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Minimize beats Solve

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]
POSTED BY: Frank Kampas
3 Replies

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

pr

Attachments:
POSTED BY: Shenghui Yang

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

POSTED BY: Neil Singer

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]}
POSTED BY: Frank Kampas
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