Chapter 2. Numerical Integration also called quadrature. 2.2 Trapezoidal Rule. 2.1 A basic principle Extending the Trapezoidal Rule DRAWINGS

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1 S Cpter Numericl Integrtion lso clled qudrture Te gol of numericl integrtion is to pproximte numericlly. f(x)dx Tis is useful for difficult integrls like sin(x) ; sin(x ); x + x 4 Or worse still for multiple-dimensionl integrls were multi could be or 0 or 0 6 etc.. Trpezoidl Rule Approximte te function between nd b by line segment ie f(x) cx re under line segment re of trpezoidl re of trpezoidl bse * eigt * f()+f(b) f() + f(b) Wic gives us te Trpezoidl Rule. Wt did we miss? f(x)dx f() + f(b) f(x)dx b f() + f(b).. Extending te Trpezoidl Rule. A bsic principle If we cnnot do f(x)dx, we pproximte f(x) wit function we cn integrte. (usully by polynomil ie f(x) x + bx + cx +...) Wen we integrte function we clculte te re below te curve. Before we took one gint step cross te intervl we now brek tis into n smll steps of size, were b n Ten we pply te trpezoidl rule t ec step 9 0

2 f(x) pproximted by series of polynomils - one for ec step. Apply trpezoidl rule to ec segment nd dd T but we ve trp. re (y 0 + y ) + (y + y ) + (y + y ) (y n + y n ) ( y 0 + y + y + y y n + y n) y 0 f(); y f(x ); y f(x );... y n f(b) And we now ve te extend trpezoidl rule ( f() + f(x ) + f(x ) + f(x ) f(x n ) + f(b)).. Error estimtes In ny subintervl, sy, x k f(x)dx pproximte by trpezoidl rule ie f(x)dx T k f() + f(x k ) Q: wts te size of te error on tis intervl? f(x)dx T k + err(x) f(x) ws pproximted by polynomil f(x) x +. We cn write f(x) s Tylor expnsion bout nerby pt. x k let x, x k nd if x x k (x k ) (x k x k ) Te pproximting polynomil is of degree x + it ccurtely represents f(x) up to te first derivtive but not beyond: d (x + ) dx 0 cnnot know f (x) f(x) f(x k nd te error strts t te next term ) + (x x k error (x x k! ) )f (x k ) f (x k ) We cnnot know f (x) so sy mx{f (x) x, x k } nd write error (x x k! ) So, trpezoidl rule fils to integrte term (x x k ) M! k Do te integrtion nd compre results from trpezoid nd true integrtion (x x k! ) dx (x x k.! (xk x k ) ( x k.!.! ( ).! + ( ).! ) x k ).4.! f(x) f(x k + (x x k ) ) + (x x k! f (x k )f (x k ) ) +...

3 Now we integrte te error by pplying te trpezoidl rule: pproximte f(x) wit polynomil of degree two (x x k ) M! k dx (xk x k ) + ( x k )!! ) + )!! 4.! Terefore te error mde by pplying Trpezoidl Rule over te intervl, x k is Error from Trp Rule True Error 4.!.4.! Now, for N subintervls te totl error is no of steps error t ec step N N (b ) M N k (b ) f N Te error formul tells us tt if we double N (number of steps) te error decreses by fctor of 4 ie N Useful to know. Sometimes you re given trget ccurcy nd rnge. You decide te stepsize, using te error formul. ie prbol. Ax + Bx + C Any noncolliner point in te plce cn be fitted wit prbol. Tus Simpson s Rule: pproximte curves wit prbols From tis we get te re of te sded region A p (y 0 + 4y + y ) Eg. pplying tis formul from x to x b we get.. Deriving A p Simplifying te previous plot f(x)dx + b (f() + 4f( ) + f(b)) vspce in Are under y Ax + Bx + C for x to is A p (Ax + Bx + C)dx Ax + frcbx + Cx A + C (A + 6C) We lso know te curve psses troug points. Simpson s Rule (, y 0 ); (0, y ); (, y ) y 0 A B + c; y C; y A + B + C Consider f(x)dx 4

4 C y A B y 0 y A + B y y A y 0 + y y expressing A p in terms of y 0, y, y From tis we get te Extended Simpson s Rule S S (y 0 + 4y + y + 4y + y y n + y n ) A p (A + 6C) ((y 0 + y y ) + 6y ) And we now ve Simpson s rule. x+ x A p ((y 0 + 4y + y ) f(x)dx (f(x ) + 4f(x) + f(x + )) Note: te re clculted, for ec subintervl is of widt... Extended Simpson s Rule We extend te formul for n subintervls. n must be even to ve ec subintervl of widt... Error of te Simpson s Rule degree Exct Simpson Rule (f() + 4f(+b ) + f(b)) 0 dx 0.5( + 4() + ) 0 xdx (0 + 4(0.5) + ) x dx 0.5 (0 + 4(0.5) + ) 0 x dx (0 + 4(0.5) + ) x4 dx (04 + 4(0.5) ) 5 4 We get n exct nswer for ny f(x) up to degree ie up to x. From te Tylor expnsion l Trpezoid rule t x, x k+. error (x x k) 4 f (4) (x) 4! Clculte ec re nd sum Let S denote ns from Simpson s rule S (y 0 + 4y + y ) + (y + 4y + y 4 ) (y n + 4y n + y n ) We now proceed s in similr fsion to te te Trpezoidl cse, to find te error. We integrte te error term over subintervls of size error (x x k) 4 f (4) (x), 4! mx{f(x) x, x k+ } 5 6

5 + (x x k ) 4 nd by Simpson s Rule dx (x x k) 5 4! 5.4! x k+ (xk+ x k ) 5 ( x k ) 5 5.4! 5.4! 5 5.4! ! 5 5.4! xk+ (x x k ) 4 dx M (xk x k ) 4 4! k 4! +4 (x k x k ) ! 4! 5 M.4! k So te error for te Simpson rule is + (x k+ x k ) 4 4! 4! 5.4! 5 5.4! 5 90 For lengt. For step of size error Terefore te error for te extended rule for N steps is N N 5 80 (b )5 N 5 80 (b )5 N 4 80 Terefore if f double N te error decreses by fctor 4 6. Tis sows tt Simpson rule is considerbly more ccurte tn Trpezoidl..4 Polynomils of low degree If f(x) is polynomil of degree less tn 4 fourt derivtive0 f (4) (x) 80 Simpson s error (b )5 N 4 (b ) 5 0(x) 0 N 4 80 Terefore no error in te Simpson s pprox of f(x)dx ie if f(x) is constnt ; liner x; qudrtic x ; cubic x. Simpson s rule give n exct nswer for f(x)dx weter te # subdivisions..5 Summry.5. Trpezoidl Rule Te Trpezoidl Rule nd te extended rule f(x)dx f() + f(b) f(x)dx f() + f(x ) + f(x ) + f(x ) f(x n ) + f(b) Te error or in oter words: te error is O( ) 7 8

6 .5. Simpson s Rule Simpson s Rule nd te extended rule f(x)dx + b f() + 4f( ) + f(b) f(x)dx f() + 4f(x ) + f(x ) + 4f(x ) +f(x 4 ) f(x n ) + f(x n ) nd te error in ec step is O( 5 ) ie error