๎บ๎Ž™๎ญ๎š๎Žฉ๎Œฏ๎ผ๎Œฆ๎ญ๎ฐ๎Žฉ๎Ž™ ๎ญ๎Žฉ ๎ข๎š๎Œผ๎Œผ๎ข๎†๎‚ก
๎›๎ฐ๎ญ๎ซ ๎‹ผ๎Ž™ ๎Œผ๎Ž•๎“๎ซ๎‹ผ๎ฃ๎ฐ๎ฃ ๎Žฉ๎Ž™ ๎“๎‹ผ๎š๎‹ผ๎Ž‹๎Ž‹๎Œผ๎Ž‹ ๎Œฆ๎Žฉ๎Ž•๎“๎ผ๎ญ๎ฐ๎Ž™๎ 
๎ƒก๎‡ซ๎†ก๎‰ญ๎‰ญ๎†ก ๎‚๎ˆถ๎ˆ๎‡ซ๎Šฝ๎†ก๎Š‹
https://joliv.et/FreeFem-tutorial
๎€ฅ๎‰ญ๎ˆถ๎‹๎‰ฝ๎†ก๎‰ญ ๎‰ฝ๎‡ ๎ˆถ๎‰ญ๎Š‹๎†ƒ๎Š ๎Š‹๎‰ฝ๎กช โ—ฆ ๎€ญ๎Š‹๎‰ญ๎ˆ ๎ค‹ ๎‡‰ ๎‹ช๎Š‹ ๎Š‹๎ˆถ ๎‹๎‡ซ๎†Ž๎Š‹๎‡ ๎กธ๎‡ ๎†ก๎‡ซ๎‡Œ๎‡ ๎Š‹
โ—ฆ
๎€ญ๎Š‹๎‰ญ๎ˆ
๎ค‹
๎‡Œ
๎‡Œ๎ˆถ ๎Š‹๎ˆถ ๎…” ๎‰ข๎…”๎‡Œ๎†ก ๎กฐ
โ—ฆ ๎€ญ๎Š‹๎‰ญ๎ˆ ๎ค‹ ๎ƒก๎‡Œ๎€ท๎‹๎ˆฅ ๎ˆฅ๎†ก๎‹‰๎Š‹ ๎‰ข๎…”๎‡Œ๎†ก
โ—ฆ ๎€ญ๎Š‹๎‰ญ๎ˆ ๎ค‹ ๎ƒก๎‡Œ๎„๎‰ข ๎‰ข๎‰ญ๎†ก๎Šฝ๎‡ซ๎ˆถ๎Š ๎‰ฝ ๎‰ข๎…”๎‡Œ๎†ก
๎Šฝ๎ ๎Ÿฟ๎ ๎ ‚๎กฒ๎Ÿฟ๎ ‡
๎บ๎Ž™๎ญ๎š๎Žฉ๎Œฏ๎ผ๎Œฆ๎ญ๎ฐ๎Žฉ๎Ž™
๎€ƒ๎Œฆ๎Ž†๎Ž™๎Žฉ๎›๎Ž‹๎Œผ๎Œฏ๎ ๎Œผ๎Ž•๎Œผ๎Ž™๎ญ๎ฃ
โ—ฆ ๎„๎ˆฅ๎‡ซ๎Šฝ๎†ก๎‰ญ๎‰ฝ๎‡ซ๎Š‹๎‹Š ๎ˆถ๎‡‰ ๎ƒพ๎‰ฝ๎Š ๎ˆ‡๎Š ๎…น๎…”๎กซ ๎‚๎…”๎‰ข๎…”๎ˆฅ
โ—ฆ ๎บ๎ˆฅ๎‰ฝ๎Š‹๎‡ซ๎Š‹๎Š ๎Š‹๎†ก ๎ˆถ๎‡‰ ๎‚ก๎…”๎Š‹๎‡ ๎†ก๎ˆŸ๎…”๎Š‹๎‡ซ๎†ƒ๎‰ฝ๎กซ ๎„๎ˆฅ๎‡ซ๎Šฝ๎†ก๎‰ญ๎‰ฝ๎‡ซ๎Š‹๎‹Š ๎ˆถ๎‡‰ ๎ƒฒ๎†ก๎Šฝ๎‡ซ๎ˆ๎ˆ๎†ก๎กซ ๎ƒฒ๎‰ข๎…”๎‡ซ๎ˆฅ
โ—ฆ ๎€ญ๎‚ก๎€ƒ๎ƒก๎กซ ๎‡๎†ƒ๎ˆถ๎ˆ๎†ก ๎ƒก๎ˆถ๎ˆ๎‹Š๎Š‹๎†ก๎†ƒ๎‡ ๎ˆฅ๎‡ซ๎‰ฉ๎Š ๎†ก๎กซ ๎ข๎‰ญ๎…”๎ˆฅ๎†ƒ๎†ก
โ—ฆ ๎€ญ๎‹Š๎…น๎†ก๎‰ญ๎ˆŸ๎†ก๎†Ž๎‡ซ๎…” ๎€ญ๎†ก๎ˆฅ๎Š‹๎†ก๎‰ญ๎กซ ๎‚ถ๎‰ฝ๎…”๎ˆ‡๎…” ๎„๎ˆฅ๎‡ซ๎Šฝ๎†ก๎‰ญ๎‰ฝ๎‡ซ๎Š‹๎‹Š๎กซ ๎‚๎…”๎‰ข๎…”๎ˆฅ
โ—ฆ ๎‚ก๎Š ๎‰ฝ๎…”๎‰ฝ๎‡ ๎‡ซ๎ˆฅ๎ˆถ ๎„๎ˆฅ๎‡ซ๎Šฝ๎†ก๎‰ญ๎‰ฝ๎‡ซ๎Š‹๎‹Š๎กซ ๎‚๎…”๎‰ข๎…”๎ˆฅ
๎ €
๎ƒก๎š๎Œผ๎š๎Œผ๎˜๎ผ๎ฐ๎ฃ๎ฐ๎ญ๎Œผ๎ฃ ๎ขŽ๎ญ๎ซ๎Œผ ๎Ž•๎Žฉ๎š๎Œผ ๎ญ๎ซ๎Œผ ๎Œž๎Œผ๎ญ๎ญ๎Œผ๎š๎ข
โ—ฆ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎ข‚๎ฑ๎†ก๎†ƒ๎‡ ๎Š‹ ๎ ๎Ÿฟ๎ €๎ ๎ขƒ https://github.com/FreeFem/FreeFem-sources
โ—ฆ ๎ฅ๎ˆŸ๎‰ฝ๎‡  ๎ข‚๎ฅ๎†ก๎Š ๎‹–๎…”๎‡ซ๎ˆฅ๎†ก ๎…”๎ˆฅ๎†Ž ๎ƒจ๎†ก๎ˆŸ๎…”๎†ƒ๎ˆ๎†ก ๎ ๎Ÿฟ๎Ÿฟ๎ ˆ๎ขƒ http://gmsh.info
โ—ฆ ๎ƒก๎…”๎‰ญ๎…”๎„ช๎‡ซ๎†ก๎‹ https://www.paraview.org
โ—ฆ ๎‚ก๎ƒก๎บ https://www.mpi-forum.org
โ—ฆ ๎ฑ๎ƒก๎€ท๎€ท๎‚ก ๎ข‚๎‚๎ˆถ๎ˆ๎‡ซ๎Šฝ๎†ก๎Š‹๎กซ ๎ฑ๎†ก๎†ƒ๎‡ ๎Š‹๎กซ ๎‚ฅ๎…”๎Š‹๎…”๎‡‰๎กซ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ https://github.com/hpddm/hpddm
โ—ฆ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ข‚๎€ฅ๎…”๎ˆ๎…”๎‹Š ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ €๎ ˆ๎ ˆ๎ †๎ขƒ https://gitlab.com/petsc/petsc
โ—ฆ ๎ƒฒ๎‚•๎†๎ƒก๎†ƒ ๎ข‚๎ฑ๎†ก๎‰ญ๎ˆฅ๎…”๎ˆฅ๎†Ž๎†ก๎‹– ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ „๎ขƒ https://gitlab.com/slepc/slepc
๎ 
๎ƒก๎š๎Žฉ๎ ๎š๎‹ผ๎Ž• ๎Žฉ๎š ๎ญ๎ซ๎Œผ ๎Ž‹๎Œผ๎Œฆ๎ญ๎ผ๎š๎Œผ
๎ €๎กฒ ๎บ๎ˆฅ๎Š‹๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ ๎กฒ ๎ข๎‡ซ๎ˆฅ๎‡ซ๎Š‹๎†ก ๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ
๎ ‚๎กฒ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก
๎ ƒ๎กฒ ๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ
๎ „๎กฒ ๎€ท๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ
๎ …๎กฒ ๎‚ถ๎Šฝ๎†ก๎‰ญ๎ˆ๎…”๎‰ข๎‰ข๎‡ซ๎ˆฅ๎‡Œ ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ
๎ †๎กฒ ๎ƒฒ๎Š ๎…น๎‰ฝ๎Š‹๎‰ญ๎Š ๎†ƒ๎Š‹๎Š ๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ
๎ ‡๎กฒ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ
๎ ˆ๎กฒ ๎ƒฒ๎‚•๎†๎ƒก๎†ƒ
๎ €๎Ÿฟ๎กฒ ๎€ƒ๎‰ข๎‰ข๎ˆ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎ข๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ๎ฃ
๎‚ก๎Žฉ๎Œฏ๎Œผ๎Ž‹ ๎“๎š๎Žฉ๎Œž๎Ž‹๎Œผ๎Ž•
โˆ’โˆ†u = f ๎‡ซ๎ˆฅ โ„ฆ
u = g
D
๎ˆถ๎ˆฅ ฮ“
D
โˆ‚
n
u = g
N
๎ˆถ๎ˆฅ ฮ“
N
x
y
โ„ฆ
โ—ฆ ๎†ก๎‰ฝ๎‰ฝ๎†ก๎ˆฅ๎Š‹๎‡ซ๎…”๎ˆ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆฅ๎…”๎Š‹๎Š ๎‰ญ๎…”๎ˆ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎ ƒ
๎‚ก๎Žฉ๎Œฏ๎Œผ๎Ž‹ ๎“๎š๎Žฉ๎Œž๎Ž‹๎Œผ๎Ž•
โˆ’โˆ†u = f ๎‡ซ๎ˆฅ โ„ฆ
u = g
D
๎ˆถ๎ˆฅ ฮ“
D
โˆ‚
n
u = g
N
๎ˆถ๎ˆฅ ฮ“
N
ฮ“
D
x
y
โ„ฆ
โ—ฆ ๎†ก๎‰ฝ๎‰ฝ๎†ก๎ˆฅ๎Š‹๎‡ซ๎…”๎ˆ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆฅ๎…”๎Š‹๎Š ๎‰ญ๎…”๎ˆ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎ ƒ
๎‚ก๎Žฉ๎Œฏ๎Œผ๎Ž‹ ๎“๎š๎Žฉ๎Œž๎Ž‹๎Œผ๎Ž•
โˆ’โˆ†u = f ๎‡ซ๎ˆฅ โ„ฆ
u = g
D
๎ˆถ๎ˆฅ ฮ“
D
โˆ‚
n
u = g
N
๎ˆถ๎ˆฅ ฮ“
N
ฮ“
D
ฮ“
N
x
y
โ„ฆ
โ—ฆ ๎†ก๎‰ฝ๎‰ฝ๎†ก๎ˆฅ๎Š‹๎‡ซ๎…”๎ˆ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆฅ๎…”๎Š‹๎Š ๎‰ญ๎…”๎ˆ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎ ƒ
๎„ช๎‹ผ๎š๎ฐ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™๎‹ผ๎Ž‹ ๎š๎Žฉ๎š๎Ž•๎ผ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™
๎ฅ๎‰ญ๎†ก๎†ก๎ˆฅ๎ขซ๎‰ฝ ๎Š‹๎‡ ๎†ก๎ˆถ๎‰ญ๎†ก๎ˆŸ
๎ข๎‡ซ๎ˆฅ๎†Ž u โˆˆ H
๎ €
(โ„ฆ) ๎‰ฝ๎Š ๎†ƒ๎‡  ๎Š‹๎‡ ๎…”๎Š‹
๎š
โ„ฆ
โˆ‡u ยท โˆ‡v =
๎š
โ„ฆ
f v +
๎š
ฮ“
N
g
N
v, โˆ€v โˆˆ H
๎ €
ฮ“
D
(โ„ฆ)
u = g
D
๎ˆถ๎ˆฅ ฮ“
D
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ u
โ—ฆ ๎Š‹๎†ก๎‰ฝ๎Š‹ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ v
๎ „
๎„ช๎‹ผ๎š๎ฐ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™๎‹ผ๎Ž‹ ๎š๎Žฉ๎š๎Ž•๎ผ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™
๎ฅ๎‰ญ๎†ก๎†ก๎ˆฅ๎ขซ๎‰ฝ ๎Š‹๎‡ ๎†ก๎ˆถ๎‰ญ๎†ก๎ˆŸ
๎ข๎‡ซ๎ˆฅ๎†Ž u โˆˆ H
๎ €
(โ„ฆ) ๎‰ฝ๎Š ๎†ƒ๎‡  ๎Š‹๎‡ ๎…”๎Š‹
๎š
โ„ฆ
โˆ‡u ยท โˆ‡v =
๎š
โ„ฆ
f v +
๎š
ฮ“
N
g
N
v, โˆ€v โˆˆ H
๎ €
ฮ“
D
(โ„ฆ)
u = g
D
๎ˆถ๎ˆฅ ฮ“
D
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ u
โ—ฆ ๎Š‹๎†ก๎‰ฝ๎Š‹ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ v
๎ „
๎‚ก๎Œผ๎ฃ๎ซ ๎‹ผ๎Ž™๎Œฏ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ๎ฃ
โ—ฆ โ„ฆ ๎†Ž๎‡ซ๎‰ฝ๎†ƒ๎‰ญ๎†ก๎Š‹๎‡ซ๎‹–๎†ก๎†Ž ๎…น๎‹Š โ„ฆ
h
๎‹๎‡ซ๎Š‹๎‡  n
h
๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ
โ—ฆ u ๎†Ž๎‡ซ๎‰ฝ๎†ƒ๎‰ญ๎†ก๎Š‹๎‡ซ๎‹–๎†ก๎†Ž ๎…น๎‹Š u
h
=
๎
N
h
i=๎ €
u
h
(i)ฯ†
i
๎กซ ๎‰ฝ๎ˆŸ๎ˆถ๎ˆถ๎Š‹๎‡  ๎‹๎กฒ๎‰ญ๎กฒ๎Š‹๎กฒ โ„ฆ
h
u
h
โˆˆ H
๎ €
(โ„ฆ
h
) โ‡โ‡’ u
h
๎‡ซ๎‰ฝ ๎†ƒ๎ˆถ๎ˆฅ๎Š‹๎‡ซ๎ˆฅ๎Š ๎ˆถ๎Š ๎‰ฝ
โ—ฆ ๎…น๎…”๎‰ฝ๎‡ซ๎‰ฝ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ {ฯ†
i
}
N
h
i=๎ €
๎ …
๎‚ก๎Œผ๎ฃ๎ซ ๎‹ผ๎Ž™๎Œฏ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ๎ฃ
โ—ฆ โ„ฆ ๎†Ž๎‡ซ๎‰ฝ๎†ƒ๎‰ญ๎†ก๎Š‹๎‡ซ๎‹–๎†ก๎†Ž ๎…น๎‹Š โ„ฆ
h
๎‹๎‡ซ๎Š‹๎‡  n
h
๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ
โ—ฆ u ๎†Ž๎‡ซ๎‰ฝ๎†ƒ๎‰ญ๎†ก๎Š‹๎‡ซ๎‹–๎†ก๎†Ž ๎…น๎‹Š u
h
=
๎
N
h
i=๎ €
u
h
(i)ฯ†
i
๎กซ ๎‰ฝ๎ˆŸ๎ˆถ๎ˆถ๎Š‹๎‡  ๎‹๎กฒ๎‰ญ๎กฒ๎Š‹๎กฒ โ„ฆ
h
u
h
โˆˆ H
๎ €
(โ„ฆ
h
) โ‡โ‡’ u
h
๎‡ซ๎‰ฝ ๎†ƒ๎ˆถ๎ˆฅ๎Š‹๎‡ซ๎ˆฅ๎Š ๎ˆถ๎Š ๎‰ฝ
โ—ฆ ๎…น๎…”๎‰ฝ๎‡ซ๎‰ฝ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ {ฯ†
i
}
N
h
i=๎ €
๎ …
๎‚ก๎Œผ๎ฃ๎ซ ๎‹ผ๎Ž™๎Œฏ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ๎ฃ
โ—ฆ โ„ฆ ๎†Ž๎‡ซ๎‰ฝ๎†ƒ๎‰ญ๎†ก๎Š‹๎‡ซ๎‹–๎†ก๎†Ž ๎…น๎‹Š โ„ฆ
h
๎‹๎‡ซ๎Š‹๎‡  n
h
๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ
โ—ฆ u ๎†Ž๎‡ซ๎‰ฝ๎†ƒ๎‰ญ๎†ก๎Š‹๎‡ซ๎‹–๎†ก๎†Ž ๎…น๎‹Š u
h
=
๎
N
h
i=๎ €
u
h
(i)ฯ†
i
๎กซ ๎‰ฝ๎ˆŸ๎ˆถ๎ˆถ๎Š‹๎‡  ๎‹๎กฒ๎‰ญ๎กฒ๎Š‹๎กฒ โ„ฆ
h
u
h
โˆˆ H
๎ €
(โ„ฆ
h
) โ‡โ‡’ u
h
๎‡ซ๎‰ฝ ๎†ƒ๎ˆถ๎ˆฅ๎Š‹๎‡ซ๎ˆฅ๎Š ๎ˆถ๎Š ๎‰ฝ
โ—ฆ ๎…น๎…”๎‰ฝ๎‡ซ๎‰ฝ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ {ฯ†
i
}
N
h
i=๎ €
๎ …
๎€ƒ๎ฃ๎ฃ๎Œผ๎Ž•๎Œž๎Ž‹๎ฃ ๎“๎š๎Žฉ๎Œฆ๎Œผ๎Œฏ๎ผ๎š๎Œผ๎ฃ
๎‚ก๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰
โˆ€(i, j) โˆˆ J๎ €; N
h
K
๎ 
, A
ij
=
๎š
โ„ฆ
h
โˆ‡ฯ†
j
ยท โˆ‡ ฯ†
i
=โ‡’ ๎ˆถ๎ˆฅ๎ˆ๎‹Š ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‡Œ๎‰ญ๎…”๎Š‹๎†ก ๎‡ซ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‡‰ ๎…น๎ˆถ๎Š‹๎‡  ๎‰ฝ๎Š ๎‰ข๎‰ข๎ˆถ๎‰ญ๎Š‹๎‰ฝ
๎ƒจ๎‡ซ๎‡Œ๎‡ ๎Š‹๎ขš๎‡ ๎…”๎ˆฅ๎†Ž ๎‰ฝ๎‡ซ๎†Ž๎†ก
โˆ€i โˆˆ J๎ €; N
h
K, b
i
=
๎š
โ„ฆ
h
f ฯ†
i
+
๎š
ฮ“
h
N
g
N
ฯ†
i
=โ‡’ ๎ˆฅ๎Š ๎ˆŸ๎†ก๎‰ญ๎‡ซ๎†ƒ๎…”๎ˆ ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‡Œ๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎‰ฉ๎Š ๎…”๎†Ž๎‰ญ๎…”๎Š‹๎Š ๎‰ญ๎†ก ๎‰ญ๎Š ๎ˆ๎†ก๎‰ฝ
