subintervals as E d2f si) It is easy to see that if f(a)is a continuous function, 'M(being the mean)must be bounded by the maximum and the minumum of f (a) on the interval [0, 1 and hence, there must exist some S E [0, 1]such that M=d2f(s)/dx2. Hence we obtain the estimate E h3d2f(5)1d2f(5) 24n2d since h= 1/n. This error estimate tells us that the scheme is again exact for constants and linear functions on the domain (no higher order polynomials and, for a smooth function, the error decays algebraically 3.3 General Quadrature Scheme 3.3.1 General 1D Form SLIDE 8 ∫(x)d ∫(x1) Free to pick the evaluation points Free to pick the weights fe An n-point formula has 2n degrees of freedom! After all the hard work we did dividing the domian into subintervals, we realize that we cannot even integrate a parabola exactly on the domain. There must e son we can back and look at the general form of the quadrature approximation sche Al approximating an integral by a weighted sum of function evaluations as shown n this slide. So far we have been choosing these weights as the subinterval lengths. We have also been choosing all the evaluation points. The weights are alizing factors which ensure that the ximation is exact if f(a)=l and the equality of areas of trapezoids and rectangles that we discussed gives us the extra polynomial accuracy of being able to obtain the area under straing line exactly. So, what would happen if ere to choose both the integration