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'''Adaptive Simpson's method''', also called '''adaptive Simpson's rule''', is a method of [[numerical integration]] proposed by G.F. Kuncir in 1962.<ref name="kuncir">{{Citation|author=G.F. Kuncir|title=Algorithm 103: Simpson's rule integrator|journal=Communications of the ACM|volume=5|issue=6|page=347|year=1962}}</ref> It is probably the first recursive adaptive algorithm for numerical integration to appear in print,<ref name="henriksson">For an earlier, non-recursive adaptive integrator more reminiscent of [[Numerical ordinary differential equations|ODE solvers]], see {{Citation|author=S. Henriksson|title=Contribution no. 2: Simpson numerical integration with variable length of step|journal=BIT Numerical Mathematics|volume=1|page=290|year=1961}}</ref> although more modern adaptive methods based on [[Gauss–Kronrod quadrature formula|Gauss–Kronrod quadrature]] and [[Clenshaw–Curtis quadrature]] are now generally preferred. Adaptive Simpson's method uses an estimate of the error we get from calculating a definite integral using [[Simpson's rule]]. If the error exceeds a user-specified tolerance, the algorithm calls for subdividing the interval of integration in two and applying adaptive Simpson's method to each subinterval in a recursive manner. The technique is usually much more efficient than [[Composite simpson's rule|composite Simpson's rule]] since it uses fewer function evaluations in places where the function is well-approximated by a [[cubic function]]. | |||
A criterion for determining when to stop subdividing an interval, suggested by J.N. Lyness,<ref name="lyness">{{Citation|author=J.N. Lyness|title=Notes on the adaptive Simpson quadrature routine|journal=Journal of the ACM|volume=16|issue=3|pages=483–495|year=1969}}</ref> is | |||
:<math>|S(a,c) + S(c,b) - S(a,b)|/15 < \epsilon \,</math> | |||
where <math>[a,b]\,\!</math> is an interval with midpoint <math>c\,\!</math>, <math>S(a,b)\,\!</math>, <math>S(a,c)\,\!</math>, and <math>S(c,b)\,\!</math> are the estimates given by Simpson's rule on the corresponding intervals and <math>\epsilon\,\!</math> is the desired tolerance for the interval. | |||
[[Simpson's rule]] is an interpolatory quadrature rule which is exact when the integrand is a polynomial of degree three or lower. Using [[Richardson extrapolation]], the more accurate Simpson estimate <math>S(a,c) + S(c,b)\,</math> for six function values is combined with the less accurate estimate <math>S(a,b)\,</math> for three function values by applying the correction <math>[S(a,c) + S(c,b) - S(a,b)]/15 \,</math>. The thus obtained estimate is exact for polynomials of degree five or less. | |||
==Sample implementations == | |||
===Python=== | |||
Here is an implementation of adaptive Simpson's method in [[Python (programming language)|Python]]. Note that this is explanatory code, without regard for efficiency. Every call to recursive_asr entails six function evaluations. For actual use, one will want to modify it so that the minimum of two function evaluations are performed. | |||
<source lang="python"> | |||
def simpsons_rule(f,a,b): | |||
c = (a+b) / 2.0 | |||
h3 = abs(b-a) / 6.0 | |||
return h3*(f(a) + 4.0*f(c) + f(b)) | |||
def recursive_asr(f,a,b,eps,whole): | |||
"Recursive implementation of adaptive Simpson's rule." | |||
c = (a+b) / 2.0 | |||
left = simpsons_rule(f,a,c) | |||
right = simpsons_rule(f,c,b) | |||
if abs(left + right - whole) <= 15*eps: | |||
return left + right + (left + right - whole)/15.0 | |||
return recursive_asr(f,a,c,eps/2.0,left) + recursive_asr(f,c,b,eps/2.0,right) | |||