๎ †
๎€ƒ๎ฃ๎ฃ๎Œผ๎Ž•๎Œž๎Ž‹๎ฃ ๎“๎š๎Žฉ๎Œฆ๎Œผ๎Œฏ๎ผ๎š๎Œผ๎ฃ
๎‚ก๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰
โˆ€(i, j) โˆˆ J๎ €; N
h
K
๎ 
, A
ij
=
๎š
โ„ฆ
h
โˆ‡ฯ†
j
ยท โˆ‡ ฯ†
i
=โ‡’ ๎ˆถ๎ˆฅ๎ˆ๎‹Š ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‡Œ๎‰ญ๎…”๎Š‹๎†ก ๎‡ซ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‡‰ ๎…น๎ˆถ๎Š‹๎‡  ๎‰ฝ๎Š ๎‰ข๎‰ข๎ˆถ๎‰ญ๎Š‹๎‰ฝ
๎ƒจ๎‡ซ๎‡Œ๎‡ ๎Š‹๎ขš๎‡ ๎…”๎ˆฅ๎†Ž ๎‰ฝ๎‡ซ๎†Ž๎†ก
โˆ€i โˆˆ J๎ €; N
h
K, b
i
=
๎š
โ„ฆ
h
f ฯ†
i
+
๎š
ฮ“
h
N
g
N
ฯ†
i
=โ‡’ ๎ˆฅ๎Š ๎ˆŸ๎†ก๎‰ญ๎‡ซ๎†ƒ๎…”๎ˆ ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‡Œ๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎‰ฉ๎Š ๎…”๎†Ž๎‰ญ๎…”๎Š‹๎Š ๎‰ญ๎†ก ๎‰ญ๎Š ๎ˆ๎†ก๎‰ฝ
๎ †
๎†๎ฃ๎ฃ๎Œผ๎Ž™๎ญ๎ฐ๎‹ผ๎Ž‹ ๎Œž๎Žฉ๎ผ๎Ž™๎Œฏ๎‹ผ๎š๎ฃ ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ
G
D
๎‰ฝ๎Š ๎…น๎‰ฝ๎†ก๎Š‹ ๎ˆถ๎‡‰ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ ๎…”๎‰ฝ๎‰ฝ๎ˆถ๎†ƒ๎‡ซ๎…”๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎€ท๎‡ซ๎‰ญ๎‡ซ๎†ƒ๎‡ ๎ˆ๎†ก๎Š‹ ๎€ฅ๎€ญ
โ—ฆ ๎ˆฅ๎ˆถ๎ˆฅ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
Ax =
๎€ข
A
G
D
G
D
A
G
D
G
D
๎Ÿฟ I
G
D
G
D
๎€ฃ๎€ข
x
G
D
x
G
D
๎€ฃ
=
๎€ข
b
G
D
g
D
๎€ฃ
โ—ฆ ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
Ax =
๎€ข
A
G
D
G
D
๎Ÿฟ
๎Ÿฟ I
G
D
G
D
๎€ฃ๎€ข
x
G
D
x
G
D
๎€ฃ
=
๎€ข
b
G
D
โˆ’ A
G
D
G
D
g
D
g
D
๎€ฃ
โ—ฆ ๎‰ข๎†ก๎ˆฅ๎…”๎ˆ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
Ax =
๎€ข
A
G
D
G
D
A
G
D
G
D
A
G
D
G
D
A
G
D
G
D
+ ๎ €๎Ÿฟ
๎ ‚๎Ÿฟ
I
G
D
G
D
๎€ฃ๎€ข
x
G
D
x
G
D
๎€ฃ
=
๎€ข
b
G
D
๎ €๎Ÿฟ
๎ ‚๎Ÿฟ
g
D
๎€ฃ
๎ ‡
๎†๎ฃ๎ฃ๎Œผ๎Ž™๎ญ๎ฐ๎‹ผ๎Ž‹ ๎Œž๎Žฉ๎ผ๎Ž™๎Œฏ๎‹ผ๎š๎ฃ ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ
G
D
๎‰ฝ๎Š ๎…น๎‰ฝ๎†ก๎Š‹ ๎ˆถ๎‡‰ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ ๎…”๎‰ฝ๎‰ฝ๎ˆถ๎†ƒ๎‡ซ๎…”๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎€ท๎‡ซ๎‰ญ๎‡ซ๎†ƒ๎‡ ๎ˆ๎†ก๎Š‹ ๎€ฅ๎€ญ
โ—ฆ ๎ˆฅ๎ˆถ๎ˆฅ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
Ax =
๎€ข
A
G
D
G
D
A
G
D
G
D
๎Ÿฟ I
G
D
G
D
๎€ฃ๎€ข
x
G
D
x
G
D
๎€ฃ
=
๎€ข
b
G
D
g
D
๎€ฃ
โ—ฆ ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
Ax =
๎€ข
A
G
D
G
D
๎Ÿฟ
๎Ÿฟ I
G
D
G
D
๎€ฃ๎€ข
x
G
D
x
G
D
๎€ฃ
=
๎€ข
b
G
D
โˆ’ A
G
D
G
D
g
D
g
D
๎€ฃ
โ—ฆ ๎‰ข๎†ก๎ˆฅ๎…”๎ˆ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
Ax =
๎€ข
A
G
D
G
D
A
G
D
G
D
A
G
D
G
D
A
G
D
G
D
+ ๎ €๎Ÿฟ
๎ ‚๎Ÿฟ
I
G
D
G
D
๎€ฃ๎€ข
x
G
D
x
G
D
๎€ฃ
=
๎€ข
b
G
D
๎ €๎Ÿฟ
๎ ‚๎Ÿฟ
g
D
๎€ฃ
๎ ‡
๎†๎ฃ๎ฃ๎Œผ๎Ž™๎ญ๎ฐ๎‹ผ๎Ž‹ ๎Œž๎Žฉ๎ผ๎Ž™๎Œฏ๎‹ผ๎š๎ฃ ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ
G
D
๎‰ฝ๎Š ๎…น๎‰ฝ๎†ก๎Š‹ ๎ˆถ๎‡‰ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ ๎…”๎‰ฝ๎‰ฝ๎ˆถ๎†ƒ๎‡ซ๎…”๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎€ท๎‡ซ๎‰ญ๎‡ซ๎†ƒ๎‡ ๎ˆ๎†ก๎Š‹ ๎€ฅ๎€ญ
โ—ฆ ๎ˆฅ๎ˆถ๎ˆฅ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
Ax =
๎€ข
A
G
D
G
D
A
G
D
G
D
๎Ÿฟ I
G
D
G
D
๎€ฃ๎€ข
x
G
D
x
G
D
๎€ฃ
=
๎€ข
b
G
D
g
D
๎€ฃ
โ—ฆ ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
Ax =
๎€ข
A
G
D
G
D
๎Ÿฟ
๎Ÿฟ I
G
D
G
D
๎€ฃ๎€ข
x
G
D
x
G
D
๎€ฃ
=
๎€ข
b
G
D
โˆ’ A
G
D
G
D
g
D
g
D
๎€ฃ
โ—ฆ ๎‰ข๎†ก๎ˆฅ๎…”๎ˆ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
Ax =
๎€ข
A
G
D
G
D
A
G
D
G
D
A
G
D
G
D
A
G
D
G
D
+ ๎ €๎Ÿฟ
๎ ‚๎Ÿฟ
I
G
D
G
D
๎€ฃ๎€ข
x
G
D
x
G
D
๎€ฃ
=
๎€ข
b
G
D
๎ €๎Ÿฟ
๎ ‚๎Ÿฟ
g
D
๎€ฃ
๎ ‡
๎ข๎š๎Œผ๎Œผ๎ข๎†๎‚ก
๎ฑ๎ฐ๎ฃ๎ญ๎Žฉ๎š๎ฃ
๎ €๎ ˆ๎ ‡๎ † ๎‚ก๎…”๎†ƒ๎ข๎†ก๎ˆŸ๎กธ๎ƒก๎€ญ๎ข๎†ก๎ˆŸ ๎…น๎‹Š ๎ƒก๎‡ซ๎‰ญ๎ˆถ๎ˆฅ๎ˆฅ๎†ก๎…”๎Š 
๎ €๎ ˆ๎ ˆ๎  ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ ๎…น๎‹Š ๎ƒก๎‡ซ๎‰ญ๎ˆถ๎ˆฅ๎ˆฅ๎†ก๎…”๎Š ๎กซ ๎€ฅ๎†ก๎‰ญ๎ˆฅ๎…”๎‰ญ๎†Ž๎‡ซ๎กซ ๎ฑ๎†ก๎†ƒ๎‡ ๎Š‹๎กซ ๎…”๎ˆฅ๎†Ž ๎ƒก๎‰ญ๎Š ๎†Ž๎ขซ๎‡ ๎ˆถ๎ˆŸ๎ˆŸ๎†ก
๎ €๎ ˆ๎ ˆ๎ … ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ค‹ ๎…น๎‹Š ๎ƒก๎‡ซ๎‰ญ๎ˆถ๎ˆฅ๎ˆฅ๎†ก๎…”๎Š ๎กซ ๎€ฅ๎†ก๎‰ญ๎ˆฅ๎…”๎‰ญ๎†Ž๎‡ซ๎กซ ๎…”๎ˆฅ๎†Ž ๎ฑ๎†ก๎†ƒ๎‡ ๎Š‹
๎ €๎ ˆ๎ ˆ๎ ‡ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ค‹๎ค‹ ๎…น๎‹Š ๎ฑ๎†ก๎†ƒ๎‡ ๎Š‹๎กซ ๎ƒก๎‡ซ๎‰ญ๎ˆถ๎ˆฅ๎ˆฅ๎†ก๎…”๎Š ๎กซ ๎…”๎ˆฅ๎†Ž ๎‚ถ๎‡ ๎Š‹๎‰ฝ๎Š ๎ˆ‡๎…”
๎ ๎Ÿฟ๎Ÿฟ๎ ‡ ๎Šฝ๎†ก๎‰ญ๎‰ฝ๎‡ซ๎ˆถ๎ˆฅ ๎ ‚
๎ ๎Ÿฟ๎ €๎ ƒ ๎Šฝ๎†ก๎‰ญ๎‰ฝ๎‡ซ๎ˆถ๎ˆฅ ๎ ‚๎กฒ๎ ‚๎ ƒ ๎‹๎‡ซ๎Š‹๎‡  ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ
๎ ๎Ÿฟ๎ €๎ ‡ ๎ˆŸ๎ˆถ๎Šฝ๎‡ซ๎ˆฅ๎‡Œ ๎Š‹๎ˆถ ๎ฅ๎‡ซ๎Š‹๎ฑ๎Š ๎…น
๎ ๎Ÿฟ๎ €๎ ˆ ๎Šฝ๎†ก๎‰ญ๎‰ฝ๎‡ซ๎ˆถ๎ˆฅ ๎ ƒ๎กซ ๎‰ญ๎†ก๎…น๎‰ญ๎…”๎ˆฅ๎†Ž๎†ก๎†Ž ๎…”๎‰ฝ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก
๎ ๎Ÿฟ๎ ๎ ‚ ๎Šฝ๎†ก๎‰ญ๎‰ฝ๎‡ซ๎ˆถ๎ˆฅ ๎ ƒ๎กฒ๎ €๎ ‚๎กซ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎†ก ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก๎‰ฝ
๎ ˆ
๎บ๎Ž™๎ฃ๎ญ๎‹ผ๎Ž‹๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™
โ—ฆ ๎‰ข๎…”๎†ƒ๎ˆ‡๎…”๎‡Œ๎†ก๎‰ฝ ๎…”๎Šฝ๎…”๎‡ซ๎ˆ๎…”๎…น๎ˆ๎†ก ๎‡‰๎ˆถ๎‰ญ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹ ๎‚ถ๎ƒฒ๎†ก๎‰ฝ
โ€ข ๎„ญ๎‡ซ๎ˆฅ๎†Ž๎ˆถ๎‹๎‰ฝ
โ€ข ๎ˆŸ๎…”๎†ƒ๎‚ถ๎ƒฒ
โ€ข ๎€ท๎†ก๎…น๎‡ซ๎…”๎ˆฅ
โ—ฆ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎‡ซ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎Š‹๎‡ ๎†ก ๎‰ฝ๎ˆถ๎Š ๎‰ญ๎†ƒ๎†ก๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎ˆŸ๎ˆถ๎‰ญ๎†ก ๎‹ซ๎†ก๎‹‰๎‡ซ๎…น๎‡ซ๎ˆ๎‡ซ๎Š‹๎‹Š
โ€ข ๎†ƒ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ๎กธ๎ƒฒ๎‚•๎†๎ƒก๎†ƒ ๎‡ซ๎ˆฅ๎‰ฝ๎Š‹๎…”๎ˆ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ€ข ๎†Ž๎†ก๎Šฝ๎†ก๎ˆ๎ˆถ๎‰ข ๎…น๎‰ญ๎…”๎ˆฅ๎†ƒ๎‡ 
โ—ฆ ๎ˆถ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆ๎ˆถ๎Š ๎†Ž
โ€ข https://www.rescale.com
โ€ข http://qarnot.com
=โ‡’ https://community.freefem.org
๎ €๎Ÿฟ
๎บ๎Ž™๎ฃ๎ญ๎‹ผ๎Ž‹๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™
โ—ฆ ๎‰ข๎…”๎†ƒ๎ˆ‡๎…”๎‡Œ๎†ก๎‰ฝ ๎…”๎Šฝ๎…”๎‡ซ๎ˆ๎…”๎…น๎ˆ๎†ก ๎‡‰๎ˆถ๎‰ญ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹ ๎‚ถ๎ƒฒ๎†ก๎‰ฝ
โ€ข ๎„ญ๎‡ซ๎ˆฅ๎†Ž๎ˆถ๎‹๎‰ฝ
โ€ข ๎ˆŸ๎…”๎†ƒ๎‚ถ๎ƒฒ
โ€ข ๎€ท๎†ก๎…น๎‡ซ๎…”๎ˆฅ
โ—ฆ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎‡ซ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎Š‹๎‡ ๎†ก ๎‰ฝ๎ˆถ๎Š ๎‰ญ๎†ƒ๎†ก๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎ˆŸ๎ˆถ๎‰ญ๎†ก ๎‹ซ๎†ก๎‹‰๎‡ซ๎…น๎‡ซ๎ˆ๎‡ซ๎Š‹๎‹Š
โ€ข ๎†ƒ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ๎กธ๎ƒฒ๎‚•๎†๎ƒก๎†ƒ ๎‡ซ๎ˆฅ๎‰ฝ๎Š‹๎…”๎ˆ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ€ข ๎†Ž๎†ก๎Šฝ๎†ก๎ˆ๎ˆถ๎‰ข ๎…น๎‰ญ๎…”๎ˆฅ๎†ƒ๎‡ 
โ—ฆ ๎ˆถ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆ๎ˆถ๎Š ๎†Ž
โ€ข https://www.rescale.com
โ€ข http://qarnot.com
=โ‡’ https://community.freefem.org
๎ €๎Ÿฟ
๎ƒฒ๎ญ๎‹ผ๎Ž™๎Œฏ๎‹ผ๎š๎Œฏ ๎Œฆ๎Žฉ๎Ž•๎“๎ฐ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎“๎š๎Žฉ๎Œฆ๎Œผ๎ฃ๎ฃ ๎บ
โ—ฆ ๎‰ญ๎†ก๎ˆŸ๎ˆถ๎Šฝ๎†ก ๎…”๎ˆฅ๎‹Š ๎‰ข๎‰ญ๎†ก๎Šฝ๎‡ซ๎ˆถ๎Š ๎‰ฝ ๎‡ซ๎ˆฅ๎‰ฝ๎Š‹๎…”๎ˆฅ๎†ƒ๎†ก ๎ˆถ๎‡‰ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก
โ—ฆ ๎‡ซ๎ˆฅ๎‰ฝ๎Š‹๎…”๎ˆ๎ˆ ๎‡Œ๎†ƒ๎†ƒ๎กธ๎†ƒ๎ˆ๎…”๎ˆฅ๎‡Œ ๎…”๎ˆฅ๎†Ž ๎‡Œ๎‡‰๎ˆถ๎‰ญ๎Š‹๎‰ญ๎…”๎ˆฅ
โ—ฆ ๎ˆŸ๎…”๎ˆ‡๎†ก ๎‰ฝ๎Š ๎‰ญ๎†ก ๎‹Š๎ˆถ๎Š  ๎‡ ๎…”๎Šฝ๎†ก ๎…” ๎‹๎ˆถ๎‰ญ๎ˆ‡๎‡ซ๎ˆฅ๎‡Œ ๎‚ก๎ƒก๎บ ๎‡ซ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ €๎ €
๎ƒฒ๎ญ๎‹ผ๎Ž™๎Œฏ๎‹ผ๎š๎Œฏ ๎Œฆ๎Žฉ๎Ž•๎“๎ฐ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎“๎š๎Žฉ๎Œฆ๎Œผ๎ฃ๎ฃ ๎บ๎บ
