A Test of the Numerical Integration Speediness of Four Calculators The four calculators are: 1. The TI-36X Pro (scientific calculator) 2. The Casio fx-115ES Plus (scientific calculator) 3. The Casio fx-991EX ClassWiz (scientific calculator) 4. The Casio fx-9750gii (graphing calculator) The problem used for test: B = 1.3271244e+20 C = 6.957e+08 D = 1.495978707e+11 Evaluate: ∫ (C, 0.999999D) 1 / √(2B(1/x − 1/D)) dx This integral returns the amount of time, in seconds, for an object to fall to the sun's photosphere (one solar radius = C) starting from a distance of 1 astronomical unit (D), assuming that the sun and the object are initially at rest with respect to each other and are acted upon by no forces other than their mutual gravitational attraction. The variable B is the solar gravitational parameter. TI-36X Pro Answer: 5570898.581 Time to Solve: 90.4 sec Casio fx-115ES Plus Answer: 5570898.583 Time to Solve: 76.6 sec Casio fx-991EX ClassWiz Answer: 5570898.583 Time to Solve: 17.3 sec Casio fx-9750gii Answer: 5570898.583 Time to Solve: 4.9 sec As you can see, the Texas Instruments device is a slow-poke. The TI-36X Pro costs anywhere from $19 to $30 on Amazon. The Casio fx-991EX ClassWiz is the official (and substantial) Casio upgrade to the earlier Casio fx-115ES Plus. For moderately complicated numerical integration chores, the Casio fx-991EX is more than four times faster than its predecessor model. Furthermore, the Casio fx-991EX is more than five times faster than its ostensible competitor from Texas Instruments, the TI-36X Pro. The Casio fx-991EX sells at prices ranging from $14 to $36 on Amazon. But the Casio fx-9750gii is more than three times faster than the Casio fx-991EX ClassWiz and is about 18 times faster than the TI-36X Pro. Reasonable prices on Amazon go from $30 to $37, but there are some sellers (that should be ignored) who are trying to gouge for more. I have solved the indefinite integral analytically. Omitting the intermediate steps, ∫ 1 / √(2B(1/x − 1/D)) dx = √[D/(2B)] { √(XD−X²) + D arctan[√(D/X−1)] } When I used the calculators to solve the integral numerically, I had to avoid entering D itself for x, since that would cause an overflow error. But in the analytical solution, when X=C, the answer is 5578001.670517. When X=0.999999D, the answer is 7103.088517. So the precise answer from the analytical solution, used twice to evaluate the integral from X=C to X=0.999999D, is 5570898.582. There is no appreciable error in the numerical results from any of the calculators.
@@Jenab7 of course, HP Prime is top of my list to get! But I need/want a Scientific Calculator as well, as a backup for when exams/tests won't allow the HP Prime. Seems like the Casio fx-991EX is my next best choice to bring along "just in case".
The HP Prime G2 calculator does the above numerical integration almost instantaneously. Also, I've noticed a simplification to the analytic solution that I didn't see earlier. ∫ 1 / √(2B(1/x − 1/D)) dx = √[D/(2B)] { √(XD−X²) + D arccos[√(X/D)] } This removes the singularity at X=0.
So glad I saw this, I thought I was going to buy the Canon, but I think I am going to revise my decision, I am a quick paced person and the canon would not be for me.
Edward Shore I think I am going with the fx 991 ex class-wiz model. Another note, I like how you replied very quickly to what I said. Being interacted with the community if part of being a good youtuber.
Considering that there is only about a $10 difference in price between the two, I don't see any advantages in buying the Canon. Thanks for this test and sharing it with us.
You are just right. But canon is slower than casio in deed. Canon does 180 times of plus calculation, while casio does 600 times. But we can over clock the canon f789sga by just taking off the back cover and writing on R2 resistance lightly with a 2B pencil, and this will make the number of resistance of R2 smaller. If you try it in a proper maner, this will make your canon much faster. My canon can calculate 625 times now.
You said twice that the last test had the greatest difference (in time I assume). However: 5.95/1.75 = 3.4x 19.09/2.75 = 6.9x 12.73/1.75 = 7.3x 138/23 = 6x The last test had actually the second smallest speed difference (hence Speed Test)! I have the very similar Canon F-789SGA, so thanks for a useful video! I think the Canon is fast enough. If I need speed, I use a smart phone or a desktop computer.
