9526's answer to his own question, which is correct, can be clarified as :

reset()
Vars = var('i_B10 i_B6 i_B11 i_C10 i_C6 i_C11 beta i_in i_out i_E10 i_E6 i_E11')
Sys = [
    i_C10 == i_B10 * beta,
    i_E10 == i_C10 + i_B10,
    i_C6 == i_B6 * beta,
    i_E6 == i_B6 + i_C6,
    i_C11 == i_B11 * beta,
    i_E11 == i_B11 + i_C11,
    i_E10 == i_B11 + i_E6,
    i_E11 + i_E6 == i_in + i_out,
    i_B11 == i_B6,
    i_in == i_B10 + i_C11,
    i_out == i_C10
]
Unk = [i_B10, i_B6, i_B11, i_C10, i_C6, i_C11, i_out, i_E10, i_E6, i_E11]
Sol =solve(Sys,Unk)

This solution is unique :

sage: len(Sol)
1

ad can be displayed as :

sage: print("\n".join(map(repr,Sol[0])))
i_B10 == (beta*i_in + 2*i_in)/(beta^2 + 2*beta + 2)
i_B6 == (beta*i_in + i_in)/(beta^2 + 2*beta + 2)
i_B11 == (beta*i_in + i_in)/(beta^2 + 2*beta + 2)
i_C10 == (beta^2*i_in + 2*beta*i_in)/(beta^2 + 2*beta + 2)
i_C6 == (beta^2*i_in + beta*i_in)/(beta^2 + 2*beta + 2)
i_C11 == (beta^2*i_in + beta*i_in)/(beta^2 + 2*beta + 2)
i_out == (beta^2*i_in + 2*beta*i_in)/(beta^2 + 2*beta + 2)
i_E10 == (beta^2*i_in + 3*beta*i_in + 2*i_in)/(beta^2 + 2*beta + 2)
i_E6 == (beta^2*i_in + 2*beta*i_in + i_in)/(beta^2 + 2*beta + 2)
i_E11 == (beta^2*i_in + 2*beta*i_in + i_in)/(beta^2 + 2*beta + 2)

HTH,

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Comments

I'm sorry, but I'm a bit confused by what you're saying. I manually derived that formula, and it appears to be the correct solution for calculating i_out. Please see my answer. Your approach of presenting all the steps in the form of a table is quite elegant - thank you for sharing it.

avi9526 gravatar imageavi9526 ( 2023-08-21 04:07:01 +0200 )edit

Your first attempt was a series of statements, assigning values to variables (through the = operator). Doing so introduced inconsistencies...

This version creates a list of equations (whose operator is ==, a. k. a.eq) , then solves it; See my edit to my anser.

I suggest perusing this excellent book.

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2023-08-21 07:58:03 +0200 )edit

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Asked: 2023-08-07 04:21:59 +0200

Seen: 254 times

Last updated: Aug 21 '23