(x)=0x=0(0)=1 Note 2I As shown in the first equation on the above shii e, the abstract operator for the secon'-kin'equation is ' enote by I+K, where I here is fist the i entity operator anK is the integramoperator To see why the spike function, oo, is not in the nullspace of the operator I+K or equivalentiy that (I+K)(00+a)垂(I+K)(σ) osier the plts on the bottom of the above slii e. If a spike is a'e to a smooth a, the(I +K)operator wimMreserve the spike. Another way to see th is to consier that since oo is in the nuLLspace of K (I+K)=(a+Ko=ao垂0 3.8 Numerical Results with Increasing n n=10 Eigenvalues do Note 21 As we note before, the matrix associate with ' iscretizing the operator I+K i enticaMito the sum of the i entity matrix anthe matrix associate with generat te by iscretizing the 1-D example problem. Discretizing using 10 subin- tervaslgenerates a matrix with 10 eigenvalues plot te in bhe. The bMe eigen vale closest to zero is x0. 2. As the ' iscretization is refine' to 20 subinterval the igenvaMe (pMtte' in re')is stiNE 0.2, an' with 40 subinterval the minimum eigenvalue(plotte' in green) is stiN Examining this ata uggests that as the’ iscret ization is refine’,an’ the generate’ matrix curately reflects the operator I+K, the matrix is not becoming more singularSMA-HPC ©1999 MIT Convergence Analysis