In your third test, the one with the matrix test, the very first term written black on white at 1:14 was 0.5 but when you entered in the calculators you typed -0.5, with the minus sign... I am sorry to be annoying like that but I simply run the tests at my end to compare and at first I did not get the same result. For the last summation test, my valorous TI-89 Titanium remained busy for 11.48 seconds. For the last summation test, the TI-84 Plus CE took 6.63 seconds. For the last summation test, the Casio fx-CG10 took 3.20 seconds. ...and for the five tests, the "beast" HP Prime gave the result instantly! But I admit, it's not a solar one and it cost a bit more. :-) Again, HP Prime rules! :-)
@@theedspage Also in 0:25 you inputted .6 on the upper bound of the integral but earlier you flashes .06 on the screen. Please be consistent and check your input and display. Thank you
the F-792SGA has more functions but it isnt for complex calculations. It even has formulas stored there such as sine law and potential energy formula. It is basically kinda like Internet Explorer back in the days where is had a lot of features but it sacrificed performance which Chrome did have the opposite. Guess what did the majority picked?
You are just right. But canon is slower than casio in deed. Canon does 180 times of plus calculation, while casio does 600 times. But we can over clock the canon f789sga by just taking off the back cover and writing on R2 resistance lightly with a 2B pencil, and this will make the number of resistance of R2 smaller. If you try it in a proper maner, this will make your canon much faster. My canon can calculate 625 times now.
HAVE you done a side by side comparison of CASIO fx-991EX Classwiz and SHARP EL-W516XBS . please we would love to see this two compared. i asked of you because i loved your comparison test video.
I own these two plus the TI-36XPRO. The TI is overall faster than the Canon but still not as fast as the Casio. Also the TI only goes up to 3rd order polynomials and Matrices. I do like the TI over the Canon because it is more user friendly. However, I have to admit the Casio beats them both.
The TI-nspire cx and TI-nspire cx cas calculators performed calcul in a fraction of a second. But score I had to convert to the decimal figure later. This is not a critique of both calculators, just a litlle curiosity. As for the Canon calculator, it is very slow. But maybe it is not annoying in simple calculations?
This is Not a fair test.... The Canon is from a previous generation of calc. In fact a closer comparison would be the older Casio FX-991ES .. They even had a very similar looking menu system and items.
Yes, i've just compare my Canon F-792SGA with my Casio fx-991ES PLUS C and i can confirm that they are in the same league. Also, the Casio has only 14 digits of precision in it's calculation while the Canon has 20. That kind of comparison video is worthless and misleading.
I managed to program in the summation on my Casio fx-10F, it's still running.. I'd test the integral but ran out of memory before I got to the end so I never got it to run.
I looked for Casio fx-991EX Claswiz in the market but I didn't find it. So I had to buy Casio fx-991 es plus. It is also fast but not as classwiz. In the last test es plus took just 8 seconds to solve it
Actually, the CLASSWIZ does not. At least as far as saving lists. For example, when you enter stat data and then switch the calculator to another mode other than statistics, the data is lost. One bad feature of the Casio they don't seem to want to correct.
+Mos Rus (mRussia) My concern is more basic. I understand (and accept) that by changing to another calculation Mode your calculation history in the first Mode is lost. Even Sharp calculators have that problem. However, when I am in a calculation Mode, and am thinking about what I am solving, or, I am explaining to my niece how a problem is solved, the Casio will auto-time out, and when I turn it back on I find that my work is lost; gone forever. Unacceptable! This is common in all Japanese scientific calculators. I cannot understand why Casio's advanced Classwiz did not fix this very simple problem. TI's calculators have always had power-off memory protection, which is why I prefer using my TI-36X Pro over my Casios. With T.I., no matter what Mode you're in, or what Mode you change, ALL of your calculation history is saved for reference! I would never go into an exam without memory-safeguard.
+George Obregon Yes. I agree. I also have a TI-36XPRO and it does save all your work. Maybe TI will come up with a new calculator that does all the CLASSWIZ does and more. The Casio graphics retain calculation history. It is beyond me why the Japanese non-graphic calculators don't do this because TI and HP calculators always have.
+George Obregon This calculator is meant to be allowed in examinations. It should not be possible to store lots of numbers when the calculator is switched off, or changed to another mode. So they deliberately don't have memory protection.
