Difference between revisions of "Simpson formula"
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A special case of the [[Newton–Cotes quadrature formula|Newton–Cotes quadrature formula]], in which three nodes are specified: | A special case of the [[Newton–Cotes quadrature formula|Newton–Cotes quadrature formula]], in which three nodes are specified: | ||
| − | + | $$ \tag{1 } | |
| + | \int\limits _ { a } ^ { b } f ( x) dx \cong \ | ||
| + | { | ||
| + | \frac{b - a }{6} | ||
| + | } | ||
| + | \left [ f ( a) + 4f \left ( { | ||
| + | \frac{a + b }{2} | ||
| + | } \right ) + f ( b) \right ] . | ||
| + | $$ | ||
| − | Let the interval | + | Let the interval $ [ a, b] $ |
| + | be broken up into an even number $ n $ | ||
| + | of subintervals $ [ x _ {i} , x _ {i + 1 } ] $, | ||
| + | $ i = 0 \dots n - 1 $, | ||
| + | of length $ h = ( b - a)/n $, | ||
| + | and calculate the integral over the interval $ [ x _ {2k} , x _ {2k + 2 } ] $ | ||
| + | by the quadrature formula (1): | ||
| − | + | $$ | |
| + | \int\limits _ {x _ {2k} } ^ { {x _ {2k} + 2 } } | ||
| + | f ( x) dx \cong { | ||
| + | \frac{h}{3} | ||
| + | } | ||
| + | [ f ( x _ {2k} ) + 4f ( x _ {2k + 1 } ) + f ( x _ {2k + 2 } )]. | ||
| + | $$ | ||
| − | Summation over | + | Summation over $ k $ |
| + | from 0 to $ ( n/2) - 1 $ | ||
| + | on both sides leads to the composite Simpson formula | ||
| − | + | $$ \tag{2 } | |
| + | \int\limits _ { a } ^ { b } f ( x) dx \cong | ||
| + | $$ | ||
| − | + | $$ | |
| + | \cong \ | ||
| + | { | ||
| + | \frac{h}{3} | ||
| + | } \{ f ( a) + f ( b) + 2 [ f ( x _ {2} ) + f ( x _ {4} ) + | ||
| + | \dots | ||
| + | + f ( x _ {n - 2 } )] + | ||
| + | $$ | ||
| − | + | $$ | |
| + | + | ||
| + | {} 4 [ f ( x _ {1} ) + | ||
| + | f ( x _ {3} ) + \dots + f ( x _ {n - 1 } )] \} , | ||
| + | $$ | ||
| − | where | + | where $ x _ {j} = a + jh $, |
| + | $ j = 0 \dots n $. | ||
| + | The quadrature formula (2) is also called Simpson's formula (that is, the word composite is dropped). The algebraic degree of accuracy of (2), and of (1), is equal to 3. | ||
| − | If the integrand | + | If the integrand $ f $ |
| + | has a continuous derivative of the fourth order on $ [ a, b] $, | ||
| + | then the error $ R ( f ) $ | ||
| + | of the quadrature formula (2) — the difference between the left-hand and right-hand members of the approximate equation (2) — can be written as | ||
| − | + | $$ | |
| + | R ( f ) = - { | ||
| + | \frac{b - a }{180} | ||
| + | } h ^ {4} f ^ { ( 4) } ( \xi ), | ||
| + | $$ | ||
| − | where | + | where $ \xi $ |
| + | is some point in the interval $ [ a, b] $. | ||
Simpson's formula was named after Th. Simpson, who obtained it in 1743, although the formula was already known, for example to J. Gregory, in 1668. | Simpson's formula was named after Th. Simpson, who obtained it in 1743, although the formula was already known, for example to J. Gregory, in 1668. | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
Latest revision as of 08:14, 6 June 2020
A special case of the Newton–Cotes quadrature formula, in which three nodes are specified:
$$ \tag{1 } \int\limits _ { a } ^ { b } f ( x) dx \cong \ { \frac{b - a }{6} } \left [ f ( a) + 4f \left ( { \frac{a + b }{2} } \right ) + f ( b) \right ] . $$
Let the interval $ [ a, b] $ be broken up into an even number $ n $ of subintervals $ [ x _ {i} , x _ {i + 1 } ] $, $ i = 0 \dots n - 1 $, of length $ h = ( b - a)/n $, and calculate the integral over the interval $ [ x _ {2k} , x _ {2k + 2 } ] $ by the quadrature formula (1):
$$ \int\limits _ {x _ {2k} } ^ { {x _ {2k} + 2 } } f ( x) dx \cong { \frac{h}{3} } [ f ( x _ {2k} ) + 4f ( x _ {2k + 1 } ) + f ( x _ {2k + 2 } )]. $$
Summation over $ k $ from 0 to $ ( n/2) - 1 $ on both sides leads to the composite Simpson formula
$$ \tag{2 } \int\limits _ { a } ^ { b } f ( x) dx \cong $$
$$ \cong \ { \frac{h}{3} } \{ f ( a) + f ( b) + 2 [ f ( x _ {2} ) + f ( x _ {4} ) + \dots + f ( x _ {n - 2 } )] + $$
$$ + {} 4 [ f ( x _ {1} ) + f ( x _ {3} ) + \dots + f ( x _ {n - 1 } )] \} , $$
where $ x _ {j} = a + jh $, $ j = 0 \dots n $. The quadrature formula (2) is also called Simpson's formula (that is, the word composite is dropped). The algebraic degree of accuracy of (2), and of (1), is equal to 3.
If the integrand $ f $ has a continuous derivative of the fourth order on $ [ a, b] $, then the error $ R ( f ) $ of the quadrature formula (2) — the difference between the left-hand and right-hand members of the approximate equation (2) — can be written as
$$ R ( f ) = - { \frac{b - a }{180} } h ^ {4} f ^ { ( 4) } ( \xi ), $$
where $ \xi $ is some point in the interval $ [ a, b] $.
Simpson's formula was named after Th. Simpson, who obtained it in 1743, although the formula was already known, for example to J. Gregory, in 1668.
Comments
Simpson's formula is also called Simpson's rule.
References
| [a1] | R. Courant, "Vorlesungen über Differential- und Integralrechnung" , 1 , Springer (1971) |
| [a2] | D.M. Young, R.T. Gregory, "A survey of numerical mathematics" , Dover, reprint (1988) pp. §7.4 |
Simpson formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simpson_formula&oldid=48712