๎ค‘ ๎‡Œ๎‡ซ๎Š‹ ๎†ƒ๎ˆ๎ˆถ๎ˆฅ๎†ก ๎‡ ๎Š‹๎Š‹๎‰ข๎‰ฝ๎กช๎กธ๎กธ๎‡Œ๎‡ซ๎Š‹๎‡ ๎Š ๎…น๎กฒ๎†ƒ๎ˆถ๎ˆŸ๎กธ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎กธ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ขš๎‰ฝ๎ˆถ๎Š ๎‰ญ๎†ƒ๎†ก๎‰ฝ
๎ค‘ ๎†ƒ๎†Ž ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ขš๎‰ฝ๎ˆถ๎Š ๎‰ญ๎†ƒ๎†ก๎‰ฝ
๎ค‘ ๎‡Œ๎‡ซ๎Š‹ ๎†ƒ๎‡ ๎†ก๎†ƒ๎ˆ‡๎ˆถ๎Š ๎Š‹ ๎†Ž๎†ก๎Šฝ๎†ก๎ˆ๎ˆถ๎‰ข
๎ค‘ ๎…”๎Š ๎Š‹๎ˆถ๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎‡‰ ๎ขš๎‡ซ
๎ค‘ ๎กฒ๎กธ๎†ƒ๎ˆถ๎ˆฅ๎‹ช๎‡Œ๎Š ๎‰ญ๎†ก ๎ขš๎ขš๎†ก๎ˆฅ๎…”๎…น๎ˆ๎†ก๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž ๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ˆถ๎Š ๎Š‹๎ขš๎‡ ๎†Ž๎‡‰๎ „
๎ขš๎ขš๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎ค๎ขผ๎ข€๎ƒก๎„ญ๎€ท๎ข
๎ค‘ ๎†ƒ๎†Ž ๎ ‚๎‰ญ๎†Ž๎‰ข๎…”๎‰ญ๎Š‹๎‹Š๎กธ๎‡‰๎‡‰๎ขš๎‰ข๎†ก๎Š‹๎‰ฝ๎†ƒ
๎ค‘ ๎ˆŸ๎…”๎ˆ‡๎†ก ๎‰ข๎†ก๎Š‹๎‰ฝ๎†ƒ๎ขš๎‰ฝ๎ˆ๎†ก๎‰ข๎†ƒ
๎ค‘ ๎†ƒ๎†Ž ๎ขš
๎ค‘ ๎กฒ๎กธ๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎‹ช๎‡Œ๎Š ๎‰ญ๎†ก
๎ค‘ ๎ˆŸ๎…”๎ˆ‡๎†ก
๎ €๎ €
๎€ฅ๎ฐ๎Ž™๎‹ผ๎š๎ฐ๎Œผ๎ฃ
โ—ฆ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ค‹๎ค‹
โ—ฆ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ค‹๎ค‹๎ขš๎ˆŸ๎‰ข๎‡ซ
โ—ฆ ๎‡‰๎‡‰๎‡Œ๎ˆ๎Š ๎Š‹
โ—ฆ ๎‡‰๎‡‰๎ˆŸ๎†ก๎†Ž๎‡ซ๎Š‹
โ—ฆ ๎‡‰๎‡‰๎ขš๎†ƒ๎ค‹๎ค‹
๎€ฅ๎…”๎‰ฝ๎‡ซ๎†ƒ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ—ฆ -ns
โ—ฆ -nw๎กธ-wg
โ—ฆ -v ๎Ÿฟ
๎‰ฝ๎†ƒ๎‰ญ๎‡ซ๎‰ข๎Š‹ ๎ˆฅ๎ˆถ๎Š‹ ๎‰ข๎‰ญ๎‡ซ๎ˆฅ๎Š‹๎†ก๎†Ž
๎‡Œ๎‰ญ๎…”๎‰ข๎‡ ๎‡ซ๎†ƒ๎…”๎ˆ ๎ˆถ๎Š ๎Š‹๎‰ข๎Š ๎Š‹ ๎†Ž๎†ก๎…”๎†ƒ๎Š‹๎‡ซ๎Šฝ๎…”๎Š‹๎†ก๎†Ž
๎ˆ๎†ก๎Šฝ๎†ก๎ˆ ๎ˆถ๎‡‰ ๎Šฝ๎†ก๎‰ญ๎…น๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‹Š
๎ €๎ 
mesh ๎‹ผ๎Ž™๎Œฏ mesh3 ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎Š‹๎‰ญ๎Š ๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎†Ž ๎ˆŸ๎†ก๎‰ฝ๎‡ ๎†ก๎‰ฝ
โ—ฆ ๎‡ซ๎ˆฅ ๎ ๎€ท๎กซ square
โ—ฆ ๎‡ซ๎ˆฅ ๎ ‚๎€ท๎กซ cube
๎„๎ˆฅ๎‰ฝ๎Š‹๎‰ญ๎Š ๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎†Ž ๎ˆŸ๎†ก๎‰ฝ๎‡ ๎†ก๎‰ฝ
โ—ฆ buildmesh
โ—ฆ ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡‰๎…”๎†ƒ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎ฅ๎ˆŸ๎‰ฝ๎‡ ๎กซ ๎ƒพ๎†ก๎Š‹๎ฅ๎†ก๎ˆฅ ๎ข‚๎ƒฒ๎‡ซ ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ๎กซ ๎…”๎ˆฅ๎†Ž ๎‚ก๎‚ก๎ฅ
๎‚ถ๎ˆฅ๎ˆ๎‡ซ๎ˆฅ๎†ก ๎Šฝ๎‡ซ๎‰ฝ๎Š ๎…”๎ˆ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‡‰๎ˆถ๎‰ญ ๎†Ž๎†ก๎…น๎Š ๎‡Œ๎‡Œ๎‡ซ๎ˆฅ๎‡Œ ๎‰ข๎Š ๎‰ญ๎‰ข๎ˆถ๎‰ฝ๎†ก๎‰ฝ
โ—ฆ ๎‰ข๎‰ญ๎†ก๎‡‰๎†ก๎‰ญ medit ๎ˆถ๎Šฝ๎†ก๎‰ญ plot ๎‡ซ๎ˆฅ ๎ ‚๎€ท
๎ €๎ ‚
fespace ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎ˆ ๎‰ญ๎†ก๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ๎‡ ๎‡ซ๎‰ข ๎…น๎†ก๎Š‹๎‹๎†ก๎†ก๎ˆฅ ๎…” mesh ๎…”๎ˆฅ๎†Ž ๎…” ๎ข๎†
โ—ฆ ๎ˆถ๎ˆฅ๎†ก ๎…”๎†ƒ๎†ƒ๎†ก๎‰ฝ๎‰ฝ๎‡ซ๎…น๎ˆ๎†ก ๎ˆŸ๎†ก๎ˆŸ๎…น๎†ก๎‰ญ .ndof
โ—ฆ ๎ˆŸ๎…”๎‹Š ๎…น๎†ก ๎Š ๎‰ฝ๎†ก๎†Ž ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก ๎‹ช๎ˆฅ๎‡ซ๎Š‹๎†ก ๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎ €๎ ƒ
fespace ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎ˆ ๎‰ญ๎†ก๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ๎‡ ๎‡ซ๎‰ข ๎…น๎†ก๎Š‹๎‹๎†ก๎†ก๎ˆฅ ๎…” mesh ๎…”๎ˆฅ๎†Ž ๎…” ๎ข๎†
โ—ฆ ๎ˆถ๎ˆฅ๎†ก ๎…”๎†ƒ๎†ƒ๎†ก๎‰ฝ๎‰ฝ๎‡ซ๎…น๎ˆ๎†ก ๎ˆŸ๎†ก๎ˆŸ๎…น๎†ก๎‰ญ .ndof
โ—ฆ ๎ˆŸ๎…”๎‹Š ๎…น๎†ก ๎Š ๎‰ฝ๎†ก๎†Ž ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก ๎‹ช๎ˆฅ๎‡ซ๎Š‹๎†ก ๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎ €๎ ƒ
varf ๎‹ผ๎Ž™๎Œฏ on ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ƒ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎…น๎‡ซ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎‰ฝ
โ—ฆ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎‰ฝ
โ—ฆ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎†Ž๎‹Š๎ˆฅ๎…”๎ˆŸ๎‡ซ๎†ƒ ๎†Ž๎†ก๎‹ช๎ˆฅ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎‡ซ๎ˆฅ๎‰ฝ๎Š‹๎…”๎ˆฅ๎Š‹๎‡ซ๎…”๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎†ก ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎†ƒ๎†ก๎‰ฝ ๎ˆถ๎‰ญ ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ
โ—ฆ qforder ๎Š‹๎ˆถ ๎†ƒ๎‡ ๎…”๎ˆฅ๎‡Œ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‡Œ๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ญ๎Š ๎ˆ๎†ก๎‰ฝ
๎ €๎ „
varf ๎‹ผ๎Ž™๎Œฏ on ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ƒ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎…น๎‡ซ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎‰ฝ
โ—ฆ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎‰ฝ
โ—ฆ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎†Ž๎‹Š๎ˆฅ๎…”๎ˆŸ๎‡ซ๎†ƒ ๎†Ž๎†ก๎‹ช๎ˆฅ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎‡ซ๎ˆฅ๎‰ฝ๎Š‹๎…”๎ˆฅ๎Š‹๎‡ซ๎…”๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎†ก ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎†ƒ๎†ก๎‰ฝ ๎ˆถ๎‰ญ ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ
โ—ฆ qforder ๎Š‹๎ˆถ ๎†ƒ๎‡ ๎…”๎ˆฅ๎‡Œ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‡Œ๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ญ๎Š ๎ˆ๎†ก๎‰ฝ
๎ €๎ „
matrix ๎‹ผ๎Ž™๎Œฏ set ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ „๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎†ก ๎…” varf ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎…” ๎‰ข๎…”๎‡ซ๎‰ญ ๎ˆถ๎‡‰ fespace๎‰ฝ
โ—ฆ ๎Šฝ๎…”๎‰ญ๎‡ซ๎…”๎…น๎ˆ๎†ก sym ๎Š‹๎ˆถ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎†ก ๎Š‹๎‡ ๎†ก ๎Š ๎‰ข๎‰ข๎†ก๎‰ญ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎Š ๎ˆ๎…”๎‰ญ ๎‰ข๎…”๎‰ญ๎Š‹
โ—ฆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹๎‰ฝ
โ—ฆ ๎‰ฝ๎‰ข๎†ก๎†ƒ๎‡ซ๎‡‰๎‹Š ๎…” ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ ๎‡‰๎ˆถ๎‰ญ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  set
๎†๎‰ฝ๎‰ฝ๎†ก๎ˆฅ๎Š‹๎‡ซ๎…”๎ˆ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ tgv ๎ค ๎ขš๎ € ๎‡‰๎ˆถ๎‰ญ ๎ˆฅ๎ˆถ๎ˆฅ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ tgv ๎ค ๎ขš๎  ๎‡‰๎ˆถ๎‰ญ ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎กฒ ๎†ก๎ˆ๎‡ซ๎ˆŸ๎กฒ ๎ขŽ๎†ƒ๎…”๎‰ญ๎†ก๎‡‰๎Š ๎ˆ ๎…”๎…น๎ˆถ๎Š ๎Š‹ ๎Š‹๎‡ ๎†ก ๎ƒจ๎ฑ๎ƒฒ๎ข
โ—ฆ tgv ๎ค ๎ €๎†ก๎ค‹๎ ‚๎Ÿฟ ๎‡‰๎ˆถ๎‰ญ ๎‰ข๎†ก๎ˆฅ๎…”๎ˆ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ €๎ …
matrix ๎‹ผ๎Ž™๎Œฏ set ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ „๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎†ก ๎…” varf ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎…” ๎‰ข๎…”๎‡ซ๎‰ญ ๎ˆถ๎‡‰ fespace๎‰ฝ
โ—ฆ ๎Šฝ๎…”๎‰ญ๎‡ซ๎…”๎…น๎ˆ๎†ก sym ๎Š‹๎ˆถ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎†ก ๎Š‹๎‡ ๎†ก ๎Š ๎‰ข๎‰ข๎†ก๎‰ญ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎Š ๎ˆ๎…”๎‰ญ ๎‰ข๎…”๎‰ญ๎Š‹
โ—ฆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹๎‰ฝ
โ—ฆ ๎‰ฝ๎‰ข๎†ก๎†ƒ๎‡ซ๎‡‰๎‹Š ๎…” ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ ๎‡‰๎ˆถ๎‰ญ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  set
๎†๎‰ฝ๎‰ฝ๎†ก๎ˆฅ๎Š‹๎‡ซ๎…”๎ˆ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ tgv ๎ค ๎ขš๎ € ๎‡‰๎ˆถ๎‰ญ ๎ˆฅ๎ˆถ๎ˆฅ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ tgv ๎ค ๎ขš๎  ๎‡‰๎ˆถ๎‰ญ ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎กฒ ๎†ก๎ˆ๎‡ซ๎ˆŸ๎กฒ ๎ขŽ๎†ƒ๎…”๎‰ญ๎†ก๎‡‰๎Š ๎ˆ ๎…”๎…น๎ˆถ๎Š ๎Š‹ ๎Š‹๎‡ ๎†ก ๎ƒจ๎ฑ๎ƒฒ๎ข