Hi there, Could u compare HP 300s+ and CASIO FX-991EX Please.🙏🏻
6 лет назад
ti83 plus took about 20.5 secs although i still would prefer the casio's vpam. so much easier than entering "sum(seq(X^-2,X,1,500))". also ( - ) minus is not negative so if i enter minus, it shows error. has to use negative button.
But speed doesn't tell the whole story. The Canon calculates everything internally to 20 DIGITS! It's just kind of cool to have one of the few scientific calculators with that accuracy. However it uses the last 4 digits as guard digits, so for complex calculations you should only trust 16 digits, and it takes a little fooling around to get all 16. The Casio (if it's like other Casios) calculates to 13 digits and uses three of them as guard digits. This is part of the reason the Canon is so much slower. I will also say the Casio has a much nicer display, with beautiful numbers -- easy on tired eyes. ;)
@@cypherf0x And it costs about 50 times as much, right? You can get an Android phone that runs Maxima on Linux, with infinite precision. Takes a lot of work, but can be done. Get a phone with real qwerty keys, and use labels to customize. Viola!
I tried this on a duplicate Casio-fx 991 ex For the 1st calculation it took:8sec For the second calculation it took:45 sec and gave an answer of 0.999999634 4th calculation:9 sec Last calc:5 sec
With my fx-991EX the e^-x integral consistently took 6.2 seconds over ten runs with different numerical settings just to check. I think that there must be a big variation in processor speed and possibly software improvements depending on the version.
I wanted to try the last summation test ruclips.net/video/39tcSQBakMo/видео.html on my calculators : 1) a CASIO fx-991ES PLUS took 1 minute 51.76 seconds 2) a CASIO fx-9860GII SD took incredibly only 1.15 seconds
My HP 50g took about 7.2 seconds in approx mode and about 49 seconds in exact mode. The exact answer was given as a fraction which has far too many digits to write down here!
+ijabbott63 I just tried the summation test on my TI-nspire cx CAS overclocked to 234Mhz with AHB at 58Mhz, result is ~ 0.73s (measured with a stopwatch)
The Canon's longer processing times will make a huge difference for a student sitting for an exam.
I would have to agree
right.. i am a Bangladeshi student.. i used casio fx 991 ex.. it is very effective
Not as much as a Casio's tendency to glitch and give wrong answers .
A Test of the Numerical Integration Speediness of Four Calculators
The four calculators are:
1. The TI-36X Pro (scientific calculator)
2. The Casio fx-115ES Plus (scientific calculator)
3. The Casio fx-991EX ClassWiz (scientific calculator)
4. The Casio fx-9750gii (graphing calculator)
The problem used for test:
B = 1.3271244e+20
C = 6.957e+08
D = 1.495978707e+11
Evaluate:
∫ (C, 0.999999D) 1 / √(2B(1/x − 1/D)) dx
This integral returns the amount of time, in seconds, for an object to fall to the sun's photosphere (one solar radius = C) starting from a distance of 1 astronomical unit (D), assuming that the sun and the object are initially at rest with respect to each other and are acted upon by no forces other than their mutual gravitational attraction. The variable B is the solar gravitational parameter.
TI-36X Pro
Answer: 5570898.581
Time to Solve: 90.4 sec
Casio fx-115ES Plus
Answer: 5570898.583
Time to Solve: 76.6 sec
Casio fx-991EX ClassWiz
Answer: 5570898.583
Time to Solve: 17.3 sec
Casio fx-9750gii
Answer: 5570898.583
Time to Solve: 4.9 sec
As you can see, the Texas Instruments device is a slow-poke. The TI-36X Pro costs anywhere from $19 to $30 on Amazon.
The Casio fx-991EX ClassWiz is the official (and substantial) Casio upgrade to the earlier Casio fx-115ES Plus. For moderately complicated numerical integration chores, the Casio fx-991EX is more than four times faster than its predecessor model. Furthermore, the Casio fx-991EX is more than five times faster than its ostensible competitor from Texas Instruments, the TI-36X Pro. The Casio fx-991EX sells at prices ranging from $14 to $36 on Amazon.
