Minimize beats Solve

Ideas
,
1

This is the second step in solving a problem Reygan Dionisio posted on LinkedIn

Solve[{res == b/a, a == 9^k, b == 15^k, a + 2 b == 25^k, a > 0, b > 0}, {a, b,
   k}]

During evaluation := Solve::nsmet: This system cannot be solved with the methods available to Solve.

Solve[{res == b/a, a == 9^k, b == 15^k, a + 2 b == 25^k, a > 0, b > 0}, {a, b,
   k}]

 MinValue[{ b/a, a == 9^k, b == 15^k, a + 2 b == 25^k, a > 0, b > 0}, {a, b, 
   k}] // FullSimplify

1 + Sqrt[2]

MaxValue[{ b/a, a == 9^k, b == 15^k, a + 2 b == 25^k, a > 0, b > 0}, {a, b, 
   k}] // FullSimplify

1 + Sqrt[2]
2

Frank,

The reason Solve fails is that you have more variables in the equations than you told Solve about. This works (by adding res to the list of variables):

Solve[{res == b/a, a == 9^k, b == 15^k, a + 2 b == 25^k, a > 0, 
  b > 0}, {a, b, k, res}]

as does this (let Solve figure out the variables):

Solve[{res == b/a, a == 9^k, b == 15^k, a + 2 b == 25^k, a > 0, 
  b > 0}]

However, the minimize approach runs faster if you know that the minimum is the solution.

Regards,

Neil

3

MinValue is 100 times faster than Solve

In[6]:= AbsoluteTiming[
 Solve[{res == b/a, a == 9^k, b == 15^k, a + 2 b == 25^k, a > 0, 
    b > 0}, {a, b, k, res}] // FullSimplify]

Out[6]= {84.9759, {{a -> 9^(ArcSinh[1]/Log[5/3]), 
   b -> 15^(ArcSinh[1]/Log[5/3]), k -> ArcSinh[1]/Log[5/3], 
   res -> 1 + Sqrt[2]}}}

In[2]:= AbsoluteTiming[
 MinValue[{ b/a, a == 9^k, b == 15^k, a + 2 b == 25^k, a > 0, 
    b > 0}, {a, b, k}] // FullSimplify]

Out[2]= {0.399868, 1 + Sqrt[2]}
4

PositiveReals puts a constraint on every variable. A notebooks is attached with code.

pr.png
pr.png982×740 29.8 KB

solve.nb (16.4 KB)

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