โ—ฆ tgv ๎ค ๎ €๎†ก๎ค‹๎ ‚๎Ÿฟ ๎‡‰๎ˆถ๎‰ญ ๎‰ข๎†ก๎ˆฅ๎…”๎ˆ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ €๎ …
real[int] ๎‹ผ๎Ž™๎Œฏ ๎Žฉ๎ญ๎ซ๎Œผ๎š ๎‹ผ๎š๎š๎‹ผ๎ฃ๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ …๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ real[int,int]
โ—ฆ real[int][int]
โ—ฆ complex[int]
โ—ฆ matrix[int]๎กซ string[int]
โ—ฆ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎ˆ ๎…”๎‰ญ๎‰ญ๎…”๎‹Š ๎ˆถ๎‡‰ ๎‹ช๎ˆฅ๎‡ซ๎Š‹๎†ก ๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎€ญ๎ˆถ๎ˆŸ๎ˆŸ๎ˆถ๎ˆฅ ๎ˆŸ๎†ก๎ˆŸ๎…น๎†ก๎‰ญ๎‰ฝ ๎…”๎ˆฅ๎†Ž ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ
โ—ฆ .n ๎…”๎ˆฅ๎†Ž .m
โ—ฆ .resize
โ—ฆ ๎ค๎กซ ๎ค‹๎ค๎กซ ๎กธ๎ค
๎ €๎ †
๎€ƒ๎Œฏ๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎‹ผ๎Ž‹ ๎“๎Ž‹๎ผ๎ ๎ฐ๎Ž™๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ †๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎ˆ‡๎†ก๎‹Š๎‹๎ˆถ๎‰ญ๎†Ž load
โ—ฆ load "something" =โ‡’ ๎‡‰๎‡‰๎ขš๎†ƒ๎ค‹๎ค‹ ๎ขš๎…”๎Š ๎Š‹๎ˆถ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก๎Š‹๎‡ ๎‡ซ๎ˆฅ๎‡Œ๎กฒ๎†ƒ๎‰ข๎‰ข
โ—ฆ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎ˆถ๎…น๎‡ฟ๎†ก๎†ƒ๎Š‹๎‰ฝ ๎ˆŸ๎…”๎ˆฅ๎‡ซ๎‰ข๎Š ๎ˆ๎…”๎Š‹๎†ก๎†Ž ๎‡ซ๎ˆฅ ๎€ญ๎ค‹๎ค‹๎กธ๎ข๎ˆถ๎‰ญ๎Š‹๎‰ญ๎…”๎ˆฅ
๎ €๎ ‡
func ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ †๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎Š ๎‰ฝ๎†ก๎‰ญ๎ขš๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก๎†Ž ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎…”๎‹Š ๎…น๎†ก ๎‰ข๎…”๎‰ฝ๎‰ฝ๎†ก๎†Ž ๎Š‹๎ˆถ ๎†ก๎‹‰๎Š‹๎†ก๎‰ญ๎ˆฅ๎…”๎ˆ ๎†ƒ๎ˆถ๎†Ž๎†ก๎‰ฝ
โ—ฆ ๎Š ๎‰ฝ๎†ก๎‡‰๎Š ๎ˆ ๎‡‰๎ˆถ๎‰ญ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ขš๎‡‰๎‰ญ๎†ก๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ LinearCG๎กซ EigenValue
๎ €๎ ˆ
macro ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ †๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎†ก๎Šฝ๎…”๎ˆ๎Š ๎…”๎Š‹๎†ก๎†Ž ๎‹๎‡ ๎†ก๎ˆฅ ๎‰ข๎…”๎‰ญ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎‡ซ๎ˆฅ๎‰ข๎Š ๎Š‹ ๎‹ช๎ˆ๎†ก๎‰ฝ
โ—ฆ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก๎†Ž ๎ˆถ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎…”๎ˆฅ๎†Ž ๎ˆ๎‡ซ๎ˆฅ๎†ก -DmacroName=value
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ฝ๎Š‹๎…”๎Š‹๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ IFMACRO
๎ ๎Ÿฟ
๎ƒฒ๎ผ๎Œž๎Ž•๎Œผ๎ฃ๎ซ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‡๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ trunc ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ new2old
โ—ฆ ๎ค‹ restrict ๎Š‹๎ˆถ ๎‡Œ๎ˆถ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎ˆถ๎ˆฅ๎†ก fespace ๎Š‹๎ˆถ ๎…”๎ˆฅ๎ˆถ๎Š‹๎‡ ๎†ก๎‰ญ
โ—ฆ ๎Š ๎‰ฝ๎†ก๎‡‰๎Š ๎ˆ ๎Š‹๎ˆถ ๎…”๎Šฝ๎ˆถ๎‡ซ๎†Ž ๎ขŽ๎†ƒ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š๎ข ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‰ข๎ˆถ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎†ก๎‰ฝ๎‡ ๎†ก๎‰ฝ ๎‡ซ๎ˆฅ ๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎‰ข๎‡ ๎‹Š๎‰ฝ๎‡ซ๎†ƒ๎‰ฝ๎กซ ๎†ก๎กฒ๎‡Œ๎กฒ๎กซ ๎‰ฝ๎ˆถ๎ˆ๎‡ซ๎†Ž ๎ค‹ ๎‹ซ๎Š ๎‡ซ๎†Ž ๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ๎‰ฝ
๎ ๎ €
๎ƒฒ๎ซ๎‹ผ๎š๎Œผ๎Œฏ๎ขš๎Ž•๎Œผ๎Ž•๎Žฉ๎š๎ฃ ๎“๎‹ผ๎š๎‹ผ๎Ž‹๎Ž‹๎Œผ๎Ž‹๎ฐ๎ฃ๎Ž•
๎‚ก๎Žฉ๎ญ๎ฐ๎˜๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™
๎‚ก๎†ก๎‰ฝ๎‡  ๎‡Œ๎†ก๎ˆฅ๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ƒจ๎‡ซ๎‡Œ๎‡ ๎Š‹๎ขš๎‡ ๎…”๎ˆฅ๎†Ž ๎‰ฝ๎‡ซ๎†Ž๎†ก
๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎‹Š
๎‚ก๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎‹Š
๎‚•๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก
๎ƒฒ๎ˆถ๎ˆ๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎†ก๎‹‰๎‰ข๎ˆถ๎‰ญ๎Š‹๎‡ซ๎ˆฅ๎‡Œ
๎Ÿฟ
๎ ๎Ÿฟ
๎ ƒ๎Ÿฟ
๎ …๎Ÿฟ
๎ ‡๎Ÿฟ
๎ƒพ๎‡ซ๎ˆŸ๎†ก ๎ขŽ๎‰ฝ๎ข
๎ ๎ 
๎‚ก๎Žฉ๎ญ๎ฐ๎˜๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™
๎‚ก๎†ก๎‰ฝ๎‡  ๎‡Œ๎†ก๎ˆฅ๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ƒจ๎‡ซ๎‡Œ๎‡ ๎Š‹๎ขš๎‡ ๎…”๎ˆฅ๎†Ž ๎‰ฝ๎‡ซ๎†Ž๎†ก
๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎‹Š
๎‚ก๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎‹Š
๎‚•๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก
๎ƒฒ๎ˆถ๎ˆ๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎†ก๎‹‰๎‰ข๎ˆถ๎‰ญ๎Š‹๎‡ซ๎ˆฅ๎‡Œ
๎Ÿฟ
๎ ๎Ÿฟ
๎ ƒ๎Ÿฟ
๎ …๎Ÿฟ
๎ ‡๎Ÿฟ
๎ƒพ๎‡ซ๎ˆŸ๎†ก ๎ขŽ๎‰ฝ๎ข
๎ƒก๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎‡ซ๎‰ฝ ๎ˆ‡๎†ก๎‹Š ๎‡‰๎ˆถ๎‰ญ ๎‰ข๎†ก๎‰ญ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎ˆฅ๎†ƒ๎†ก
๎ ๎ 
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ๎กช ๎ €๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ๎กช ๎ €๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‚๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ๎กช ๎ €๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‚๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ƒ๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ๎กช ๎ €๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‚๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ƒ๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ „๎กช ๎ ˆ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ๎กช ๎ €๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‚๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ƒ๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ „๎กช ๎ ˆ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ …๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ๎กช ๎ €๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‚๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ƒ๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ „๎กช ๎ ˆ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ …๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ †๎กช ๎ ƒ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ๎กช ๎ €๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‚๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ƒ๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ „๎กช ๎ ˆ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ …๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ †๎กช ๎ ƒ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‡๎กช ๎ … ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ๎กช ๎ €๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‚๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ƒ๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ „๎กช ๎ ˆ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ …๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ †๎กช ๎ ƒ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‡๎กช ๎ … ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ˆ๎กช ๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ๎กช ๎ €๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‚๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ƒ๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ „๎กช ๎ ˆ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ …๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ †๎กช ๎ ƒ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‡๎กช ๎ … ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ˆ๎กช ๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎Ÿฟ๎กช ๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก
