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### numerical_integration_2

Course: PH 4390, Fall 2008
School: Mich Tech
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Word Count: 1962

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Integration Romberg Numerical Integration cont The two indices within the Romberg scheme describe: the number of segments used m #evaluation points as 2 within the numerical integration the order of combinations of neighboring approximations for the integral Tm, l order of approximation A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU...

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Integration Romberg Numerical Integration cont The two indices within the Romberg scheme describe: the number of segments used m #evaluation points as 2 within the numerical integration the order of combinations of neighboring approximations for the integral Tm, l order of approximation A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.1/13 Numerical Integration Romberg Integration cont T2,0 = h f0 2 + fN 2 +f1 + f2 + f3 h2 /h4 /h6 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/13 Numerical Integration Romberg Integration cont T2,0 = h f0 2 + fN 2 +f1 + f2 + f3 h2 /h4 /h6 T3,0 = h 2 f0 2 + fN 2 +f1 + f2 + f3 +f5 + f6 + f7 + f8 h2 h4 h6 4 / 16 / 64 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/13 Numerical Integration Romberg Integration cont T2,0 = h f0 2 + fN 2 +f1 + f2 + f3 h2 /h4 /h6 T3,0 = T2,0 2 + h 2 + f5 + f6 +f7 + f8 h2 h4 h6 / / 4 16 64 1.) only a few additional terms ! A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/13 Numerical Integration Romberg Integration cont T2,0 = h f0 2 + fN 2 +f1 + f2 + f3 h2 /h4 /h6 T3,0 = T2,0 2 + h 2 + f5 + f6 +f7 + f8 h2 4 h4 h6 / 16 / 64 2.) comparing errors A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/13 Numerical Integration Romberg Integration cont T2,0 T3,0 = h2 /h4 /h6 T2,0 2 + h 2 + f5 + f6 +f7 + f8 T4,0 = T3,0 2 h2 4 h4 h6 / 16 / 64 h 4 + + f9 + f10 +f11 + f12 + f13 +f14 + f15 + f16 h2 16 h4 h6 / 256 / 4096 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/13 Numerical Integration Romberg Integration cont T3,1 = 4T3,0 T2,0 3 h4 5h6 4 / 16 4T4,0 T3,0 3 h4 5h6 64 / 1024 T4,1 = A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/13 Numerical Integration Romberg Integration cont T4,2 = 16T4,1 T3,1 15 h6 64 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.2/13 Numerical Integration Romberg Integration cont Computing Sequence T2,0 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T2,0 T3,0 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T3,1 T2,0 T3,0 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T3,1 ? T2,0 T3,0 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T3,1 T2,0 T3,0 T4,0 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T3,1 T4,1 T2,0 T3,0 T4,0 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T4,2 T3,1 T4,1 T2,0 T3,0 T4,0 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T4,2 ? T3,1 T4,1 T2,0 T3,0 T4,0 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T4,2 T3,1 T4,1 T2,0 T3,0 T4,0 T5,0 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T4,2 T3,1 T4,1 T5,1 T2,0 T3,0 T4,0 T5,0 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T4,2 T5,2 T3,1 T4,1 T5,1 T2,0 T3,0 T4,0 T5,0 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T4,2 T5,2 T3,1 T4,1 T5,1 T2,0 T3,0 T4,0 T5,0 T5,3 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T4,2 T5,2 T3,1 T4,1 T5,1 T2,0 T3,0 T4,0 T5,0 T5,3 ? A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T4,2 T5,2 T3,1 T4,1 T5,1 T2,0 T3,0 T4,0 T5,0 T5,3 ? For a given m we can get a number of m 1 improvements starting with 0, the bare numerical integration. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T4,2 T5,2 T3,1 T4,1 T5,1 T2,0 T3,0 T4,0 T5,0 T5,3 ? At the endpoint (bottom line, leftmost data point) we have to decide ? whether the remaining error h2m2 4m1 is small enough to abort the series. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Computing Sequence T4,2 T5,2 T3,1 T4,1 T5,1 T2,0 T3,0 T4,0 T5,0 T5,3 ? If we need to continue, we need to increment the value m do one more integral Tm,0 and derive all other Tm,l values towards the left end by combination with items from the previous line, Tm,l+1 = 4l+1 Tm,l Tm1,l 4l+1 1 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.3/13 Numerical Integration Romberg Integration cont Successive Tm,0 Calculation 1 m=0 Incr = ba 20 f (x0 ) 2 T0,0 = Incr + f (xN ) 2 A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.4/13 Numerical Integration Romberg Integration cont Successive Tm,0 Calculation 2 m=1 Incr = T1,0 = Incr 2 ba 21 i f (xi ) = T0,0 2 + Incr = T0,0 2 Do i = 1, 21 , 2 xi = x0 + (i Incr) + Incr f (xi ) Enddo A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.4/13 Numerical Integration Romberg Integration cont Successive Tm,0 Calculation 3 m=2 Incr = T2,0 = Incr 2 ba (called h in graphical schemes) 22 i f (xi ) = T1,0 2 + Incr = T1,0 2 Do i = 1, 22 , 2 xi = x0 + (i Incr) + Incr f (xi ) Enddo A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.4/13 Numerical Integration Romberg Integration cont Successive Tm,0 Calculation 4 m=3 Incr = T3,0 = Incr 2 ba 23 i f (xi ) = T2,0 2 + Incr = T2,0 2 Do i = 1, 23 , 2 xi = x0 + (i Incr) + Incr f (xi ) Enddo A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.4/13 Numerical Integration Romberg Integration cont Programming scheme for the Romberg integration rule MROMBG INITIALIZE T(), TOL, ERR M=1 CALL ROMBG(T,M,TOL..) ITER(TOL,ERR..) M = M+1 CALL CLCTM0(T,M...) CALL LNFLLT(T,M...) CALC ERR CALL ROMBG(T,M,TOL...) END CLCTM0(T,M,...) END T(M,0) = 0.5 T(M1,0) + PRINT RESULT END ROMBG(T,M,TOL..) IF ( ERR > TOL ) THEN CALL ITER(T,M,TOL..) ELSE DONE END IF END f_i LNFLLT(T,M,...) DO L=0, M1 T(M+1,L) = [C1*T(M,L)T(M1,L)]/C2 ENDDO END F(X) F(X) = SIN(X) END A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.5/13 Numerical Integration Romberg Integration cont Although this Romberg scheme seems to work ne, we need to be careful with special cases. Consider for example A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.6/13 Numerical Integration Romberg Integration cont If we are interested in the integral 1 1 1 x2 dx then we run into troubles when using the Romberg scheme... None of the approximations within the Romberg scheme are converging rapidly. The reason for this can be explained again from looking into the Euler-McClaurin formula. The error is expressed there in terms of derivatives evaluated only at the endpoints ! But exactly at the endpoints the derivatives become innite in this special case ! A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.7/13 Numerical Integration Romberg Integration cont We recognize the problem already in the difference between actual and theoretical error during individual Romberg iterations. A possible cure to such problems can simply be changing to different variables. If we substitute for example, x = cos we obtain a different integral, 1 1 1 x2 dx = 0 sin2 () d which is a perfect task for Romberg integration. A First Course in Computational Physics, PL Vries, Chp 4, ISBN 0-471548689-3 PH4390: Computational Methods in Physics, MTU Fall 2006 p.8/13 Numerical Integration Improper Integrals Integrals of the form 0 f (x) dx are common in physics and called improper integrals. Some...

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