How to check if a sequence of numbers is monotonically increasing (or decreasing)?

Question

We are given a sequence of numbers, as a vector foo. The task is to find is foo is monotonically increasing - every item is less than or equal to the next one - or monotonically decreasing - every item greater than or equal than the next one.

For sure this can be found through a loop, but any more creative ideas?

Answer 1

Another one: check if

all(x == cummax(x))

or

all(x == cummin(x))

for monotonously increasing or decreasing respectively. It seems that cummax is a lot faster than diff and also uses less memory:

> x <- seq_len(1e7)
> system.time(all(x == cummax(x)))
   user  system elapsed 
   0.11    0.00    0.11 
> system.time(all(diff(x) >= 0))
   user  system elapsed 
   0.47    0.13    0.59

> x <- seq_len(1e8)
> system.time(all(x == cummax(x)))
   user  system elapsed 
   1.06    0.09    1.16 
> system.time(all(diff(x) >= 0))
Error: cannot allocate vector of size 381.5 Mb
In addition: Warning messages:
1: Reached total allocation of 1535Mb: see help(memory.size) 
2: Reached total allocation of 1535Mb: see help(memory.size) 
3: Reached total allocation of 1535Mb: see help(memory.size) 
4: Reached total allocation of 1535Mb: see help(memory.size) 
Timing stopped at: 1.96 0.38 2.33

My bet as to why cummax is faster than diff is because it only has to compare numbers which is faster than having to compute differences.

Edit: at your (Ali's) request, additional tests including your answer (Note that I am now running from a different machine so the following results should not be compared with the ones above)

> x <- seq_len(1e7)
> system.time(x == cummax(x))
   user  system elapsed 
  0.316   0.096   0.416 
> system.time(all(diff(x) >= 0))
   user  system elapsed 
  4.364   0.240   4.632 
> system.time(x[-1] - x[-length(x)] >= 0)
   user  system elapsed 
  3.828   0.380   4.227
> system.time(all(x[-1] >= x[-length(x)]))
   user  system elapsed 
  2.572   0.288   2.865