๎ ๎ ‚
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‰ข๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กซ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก๎ข
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎‰ญ๎‡ซ๎Š‹๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎ˆถ๎‰ฝ๎Š‹๎ˆ๎‹Š ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎ˆŸ๎…”๎ˆ๎ˆ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎…”๎‰ญ๎†ƒ๎‡ ๎‡ซ๎Š‹๎†ก๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ๎กช ๎ €๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‚๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ƒ๎กช ๎ €๎Ÿฟ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ „๎กช ๎ ˆ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ …๎กช ๎ ‡ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ †๎กช ๎ ƒ ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ‡๎กช ๎ … ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ ˆ๎กช ๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก
๎€ญ๎ˆถ๎ˆ๎ˆถ๎‰ญ ๎กฐ๎ €๎Ÿฟ๎กช ๎ € ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก
๎ ๎ ‚
๎‚ถ๎ญ๎ซ๎Œผ๎š ๎Ž†๎Œผ๎š๎Ž™๎Œผ๎Ž‹๎ฃ
โ—ฆ ๎†ก๎‡‰๎‹ช๎†ƒ๎‡ซ๎†ก๎ˆฅ๎Š‹ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎‹Š ๎‡ซ๎‰ฝ ๎‡ ๎…”๎‰ญ๎†Ž ๎ขŽ๎†ก๎‰ฝ๎‰ข๎†ก๎†ƒ๎‡ซ๎…”๎ˆ๎ˆ๎‹Š ๎‡‰๎ˆถ๎‰ญ ๎ˆ๎ˆถ๎‹๎ขš๎ˆถ๎‰ญ๎†Ž๎†ก๎‰ญ ๎ข๎†๎ข
โ—ฆ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎…”๎ˆ๎‡Œ๎†ก๎…น๎‰ญ๎…”
โ—ฆ ๎†ก๎‹‰๎…”๎†ƒ๎Š‹ ๎‡‰๎…”๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ขŽLU ๎ˆถ๎‰ญ LDL
H
๎ข
๎ ๎ ƒ
๎€ท๎ฐ๎š๎Œผ๎Œฆ๎ญ ๎ฃ๎Žฉ๎Ž‹๎˜๎Œผ๎š๎ฃ ๎‹ผ๎Ž™๎Œฏ ๎€ฅ๎‚•๎€ƒ๎ƒฒ
โ—ฆ ๎Š‹๎‡ ๎‰ญ๎†ก๎†ก ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ€ข ๎‚ก๎‚‘๎‚• ๎ƒก๎€ƒ๎ƒจ๎€ท๎บ๎ƒฒ๎‚ถ๎กซ ๎บ๎ˆฅ๎Š‹๎†ก๎ˆ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ€ข ๎€ท๎‡ซ๎‰ฝ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ข‚๎ƒฒ๎Š ๎‹–๎Š ๎ˆ‡๎‡ซ ๎…”๎ˆฅ๎†Ž ๎ƒจ๎ˆถ๎Š ๎‹‰ ๎ ๎Ÿฟ๎ €๎ ƒ๎ขƒ
โ€ข ๎‚ก๎„๎‚ก๎ƒก๎ƒฒ๎กผ๎‰ฝ๎†ก๎‰ฉ
โ—ฆ ๎‚ก๎‚‘๎‚• ๎‡‰๎ˆถ๎‰ญ ๎†Ž๎†ก๎ˆฅ๎‰ฝ๎†ก ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎…”๎ˆ๎‡Œ๎†ก๎…น๎‰ญ๎…” ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎…น๎†ก ๎†ƒ๎…”๎‰ญ๎†ก๎‡‰๎Š ๎ˆ ๎‹๎‡ซ๎Š‹๎‡  OMP_NUM_THREADS
๎ ๎ „
๎€ท๎ฐ๎ฃ๎ญ๎š๎ฐ๎Œž๎ผ๎ญ๎Œผ๎Œฏ๎ขš๎Ž•๎Œผ๎Ž•๎Žฉ๎š๎ฃ ๎“๎‹ผ๎š๎‹ผ๎Ž‹๎Ž‹๎Œผ๎Ž‹๎ฐ๎ฃ๎Ž•
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ท๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‚ก๎ƒก๎บ๎ข
โ—ฆ ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก ๎†Ž๎…”๎Š‹๎…” ๎†ก๎‡‰๎‹ช๎†ƒ๎‡ซ๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎Š‹๎ˆถ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎Š ๎ˆฅ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎ˆถ๎‰ญ๎Š‹๎‡ ๎ˆถ๎‡Œ๎ˆถ๎ˆฅ๎…”๎ˆ ๎Š‹๎ˆถ ๎‰ฝ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ
๎ ๎ …
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ท๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‚ก๎ƒก๎บ๎ข
โ—ฆ ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก ๎†Ž๎…”๎Š‹๎…” ๎†ก๎‡‰๎‹ช๎†ƒ๎‡ซ๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎Š‹๎ˆถ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎Š ๎ˆฅ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎ˆถ๎‰ญ๎Š‹๎‡ ๎ˆถ๎‡Œ๎ˆถ๎ˆฅ๎…”๎ˆ ๎Š‹๎ˆถ ๎‰ฝ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ
๎ ๎ …
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ท๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‚ก๎ƒก๎บ๎ข
โ—ฆ ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก ๎†Ž๎…”๎Š‹๎…” ๎†ก๎‡‰๎‹ช๎†ƒ๎‡ซ๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎Š‹๎ˆถ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎Š ๎ˆฅ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎ˆถ๎‰ญ๎Š‹๎‡ ๎ˆถ๎‡Œ๎ˆถ๎ˆฅ๎…”๎ˆ ๎Š‹๎ˆถ ๎‰ฝ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ €๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ …
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ท๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‚ก๎ƒก๎บ๎ข
โ—ฆ ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก ๎†Ž๎…”๎Š‹๎…” ๎†ก๎‡‰๎‹ช๎†ƒ๎‡ซ๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎Š‹๎ˆถ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎Š ๎ˆฅ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎ˆถ๎‰ญ๎Š‹๎‡ ๎ˆถ๎‡Œ๎ˆถ๎ˆฅ๎…”๎ˆ ๎Š‹๎ˆถ ๎‰ฝ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ €๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ ๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ …
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ท๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‚ก๎ƒก๎บ๎ข
โ—ฆ ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก ๎†Ž๎…”๎Š‹๎…” ๎†ก๎‡‰๎‹ช๎†ƒ๎‡ซ๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎Š‹๎ˆถ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎Š ๎ˆฅ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎ˆถ๎‰ญ๎Š‹๎‡ ๎ˆถ๎‡Œ๎ˆถ๎ˆฅ๎…”๎ˆ ๎Š‹๎ˆถ ๎‰ฝ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ €๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ ๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ ‚๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ …
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ท๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‚ก๎ƒก๎บ๎ข
โ—ฆ ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก ๎†Ž๎…”๎Š‹๎…” ๎†ก๎‡‰๎‹ช๎†ƒ๎‡ซ๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎Š‹๎ˆถ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎Š ๎ˆฅ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎ˆถ๎‰ญ๎Š‹๎‡ ๎ˆถ๎‡Œ๎ˆถ๎ˆฅ๎…”๎ˆ ๎Š‹๎ˆถ ๎‰ฝ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ €๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ ๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ ‚๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ ƒ๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ …
๎บ๎Ž•๎“๎‹ผ๎Œฆ๎ญ ๎Žฉ๎Ž™ ๎ญ๎ซ๎Œผ ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œผ๎Ž‹๎Œผ๎Ž•๎Œผ๎Ž™๎ญ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ท๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎ขŽ๎‚ก๎ƒก๎บ๎ข
โ—ฆ ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎…”๎†Ž๎†Ž๎‰ญ๎†ก๎‰ฝ๎‰ฝ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก
โ—ฆ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก ๎†Ž๎…”๎Š‹๎…” ๎†ก๎‡‰๎‹ช๎†ƒ๎‡ซ๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎Š‹๎ˆถ ๎ˆŸ๎‡ซ๎ˆฅ๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎Š ๎ˆฅ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎ˆถ๎‰ญ๎Š‹๎‡ ๎ˆถ๎‡Œ๎ˆถ๎ˆฅ๎…”๎ˆ ๎Š‹๎ˆถ ๎‰ฝ๎‡ ๎…”๎‰ญ๎†ก๎†Ž๎ขš๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ €๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ ๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ ‚๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎กฐ๎ ƒ๎กช ๎ €๎ † ๎Š‹๎‰ญ๎‡ซ๎…”๎ˆฅ๎‡Œ๎ˆ๎†ก๎‰ฝ