But the Casio fx-9750gii is more than three times faster than the Casio fx-991EX ClassWiz and is about 18 times faster than the TI-36X Pro. Reasonable prices on Amazon go from $30 to $37, but there are some sellers (that should be ignored) who are trying to gouge for more.
I have solved the indefinite integral analytically. Omitting the intermediate steps,
∫ 1 / √(2B(1/x − 1/D)) dx
= √[D/(2B)] { √(XD−X²) + D arctan[√(D/X−1)] }
When I used the calculators to solve the integral numerically, I had to avoid entering D itself for x, since that would cause an overflow error. But in the analytical solution, when X=C, the answer is 5578001.670517. When X=0.999999D, the answer is 7103.088517. So the precise answer from the analytical solution, used twice to evaluate the integral from X=C to X=0.999999D, is 5570898.582. There is no appreciable error in the numerical results from any of the calculators.
Thank you! These speed tests are very helpful.
@@SoundSpeeding The HP Prime G2 calculator beats them all. There's no discernable delay between execution and the output of the answer.
@@Jenab7 of course, HP Prime is top of my list to get!
But I need/want a Scientific Calculator as well, as a backup for when exams/tests won't allow the HP Prime.
Seems like the Casio fx-991EX is my next best choice to bring along "just in case".
I just tried this on my classpad2, cp-400 and it took 85 seconds.
The HP Prime G2 calculator does the above numerical integration almost instantaneously.
Also, I've noticed a simplification to the analytic solution that I didn't see earlier.
∫ 1 / √(2B(1/x − 1/D)) dx
= √[D/(2B)] { √(XD−X²) + D arccos[√(X/D)] }
This removes the singularity at X=0.
So glad I saw this, I thought I was going to buy the Canon, but I think I am going to revise my decision, I am a quick paced person and the canon would not be for me.
+Kyle Bell That's cool. I hope you find the calculator(s) that does work for you! :)
Edward Shore I think I am going with the fx 991 ex class-wiz model. Another note, I like how you replied very quickly to what I said. Being interacted with the community if part of being a good youtuber.
Considering that there is only about a $10 difference in price between the two, I don't see any advantages in buying the Canon. Thanks for this test and sharing it with us.
For reference, Casio FX9860-GII did the summation in 5 seconds. The summation is the one from the video at 2:39
the fx991ex is so good that it takes you off from apps and other math-software
You are just right. But canon is slower than casio in deed. Canon does 180 times of plus calculation, while casio does 600 times. But we can over clock the canon f789sga by just taking off the back cover and writing on R2 resistance lightly with a 2B pencil, and this will make the number of resistance of R2 smaller. If you try it in a proper maner, this will make your canon much faster. My canon can calculate 625 times now.
and you think people will gonna want to go through all that to save 10$ ?
Casio 991 ex is like usain bolt
And canon is like fat joe
You said twice that the last test had the greatest difference (in time I assume). However:
5.95/1.75 = 3.4x
19.09/2.75 = 6.9x
12.73/1.75 = 7.3x
138/23 = 6x
The last test had actually the second smallest speed difference (hence Speed Test)!
I have the very similar Canon F-789SGA, so thanks for a useful video!
I think the Canon is fast enough. If I need speed, I use a smart phone or a desktop computer.
You should've have subtracted not divided
He said difference not quotient so he is right
In your third test, the one with the matrix test, the very first term written black on white at 1:14 was 0.5 but when you entered in the calculators you typed -0.5, with the minus sign...
I am sorry to be annoying like that but I simply run the tests at my end to compare and at first I did not get the same result.
For the last summation test, my valorous TI-89 Titanium remained busy for 11.48 seconds.
For the last summation test, the TI-84 Plus CE took 6.63 seconds.
For the last summation test, the Casio fx-CG10 took 3.20 seconds.
...and for the five tests, the "beast" HP Prime gave the result instantly! But I admit, it's not a solar one and it cost a bit more. :-)
Again, HP Prime rules! :-)
Thanks. I hate it when I miss those typos.
+Edward Shore First test: limits on display are from 0.001 to 0.6, but the first screen said 0.001 to 0.06.
The Sharp wl-516x takes only 15 secs for the last test...(although my answer was 1.642936065)....strange as it was slower on all the other tests...