points and the weights intelligently? For an n-point formula we have➻➷➃➻✑❅❇❡✑♣✺✾❭❊❑●❖❀✗◆❖❨✻❍✮❴❁❅❄❍❋❅ ● ✽ ✙ ➷✍★■❈ ✗❊❉ ￾ ✴ ➷ ✫✽ ✃✿ ✢ ✫ ✯ ✒ ✕❋✷✗✃✒✗ ✫❋Ï✯✂✁ ❈ ❉✝❊ ❋ ✷ ◆❇❖ ❖✹❒✥❮ ✃☎✄✝✆✞✠✟ ❤✫●➃✾❁❅✿❀✗❍✮❅❖❛❻●❖❉❈❅❖❀✓❀⑨●❖P✭❍✻●✿✾❭❵ ✒ ✕✥Ï✘✗✣✾❆❅➃❍✎❂✓❉✮❊❑●❖✾❁❊✙❡✑❉✮❡✭❅✴❵r❡✺❊✺❂➧●■✾❭❉❋❊➣➚ ✙☛✡➻✳❡r♣✰❀✓✾❁❊✑❩❻●❖P✺❀❄❜➫❀✵❍❏❊❵❢✞❜➭❡✺❅q● ♣✰❀❦♣✭❉❋❡✑❊✺❣❲❀✵❣➩♣✙❛❢●■P✑❀❦❜❦❍✻Ñ❲✾❭❜➭❡✑❜ ❍✮❊✺❣✩●■P✑❀❦❜❝✾❭❊✙❡✑❜➭❡✑❜ ❉✮❵ ✒ ✕✥Ï✘✗✎❉✮❊✷●❖P✺❀❦✾❭❊❑●❖❀✗◆❖❨✻❍✮❴☞✱✲ ✦❳✴✝✶ ❍❏❊✭❣♥P✑❀✓❊✺❂✓❀✮➚✰●❖P✺❀✓◆■❀③❜➵❡✺❅❇●➉❀✓Ñ❲✾❁❅❇●✎❅❖❉✮❜❝❀✜✷ ✯✏✱✲✳✦❳✴❳✶✞❅❖❡✺❂◗P❢●❖P✭❍✻● ✡ ✙✡✫✯ ✒ ✕❋✷❜✗✌☞❘✫❋Ï✯ ✈✌✦▲❀✗❊✺❂✠❀ ❃⑨❀✎❉❋♣❲●■❍✮✾❭❊♦●❖P✺❀❈❀✗❅❇●❖✾❁❜❝❍❏●❖❀ ● ✽ ✙ ➷✍★☛❈ ✗❊❉ ✫ ✯ ✒ ✕❋✷❜✗ ✫❋Ï✯ ✙ ✴ ✗❊❉➷ ✯ ✫ ✯ ✒ ✕❋✷❜✗ ✫❋Ï✯ ❅❖✾❭❊✺❂✓❀ ★✔✙ ✴✍☞✻➷➃✈③➾✉P✑✾❁❅❻❀✗◆❖◆■❉✮◆➉❀✗❅❇●❖✾❁❜❦❍✻●❖❀➭●❖❀✗❴❭❴❆❅❻❡✺❅➉●❖P✭❍✻●❻●❖P✑❀❦❅■❂◗P✑❀✓❜❝❀➫✾❁❅❻❍✮❩❋❍❏✾❁❊➩❀✓Ñ✑❍✮❂➧●✎❵r❉✮◆ ❂✠❉❋❊✺❅❇●■❍❏❊❑●◗❅➭❍✮❊✺❣➆❴❁✾❭❊✺❀✗❍❏◆➭❵r❡✑❊✺❂✠●❖✾❁❉✮❊✺❅➫❉❋❊➆●❖P✺❀♥❣✑❉✮❜❦❍❏✾❁❊✥❡r❊✺❉➤P✺✾❭❩❋P✑❀✓◆➫❉✮◆◗❣❲❀✓◆➫▼✰❉✮❴❁❛❑❊✺❉✮❜❝✾❁❍✮❴❁❅✏✎ ❢ ❍❏❊✭❣✧➚✙❵r❉✮◆▲❍❝❅❖❜➫❉✙❉✮●❖P☞❵r❡✑❊✺❂✠●❖✾❁❉✮❊➣➚❲●■P✑❀❈❀✓◆■◆■❉✮◆✉❣❲❀✗❂✗❍✛❛✙❅❄✍✺✼✥✶✰✖✟✣✬✻✍✑✢✥✘❋✍✑✼✥✼✥✒ → ✌✉❾✒✌ ✑✆✴➈✆➣➒✺➁✮✄ ✞➇▲➁✡✠❬➒✑➁✣➋❲➇❬➒✆✎￾✉➊✭➓✆✍✁✎✆ ✍✴→ ✍✴→✥↔ ❘✖✙✜✣✖❋✬✻✍✑✼ ↔✎ ★✴✪✭✬✻✯ ➙✰➛❑➜➞➝✴➟✓✒ ✛✢ ✣ ✒ ✕rÏ ✗✬✫❋Ï❂✔ ✫✽ ✃✿ ✢✕✔ ✃ ❈❉✝❊❋ ✖ ❏ ✃✘✗ ❮✗❒ ✒ ✕✥Ï✭✃✹✗ ❈ ❉❊ ❋ ✙▲✝◆❇❖ P◆➧❒ ✃ ●✽✛✚● ✃✽❒ ✘✑◆■❀✓❀✎●■❉➫▼✺✾❁❂◗➴✆●❖P✑❀③✖✄✑✍✑✼✸✦✞✍❲✔✵✢✥✪✰✜ ✱❫✪✭✢✸✜✹✔✗✏❋✈ ✘✑◆■❀✓❀✎●■❉➫▼✺✾❁❂◗➴✆●❖P✑❀ ✄➵✖✙✢✸✶✭✚✹✔✵✏✎❵r❉✮◆❄❀✗❍❋❂◗P✆▼✰❉✮✾❁❊❑●✗✈ ✆✆✜❚✜✥✤❖✱➍✪✭✢✸✜✹✔➫★✫✪✭✬✛✯➤✦✣✼✥✍➩✚✣✍✑✏ ➔ ✜➑❱✞✖✙✶✭✬✛✖✙✖✙✏❝✪✭★❬★✸✬✻✖✙✖✙❱✞✪✰✯✢✜ ➡➤➢✧➥➧➦✤✣ ♠❵➞●■❀✓◆✉❍❏❴❁❴✺●■P✑❀❬P✭❍❏◆◗❣❝❃➍❉❋◆❖➴➫❃➍❀➉❣❲✾❁❣♦❣❲✾❁❨❑✾❆❣❲✾❁❊✑❩③●■P✑❀➉❣✑❉✮❜❝✾❁❍✮❊❦✾❭❊❑●■❉➭❅❖❡✑♣✑✾❁❊❋●■❀✓◆■❨✻❍❏❴❆❅✓➚❑❃➍❀❬◆❖❀✵❍❏❴❁✾❭➲✗❀ ●❖P✭❍✻●❻❃➍❀❝❂✓❍✮❊✑❊✑❉❏●❻❀✓❨❋❀✓❊✩✾❁❊❑●❖❀✗❩✮◆◗❍✻●❖❀➫❍✆▼✭❍❏◆◗❍❏♣✰❉✮❴❆❍❦❀✠Ñ✑❍✮❂✠●❖❴❁❛❢❉✮❊➩●❖P✑❀❝❣❲❉✮❜❦❍✮✾❭❊➣✈③➾✉P✺❀✓◆■❀➭❜➭❡✺❅q● ♣✰❀➵❅❇❉❋❜❝❀✠●❖P✺✾❭❊✑❩❯●❖P✺❍❏●❬❃➍❀➭❂✓❍❏❊✩❣✑❉❦●❖❉❯✾❁❜❝▼✑◆■❉✻❨✮❀✎●❖P✑✾❆❅✎❅❖❂◗P✑❀✗❜❝❀✮✈✉➸✷❀➵❩✮❉❯♣✺❍❋❂◗➴♦❍✮❊✺❣♥❴❭❉✙❉❋➴♦❍✻● ●❖P✺❀➵❩✮❀✗❊✑❀✓◆◗❍❏❴➣❵r❉✮◆■❜ ❉✮❵✣●■P✑❀➭t❑❡✺❍❋❣❲◆◗❍✻●❖❡✺◆❖❀③❍✮▼✑▼✑◆■❉✛Ñ✙✾❁❜❦❍✻●■✾❭❉❋❊☞❅■❂◗P✑❀✓❜❝❀❋✈ ♠❴❁❴✞❃⑨❀➵❍❏◆■❀③❣❲❉❋✾❭❊✑❩✆✾❁❅ ❍❏▼✺▼✑◆❖❉✛Ñ❲✾❁❜❦❍✻●❖✾❁❊✑❩❝❍❏❊♥✾❁❊❋●■❀✓❩❋◆■❍✮❴➣♣✙❛♦❍❝❃⑨❀✓✾❁❩✮P❑●❖❀✵❣☞❅❖❡✑❜ ❉❏❵✳❵r❡✑❊✺❂✠●❖✾❁❉✮❊❢❀✓❨✻❍✮❴❭❡✺❍❏●❖✾❁❉✮❊✺❅❄❍❋❅▲❅❖P✑❉✻❃❄❊ ✾❁❊ ●■P✑✾❁❅✆❅❇❴❁✾❁❣✑❀✮✈ s❉➤❵✥❍❏◆❦❃⑨❀☞P✭❍✛❨✮❀☞♣✭❀✗❀✓❊➑❂◗P✑❉✙❉❋❅❖✾❭❊✑❩✷●❖P✺❀✗❅❖❀☞❃⑨❀✓✾❁❩✮P❑●◗❅❦❍✮❅➭●❖P✑❀✩❅❖❡✑♣✑✾❁❊❑●❖❀✗◆❖❨✻❍❏❴ ❴❁❀✓❊✑❩✮●❖P✺❅✗✈✳➸✷❀❈P✺❍✛❨❋❀✎❍✮❴❁❅❖❉➫♣✭❀✗❀✓❊♥❂◗P✑❉✙❉❋❅❖✾❭❊✑❩❝❍❏❴❁❴✹●❖P✺❀❈❀✓❨✻❍❏❴❁❡✺❍✻●■✾❭❉❋❊✆▼✰❉✮✾❁❊❑●■❅✗✈✿➾✉P✑❀❻❃➍❀✗✾❭❩❋P❋●◗❅✉❍❏◆■❀ ✏q❡✺❅❇●③❅❇❉❋❜➫❀➫❊✑❉❋◆❖❜❦❍❏❴❁✾❁➲✓✾❁❊✑❩✆❵✥❍❋❂➧●■❉✮◆◗❅❬❃❄P✑✾❆❂◗P➤❀✓❊✺❅❖❡✑◆■❀➵●■P✺❍✻●❻●❖P✺❀❝❍✮▼✑▼✑◆■❉✛Ñ❲✾❭❜❦❍✻●■✾❭❉❋❊✩✾❆❅➉❀✓Ñ✑❍✮❂➧●❈✾❭❵ ✒ ✕rÏ ✗✚✙❛✴⑨❍❏❊✺❣③●❖P✑❀⑨❀✗t❑❡✺❍✮❴❭✾❭●q❛❻❉❏❵✺❍✮◆❖❀✵❍✮❅➣❉✮❵❲●❖◆◗❍❏▼✰❀✓➲✗❉✮✾❆❣✑❅✞❍✮❊✺❣❈◆■❀✗❂✠●■❍❏❊✺❩✮❴❁❀✗❅✴●❖P✺❍❏●✞❃⑨❀⑨❣❲✾❆❅■❂✠❡✺❅■❅❇❀✵❣ ❩✮✾❁❨✮❀✵❅▲❡✺❅➉●■P✑❀➭❀✓Ñ✙●❖◆◗❍❯▼✰❉✮❴❁❛❑❊✺❉✮❜❝✾❁❍✮❴✣❍❋❂✓❂✓❡✑◆■❍❋❂✠❛♦❉❏❵❫♣✰❀✓✾❁❊✑❩♥❍❏♣✑❴❁❀③●■❉♦❉❋♣❲●■❍✮✾❭❊❢●■P✑❀❝❍❏◆■❀✗❍❦❡✑❊✭❣❲❀✓◆ ❍❢❅❇●❖◆◗❍❏✾❁❊✑❩❢❴❭✾❁❊✑❀❯❀✓Ñ✑❍✮❂➧●■❴❭❛❋✈ s ❉✺➚✳❃❄P✺❍✻●③❃⑨❉✮❡✺❴❁❣⑥P✺❍✮▼✑▼✰❀✓❊✲✾❭❵✉❃⑨❀❯❃⑨❀✓◆■❀❝●❖❉➩❂◗P✺❉❑❉❑❅❇❀❝♣✰❉❏●■P✲●❖P✑❀ ✾❁❊❋●■❀✓❩❋◆■❍❏●❖✾❁❉✮❊③▼✰❉✮✾❁❊❑●■❅➃❍❏❊✭❣③●■P✑❀✉❃➍❀✗✾❭❩❋P❋●◗❅✣✾❭❊❑●❖❀✗❴❭❴❁✾❁❩✮❀✓❊❑●■❴❭❛✛✒✘✑❉❋◆✳❍✮❊➵➷✹➹✫▼✭❉❋✾❭❊❑●✳❵r❉✮◆■❜➵❡✺❴❁❍➉❃➍❀⑨P✺❍✛❨❋❀ ✥