๎ ๎ …
๎‚ก๎Œผ๎ฃ๎ฃ๎‹ผ๎ ๎Œผ ๎ƒก๎‹ผ๎ฃ๎ฃ๎ฐ๎Ž™๎  ๎บ๎Ž™๎ญ๎Œผ๎š๎š๎‹ผ๎Œฆ๎Œผ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ค‹๎ค‹๎ขš๎ˆŸ๎‰ข๎‡ซ
โ—ฆ ๎‡ฟ๎Š ๎‰ฝ๎Š‹ ๎ˆ๎‡ซ๎ˆ‡๎†ก ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ค‹๎ค‹๎กซ ๎…น๎Š ๎Š‹ ๎‹๎‡ซ๎Š‹๎‡  ๎‚ก๎ƒก๎บ
โ—ฆ ๎…” ๎‡‰๎‰ญ๎‡ซ๎†ก๎ˆฅ๎†Ž๎ˆ๎‹Š ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎…”๎†ƒ๎‡  ๎Š‹๎ˆถ ๎ˆŸ๎†ก๎‰ฝ๎‰ฝ๎…”๎‡Œ๎†ก ๎‰ข๎…”๎‰ฝ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎ขŽ๎ˆ๎‡ซ๎ˆ‡๎†ก ๎ˆŸ๎‰ข๎‡ซ๎ ƒ๎‰ข๎‹Š๎ข
โ—ฆ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก ๎ˆฅ๎†ก๎‹ ๎Š‹๎‹Š๎‰ข๎†ก๎‰ฝ ๎ˆ๎‡ซ๎ˆ‡๎†ก mpiComm ๎ˆถ๎‰ญ mpiRequest
โ—ฆ ๎ˆฅ๎ˆถ ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ ๎‹๎‡ซ๎Š‹๎‡  mesh๎กซ matrix๎กซ ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎‰ฝ
๎ ๎ †
๎€ƒ๎ฃ๎ฃ๎Œผ๎Ž•๎Œž๎Ž‹๎ฃ ๎‹ผ๎Ž™๎Œฏ ๎Œฏ๎ฐ๎š๎Œผ๎Œฆ๎ญ ๎ฃ๎Žฉ๎Ž‹๎˜๎Œผ๎š๎ฃ
โ—ฆ ๎ˆฅ๎…”๎‡ซ๎Šฝ๎†ก ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎…”๎†ƒ๎‡  ๎Š‹๎ˆถ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎…”๎‰ฝ๎‰ฝ๎Š ๎ˆŸ๎‡ซ๎ˆฅ๎‡Œ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎Šฝ๎…”๎‰ญ๎‡ซ๎…”๎…น๎ˆ๎†ก๎‰ฝ ๎ˆŸ๎…”๎‹Š ๎…น๎†ก ๎‰ญ๎†ก๎‰ข๎ˆ๎‡ซ๎†ƒ๎…”๎Š‹๎†ก๎†Ž
๎‚•๎†ก๎‡Œ๎…”๎†ƒ๎‹Š ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡‰๎…”๎†ƒ๎†ก๎‰ฝ
โ—ฆ ๎‚ก๎„๎‚ก๎ƒก๎ƒฒ ๎ข‚๎€ƒ๎ˆŸ๎†ก๎‰ฝ๎Š‹๎ˆถ๎‹Š ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ €๎ขƒ
โ—ฆ ๎ƒก๎…”๎ƒฒ๎Š‹๎‡ซ๎„ด ๎ข‚๎ฑ๎†ข๎ˆฅ๎ˆถ๎ˆฅ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ ๎ขƒ
โ—ฆ ๎ƒฒ๎Š ๎‰ข๎†ก๎‰ญ๎‚•๎„๎กผ๎€ท๎บ๎ƒฒ๎ƒพ ๎ข‚๎‚•๎‡ซ ๎ ๎Ÿฟ๎Ÿฟ๎ „๎ขƒ
๎ ๎ ‡
๎€ƒ๎ฃ๎ฃ๎Œผ๎Ž•๎Œž๎Ž‹๎ฃ ๎‹ผ๎Ž™๎Œฏ ๎Œฏ๎ฐ๎š๎Œผ๎Œฆ๎ญ ๎ฃ๎Žฉ๎Ž‹๎˜๎Œผ๎š๎ฃ
โ—ฆ ๎ˆฅ๎…”๎‡ซ๎Šฝ๎†ก ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎…”๎†ƒ๎‡  ๎Š‹๎ˆถ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎…”๎‰ฝ๎‰ฝ๎Š ๎ˆŸ๎‡ซ๎ˆฅ๎‡Œ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎Šฝ๎…”๎‰ญ๎‡ซ๎…”๎…น๎ˆ๎†ก๎‰ฝ ๎ˆŸ๎…”๎‹Š ๎…น๎†ก ๎‰ญ๎†ก๎‰ข๎ˆ๎‡ซ๎†ƒ๎…”๎Š‹๎†ก๎†Ž
๎‚•๎†ก๎‡Œ๎…”๎†ƒ๎‹Š ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡‰๎…”๎†ƒ๎†ก๎‰ฝ
โ—ฆ ๎‚ก๎„๎‚ก๎ƒก๎ƒฒ ๎ข‚๎€ƒ๎ˆŸ๎†ก๎‰ฝ๎Š‹๎ˆถ๎‹Š ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ €๎ขƒ
โ—ฆ ๎ƒก๎…”๎ƒฒ๎Š‹๎‡ซ๎„ด ๎ข‚๎ฑ๎†ข๎ˆฅ๎ˆถ๎ˆฅ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ ๎ขƒ
โ—ฆ ๎ƒฒ๎Š ๎‰ข๎†ก๎‰ญ๎‚•๎„๎กผ๎€ท๎บ๎ƒฒ๎ƒพ ๎ข‚๎‚•๎‡ซ ๎ ๎Ÿฟ๎Ÿฟ๎ „๎ขƒ
๎ ๎ ‡
๎€ƒ๎ฃ๎ฃ๎Œผ๎Ž•๎Œž๎Ž‹๎ฃ ๎‹ผ๎Ž™๎Œฏ ๎Œฏ๎ฐ๎š๎Œผ๎Œฆ๎ญ ๎ฃ๎Žฉ๎Ž‹๎˜๎Œผ๎š๎ฃ
โ—ฆ ๎ˆฅ๎…”๎‡ซ๎Šฝ๎†ก ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎…”๎†ƒ๎‡  ๎Š‹๎ˆถ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎…”๎‰ฝ๎‰ฝ๎Š ๎ˆŸ๎‡ซ๎ˆฅ๎‡Œ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎Šฝ๎…”๎‰ญ๎‡ซ๎…”๎…น๎ˆ๎†ก๎‰ฝ ๎ˆŸ๎…”๎‹Š ๎…น๎†ก ๎‰ญ๎†ก๎‰ข๎ˆ๎‡ซ๎†ƒ๎…”๎Š‹๎†ก๎†Ž
๎‚•๎†ก๎‡Œ๎…”๎†ƒ๎‹Š ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡‰๎…”๎†ƒ๎†ก๎‰ฝ
โ—ฆ ๎‚ก๎„๎‚ก๎ƒก๎ƒฒ ๎ข‚๎€ƒ๎ˆŸ๎†ก๎‰ฝ๎Š‹๎ˆถ๎‹Š ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ €๎ขƒ
โ—ฆ ๎ƒก๎…”๎ƒฒ๎Š‹๎‡ซ๎„ด ๎ข‚๎ฑ๎†ข๎ˆฅ๎ˆถ๎ˆฅ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ ๎ขƒ
โ—ฆ ๎ƒฒ๎Š ๎‰ข๎†ก๎‰ญ๎‚•๎„๎กผ๎€ท๎บ๎ƒฒ๎ƒพ ๎ข‚๎‚•๎‡ซ ๎ ๎Ÿฟ๎Ÿฟ๎ „๎ขƒ
๎ ๎ ‡
๎€ƒ๎ฃ๎ฃ๎Œผ๎Ž•๎Œž๎Ž‹๎ฃ ๎‹ผ๎Ž™๎Œฏ ๎Œฏ๎ฐ๎š๎Œผ๎Œฆ๎ญ ๎ฃ๎Žฉ๎Ž‹๎˜๎Œผ๎š๎ฃ
โ—ฆ ๎ˆฅ๎…”๎‡ซ๎Šฝ๎†ก ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎…”๎†ƒ๎‡  ๎Š‹๎ˆถ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎…”๎‰ฝ๎‰ฝ๎Š ๎ˆŸ๎‡ซ๎ˆฅ๎‡Œ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎Šฝ๎…”๎‰ญ๎‡ซ๎…”๎…น๎ˆ๎†ก๎‰ฝ ๎ˆŸ๎…”๎‹Š ๎…น๎†ก ๎‰ญ๎†ก๎‰ข๎ˆ๎‡ซ๎†ƒ๎…”๎Š‹๎†ก๎†Ž
๎‚•๎†ก๎‡Œ๎…”๎†ƒ๎‹Š ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡‰๎…”๎†ƒ๎†ก๎‰ฝ โ‡ ๎Š ๎‰ฝ๎†ก ๎…”๎Š‹ ๎‹Š๎ˆถ๎Š ๎‰ญ ๎ˆถ๎‹๎ˆฅ ๎‰ญ๎‡ซ๎‰ฝ๎ˆ‡๎กฎ
โ—ฆ ๎‚ก๎„๎‚ก๎ƒก๎ƒฒ ๎ข‚๎€ƒ๎ˆŸ๎†ก๎‰ฝ๎Š‹๎ˆถ๎‹Š ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ €๎ขƒ
โ—ฆ ๎ƒก๎…”๎ƒฒ๎Š‹๎‡ซ๎„ด ๎ข‚๎ฑ๎†ข๎ˆฅ๎ˆถ๎ˆฅ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ ๎ขƒ
โ—ฆ ๎ƒฒ๎Š ๎‰ข๎†ก๎‰ญ๎‚•๎„๎กผ๎€ท๎บ๎ƒฒ๎ƒพ ๎ข‚๎‚•๎‡ซ ๎ ๎Ÿฟ๎Ÿฟ๎ „๎ขƒ
๎ ๎ ‡
๎ƒก๎‹ผ๎š๎‹ผ๎„ช๎ฐ๎Œผ๎›
โ—ฆ ๎ˆŸ๎ˆถ๎‰ญ๎†ก ๎‰ข๎ˆถ๎‹๎†ก๎‰ญ๎‡‰๎Š ๎ˆ ๎‰ข๎ˆถ๎‰ฝ๎Š‹๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎Š‹๎ˆถ๎ˆถ๎ˆ
โ—ฆ ๎‡ ๎…”๎ˆฅ๎†Ž๎ˆ๎†ก ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž ๎‰ฝ๎ˆถ๎ˆ๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎‡Œ๎†ก๎ˆฅ๎†ก๎‰ญ๎…”๎Š‹๎†ก ๎ˆŸ๎ˆถ๎Šฝ๎‡ซ๎†ก๎‰ฝ ๎‡ซ๎‡‰ ๎ˆฅ๎†ก๎†ก๎†Ž๎†ก๎†Ž
โ—ฆ savevtk("sol.vtu",Th,sol)
๎ ๎ ˆ
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ ‚๎กช ๎ซ๎Œผ๎‹ผ๎ญ ๎Œผ๎˜๎ผ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎Š‹๎‡ ๎†ก ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‡ซ๎†ก๎ˆฅ๎Š‹ ๎ƒก๎€ท๎†
โˆ‚
u
โˆ‚t
โˆ’ โˆ†u = ๎ € ๎‡ซ๎ˆฅ โ„ฆ ร— [๎Ÿฟ; T]
u(x, y, z, ๎Ÿฟ) = ๎ € ๎‡ซ๎ˆฅ โ„ฆ
u(x, y, z, t) = ๎ € ๎ˆถ๎ˆฅ ฮ“ ร— [๎Ÿฟ; T]
๎บ๎ˆŸ๎‰ข๎ˆ๎‡ซ๎†ƒ๎‡ซ๎Š‹ ๎†๎Š ๎ˆ๎†ก๎‰ญ ๎‰ฝ๎†ƒ๎‡ ๎†ก๎ˆŸ๎†ก
๎š
โ„ฆ
u
n+๎ €
โˆ’ u
n
๎†Žt
w + โˆ‡u
n
โˆ‡w =
๎š
โ„ฆ
w
=โ‡’
(M + ๎†Žt ยท A)u
n+๎ €
= Mu
n
+ ๎†Žt ยท b
๎ ‚๎Ÿฟ
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ ‚๎กช ๎ซ๎Œผ๎‹ผ๎ญ ๎Œผ๎˜๎ผ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎Š‹๎‡ ๎†ก ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‡ซ๎†ก๎ˆฅ๎Š‹ ๎ƒก๎€ท๎†
โˆ‚
u
โˆ‚t
โˆ’ โˆ†u = ๎ € ๎‡ซ๎ˆฅ โ„ฆ ร— [๎Ÿฟ; T]
u(x, y, z, ๎Ÿฟ) = ๎ € ๎‡ซ๎ˆฅ โ„ฆ
u(x, y, z, t) = ๎ € ๎ˆถ๎ˆฅ ฮ“ ร— [๎Ÿฟ; T]
๎บ๎ˆŸ๎‰ข๎ˆ๎‡ซ๎†ƒ๎‡ซ๎Š‹ ๎†๎Š ๎ˆ๎†ก๎‰ญ ๎‰ฝ๎†ƒ๎‡ ๎†ก๎ˆŸ๎†ก
๎š
โ„ฆ
u
n+๎ €
โˆ’ u
n
๎†Žt
w + โˆ‡u
n
โˆ‡w =
๎š
โ„ฆ
w
=โ‡’
(M + ๎†Žt ยท A)u
n+๎ €
= Mu
n
+ ๎†Žt ยท b
๎ ‚๎Ÿฟ
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ ‚๎กช ๎ซ๎Œผ๎‹ผ๎ญ ๎Œผ๎˜๎ผ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎Š‹๎‡ ๎†ก ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‡ซ๎†ก๎ˆฅ๎Š‹ ๎ƒก๎€ท๎†
โˆ‚
u
โˆ‚t
โˆ’ โˆ†u = ๎ € ๎‡ซ๎ˆฅ โ„ฆ ร— [๎Ÿฟ; T]
u(x, y, z, ๎Ÿฟ) = ๎ € ๎‡ซ๎ˆฅ โ„ฆ
u(x, y, z, t) = ๎ € ๎ˆถ๎ˆฅ ฮ“ ร— [๎Ÿฟ; T]
๎บ๎ˆŸ๎‰ข๎ˆ๎‡ซ๎†ƒ๎‡ซ๎Š‹ ๎†๎Š ๎ˆ๎†ก๎‰ญ ๎‰ฝ๎†ƒ๎‡ ๎†ก๎ˆŸ๎†ก
๎š
โ„ฆ
u
n+๎ €
โˆ’ u
n
๎†Žt
w + โˆ‡u
n
โˆ‡w =
๎š
โ„ฆ
w
=โ‡’
(M + ๎†Žt ยท A)u
n+๎ €
= Mu
n
+ ๎†Žt ยท b
๎ ‚๎Ÿฟ
๎‚ถ๎˜๎Œผ๎š๎Ž‹๎‹ผ๎“๎“๎ฐ๎Ž™๎  ๎ƒฒ๎Œฆ๎ซ๎›๎‹ผ๎š๎ฎ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
๎ฑ๎ฐ๎ฃ๎ญ๎Žฉ๎š๎ฃ
โ—ฆ ๎‡ซ๎ˆฅ๎‡ซ๎Š‹๎‡ซ๎…”๎ˆ๎ˆ๎‹Š ๎‡‰๎ˆถ๎†ƒ๎Š ๎‰ฝ๎†ก๎†Ž ๎ˆถ๎ˆฅ ๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ
โ—ฆ ๎ˆฅ๎†ก๎‹ ๎†Ž๎†ก๎Šฝ๎†ก๎ˆ๎ˆถ๎‰ข๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ ๎…”๎‰ญ๎ˆถ๎Š ๎ˆฅ๎†Ž ๎‡ซ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎Šฝ๎†ก ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ ๎ข‚๎‚๎ˆถ๎ˆ๎‡ซ๎Šฝ๎†ก๎Š‹
๎…”๎ˆฅ๎†Ž ๎ƒพ๎ˆถ๎Š ๎‰ญ๎ˆฅ๎‡ซ๎†ก๎‰ญ ๎ ๎Ÿฟ๎ €๎ …๎ขƒ
โ—ฆ ๎ˆถ๎ˆฅ๎ˆ๎‹Š ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡‰๎…”๎†ƒ๎†ก๎†Ž ๎‹๎‡ซ๎Š‹๎‡  ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎…”๎Š‹ ๎‹ช๎‰ญ๎‰ฝ๎Š‹
โ—ฆ ๎ˆ๎ˆถ๎‹๎ขš๎ˆ๎†ก๎Šฝ๎†ก๎ˆ ๎ˆ๎…”๎ˆฅ๎‡Œ๎Š ๎…”๎‡Œ๎†ก๎‰ฝ ๎ขŽ๎€ญ๎กธ๎€ญ๎ค‹๎ค‹๎กซ ๎ข๎ˆถ๎‰ญ๎Š‹๎‰ญ๎…”๎ˆฅ๎กซ ๎ƒก๎‹Š๎Š‹๎‡ ๎ˆถ๎ˆฅ๎ข
โ—ฆ ๎‹ช๎‰ญ๎‰ฝ๎Š‹ ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎‡ซ๎Š‹ ๎‡ซ๎ˆฅ ๎ˆ๎…”๎Š‹๎†ก ๎ ๎Ÿฟ๎ €๎ €
โ—ฆ ๎ˆถ๎‰ข๎†ก๎ˆฅ๎ขš๎‰ฝ๎ˆถ๎Š ๎‰ญ๎†ƒ๎†ก๎†Ž ๎‡ซ๎ˆฅ ๎ˆ๎…”๎Š‹๎†ก ๎ ๎Ÿฟ๎ €๎ ƒ
