Improper integral

[parisse]

Most CAS are using the standard definition for fractional powers, i.e. complex value for negative argument, giac does the same, that's why you get a complex result, since argument is negative in 0..1.

[Wes Loewer]

(06-11-2014 07:22 PM)Mark Hardman Wrote:  

(06-11-2014 07:01 PM)HP67 Wrote:  Why does the op say it should be 75/4?

The asymptote is at x=1, I'm not sure why the OP thinks there would be a real solution in the interval 0..1.

Because using the real root definition, the 5th root of a negative real is a negative real.

   

This improper integral can be evaluated be splitting the interval at the asymptote.
Integrating from 0 to 1 gives a result of -5/4
Integrating from 1 to 33 gives a result of 20
20-5/4 = 75/4

[Wes Loewer]

(06-12-2014 06:49 AM)parisse Wrote:  Most CAS are using the standard definition for fractional powers, i.e. complex value for negative argument, giac does the same, that's why you get a complex result, since argument is negative in 0..1.

Since this seems to be the source of a significant amount of confusion for the experienced users of this forum, I can only imagine the confusion for students.

from CAS:
5 NTHROOT -1 = -1
(-1)^(1/5) = 0.809016994375+0.587785252292*i
These are consistent with the stated CAS definitions of NTHROOT (real) and fractional powers (complex).

from Home:
5 NTHROOT -1 = -1
(-1)^(1/5) = 0.809016994375+0.587785252292*i (in complex mode)
(-1)^(1/5) = error (in real mode)
I'm guessing this error is the result of using exp(y*ln(x)) to evaluate x^y.

I understand why each of these answers is given, but I can't help but think that the average user is going to be confused.

I realize that the Prime CAS is consistent with other CAS's, but it's quite different from other calculators. Both TI and Casio treat NTHROOT and fractional powers the same and use the Mode Complex settings to determine whether to use the real root or the complex root.

At least in American schools, students are taught that n NTHROOT x is the same as x^(1/n). The Prime's target consumer is going to expect this same behavior on their calculators.

Also, all the other graphing calculators that I'm aware of give (-1)^(1/5) = -1 in real mode. The Prime is the only one I know that gives an error. The error message alone (Error: (X<0)^(∉Z)) is enough to scare away new users.

-wes

[Angus]

I tried in cas sqrt(-1) wich gives i as result.
When I type (-1)^0.5 I get undef. The term (-1)^0.5 however is rewritten as sqrt(-1) on the left side. That means sqrt(-1) is on the left side. One having i as result and one undef.
What is about that?

[parisse]

Looks like a print bug, (-1)^.5 should not be converted to sqrt(-1). I will change that at least for giac.
Regarding fractional powers, there is no good solution, but I strongly believe the current behavior is the best compromise (of course the error message can be changed if you have a good suggestion, perhaps something that would suggest using surd/NTHROOT). I'm fixing a small bug, so that int(1/surd(x-1,5),x,1,33) will return 20, and int(1/surd(x-1,5),x,0,33) will return 75/4.

[John Colvin]

After reading all of these posts and some of the weird answers to this simple
improper integral, I put down my Prime and solved the integral the old-fashioned way-
with pencil and paper and had no problem getting the correct result of 18.75 or 75/4.
Sometimes it's better to rely on one's understanding of calculus rather than the inconsistent results given by a calculator.

I was able to get the correct result on the Prime in Home by using fractional power-
1/(x-1)^(1/5).

[alexzkter]

(06-12-2014 06:27 PM)John Colvin Wrote:  I was able to get the correct result on the Prime in Home by using fractional power-
1/(x-1)^(1/5).

I'm not an experienced user, but in Home, regardless off the CAS settings I get:
18.75:


On the virtual calculator HOWEVER, in Home too, I get:


On CAS, if exact and complex mode settings are not selected I get on both calculators 18.75 as well.

So either the physical and virtual calculators are not identically working, or I did something wrong, or the Prime I paid for is smarter than all of yours. Joking...

I hope I was helpful somehow.

P.S. Revision A, latest firmware at present, standard digit grouping, Spanish language on physical calculator.

[John Colvin]

(06-12-2014 07:01 PM)alexzkter Wrote:  

(06-12-2014 06:27 PM)John Colvin Wrote:  I was able to get the correct result on the Prime in Home by using fractional power-
1/(x-1)^(1/5).

I'm not an experienced user, but in Home, regardless off the CAS settings I get:
18.75:


On the virtual calculator HOWEVER, in Home too, I get:


On CAS, if exact and complex mode settings are not selected I get on both calculators 18.75 as well.

So either the physical and virtual calculators are not identically working, or I did something wrong, or the Prime I paid for is smarter than all of yours. Joking...

I hope I was helpful somehow.

P.S. Revision A, latest firmware at present, standard digit grouping, Spanish language on physical calculator.

This is what I don't like about the Prime. Inconsistencies even between calculators.
If I express the integrand using the radical sign instead of the fractional power, I
get the approximate answer 18.74795... instead of the correct answer of 18.75.
Could be a settings issue-I used textbook mode in Home which I think is what you used.

