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09-25-09, 15:27
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Registered User
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Join Date: Jul 2009
Location: NY
Posts: 886
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Definite Integral Calculation
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Definite Integral interval [a, b]: Sf(x)dx = G(b) - G(a) where G'(x) = f(x) , or differential of G(x) = f(x).
Integrals appear in many practical situations.
Consider a swimming pool. If it is rectangular, then from its length, width, and depth we can easily determine the volume of water it can contain (to fill it), the area of its surface (to cover it), and the length of its edge (to rope it).
But if it is oval with a rounded bottom, all of these quantities call for integrals. Practical approximations may suffice for such trivial examples, but precision engineering (of any discipline) requires exact and rigorous values for these elements.
I create the query which calculate integral of 3 * X^2 - 2 * X in interval [2, 4]....
S(3 * X^2 - 2 * X)dx = X^3 - X^2 and exact value of integral will be
S = (4^3 - 4^2) - (2^3 - 2^2) = 48 - 4 = 44.
We know nothing in our solution about exact value and differential and suppose to calculate approximate value of integral.
Process will stop when the absolute difference between current value of integral and previous become less or equal some eps real number.
Code:
With
Source (Xstart, Xfinish, imgFunc, eps) as
(select double(2), double(4), '3 * X^2 - 2 * X', double(1.e-3)
from sysibm.sysdummy1
)
,
Integral_calc (Xs, Xf, Xc, step, curint, prevint, eps, iterno) as
(select Xstart, Xfinish, double(Xstart - (Xfinish - Xstart) / 10.) Xc,
double((Xfinish - Xstart) / 10.) step, double(0), double(0), eps, int(0)
from Source
union All
select Xs, Xf, Xc + step, step, curint + Fc * step, prevint, eps, iterno + 1
from Integral_calc, table
(select 3 * power(Xc + step, 2) - 2 * (Xc + step) Fc
from sysibm.sysdummy1 ) it
where Xc + step <= Xf
Union All
select Xs, Xf, Xs - (step / 2.), step / 2., 0., curint, eps, iterno + 1
from Integral_calc
where Xc + step > Xf
and abs(curint - prevint) > eps
)
,
Integral(integral_value, integral_image) as
(select curint, 'interval: [' || varchar(Xs) || ', ' || varchar(Xf) || ']: S' || '('
|| imgFunc || ')dx = ' || varchar(round(curint, 3))
from Integral_calc, Source
where iterno = (select max(iterno) from Integral_calc)
)
select integral_value, integral_image from Integral
Result of calculation:
Quote:
INTEGRAL_VALUE............................ INTEGRAL_IMAGE
4.40005859378289E+001............... interval: [2.0E0, 4.0E0]: S(3 * X^2 - 2 * X)dx = 4.4001E1
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Could be very useful for students and engineers.
Lenny
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09-25-09, 15:50
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Registered User
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Join Date: Jul 2009
Location: NY
Posts: 886
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This query is working fast for short intervals.
Do not use it for intervals with length > 10....
Also, if you want calculate yours integral, you have to change
Code:
table (select 3 * power(Xc + step, 2) - 2 * (Xc + step) Fc
from sysibm.sysdummy1 ) it
and
Code:
Source (Xstart, Xfinish, imgFunc, eps) as
(select double(1), double(12), '3 * X^2 - 2 * X', double(1.e-3)
from sysibm.sysdummy1
Lenny
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09-25-09, 23:42
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Registered User
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Join Date: Jul 2009
Posts: 150
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Lenny, do you think somebody here remember what is it "definite integral" ?
Kara S.