โ—ฆ https://github.com/hpddm/hpddm
โ—ฆ ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‡Œ๎‰ญ๎…”๎Š‹๎†ก๎†Ž ๎‡ซ๎ˆฅ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎‡ซ๎ˆฅ ๎ ๎Ÿฟ๎ €๎ ˆ
๎ ‚๎ €
๎ฑ๎ฐ๎ฃ๎ญ๎Žฉ๎š๎ฐ๎Œฆ๎‹ผ๎Ž‹ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ
โ—ฆ ๎†Ž๎Š ๎†ก ๎Š‹๎ˆถ ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎ขŽ๎ €๎ ‡๎ †๎Ÿฟ๎ข
โ—ฆ ๎‡ ๎ˆถ๎‹ ๎Š‹๎ˆถ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก ๎ƒก๎ˆถ๎‡ซ๎‰ฝ๎‰ฝ๎ˆถ๎ˆฅ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎ˆฅ ๎ขจ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ขฉ ๎‡Œ๎†ก๎ˆถ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ก๎‰ฝ๎กณ
โ„ฆ
โ—ฆ ๎…น๎‹Š ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ๎‰ฝ ๎Š‹๎‡ ๎…”๎Š‹ ๎‹๎ˆถ๎‰ญ๎ˆ‡ ๎ˆถ๎ˆฅ ๎‰ฝ๎‡ซ๎ˆŸ๎‰ข๎ˆ๎†ก๎‰ญ ๎‰ฝ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ๎‰ฝ
๎ ‚๎ 
๎ฑ๎ฐ๎ฃ๎ญ๎Žฉ๎š๎ฐ๎Œฆ๎‹ผ๎Ž‹ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ
โ—ฆ ๎†Ž๎Š ๎†ก ๎Š‹๎ˆถ ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎ขŽ๎ €๎ ‡๎ †๎Ÿฟ๎ข
โ—ฆ ๎‡ ๎ˆถ๎‹ ๎Š‹๎ˆถ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก ๎ƒก๎ˆถ๎‡ซ๎‰ฝ๎‰ฝ๎ˆถ๎ˆฅ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎ˆฅ ๎ขจ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ขฉ ๎‡Œ๎†ก๎ˆถ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ก๎‰ฝ๎กณ
โ„ฆ
๎ 
โ„ฆ
๎ €
โ—ฆ ๎…น๎‹Š ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ๎‰ฝ ๎Š‹๎‡ ๎…”๎Š‹ ๎‹๎ˆถ๎‰ญ๎ˆ‡ ๎ˆถ๎ˆฅ ๎‰ฝ๎‡ซ๎ˆŸ๎‰ข๎ˆ๎†ก๎‰ญ ๎‰ฝ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ๎‰ฝ
๎ ‚๎ 
๎€ท๎ฐ๎ฃ๎Œฆ๎š๎Œผ๎ญ๎ฐ๎ฎ๎Œผ๎Œฏ ๎Ž•๎Žฉ๎Œฏ๎Œผ๎Ž‹ ๎“๎š๎Žฉ๎Œž๎Ž‹๎Œผ๎Ž• ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎ˆฅ๎ˆถ ๎ˆฅ๎†ก๎†ก๎†Ž ๎‡‰๎ˆถ๎‰ญ ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎Š‹๎†ก ๎ˆŸ๎†ก๎‰ฝ๎‡ 
โ—ฆ ๎ˆฅ๎†ก๎‡ซ๎‡Œ๎‡ ๎…น๎ˆถ๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ ๎ˆถ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎ˆถ๎Šฝ๎†ก๎‰ญ๎ˆ๎…”๎‰ข๎‰ฝ
๎ ‚๎ ‚
๎€ท๎ฐ๎ฃ๎Œฆ๎š๎Œผ๎ญ๎ฐ๎ฎ๎Œผ๎Œฏ ๎Ž•๎Žฉ๎Œฏ๎Œผ๎Ž‹ ๎“๎š๎Žฉ๎Œž๎Ž‹๎Œผ๎Ž• ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎ˆฅ๎ˆถ ๎ˆฅ๎†ก๎†ก๎†Ž ๎‡‰๎ˆถ๎‰ญ ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎Š‹๎†ก ๎ˆŸ๎†ก๎‰ฝ๎‡ 
โ—ฆ ๎ˆฅ๎†ก๎‡ซ๎‡Œ๎‡ ๎…น๎ˆถ๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ ๎ˆถ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎ˆถ๎Šฝ๎†ก๎‰ญ๎ˆ๎…”๎‰ข๎‰ฝ
๎ ‚๎ ‚
๎€ท๎ฐ๎ฃ๎Œฆ๎š๎Œผ๎ญ๎ฐ๎ฎ๎Œผ๎Œฏ ๎Ž•๎Žฉ๎Œฏ๎Œผ๎Ž‹ ๎“๎š๎Žฉ๎Œž๎Ž‹๎Œผ๎Ž• ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎ˆฅ๎ˆถ ๎ˆฅ๎†ก๎†ก๎†Ž ๎‡‰๎ˆถ๎‰ญ ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎Š‹๎†ก ๎ˆŸ๎†ก๎‰ฝ๎‡ 
โ—ฆ ๎ˆฅ๎†ก๎‡ซ๎‡Œ๎‡ ๎…น๎ˆถ๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ ๎ˆถ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎ˆถ๎Šฝ๎†ก๎‰ญ๎ˆ๎…”๎‰ข๎‰ฝ
๎ ‚๎ ‚
๎€ท๎ฐ๎ฃ๎Œฆ๎š๎Œผ๎ญ๎ฐ๎ฎ๎Œผ๎Œฏ ๎Ž•๎Žฉ๎Œฏ๎Œผ๎Ž‹ ๎“๎š๎Žฉ๎Œž๎Ž‹๎Œผ๎Ž• ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
R
๎ €
R
๎ 
โ—ฆ ๎ˆฅ๎ˆถ ๎ˆฅ๎†ก๎†ก๎†Ž ๎‡‰๎ˆถ๎‰ญ ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎Š‹๎†ก ๎ˆŸ๎†ก๎‰ฝ๎‡ 
โ—ฆ ๎ˆฅ๎†ก๎‡ซ๎‡Œ๎‡ ๎…น๎ˆถ๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ ๎ˆถ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎ˆถ๎Šฝ๎†ก๎‰ญ๎ˆ๎…”๎‰ข๎‰ฝ
๎ ‚๎ ‚
๎€ท๎ฐ๎ฃ๎Œฆ๎š๎Œผ๎ญ๎ฐ๎ฎ๎Œผ๎Œฏ ๎Ž•๎Žฉ๎Œฏ๎Œผ๎Ž‹ ๎“๎š๎Žฉ๎Œž๎Ž‹๎Œผ๎Ž• ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
R
๎ €
R
๎ 
R
T
๎ 
R
T
๎ €
โ—ฆ ๎ˆฅ๎ˆถ ๎ˆฅ๎†ก๎†ก๎†Ž ๎‡‰๎ˆถ๎‰ญ ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎Š‹๎†ก ๎ˆŸ๎†ก๎‰ฝ๎‡ 
โ—ฆ ๎ˆฅ๎†ก๎‡ซ๎‡Œ๎‡ ๎…น๎ˆถ๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ ๎ˆถ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎ˆถ๎Šฝ๎†ก๎‰ญ๎ˆ๎…”๎‰ข๎‰ฝ
๎ ‚๎ ‚
๎€ท๎ฐ๎ฃ๎Œฆ๎š๎Œผ๎ญ๎ฐ๎ฎ๎Œผ๎Œฏ ๎Ž•๎Žฉ๎Œฏ๎Œผ๎Ž‹ ๎“๎š๎Žฉ๎Œž๎Ž‹๎Œผ๎Ž• ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
R
๎ €
R
๎ 
R
T
๎ 
R
T
๎ €
โ—ฆ ๎ˆฅ๎ˆถ ๎ˆฅ๎†ก๎†ก๎†Ž ๎‡‰๎ˆถ๎‰ญ ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎Š‹๎†ก ๎ˆŸ๎†ก๎‰ฝ๎‡ 
โ—ฆ ๎ˆฅ๎†ก๎‡ซ๎‡Œ๎‡ ๎…น๎ˆถ๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ ๎ˆถ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎ˆถ๎Šฝ๎†ก๎‰ญ๎ˆ๎…”๎‰ข๎‰ฝ
๎ ‚๎ ‚
๎„๎Ž™๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎Œฏ ๎ฐ๎ญ๎Œผ๎š๎‹ผ๎ญ๎ฐ๎˜๎Œผ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
โ—ฆ ๎‰ข๎…”๎‰ญ๎Š‹๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‡‰ ๎Š ๎ˆฅ๎‡ซ๎Š‹๎‹Š I =
N
๎˜
i=๎ €
R
T
i
D
i
R
i
โ—ฆ ๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ (u, v) =
N
๎˜
i=๎ €
(
R
i
u
,
D
i
R
i
v
)
โ—ฆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ R
i
Au = R
i
N
๎˜
j=๎ €
R
T
j
R
j
AR
T
j
D
j
R
j
u
=
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๎‰ฝ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ๎‰ฝ ๎ˆถ๎ˆฅ๎ˆ๎‹Š ๎‰ญ๎†ก๎‰ฉ๎Š ๎‡ซ๎‰ญ๎†ก ๎ขจ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎†Ž๎…”๎Š‹๎…”๎ขฉ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ ‚๎ ƒ
๎„๎Ž™๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎Œฏ ๎ฐ๎ญ๎Œผ๎š๎‹ผ๎ญ๎ฐ๎˜๎Œผ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
โ—ฆ ๎‰ข๎…”๎‰ญ๎Š‹๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‡‰ ๎Š ๎ˆฅ๎‡ซ๎Š‹๎‹Š I =
N
๎˜
i=๎ €
R
T
i
D
i
R
i
โ—ฆ ๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ (u, v) =
N
๎˜
i=๎ €
(
R
i
u
,
D
i
R
i
v
)
โ—ฆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ R
i
Au = R
i
N
๎˜
j=๎ €
R
T
j
R
j
AR
T
j
D
j
R
j
u
=
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๎‰ฝ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ๎‰ฝ ๎ˆถ๎ˆฅ๎ˆ๎‹Š ๎‰ญ๎†ก๎‰ฉ๎Š ๎‡ซ๎‰ญ๎†ก ๎ขจ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎†Ž๎…”๎Š‹๎…”๎ขฉ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ ‚๎ ƒ
๎„๎Ž™๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎Œฏ ๎ฐ๎ญ๎Œผ๎š๎‹ผ๎ญ๎ฐ๎˜๎Œผ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
โ—ฆ ๎‰ข๎…”๎‰ญ๎Š‹๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‡‰ ๎Š ๎ˆฅ๎‡ซ๎Š‹๎‹Š I =
N
๎˜
i=๎ €
R
T
i
D
i
R
i
โ—ฆ ๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ (u, v) =
N
๎˜
i=๎ €
(
R
i
u
,
D
i
R
i
v
)
โ—ฆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ R
i
Au = R
i
N
๎˜
j=๎ €
R
T
j
R
j
AR
T
j
D
j
R
j
u
=
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๎‰ฝ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ๎‰ฝ ๎ˆถ๎ˆฅ๎ˆ๎‹Š ๎‰ญ๎†ก๎‰ฉ๎Š ๎‡ซ๎‰ญ๎†ก ๎ขจ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎†Ž๎…”๎Š‹๎…”๎ขฉ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ ‚๎ ƒ
๎„๎Ž™๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎Œฏ ๎ฐ๎ญ๎Œผ๎š๎‹ผ๎ญ๎ฐ๎˜๎Œผ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
โ—ฆ ๎‰ข๎…”๎‰ญ๎Š‹๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‡‰ ๎Š ๎ˆฅ๎‡ซ๎Š‹๎‹Š I =
N
๎˜
i=๎ €
R
T
i
D
i
R
i
โ—ฆ ๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ (u, v) =
N
๎˜
i=๎ €
(
R
i
u
,
D
i
R
i
v
)
โ—ฆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ R
i
Au = R
i
N
๎˜
j=๎ €
R
T
j
R
j
AR
T
j
D
j
R
j
u
=