def adaptive_simpsons_rule(f,a,b,eps): | |||
"Calculate integral of f from a to b with max error of eps." | |||
return recursive_asr(f,a,b,eps,simpsons_rule(f,a,b)) | |||
from math import sin | |||
print adaptive_simpsons_rule(sin,0,1,.000000001) | |||
</source> | |||
===C=== | |||
Here is an implementation of the adaptive Simpson's method in C99 that avoids redundant evaluations of f and quadrature computations. | |||
The amount of memory used is O(D) where D is the maximum recursion depth. Each stack frame caches computed values | |||
that may be needed in subsequent calls. | |||
<source lang="C"> | |||
#include <math.h> // include file for fabs and sin | |||
#include <stdio.h> // include file for printf | |||
// | |||
// Recursive auxiliary function for adaptiveSimpsons() function below | |||
// | |||
double adaptiveSimpsonsAux(double (*f)(double), double a, double b, double epsilon, | |||
double S, double fa, double fb, double fc, int bottom) { | |||
double c = (a + b)/2, h = b - a; | |||
double d = (a + c)/2, e = (c + b)/2; | |||
double fd = f(d), fe = f(e); | |||
double Sleft = (h/12)*(fa + 4*fd + fc); | |||
double Sright = (h/12)*(fc + 4*fe + fb); | |||
double S2 = Sleft + Sright; | |||
if (bottom <= 0 || fabs(S2 - S) <= 15*epsilon) | |||
return S2 + (S2 - S)/15; | |||
return adaptiveSimpsonsAux(f, a, c, epsilon/2, Sleft, fa, fc, fd, bottom-1) + | |||
adaptiveSimpsonsAux(f, c, b, epsilon/2, Sright, fc, fb, fe, bottom-1); | |||
} | |||
// | |||
// Adaptive Simpson's Rule | |||
// | |||
double adaptiveSimpsons(double (*f)(double), // ptr to function | |||
double a, double b, // interval [a,b] | |||
double epsilon, // error tolerance | |||
int maxRecursionDepth) { // recursion cap | |||
double c = (a + b)/2, h = b - a; | |||
double fa = f(a), fb = f(b), fc = f(c); | |||
double S = (h/6)*(fa + 4*fc + fb); | |||
return adaptiveSimpsonsAux(f, a, b, epsilon, S, fa, fb, fc, maxRecursionDepth); | |||
} | |||
int main(){ | |||
double I = adaptiveSimpsons(sin, 0, 1, 0.000000001, 10); // compute integral of sin(x) | |||
// from 0 to 1 and store it in | |||
// the new variable I | |||
printf("I = %lf\n",I); // print the result | |||
return 0; | |||
} | |||
</source> | |||
===Racket=== | |||
Here is an implementation of the adaptive Simpson method in [[Racket (programming language)|Racket]] with a behavioral software contract. The exported function computes the indeterminate integral for some given function ''f''. | |||
<source lang="Lisp"> | |||
;; ----------------------------------------------------------------------------- | |||
;; interface, with contract | |||
;; Simpson's rule for approximating an integral | |||
(define (simpson f L R) | |||
(* (/ (- R L) 6) (+ (f L) (* 4 (f (mid L R))) (f R)))) | |||
(provide/contract | |||
[adaptive-simpson | |||
(->i ((f (-> real? real?)) (L real?) (R (L) (and/c real? (>/c L)))) | |||
(#:epsilon (ε real?)) | |||
(r real?))] | |||
[step (-> real? real?)]) | |||
;; ----------------------------------------------------------------------------- | |||
;; implementation | |||
(define (adaptive-simpson f L R #:epsilon [ε .000000001]) | |||
(define f@L (f L)) | |||
(define f@R (f R)) | |||
(define-values (M f@M whole) (simpson-1call-to-f f L f@L R f@R)) | |||
(asr f L f@L R f@R ε whole M f@M)) | |||
;; computationally efficient: 2 function calls per step | |||
(define (asr f L f@L R f@R ε whole M f@M) | |||
(define-values (leftM f@leftM left*) (simpson-1call-to-f f L f@L M f@M)) | |||
(define-values (rightM f@rightM right*) (simpson-1call-to-f f M f@M R f@R)) | |||
(define delta* (- (+ left* right*) whole)) | |||
(cond | |||