Example Problems 2-D Kind Equation has fewer problems σ σ 0 0 ( ) x x = ≠ 0, 0, (0) =1 -1 1 ( ) I + Κ -1 1 + = -1 1 ≠ Ψ Ô➥Õ✮ÖØ×✏✎✝✠ Û❣✿❣Pt★●✦Ý✿❃❘❁◆❃❘❏Pt★✾ ✚❆❇❤❣✐❏✪✾✜♠✏❍❆❈✦❏r❁◆●❥❃✆●■❃❘❏Pt❆✾▲❈■❊✤●✦♦❥✾✬❣P▼◆❁⑤❅✰✾❥Ü◗❏Pt★✾✉❈■❊❆❣⑥❏r❇r❈❥❱Ø❏Þ●■♣✴✾❲❇❤❈✦❏P●❥❇➞ß➯●❥❇✪❏Pt★✾ ❣P✾✹❱✠●❥❃❆❅◗à③❖◗❁◆❃❆❅ ✾✹♠✏❍❆❈❋❏P❁❄●■❃ ❁⑤❣✇❅✰✾✹❃★●❋❏r✾✹❅✖❊✏⑦ ✠ ✌ ☎❿Ü❯Ý✿t★✾❲❇r✾✞✠➎t❆✾❲❇r✾☞❁⑤❣ ￾⑥❍❆❣✐❏✇❏Pt★✾❪❁⑤❅✰✾❲❃✏❏P❁◆❏⑥⑦ ●■♣✴✾❲❇❤❈✦❏r●■❇✿❈❋❃✤❅ ☎ ❁⑤❣Þ❏Pt★✾❼❁❄❃✏❏P✾❲❧❥❇r❈■▼✮●■♣✴✾❲❇❤❈✦❏r●■❇✜ã ä✞●✉❣✐✾✹✾❉Ý✿t◗⑦✉❏Pt❆✾❵❣✐♣❆❁◆❖❥✾✪ß➯❍★❃❆❱Ø❏r❁◆●❥❃✻Ü◗➼✑ Ü❥❁⑤❣➐❃★●■❏➞❁◆❃q❏rt★✾❉❃◗❍★▼❄▼✤❣P♣❆❈■❱❲✾✿●❋ß✚❏Pt❆✾❉●■♣✴✾❲❇❤❈✦❏r●■❇ ✠ ✌ ☎❿Ü ●■❇✪✾✜♠❥❍❆❁◆♦✦❈❋▼❄✾❲❃✏❏r▼◆⑦✆❏Pt❆❈❋❏ ➫ ✠ ✌ ☎è➲❲➫✳➼✑ ✌❿➼✻➲ ➳✂ ➫ ✠✏✌ ☎➥➲✠➫❜➼✻➲ ❱✠●❥❃❆❣P❁❄❅✰✾✹❇❼❏Pt❆✾❦♣★▼❄●❋❏r❣➃●■❃➎❏rt★✾✆❊✴●❋❏P❏P●■❀ ●❋ß✪❏rt★✾✆❈❋❊✴●✦♦■✾❦❣P▼❄❁❄❅✰✾❥ãs➦③ß❉❈❚❣✐♣❆❁◆❖❥✾❦❁❄❣❹❈■❅★❅★✾✹❅è❏P●Ú❈ ❣P❀❂●◗●■❏Pt❦➼❭Ü✏❏Pt★✾❂➫ ✠✥✌ ☎è➲❭●❥♣✤✾✹❇r❈❋❏P●❥❇êÝ✿❁◆▼❄▼✴♣★❇P✾✜❣✐✾✹❇P♦❥✾Þ❏Pt★✾á❣✐♣★❁❄❖■✾❥ã Û❃★●❋❏rt★✾❲❇ÞÝ✪❈②⑦✉❏P●❂❣✐✾✹✾✿❏Pt❆❁❄❣ ❁⑤❣Þ❏P●❦❱✠●❥❃❆❣P❁❄❅✰✾✹❇Þ❏Pt❆❈❋❏❵❣✐❁❄❃❆❱✠✾❼➼✑ ❁❄❣✪❁❄❃☞❏Pt★✾❼❃◗❍★▼❄▼✔❣P♣❆❈■❱❲✾✬●■ß ☎Ü ➫ ✠ ✌ ☎è➲✐➼✑ ➳ ➫ ✠◗➲⑥➼✑ ✌ ☎Ú➼✑ ➳æ➼✑ ➳✂ ✵▲✓ ✖ ë■ë ✘ ✢✪➊✁￾ ￾➓✥✩✻➝❆➑P→✚➍✦✧✂✁✩✔➏◗➓✧➧➣★➏☎✄✖➑⑥➣★➟✝✆✏➔❵→✚➝✩✔➍❭➏◗➑P➔✟✞ ➔ ✌ ✍✏✎✒✑✔✓➥➤✮✕ SMA-HPC ©1999 MIT n = 10 n = 20 n = 40 Eigenvalues do not get closer to zero. Ô➥Õ✮ÖØ×✏✎✔Ù Û❣ÞÝÞ✾✬❃❆●❋❏P✾✜❅❦❊✴✾✠ß➯●❥❇P✾❥Ü◗❏Pt★✾▲❀✇❈✦❏r❇P❁◆å❦❈■❣r❣✐●✰❱✠❁⑤❈✦❏r✾✹❅✇Ý✿❁◆❏Pt❪❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑❲❁❄❃★❧❹❏Pt★✾á●■♣✴✾❲❇❤❈✦❏r●■❇ ✠ ✌ ☎ ❁⑤❣❼❁❄❅★✾❲❃✏❏P❁⑤❱❲❈■▼ê❏P●❚❏Pt★✾✆❣P❍★❀ ●❋ß✪❏Pt❆✾❦❁❄❅★✾❲❃✏❏P❁◆❏⑥⑦è❀✇❈❋❏P❇r❁éå➎❈❋❃❆❅➥❏rt★✾❦❀✇❈✦❏r❇P❁◆å➥❈❥❣P❣P●✰❱✠❁⑤❈✦❏P✾✜❅➥Ý✿❁é❏rt ❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑❲❁❄❃★❧ ☎ ❈■▼◆●❥❃★✾■ã✿➦➧❃❚❏Pt★✾❹♣★▼◆●■❏á❈■❊✤●✦♦❥✾■Ü❆ÝÞ✾✉●❥❃❆❱✠✾❂❈❋❧❥❈■❁◆❃❪♣★❇r✾✹❣P✾❲❃✏❏❉❏rt★✾➃✾✹❁◆❧❥✾❲❃◗♦✦❈❋▼❄❍★✾✹❣ ❧■✾✹❃★✾❲❇❤❈✦❏r✾✹❅❼❊◗⑦✉❅✰❁⑤❣r❱✠❇r✾✠❏P❁✕❑❲❁❄❃★❧✬❏Pt★✾ ë à➧♥➜✾✠å★❈■❀❂♣❆▼◆✾❯♣★❇P●❥❊★▼❄✾❲❀❪ã✯♥✬❁⑤❣P❱❲❇P✾❲❏P❁✕❑❲❁❄❃★❧✬❍✤❣✐❁❄❃★❧ ✳✦✵❵❣P❍★❊★❁❄❃✰à ❏P✾✹❇P♦✦❈■▼❄❣✪❧❥✾❲❃★✾✹❇r❈❋❏P✾✹❣✿❈❂❀✇❈✦❏r❇P❁◆å❘Ý✿❁◆❏Pt ë ✎ ✾❲❁❄❧■✾❲❃◗♦✦❈❋▼❄❍★✾✜❣✪♣★▼◆●■❏✐❏r✾✹❅☞❁◆❃s❊❆▼◆❍★✾❥ã➞ä✪t★✾❼❊★▼❄❍★✾✉✾✹❁◆❧❥✾❲❃✰à ♦✦❈❋▼❄❍★✾✬❱❲▼◆●✏❣✐✾✜❣⑥❏❯❏P● ❑❲✾❲❇r●✉❁⑤❣ ✤ ✵▲✓ ✠✰ã Û❣❯❏rt★✾✬❅✰❁⑤❣r❱✠❇r✾✠❏P❁✕❑✹❈❋❏P❁❄●■❃❦❁⑤❣❯❇P✾✔✚❆❃★✾✹❅✇❏P●❂ç ✎ ❣P❍★❊★❁❄❃✏❏P✾❲❇r♦✦❈❋▼⑤❣❲Ü ❏Pt❆✾✬❀q❁❄❃★❁◆❀❹❍★❀✄✾✹❁◆❧❥✾❲❃◗♦✦❈❋▼❄❍★✾ ✖✳♣★▼◆●■❏✐❏r✾✹❅❘❁◆❃❘❇P✾✜❅✘ê❁⑤❣✪❣⑥❏r❁◆▼❄▼ ✤✭✵ ✓ ✠★Ü✏❈■❃❆❅❦Ý✿❁◆❏Pt ❴✎ ❣✐❍❆❊★❁◆❃✏❏r✾❲❇r♦②❈■▼❄❣ ❏Pt❆✾❹❀q❁❄❃★❁◆❀❹❍★❀â✾❲❁❄❧■✾✹❃◗♦②❈■▼◆❍❆✾✆✖➯♣★▼❄●❋❏P❏P✾✜❅Ú❁❄❃❚❧■❇r✾❲✾✹❃✘✿❁⑤❣▲❣⑥❏r❁◆▼❄▼ ✤ ✵ ✓ ✠★ã✪❖➐å★❈■❀❂❁❄❃★❁❄❃★❧❘❏Pt★❁⑤❣▲❅★❈✦❏❤❈ ❣P❍★❧■❧❥✾✹❣✐❏r❣✯❏Pt✤❈✦❏Þ❈❥❣❭❏Pt❆✾❵❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑✹❈✦❏r❁◆●❥❃q❁❄❣➞❇P✾✔✚❆❃★✾✹❅✻Ü❥❈❋❃✤❅❂❏Pt★✾✬❧■✾❲❃❆✾❲❇❤❈✦❏P✾✜❅➃❀✇❈❋❏P❇r❁éåq❀q●■❇r✾❉❈■❱Øà ❱✠❍❆❇r❈❋❏P✾❲▼❄⑦q❇P✾✝❆✾✹❱✠❏r❣❯❏Pt❆✾✬●❥♣✤✾✹❇r❈❋❏P●■❇ ✠ ✌ ☎❿Ü✏❏Pt❆✾✬❀✇❈✦❏r❇P❁◆å❦❁⑤❣❯❃❆●❋❏✪❊✤✾✜❱✠●■❀q❁❄❃★❧❂❀❂●❥❇P✾á❣✐❁❄❃★❧❥❍★▼❄❈■❇✹ã ë ✦