@@theedspage Also in 0:25 you inputted .6 on the upper bound of the integral but earlier you flashes .06 on the screen. Please be consistent and check your input and display. Thank you
Casio: Core i7
Canon: Intel Celeron XD
lol
the F-792SGA has more functions but it isnt for complex calculations. It even has formulas stored there such as sine law and potential energy formula. It is basically kinda like Internet Explorer back in the days where is had a lot of features but it sacrificed performance which Chrome did have the opposite. Guess what did the majority picked?
Canon might be slower due to its higher internal precisions ? (18 digits vs 15 for Casio)
You are just right. But canon is slower than casio in deed. Canon does 180 times of plus calculation, while casio does 600 times. But we can over clock the canon f789sga by just taking off the back cover and writing on R2 resistance lightly with a 2B pencil, and this will make the number of resistance of R2 smaller. If you try it in a proper maner, this will make your canon much faster. My canon can calculate 625 times now.
Terminator Brown Can you share me a link on how to overclock the 789sga?. Thanks
Yes how to overclock it bro?
So the Casio is faster with inaccurate calculations.
HAVE you done a side by side comparison of CASIO fx-991EX Classwiz and SHARP EL-W516XBS . please we would love to see this two compared. i asked of you because i loved your comparison test video.
516T is quiet a bit faster than the 516x...
I own these two plus the TI-36XPRO. The TI is overall faster than the Canon but still not as fast as the Casio. Also the TI only goes up to 3rd order polynomials and Matrices. I do like the TI over the Canon because it is more user friendly. However, I have to admit the Casio beats them both.
Casio every day of the week
The TI-nspire cx and TI-nspire cx cas calculators performed calcul in a fraction of a second. But score I had to convert to the decimal figure later. This is not a critique of both calculators, just a litlle curiosity. As for the Canon calculator, it is very slow. But maybe it is not annoying in simple calculations?
This is Not a fair test.... The Canon is from a previous generation of calc. In fact a closer comparison would be the older Casio FX-991ES .. They even had a very similar looking menu system and items.
Yes, i've just compare my Canon F-792SGA with my Casio fx-991ES PLUS C and i can confirm that they are in the same league. Also, the Casio has only 14 digits of precision in it's calculation while the Canon has 20. That kind of comparison video is worthless and misleading.
I managed to program in the summation on my Casio fx-10F, it's still running..
I'd test the integral but ran out of memory before I got to the end so I never got it to run.
I looked for Casio fx-991EX Claswiz in the market but I didn't find it. So I had to buy Casio fx-991 es plus. It is also fast but not as classwiz. In the last test es plus took just 8 seconds to solve it
I don't know what's more interesting. The comparison results, or the fact that a scientific calculator showdown got 473 up votes. 474 (for mine).
For the last calculation my fake calculator was even faster than the original 😂
😂😂
I think, Canons F-792SGA is a re-branded Casio 991ES.
Does this very advanced Casio have simple memory safeguard like the TI-36X Pro? --I would hate to lose what I'm working on when it auto times-out!
It should.
Actually, the CLASSWIZ does not. At least as far as saving lists. For example, when you enter stat data and then switch the calculator to another mode other than statistics, the data is lost. One bad feature of the Casio they don't seem to want to correct.
+Mos Rus (mRussia) My concern is more basic. I understand (and accept) that by changing to another calculation Mode your calculation history in the first Mode is lost. Even Sharp calculators have that problem.
However, when I am in a calculation Mode, and am thinking about what I am solving, or, I am explaining to my niece how a problem is solved, the Casio will auto-time out, and when I turn it back on I find that my work is lost; gone forever. Unacceptable!
This is common in all Japanese scientific calculators. I cannot understand why Casio's advanced Classwiz did not fix this very simple problem.
TI's calculators have always had power-off memory protection, which is why I prefer using my TI-36X Pro over my Casios. With T.I., no matter what Mode you're in, or what Mode you change, ALL of your calculation history is saved for reference!
I would never go into an exam without memory-safeguard.
+George Obregon Yes. I agree. I also have a TI-36XPRO and it does save all your work. Maybe TI will come up with a new calculator that does all the CLASSWIZ does and more. The Casio graphics retain calculation history. It is beyond me why the Japanese non-graphic calculators don't do this because TI and HP calculators always have.
+George Obregon This calculator is meant to be allowed in examinations. It should not be possible to store lots of numbers when the calculator is switched off, or changed to another mode. So they deliberately don't have memory protection.