[(<= (abs delta*) (* 15 ε)) (+ left* right* (/ delta* 15))] | |||
[else (define epsilon1 (/ ε 2)) | |||
(+ (asr f L f@L M f@M epsilon1 left* leftM f@leftM) | |||
(asr f M f@M R f@R epsilon1 right* rightM f@rightM))])) | |||
(define (simpson-1call-to-f f L f@L R f@R) | |||
(define M (mid L R)) | |||
(define f@M (f M)) | |||
(values M f@M (* (/ (abs (- R L)) 6) (+ f@L (* 4 f@M) f@R)))) | |||
(define (mid L R) (/ (+ L R) 2.)) | |||
</source> | |||
The code is an excerpt of a "#lang racket" module and that includes a (require rackunit) line. | |||
==Bibliography== | |||
<references/> | |||
==External links== | |||
*[http://math.fullerton.edu/mathews/n2003/AdaptiveQuadMod.html Module for Adaptive Simpson's Rule] | |||
[[Category:Numerical integration (quadrature)]] |
Revision as of 03:17, 12 December 2013
Adaptive Simpson's method, also called adaptive Simpson's rule, is a method of numerical integration proposed by G.F. Kuncir in 1962.[1] It is probably the first recursive adaptive algorithm for numerical integration to appear in print,[2] although more modern adaptive methods based on Gauss–Kronrod quadrature and Clenshaw–Curtis quadrature are now generally preferred. Adaptive Simpson's method uses an estimate of the error we get from calculating a definite integral using Simpson's rule. If the error exceeds a user-specified tolerance, the algorithm calls for subdividing the interval of integration in two and applying adaptive Simpson's method to each subinterval in a recursive manner. The technique is usually much more efficient than composite Simpson's rule since it uses fewer function evaluations in places where the function is well-approximated by a cubic function.
A criterion for determining when to stop subdividing an interval, suggested by J.N. Lyness,[3] is
where is an interval with midpoint , , , and are the estimates given by Simpson's rule on the corresponding intervals and is the desired tolerance for the interval.
Simpson's rule is an interpolatory quadrature rule which is exact when the integrand is a polynomial of degree three or lower. Using Richardson extrapolation, the more accurate Simpson estimate for six function values is combined with the less accurate estimate for three function values by applying the correction . The thus obtained estimate is exact for polynomials of degree five or less.
Sample implementations
Python
Here is an implementation of adaptive Simpson's method in Python. Note that this is explanatory code, without regard for efficiency. Every call to recursive_asr entails six function evaluations. For actual use, one will want to modify it so that the minimum of two function evaluations are performed.
def simpsons_rule(f,a,b):
c = (a+b) / 2.0
h3 = abs(b-a) / 6.0
return h3*(f(a) + 4.0*f(c) + f(b))
def recursive_asr(f,a,b,eps,whole):
"Recursive implementation of adaptive Simpson's rule."
c = (a+b) / 2.0
left = simpsons_rule(f,a,c)
right = simpsons_rule(f,c,b)
if abs(left + right - whole) <= 15*eps:
return left + right + (left + right - whole)/15.0
return recursive_asr(f,a,c,eps/2.0,left) + recursive_asr(f,c,b,eps/2.0,right)
def adaptive_simpsons_rule(f,a,b,eps):
"Calculate integral of f from a to b with max error of eps."
return recursive_asr(f,a,b,eps,simpsons_rule(f,a,b))
from math import sin
print adaptive_simpsons_rule(sin,0,1,.000000001)
C
Here is an implementation of the adaptive Simpson's method in C99 that avoids redundant evaluations of f and quadrature computations. The amount of memory used is O(D) where D is the maximum recursion depth. Each stack frame caches computed values that may be needed in subsequent calls.
