is refined to 2( subintervals, the minimum eigenvalue(plotted in red) drops A0.003, and with 4 subintervals the minimum eigenvalue(plotted in green) drops to N 0.000*. Examining this data suggests that as the discretization refined, the generated matrix more accurately reflects the operator K, and therefore the matrix is becoming closer to being singular As the discretization is refined, the matrix is larger and has more eigenvalues Notice that as the discretization is refined from n=10 to n=_0 to n=40, all he additional eigenvalues are closer to zero 3.0 Example Problems 3.7.1 Intuition About the eigenvalues SLIDE 19 As the discretization is refined, go(a) becomes better approximated 十十十 H十十十}十十十十H十 As the discretization is refined, Ks null space can be more accurately represented o give a different view of why refining the discretization for the first kind equation produces a matrix with more and more smaller eigenvalues, consider the plots in the slide above. In the top plot, one of the basis functions is plotted for a coarse discretization. In the bot tom plot, one of the basis functions is plot ted for a finer discret ization. What is not able about these two plots is that the discretization is refined, these basis functions look progressively more like he spike function mentioned above. And since the spike function is in the nul space of K, one would expect that finer discretizations would generate"spikier basis functions whose associated eigenvalues would be near zero 3.7.2 Second kind Equation has Fewer Problems