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09-26-09, 23:15
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Registered User
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Join Date: Jul 2009
Posts: 150
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Just better look
For me looks better if looks so:
Code:
With
Source (Xstart, Xfinish, imgFunc, eps) as
(select double(2), double(4), '3 * X^2 - 2 * X', double(1.e-3)
from sysibm.sysdummy1
)
,
Integral_calc (Xs, Xf, Xc, step, CurInt, PrevInt, eps, iterno) as
(select Xstart, Xfinish, double(Xstart - (Xfinish - Xstart) / 10.) Xc,
double((Xfinish - Xstart) / 10.) step, double(0), double(0), eps, int(0)
from Source
union All
select Xs, Xf, Xc + step, step, CurInt + FuncVal * step, PrevInt, eps,
iterno + 1
From Integral_calc,
table (select 3 * power(X, 2) - 2 * X as FuncVal
From
(select (Xc + step) as X rom sysibm.sysdummy1 ) ii ) it
where Xc + step <= Xf
Union All
select Xs, Xf, Xs - (step / 2.), step / 2., 0., CurInt, eps, iterno + 1
from Integral_calc
where Xc + step > Xf
and abs(curint - prevint) > eps
)
,
Integral(integral_value, integral_image) as
(select curint, 'interval: [' || varchar(Xs) || ', ' || varchar(Xf) || ']: S' || '('
|| imgFunc || ')dx = ' || varchar(round(curint, 3))
from Integral_calc, Source
where iterno = (select max(iterno) from Integral_calc)
)
select integral_value, integral_image from Integral
Kara S.
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09-27-09, 00:00
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:-)
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Join Date: Jun 2003
Location: Toronto, Canada
Posts: 4,449
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Quote:
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Originally Posted by DB2Plus
Lenny, do you think somebody here remember what is it "definite integral" ?
Kara S.
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No, of course not. You two guys are the only ones to carry the torch of knowledge. All the rest are just some lowly yeomen capable of no more than herding tablespaces...
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09-27-09, 02:03
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SPAMMER
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Join Date: Sep 2009
Posts: 4
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wow that is so difficul for me...to bad
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09-27-09, 08:14
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Registered User
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Join Date: Jul 2009
Posts: 150
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Quote:
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Originally Posted by n_i
No, of course not. You two guys are the only ones to carry the torch of knowledge. All the rest are just some lowly yeomen capable of no more than herding tablespaces...
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Sorry, if I did make mistake. 
Kara S.
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09-27-09, 12:39
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Registered User
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Join Date: Jul 2009
Posts: 150
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In my opinion when interval includes special numbers pi or e, we have to use something special,
because we don't know exact values of these two numbers and have to use scalar functions, e.g.
pi = 2 * asin(1.0) , e = exp(1.)
Kara S.
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Last edited by DB2Plus; 09-27-09 at 15:49.
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09-27-09, 13:15
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Registered User
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Join Date: Jul 2009
Posts: 150
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Example with trigonometric function
For example: interval [0, 3/4pi] S sin(x)dx:
Code:
With
Source (Xstart, Xfinish, imgFunc, eps) as
(select double(0), 1.5 * asin(1.), 'sin(x)', double(1.e-4)
from sysibm.sysdummy1
)
,
Integral_calc (Xs, Xf, Xc, step, CurInt, PrevInt, eps, iterno) as
(select Xstart, Xfinish, double(Xstart - (Xfinish - Xstart) / 10.) Xc,
double((Xfinish - Xstart) / 10.) step, double(0), double(0), eps, int(0)
from Source
union All
select Xs, Xf, Xc + step, step, CurInt + FuncVal * step, PrevInt, eps,
iterno + 1
From Integral_calc,
table (select sin(x) as FuncVal
From
(select (Xc + step) as X rom sysibm.sysdummy1 ) ii ) it
where Xc + step <= Xf
Union All
select Xs, Xf, Xs - (step / 2.), step / 2., 0., CurInt, eps, iterno + 1
from Integral_calc
where Xc + step > Xf
and abs(curint - prevint) > eps
)
,
Integral(integral_value, integral_image) as
(select curint, 'interval: [' || varchar(Xs) || ', ' || varchar(Xf) || ']: S' || '('
|| imgFunc || ')dx = ' || varchar(round(curint, 4))
from Integral_calc, Source
where iterno = (select max(iterno) from Integral_calc)
)
select integral_value, integral_image from Integral
Kara S.