#include <math.h> // include file for fabs and sin
#include <stdio.h> // include file for printf
//
// Recursive auxiliary function for adaptiveSimpsons() function below
//
double adaptiveSimpsonsAux(double (*f)(double), double a, double b, double epsilon,
double S, double fa, double fb, double fc, int bottom) {
double c = (a + b)/2, h = b - a;
double d = (a + c)/2, e = (c + b)/2;
double fd = f(d), fe = f(e);
double Sleft = (h/12)*(fa + 4*fd + fc);
double Sright = (h/12)*(fc + 4*fe + fb);
double S2 = Sleft + Sright;
if (bottom <= 0 || fabs(S2 - S) <= 15*epsilon)
return S2 + (S2 - S)/15;
return adaptiveSimpsonsAux(f, a, c, epsilon/2, Sleft, fa, fc, fd, bottom-1) +
adaptiveSimpsonsAux(f, c, b, epsilon/2, Sright, fc, fb, fe, bottom-1);
}
//
// Adaptive Simpson's Rule
//
double adaptiveSimpsons(double (*f)(double), // ptr to function
double a, double b, // interval [a,b]
double epsilon, // error tolerance
int maxRecursionDepth) { // recursion cap
double c = (a + b)/2, h = b - a;
double fa = f(a), fb = f(b), fc = f(c);
double S = (h/6)*(fa + 4*fc + fb);
return adaptiveSimpsonsAux(f, a, b, epsilon, S, fa, fb, fc, maxRecursionDepth);
}
int main(){
double I = adaptiveSimpsons(sin, 0, 1, 0.000000001, 10); // compute integral of sin(x)
// from 0 to 1 and store it in
// the new variable I
printf("I = %lf\n",I); // print the result
return 0;
}
Racket
Here is an implementation of the adaptive Simpson method in Racket with a behavioral software contract. The exported function computes the indeterminate integral for some given function f.
;; -----------------------------------------------------------------------------
;; interface, with contract
;; Simpson's rule for approximating an integral
(define (simpson f L R)
(* (/ (- R L) 6) (+ (f L) (* 4 (f (mid L R))) (f R))))
(provide/contract
[adaptive-simpson
(->i ((f (-> real? real?)) (L real?) (R (L) (and/c real? (>/c L))))
(#:epsilon (ε real?))
(r real?))]
[step (-> real? real?)])
;; -----------------------------------------------------------------------------
;; implementation
(define (adaptive-simpson f L R #:epsilon [ε .000000001])
(define f@L (f L))
(define f@R (f R))
(define-values (M f@M whole) (simpson-1call-to-f f L f@L R f@R))
(asr f L f@L R f@R ε whole M f@M))
;; computationally efficient: 2 function calls per step
(define (asr f L f@L R f@R ε whole M f@M)
(define-values (leftM f@leftM left*) (simpson-1call-to-f f L f@L M f@M))
(define-values (rightM f@rightM right*) (simpson-1call-to-f f M f@M R f@R))
(define delta* (- (+ left* right*) whole))
(cond
[(<= (abs delta*) (* 15 ε)) (+ left* right* (/ delta* 15))]
[else (define epsilon1 (/ ε 2))
(+ (asr f L f@L M f@M epsilon1 left* leftM f@leftM)
(asr f M f@M R f@R epsilon1 right* rightM f@rightM))]))
(define (simpson-1call-to-f f L f@L R f@R)
(define M (mid L R))
(define f@M (f M))
(values M f@M (* (/ (abs (- R L)) 6) (+ f@L (* 4 f@M) f@R))))
(define (mid L R) (/ (+ L R) 2.))
The code is an excerpt of a "#lang racket" module and that includes a (require rackunit) line.
Bibliography
- ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ For an earlier, non-recursive adaptive integrator more reminiscent of ODE solvers, see Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010