SLIDE 20 Second Kind equation (I+K)a≡a(x) x-x(x)ds→(I+K)a=重 )≠(I+K)a❁⑤❣✿❇P✾✔✚❆❃★✾✜❅☞❏P●✆ç ✎ ❣P❍★❊★❁❄❃✏❏P✾✹❇P♦✦❈❋▼⑤❣✹Ü✰❏Pt★✾➃❀❂❁❄❃★❁❄❀➃❍★❀ ✾✹❁◆❧❥✾❲❃◗♦✦❈❋▼❄❍★✾☎✖➯♣★▼❄●❋❏P❏P✾✜❅❪❁◆❃s❇r✾✹❅✘✪❅★❇P●❥♣❆❣✪❏P● ✤ ✵ ✓ ✵✼✵✁￾❆Ü✻❈❋❃❆❅➥Ý✿❁◆❏Pt ❴✎ ❣P❍★❊★❁❄❃✏❏P✾❲❇r♦✦❈❋▼⑤❣á❏Pt★✾✆❀❂❁❄❃★❁❄❀➃❍★❀ ✾❲❁❄❧■✾✹❃◗♦②❈■▼◆❍❆✾❇✖✳♣★▼❄●❋❏✐❏r✾✹❅➥❁❄❃❬❧■❇r✾❲✾❲❃✒✘ ❅✰❇r●■♣❆❣✇❏P● ✤ ✵ ✓ ✵✼✵❆✵✄✂★ã ❖êå✰❈■❀q❁◆❃★❁❄❃★❧➎❏rt★❁❄❣☞❅★❈❋❏r❈❬❣P❍★❧■❧❥✾✹❣✐❏r❣❂❏Pt✤❈✦❏☞❈❥❣q❏Pt❆✾Ú❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑✹❈✦❏r❁◆●❥❃ ❁⑤❣❼❇P✾✔✚❆❃★✾✜❅✮Ü✔❏rt★✾✇❧■✾✹❃★✾❲❇❤❈✦❏r✾✹❅➥❀✇❈✦❏r❇P❁◆åÚ❀q●■❇r✾❦❈■❱✹❱✠❍★❇❤❈✦❏r✾❲▼❄⑦❚❇P✾✝❆✾✹❱✠❏r❣▲❏rt★✾✇●■♣✴✾❲❇❤❈✦❏r●■❇ ☎Ü✞❈❋❃✤❅ ❏Pt❆✾❲❇r✾✠ß➯●■❇r✾✬❏Pt★✾❼❀✇❈✦❏r❇P❁◆å✆❁⑤❣✿❊✴✾✹❱✠●❥❀q❁◆❃★❧✇❱❲▼◆●✏❣✐✾✹❇❯❏r●✇❊✤✾✹❁◆❃❆❧q❣P❁❄❃★❧■❍★▼⑤❈❋❇✜ã Û❣❵❏Pt❆✾❂❅✰❁⑤❣r❱✠❇r✾✠❏P❁✕❑✹❈❋❏P❁❄●■❃s❁⑤❣❵❇r✾✛✚✤❃★✾✹❅✮Ü✴❏rt★✾❹❀✇❈❋❏P❇r❁éå❪❁❄❣á▼⑤❈❋❇r❧■✾✹❇❵❈❋❃❆❅❚t❆❈■❣✬❀q●■❇r✾✉✾❲❁❄❧■✾✹❃✏♦✦❈■▼◆❍★✾✜❣❲ã ✔❉●■❏P❁⑤❱✠✾✬❏Pt❆❈❋❏❉❈■❣❯❏Pt★✾❼❅✰❁⑤❣r❱✠❇r✾✠❏P❁✕❑✹❈❋❏P❁❄●■❃❘❁❄❣Þ❇r✾✛✚❆❃★✾✜❅✇ß➯❇r●■❀ ✜s➳ ✳❉✵✉❏r● ✜❚➳✡✠❈✵❼❏r●✗✜s➳☞☛✼✵★Ü✰❈■▼◆▼ ❏Pt❆✾✉❈■❅★❅✰❁◆❏P❁❄●■❃✤❈❋▼✮✾❲❁❄❧■✾✹❃◗♦②❈■▼◆❍❆✾✹❣✪❈❋❇r✾❼❱✠▼❄●❥❣P✾❲❇Þ❏P●✗❑❲✾✹❇P●✤ã ✢✪➊✆☎ ✣✒✤✿➍✦✥➛★✧✞✩✫✪è➝★↔✭✬★✧✞✩✮✥➏ ☎✔➡✞✝✻➡ ❝ ✗✠✘✚✙✜✭✞✱✳✙✹✱❜✲✤✘✠✟☛✡➞✲✤✭✞✙❂✙✹✸✯✛❚✩✬✱❜✢✤✛◗✘✚❛★✥★✧❜✭✞✛◗✼ ✌ ✍✏✎✒✑✔✓✖✕ ✆ Û❣Þ❏rt★✾✉❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑✹❈❋❏P❁❄●■❃☞❁❄❣✪❇r✾✛✚❆❃❆✾✹❅✮Ü❆➼✑ ➫✳➭✴➲✪❊✴✾✹❱❲●■❀q✾✹❣✪❊✴✾✠❏P❏P✾❲❇✬❈❋♣★♣★❇r●②å✰❁◆❀✇❈❋❏P✾✹❅ SMA-HPC ©1999 MIT Convergence Analysis Example Problems Intuition about the Eigenvalues x0 = −1 xn =1 1 x n 1 x x2 − x0 = −1 x1 x2n =1 Û❣Þ❏rt★✾✉❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑✹❈✦❏r❁◆●❥❃❘❁⑤❣✪❇r✾✛✚❆❃★✾✜❅✮Ü ☎❁❅ ❣✪❃◗❍★▼❄▼✔❣✐♣✤❈■❱✠✾❼❱✹❈❋❃☞❊✤✾❼❀q●■❇r✾❼❈■❱✹❱✠❍★❇❤❈✦❏r✾❲▼❄⑦ ❇P✾✹♣★❇r✾✹❣P✾❲❃✏❏P✾✜❅✮ã Ô➥Õ✮ÖØ×ÚÙ☎☛ ä✞●❿❧■❁❄♦■✾Ú❈ ❅✰❁✁￾✚✾❲❇r✾❲❃✏❏☞♦✏❁❄✾❲Ý ●■ß✉Ý✿t✏⑦ ❇r✾✛✚❆❃★❁❄❃★❧❬❏rt★✾➎❅✰❁❄❣r❱✠❇r✾✠❏r❁✧❑✜❈✦❏r❁◆●❥❃✖ß➯●❥❇❦❏rt★✾❍✚❆❇❤❣✐❏❘❖◗❁◆❃✤❅ ✾✹♠✏❍❆❈❋❏P❁❄●■❃❚♣❆❇P●✰❅✰❍❆❱❲✾✹❣▲❈❦❀✇❈❋❏P❇r❁éå❚Ý✿❁◆❏Pt➥❀q●■❇r✾❹❈❋❃✤❅❚❀q●■❇r✾❹❣P❀✇❈❋▼❄▼◆✾✹❇✬✾✹❁◆❧❥✾❲❃◗♦✦❈❋▼❄❍★✾✹❣✹Ü✮❱✠●■❃✤❣✐❁⑤❅✰✾❲❇ ❏Pt❆✾❉♣★▼❄●❋❏r❣ê❁◆❃q❏rt★✾❵❣P▼◆❁⑤❅✰✾❵❈■❊✤●✦♦❥✾■ã✔➦➧❃q❏rt★✾✿❏r●■♣q♣★▼❄●❋❏✹Ü✏●■❃❆✾✿●❋ß✴❏Pt★✾❉❊✤❈■❣P❁❄❣❭ß➯❍❆❃❆❱Ø❏r❁◆●❥❃❆❣ê❁⑤❣ê♣★▼❄●❋❏P❏P✾✜❅ ß➯●■❇✇❈è❱✠●❥❈■❇r❣P✾❘❅★❁❄❣r❱✠❇r✾✠❏r❁✧❑✜❈✦❏P❁❄●■❃✔ã❬➦➧❃❿❏Pt❆✾☞❊✴●❋❏P❏P●❥❀ ♣★▼❄●❋❏✹Ü➞●■❃★✾☞●❋ß❉❏rt★✾❪❊❆❈■❣P❁⑤❣➃ß➯❍❆❃❆❱Ø❏r❁◆●❥❃❆❣❂❁❄❣ ♣★▼❄●❋❏P❏P✾✹❅✆ß➯●■❇✿❈ ✚❆❃★✾✹❇✿❅✰❁❄❣r❱✠❇r✾✠❏r❁✧❑✜❈✦❏r❁◆●❥❃✻ã ■✶t❆❈✦❏✿❁⑤❣✪❃★●❋❏❤❈❋❊★▼❄✾á❈■❊✤●❥❍✰❏✪❏Pt★✾✜❣✐✾✬❏⑥ÝÞ●❂♣❆▼◆●■❏r❣Þ❁❄❣Þ❏rt❆❈✦❏ ❈■❣✯❏rt★✾❉❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑✹❈❋❏P❁❄●■❃❂❁❄❣➐❇r✾✛✚✤❃★✾✹❅✮Ü❋❏Pt★✾✜❣✐✾✿❊❆❈❥❣✐❁⑤❣✯ß➯❍★❃❆❱✠❏P❁❄●■❃❆❣➞▼◆●◗●❥❖✉♣★❇r●■❧■❇r✾✹❣r❣P❁◆♦❥✾❲▼❄⑦✬❀q●■❇r✾✪▼◆❁❄❖■✾ ❏Pt❆✾▲❣P♣★❁❄❖■✾áß➯❍★❃❆❱✠❏P❁❄●■❃❪❀q✾❲❃✏❏P❁❄●■❃❆✾✹❅☞❈❋❊✴●✦♦■✾■ã Û❃✤❅☞❣P❁❄❃❆❱✠✾á❏Pt❆✾✉❣✐♣★❁❄❖■✾áß➯❍★❃✤❱Ø❏P❁❄●■❃❪❁⑤❣✪❁◆❃❘❏Pt❆✾❼❃✏❍❆▼◆▼ ❣P♣❆❈■❱❲✾❉●❋ß ☎Ü✏●❥❃★✾❉ÝÞ●■❍★▼⑤❅✇✾❲å◗♣✴✾✹❱✠❏❯❏rt❆❈✦❏✱✚❆❃★✾✹❇Þ❅✰❁⑤❣r❱✠❇r✾✠❏P❁✕❑✹❈❋❏P❁❄●■❃❆❣➞Ý❯●❥❍★▼⑤❅✇❧■✾✹❃★✾❲❇❤❈✦❏r✾ ❱✐❣P♣★❁❄❖◗❁◆✾✹❇✕❳ ❊❆❈❥❣✐❁⑤❣Þß➯❍★❃❆❱✠❏P❁❄●■❃❆❣✪Ý✿t★●✏❣✐✾❼❈❥❣P❣P●✰❱✠❁⑤❈✦❏P✾✜❅❦✾✹❁◆❧❥✾❲❃◗♦✦❈❋▼❄❍★✾✹❣✪ÝÞ●■❍★▼⑤❅❘❊✤✾❼❃★✾✜❈❋❇ ❑❲✾✹❇P●✤ã ☎✔➡✞✝✻➡⑤➠ ✄✔✛◗❴■✲✴✘✞✺➜④✻✱❜✘✞✺✶✩á✫✮✭✞✥★✙✹✱✳✲✴✘✶✸✯✥★✼ ❅✔✛✂✁➃✛✏✣ ￾✣✦✲☞✡✯✧❜✛✏❨✼ ✌ ✍✏✎✒✑✔✓➥➤ ✞ ●◗✾✜❱✠●■❃✤❅ ￾▲❁❄❃❆❅☞✾✹♠✏❍❆❈❋❏P❁❄●■❃ ➫ ✠✏✌ ☎➥➲⑥➼✮✍➁➼➐➫✳➭✴➲ ✌ ✒ ➴ ✳ ➴ ✴ ➭➜✏è➭➻ ✴ ➼➐➫➯➭➻ ➲⑥➽◗➾➻ ❋ ➫ ✠ ✌ ☎è➲✐➼❪➳ ➩ ➫ ✠✏✌ ☎➥➲✠➫❜➼✑ ✌ ➼✻➲ ➳✂ ➫ ✠ ✌ ☎è➲✐➼ ë ✼