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09-27-09, 17:20
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Registered User
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Join Date: Jul 2009
Posts: 150
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Some notes
Quote:
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Originally Posted by Lenny77
This query is working fast for short intervals.
Do not use it for intervals with length > 10....
Also, if you want calculate yours integral, you have to change
Code:
table (select 3 * power(Xc + step, 2) - 2 * (Xc + step) Fc
from sysibm.sysdummy1 ) it
and
Code:
Source (Xstart, Xfinish, imgFunc, eps) as
(select double(1), double(12), '3 * X^2 - 2 * X', double(1.e-3)
from sysibm.sysdummy1
Lenny |
Hi, Lenny !
We have to change
Code:
Integral(integral_value, integral_image) as
(select curint, 'interval: [' || varchar(Xs) || ', ' || varchar(Xf) || ']: S'
|| '(' || imgFunc || ')dx = '
|| varchar(round(curint, 3))
from Integral_calc, Source
where iterno = (select max(iterno) from Integral_calc)
Because of depends on EPS 
Kara S.
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09-29-09, 11:20
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Registered User
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Join Date: Jul 2009
Location: NY
Posts: 886
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Small correction to trigonometric integral
Thank you Kara for yours trigonometric example !
One small correction:
Code:
With
Source (Xstart, Xfinish, imgFunc, eps) as
(select double(0), 1.5 * asin(1.), 'sin(x)', double(1.e-4)
from sysibm.sysdummy1
)
,
Integral_calc (Xs, Xf, Xc, step, CurInt, PrevInt, eps, iterno) as
(select Xstart, Xfinish, double(Xstart - (Xfinish - Xstart) / 10.) Xc,
double((Xfinish - Xstart) / 10.) step, double(0), double(0), eps, int(0)
from Source
union All
select Xs, Xf, Xc + step, step, CurInt + FuncVal * step, PrevInt, eps,
iterno + 1
From Integral_calc,
table (select sin(x) as FuncVal
From
(select (Xc + step) as X from sysibm.sysdummy1 ) ii ) it
where Xc + step <= Xf
Union All
select Xs, Xf, Xs - (step / 2.), step / 2., 0., CurInt, eps, iterno + 1
from Integral_calc
where Xc + step > Xf
and abs(curint - prevint) > eps
)
,
Integral(integral_value, integral_image) as
(select curint, 'interval: [' || varchar(Xs) || ', ' || varchar(Xf) || ']: S' || '('
|| imgFunc || ')dx = ' || varchar(round(curint, 4))
from Integral_calc, Source
where iterno = (select max(iterno) from Integral_calc)
)
select integral_value, integral_image from Integral
and it works !
Result:
Quote:
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interval: [0E0, 2.35619449019234E0]: S(sin(x))dx = 1.7072E0
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Lenny
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09-29-09, 11:57
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Registered User
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Join Date: Jul 2009
Location: NY
Posts: 886
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Check the result
We can check the result of calculation (less then 1 second).
We know, or you can find in the smart books:
So, on interval: [0E0, 2.35619449019234E0] exact value of integral will be:
Code:
select -cos(2.35619449019234E0) - (-cos(0E0))
from sysibm.sysdummy1
E.g.
Quote:
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S(six(x))dx = 1.70710678118654E+000
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Lenny
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09-29-09, 21:16
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Registered User
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Join Date: Jul 2009
Posts: 150
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Thank you, Lenny.
But where is my mistake ?
Kara S.
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Last edited by DB2Plus; 09-29-09 at 21:21.
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09-29-09, 21:19
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Registered User
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Join Date: Jul 2009
Posts: 150
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Lenny, thanks.
Now I found my mistake:
(select (Xc + step) as X rom sysibm.sysdummy1 ) ii ....
where has to be:
Code:
(select (Xc + step) as X From sysibm.sysdummy1 ) ii ....
Sorry, everybody! 
I have made this query without test, because I don't have DB2 at my home.
Kara S.
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Definite Integral Calculation