Hi there, Could u compare HP 300s+ and CASIO FX-991EX Please.🙏🏻
ti83 plus took about 20.5 secs although i still would prefer the casio's vpam. so much easier than entering "sum(seq(X^-2,X,1,500))". also ( - ) minus is not negative so if i enter minus, it shows error. has to use negative button.
im only here to test the speed of my new classwiz calculator. im never going to calculate such calculations any more in my life though
I hope the biggest worry I have on exam day is whether my calculator is allowed in.
My faithful fx-991ES did the summation in 2 mins.
Hello !:) i have a question ! does the battery of the Casio go down easily ? if you use it on a daily basis?
It has solar power so not really.
It almost never uses its battery, it uses the solar cell wich is super sensible.
But speed doesn't tell the whole story. The Canon calculates everything internally to 20 DIGITS! It's just kind of cool to have one of the few scientific calculators with that accuracy. However it uses the last 4 digits as guard digits, so for complex calculations you should only trust 16 digits, and it takes a little fooling around to get all 16. The Casio (if it's like other Casios) calculates to 13 digits and uses three of them as guard digits. This is part of the reason the Canon is so much slower. I will also say the Casio has a much nicer display, with beautiful numbers -- easy on tired eyes. ;)
And the DM42 calculates with 34 digits of decimal precision.
@@cypherf0x And it costs about 50 times as much, right? You can get an Android phone that runs Maxima on Linux, with infinite precision. Takes a lot of work, but can be done. Get a phone with real qwerty keys, and use labels to customize. Viola!
i thought the upper limit of the first equation is 0.06
Thanks for this video
I tried this on a duplicate Casio-fx 991 ex
For the 1st calculation it took:8sec
For the second calculation it took:45 sec and gave an answer of 0.999999634
4th calculation:9 sec
Last calc:5 sec
With my fx-991EX the e^-x integral consistently took 6.2 seconds over ten runs with different numerical settings just to check. I think that there must be a big variation in processor speed and possibly software improvements depending on the version.
my hp prime G1 gives all the results with no visible delay. it's all instantaneous ! even the last test...
The display on the canon looks very similar to the casio fx-115es plus. Is it the same processor?
Both use the same software base, on slightly different processors
Not same. The canon does 4x4 matrices, Casio ES can do only 3x3.
To solving SUM function, go to coffee brake (Canon calculator) 😂
Perfect tests!
I'm glad I bought the classwiz tho the canon has some good features.
Casio is just built different😤😤
only the fastest thing canon sells is their EOS/Rebel Camera's
This problem is, canin is program to the sleep for options ther fuctions.
En el primer elemento se equivocó es 0.5 y puso -0.5
Nice. Where do you use these calculations in the real life?
Urm exams, if your a mathematician, mechanics, statistics, physics, game development, quite alot of areas so stfu
I wanted to try the last summation test ruclips.net/video/39tcSQBakMo/видео.html on my calculators :
1) a CASIO fx-991ES PLUS took 1 minute 51.76 seconds
2) a CASIO fx-9860GII SD took incredibly only 1.15 seconds
My HP 50g took about 7.2 seconds in approx mode and about 49 seconds in exact mode. The exact answer was given as a fraction which has far too many digits to write down here!
Another data point: My TI-36X Pro took about 35 seconds.
+ijabbott63 I have clocked the fx-9860GII SD a bit more, with a better misuration I obtain only 0.86s on the summation test previously calculated!
+Matteo Barbaro I bought a TI nSpire CX CAS a few days ago and tried the summation test, but it was too short for me to measure it.
+ijabbott63 I just tried the summation test on my TI-nspire cx CAS overclocked to 234Mhz with AHB at 58Mhz, result is ~ 0.73s (measured with a stopwatch)
If Usain Bolt was a calculator he'd be the fx-991ex! :)
Oh so the newest most up to date calculator is faster? Surprise surprise...
Sure glad I bought the Casio and not that canon.
My cheap 5€ scientific calculator needs 2 minutes for the second test... But I don't even care...
Casio always beats everyone
i bought the cannon : (
Canon may be slower but still good
y tho
my phone is almost instantaneous
My casio.says answer is 0.01045445842
calculadora canom usa chips de inferior qualidade.
Even canon printers are slow
That canon calculator is kinda ..... trash