[alexzkter]

(06-12-2014 07:25 PM)John Colvin Wrote:  Could be a settings issue-I used textbook mode in Home which I think is what you used.

Yes, I used the textbook mode.

[parisse]

18.748 instead of 18.75 for an integrand that has a singularity at x=1 is not that bad for a numeric approximation, you have 4 correct digits.

[Maro]

Hello Bernard,

it seems that CAS indeed has a problem with NthROOT command. You can simply check it by using the „int(...)“ command for determining the antiderivative of the function f(x)=1/(5thROOT(x-1)). If you evaluate the result at x=33 you will get 40, which is wrong (it should be 20)

The antiderivative given by „int(1/5thROOT(x-1)) is in principle correct, but in the non simplified result there is a (-5)thROOT expression used, which is obviously evaluated wrong by the CAS.

When doing the same with fractional power 1/(x-1)^1/5 the antiderivative and the evaluation at x=33 is fine. It gives 20.

Compare CAS and Home with calculating (-5)thROOT(32), which results in „1“ (CAS) or in „0.5“ (HOME, which is correct). (-N)thROOT(x) should always be the same as x^(-1/N)

Regards
Maro

[parisse]

Yes, as I said in a previous message, it is already fixed in Xcas.

[Maro]

(06-13-2014 08:13 AM)parisse Wrote:  Yes, as I said in a previous message, it is already fixed in Xcas.

any idea from HP when such things will be fixed in the Prime?
As long as other calculators are needed in parallel to check reliability of results, working with the Prime is quite unsatisfactory …

[alexzkter]

(06-13-2014 08:13 AM)parisse Wrote:  Yes, as I said in a previous message, it is already fixed in Xcas.

What is the Xcas? I keep reading this I don't know what it is...or where it is ;D

[jebem]

(06-13-2014 09:21 AM)alexzkter Wrote:  

(06-13-2014 08:13 AM)parisse Wrote:  Yes, as I said in a previous message, it is already fixed in Xcas.

What is the Xcas? I keep reading this I don't know what it is...or where it is ;D

Pls see point (13) here.

[parisse]

(06-13-2014 08:29 AM)Maro Wrote:  any idea from HP when such things will be fixed in the Prime?

I have no idea, and even if I had, it's not difficult to imagine that I could not talk publicly about :-)

Quote:As long as other calculators are needed in parallel to check reliability of results, working with the Prime is quite unsatisfactory …

Well, here at least, you get a warning that numeric and exact integration do not match. You don't need another calculator to find the right answer, just a little reflexion. If numeric and exact answers don't match, it's most probably the exact answer that will be wrong (especially if you are using HP-specific functions like NTHROOT because other functions have probably been tested by Xcas users). The numeric integrator is more robust, but may fail if there are singular points on the integration domain.

[Maro]

(06-13-2014 11:02 AM)parisse Wrote:  Well, here at least, you get a warning that numeric and exact integration do not match.

Yes, this "self-reflection" feature of the Prime is really nice … ?

[rprosperi]

(06-12-2014 07:01 PM)alexzkter Wrote:  P.S. Revision A, latest firmware at present, standard digit grouping, Spanish language on physical calculator.

At first glance this made me chuckle, but after some thought, having a brief summary like this for Postings about Prime issues would probably be quite helpful. Many, many postings of 'erroneous' or 'confusing' or 'unexpected' results have solutions that are one form or another of 'you need to have this mode enabled, that setting set to Textbook, be sure you are in CAS (or HOME) mode' etc. If your OP has some core modes/settings documented, it will likely help the folks helping you better and more quickly understand the problem and find a solution.

The Prime has soo many features and capabilities with soo many options to display, calculate, or solve a problem, that narrowing the domain of your issue by documenting these settings also reduces the complexity for a helper finding possible causes.

Just sayin... the easier you make it for someone to help you, the quicker you are likely to get helped.

[Joe Horn]

(06-12-2014 07:01 PM)alexzkter Wrote:  I'm not an experienced user, but in Home, regardless off the CAS settings I get:
18.75:


On the virtual calculator HOWEVER, in Home too, I get:


On CAS, if exact and complex mode settings are not selected I get on both calculators 18.75 as well.

So either the physical and virtual calculators are not identically working, or I did something wrong, or the Prime I paid for is smarter than all of yours. Joking...

Please look at your CAS Settings, page 1, end of 3rd line. Whether that checkbox is checked or not ("Change apparent integers into exact integers") changes the results for this calculation in Home view.

[alexzkter]

(06-14-2014 03:51 AM)Joe Horn Wrote:  Please look at your CAS Settings, page 1, end of 3rd line. Whether that checkbox is checked or not ("Change apparent integers into exact integers") changes the results for this calculation in Home view.

Yes, you're right.
After more than 5 seconds of processing it threw 18.74795.......

Is it the way it's meant to be? I mean, if you change a setting on CAS, shouldn't it just affect the CAS mode?