Guss Qudrture Rule o Integrtion Computer Engineering Mjors Authors: Autr Kw, Chrlie Brker http://numerilmethods.eng.us.edu Trnsorming Numeril Methods Edution or STEM Undergrdutes /0/00 http://numerilmethods.eng.us.edu
Guss Qudrture Rule o Integrtion http://numerilmethods.eng.us.edu
Wht is Integrtion? Integrtion The proess o mesuring the re under urve. y )d ) I )d Where: ) is the integrnd lower limit o integrtion upper limit o integrtion http://numerilmethods.eng.us.edu
Two-Point Gussin Qudrture Rule 4 http://numerilmethods.eng.us.edu
http://numerilmethods.eng.us.edu 5 Bsis o the Gussin Qudrture Rule Previously, the Trpezoidl Rule ws developed y the method o undetermined oeiients. The result o tht development is summrized elow. ) ) ) ) ) d
Bsis o the Gussin Qudrture Rule The two-point Guss Qudrture Rule is n etension o the Trpezoidl Rule pproimtion where the rguments o the untion re not predetermined s nd ut s unknowns nd. In the two-point Guss Qudrture Rule, the integrl is pproimted s I )d ) ) 6 http://numerilmethods.eng.us.edu
http://numerilmethods.eng.us.edu 7 Bsis o the Gussin Qudrture Rule The our unknowns,, nd re ound y ssuming tht the ormul gives et results or integrting generl third order polynomil,. ) 0 Hene ) d )d 0 4 4 0 ) 4 4 4 0
http://numerilmethods.eng.us.edu 8 Bsis o the Gussin Qudrture Rule It ollows tht ) ) 0 0 )d Equting Equtions the two previous two epressions yield ) 4 4 4 0 ) ) 0 0 ) ) ) ) 0
http://numerilmethods.eng.us.edu 9 Bsis o the Gussin Qudrture Rule Sine the onstnts 0,,, re ritrry 4 4 4
http://numerilmethods.eng.us.edu 0 Bsis o Guss Qudrture The previous our simultneous nonliner Equtions hve only one eptle solution,
http://numerilmethods.eng.us.edu Bsis o Guss Qudrture Hene Two-Point Gussin Qudrture Rule ) ) ) d
Higher Point Gussin Qudrture Formuls http://numerilmethods.eng.us.edu
http://numerilmethods.eng.us.edu Higher Point Gussin Qudrture Formuls ) ) ) ) d is lled the three-point Guss Qudrture Rule. The oeiients,, nd, nd the untionl rguments,, nd re lulted y ssuming the ormul gives et epressions or ) d 5 5 4 4 0 Generl n-point rules would pproimte the integrl )....... ) ) )d n n integrting ith order polynomil
Arguments nd Weighing Ftors or n-point Guss Qudrture Formuls In hndooks, oeiients nd rguments given or n-point Guss Qudrture Rule re given or integrls g )d n i s shown in Tle. i g i ) Tle : Weighting tors nd untion rguments used in Guss Qudrture Formuls. Points Weighting Ftors.000000000.000000000 0.555555556 0.888888889 0.555555556 4 0.47854845 0.654555 0.654555 4 0.47854845 Funtion Arguments -0.5775069 0.5775069-0.774596669 0.000000000 0.774596669-0.866-0.998044 0.998044 4 0.866 4 http://numerilmethods.eng.us.edu
Arguments nd Weighing Ftors or n-point Guss Qudrture Formuls Tle ont.) : Weighting tors nd untion rguments used in Guss Qudrture Formuls. Points Weighting Ftors 5 0.696885 0.47868670 0.568888889 4 0.47868670 5 0.696885 6 0.7449 0.607657 0.467995 4 0.467995 5 0.607657 6 0.7449 Funtion Arguments -0.90679846-0.584690 0.000000000 4 0.584690 5 0.90679846-0.946954-0.660986-0.869860 4 0.869860 5 0.660986 6 0.946954 5 http://numerilmethods.eng.us.edu
Arguments nd Weighing Ftors or n-point Guss Qudrture So i the tle is given or g Formuls )d integrls, how does one solve )d? The nswer lies in tht ny integrl with limits o [, ] n e onverted into n integrl with limits [,] mt Let I, then t I then t, Suh tht: 6 m http://numerilmethods.eng.us.edu
Arguments nd Weighing Ftors or n-point Guss Qudrture Formuls Then Hene t d dt Sustituting our vlues o, nd d into the integrl gives us ) d t dt 7 http://numerilmethods.eng.us.edu
Emple For n integrl Rule. )d, derive the one-point Gussin Qudrture Solution The one-point Gussin Qudrture Rule is )d ) 8 http://numerilmethods.eng.us.edu
Solution The two unknowns, nd re ound y ssuming tht the ormul gives et results or integrting generl irst order polynomil, ) 0. ) d ) 0 d 0 0 ) 9 http://numerilmethods.eng.us.edu
Solution It ollows tht ) ) d Equting Equtions, the two previous two epressions yield 0 ) 0 ) 0 ) ) 0 0 http://numerilmethods.eng.us.edu
Bsis o the Gussin Qudrture Rule Sine the onstnts 0, nd re ritrry giving http://numerilmethods.eng.us.edu
Solution Hene One-Point Gussin Qudrture Rule ) d ) ) http://numerilmethods.eng.us.edu
Emple Humn vision hs the remrkle ility to iner D shpes rom D imges. The intriguing question is: n we replite some o these ilities on omputer? Yes, it n e done nd to do this, integrtion o vetor ields is required. The ollowing integrl needs to integrted. 00 I ) d where ) 0, 0 < < 0 0 6 9688. 0 0 7 0, 7 < < 00. 796 0. 8487 0 Use two-point Guss Qudrture Rule to ind the vlue o the integrl. Also, ind the solute reltive true error. 9. 6778, http://numerilmethods.eng.us.edu
Solution First, hnge the limits o integrtion rom [0,00] to [-,] y previous reltions s ollows 00 00 0 00 0 00 0 ) d d 0 50 50) 50 d 4 http://numerilmethods.eng.us.edu
Solution ont). Net, get weighting tors nd untion rgument vlues rom Tle or the two point rule,..0000 0. 5775.0000 0.5775 5 http://numerilmethods.eng.us.edu
Solution ont.) Now we n use the Guss Qudrture ormul 50 50 50 50) [ 50 50) 50 50) ] d [ 50 0.5775) 50) 50 0.5775) 50) ] [.) 78.868) ] [ 0) 0.049) ] 50 50 50 5.460 6 http://numerilmethods.eng.us.edu
Solution ont) sine.) 0 6 78.868) 9.688 0 78.868).796 0 78.868).8487 0 78.868) 9. 6778 0.049 7 http://numerilmethods.eng.us.edu
Solution ont) ) True error is True Vlue Approimte E t 60.79 55.546 5.460 Vlue ) The solute reltive true error,, is Et vlue 60.79) t 60.79 5.460 60.79 t 00% 9.7% 8 http://numerilmethods.eng.us.edu
Additionl Resoures For ll resoures on this topi suh s digitl udiovisul letures, primers, tetook hpters, multiple-hoie tests, worksheets in MATLAB, MATHEMATICA, MthCd nd MAPLE, logs, relted physil prolems, plese visit http://numerilmethods.eng.us.edu/topis/guss_qu drture.html
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