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๎‰ฝ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ๎‰ฝ ๎ˆถ๎ˆฅ๎ˆ๎‹Š ๎‰ญ๎†ก๎‰ฉ๎Š ๎‡ซ๎‰ญ๎†ก ๎ขจ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎†Ž๎…”๎Š‹๎…”๎ขฉ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ ‚๎ ƒ
๎‚ถ๎˜๎Œผ๎š๎Ž‹๎‹ผ๎“๎“๎ฐ๎Ž™๎  ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹–
M
โˆ’๎ €
๎€ƒ๎ƒฒ๎‚ก
=
N
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i=๎ €
R
T
i
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R
i
AR
T
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๎€
โˆ’๎ €
R
i
โ—ฆ ๎‰ญ๎†ก๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎†ƒ๎Š‹๎†ก๎†Ž ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎ข‚๎€ญ๎…”๎‡ซ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ ‚๎ขƒ
M
โˆ’๎ €
๎ƒจ๎€ƒ๎ƒฒ
=
N
๎˜
i
=
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R
T
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D
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R
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AR
T
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โ—ฆ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก๎†Ž ๎‰ญ๎†ก๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎†ƒ๎Š‹๎†ก๎†Ž ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎ข‚๎ƒฒ๎Š‹๎ขš๎€ญ๎‹Š๎‰ญ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ †๎ขƒ
M
โˆ’๎ €
๎‚ถ๎ƒจ๎€ƒ๎ƒฒ
=
N
๎˜
i=๎ €
R
T
i
D
i
B
โˆ’๎ €
i
R
i
๎ ‚๎ „
๎‚ถ๎˜๎Œผ๎š๎Ž‹๎‹ผ๎“๎“๎ฐ๎Ž™๎  ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹–
M
โˆ’๎ €
๎€ƒ๎ƒฒ๎‚ก
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N
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R
T
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R
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โ—ฆ ๎‰ญ๎†ก๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎†ƒ๎Š‹๎†ก๎†Ž ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎ข‚๎€ญ๎…”๎‡ซ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ ‚๎ขƒ
M
โˆ’๎ €
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=
N
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R
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R
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AR
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โ—ฆ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก๎†Ž ๎‰ญ๎†ก๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎†ƒ๎Š‹๎†ก๎†Ž ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎ข‚๎ƒฒ๎Š‹๎ขš๎€ญ๎‹Š๎‰ญ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ †๎ขƒ
M
โˆ’๎ €
๎‚ถ๎ƒจ๎€ƒ๎ƒฒ
=
N
๎˜
i=๎ €
R
T
i
D
i
B
โˆ’๎ €
i
R
i
๎ ‚๎ „
๎‚ถ๎˜๎Œผ๎š๎Ž‹๎‹ผ๎“๎“๎ฐ๎Ž™๎  ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹–
M
โˆ’๎ €
๎€ƒ๎ƒฒ๎‚ก
=
N
๎˜
i=๎ €
R
T
i
๎€€
R
i
AR
T
i
๎€
โˆ’๎ €
R
i
โ—ฆ ๎‰ญ๎†ก๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎†ƒ๎Š‹๎†ก๎†Ž ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎ข‚๎€ญ๎…”๎‡ซ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ ‚๎ขƒ
M
โˆ’๎ €
๎ƒจ๎€ƒ๎ƒฒ
=
N
๎˜
i
=
๎ €
R
T
i
D
i
๎€€
R
i
AR
T
i
๎€
โˆ’๎ €
R
i
โ—ฆ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก๎†Ž ๎‰ญ๎†ก๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎†ƒ๎Š‹๎†ก๎†Ž ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎ข‚๎ƒฒ๎Š‹๎ขš๎€ญ๎‹Š๎‰ญ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ †๎ขƒ
M
โˆ’๎ €
๎‚ถ๎ƒจ๎€ƒ๎ƒฒ
=
N
๎˜
i=๎ €
R
T
i
D
i
B
โˆ’๎ €
i
R
i
๎ ‚๎ „
๎ƒฒ๎Žฉ๎Ž•๎Œผ ๎ผ๎ญ๎ฐ๎Ž‹๎ฐ๎ญ๎ฃ ๎š๎Žฉ๎ผ๎ญ๎ฐ๎Ž™๎Œผ๎ฃ
โ—ฆ exchange ๎‡‰๎ˆถ๎‰ญ ๎†ƒ๎ˆถ๎ˆฅ๎‰ฝ๎‡ซ๎‰ฝ๎Š‹๎†ก๎ˆฅ๎†ƒ๎‹Š ๎‡ซ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎ˆถ๎Šฝ๎†ก๎‰ญ๎ˆ๎…”๎‰ข
โ—ฆ statistics ๎Š‹๎ˆถ ๎‡Œ๎†ก๎Š‹ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ๎‰ฝ ๎…”๎…น๎ˆถ๎Š ๎Š‹ ๎Š‹๎‡ ๎†ก ๎€ท๎€ท
โ—ฆ A(u,v) ๎Š‹๎ˆถ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎†ก ๎‹๎†ก๎‡ซ๎‡Œ๎‡ ๎Š‹๎†ก๎†Ž ๎†Ž๎ˆถ๎Š‹ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹๎‰ฝ
โ—ฆ ChangeOperator ๎Š‹๎ˆถ ๎Š ๎‰ข๎†Ž๎…”๎Š‹๎†ก ๎…” ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰
๎ ‚๎ …
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎Ÿฟ.๎Ÿฟ
โˆ’
๎†Ž
๎ 
u
๎†Žx
๎ 
= ๎ €
u(๎Ÿฟ) = ๎Ÿฟ
๎…๎†ก๎‰ญ๎ˆถ ๎‡ซ๎ˆฅ๎‡ซ๎Š‹๎‡ซ๎…”๎ˆ ๎‡Œ๎Š ๎†ก๎‰ฝ๎‰ฝ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎Ÿฟ.๎ ‡
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ €
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ €.๎ €
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ 
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ €.๎ …
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ ‚
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ €.๎ ˆ
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ ƒ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ .๎ ƒ
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ „
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ €๎ ….๎ …
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ „๎Ÿฟ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ ๎ †.๎ ‡
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ €๎Ÿฟ๎Ÿฟ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ ‚๎ „.๎ ‚
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ €๎ „๎Ÿฟ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ ƒ๎Ÿฟ.๎ 
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ ๎Ÿฟ๎Ÿฟ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ ƒ๎ ‚.๎ „
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ ๎ „๎Ÿฟ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ ƒ๎ „.๎ †
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ ‚๎Ÿฟ๎Ÿฟ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ ƒ๎ †.๎ €
๎บ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎กฐ๎ ‚๎ „๎Ÿฟ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ „๎Ÿฟ.๎Ÿฟ
๎†๎‹‰๎…”๎†ƒ๎Š‹ ๎‰ฝ๎ˆถ๎ˆ๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎€ญ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎ ๎š๎ฐ๎Œฏ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ
โ—ฆ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ „๎Ÿฟ.๎Ÿฟ
๎†๎‹‰๎…”๎†ƒ๎Š‹ ๎‰ฝ๎ˆถ๎ˆ๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
x
u(x)
โ—ฆ AttachCoarseOperator ๎ˆฅ๎ˆถ๎Š‹ ๎…”๎ˆ๎‹๎…”๎‹Š๎‰ฝ ๎Š‹๎‰ญ๎‡ซ๎Šฝ๎‡ซ๎…”๎ˆ ๎Š‹๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก
๎ ‚๎ †
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ ƒ๎กช ๎ฅ๎Œผ๎Ž™๎†๎‚ถ ๎Œฆ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎Žฉ๎“๎Œผ๎š๎‹ผ๎ญ๎Žฉ๎š ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ƒ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
โ—ฆ ๎‰ญ๎ˆถ๎…น๎Š ๎‰ฝ๎Š‹ ๎€ท๎€ท๎‚ก ๎ข‚๎ƒฒ๎‰ข๎‡ซ๎ˆ๎ˆ๎…”๎ˆฅ๎†ก ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ
โ—ฆ โˆผ ๎‡ซ๎ˆฅ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎‰ญ๎†ก๎‰ฝ๎‰ฝ๎‡ซ๎…น๎ˆ๎†ก ๎†ก๎ˆ๎…”๎‰ฝ๎Š‹๎‡ซ๎†ƒ๎‡ซ๎Š‹๎‹Š ๎ข‚๎ฑ๎…”๎‡‰๎†ก๎‰ญ๎‰ฝ๎‰ฝ๎…”๎‰ฝ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎ €๎ †๎ขƒ
โ—ฆ ๎‡ ๎‡ซ๎‡Œ๎‡ ๎ˆ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎‹–๎…”๎…น๎ˆ๎†ก ๎ข‚๎‚๎ˆถ๎ˆ๎‡ซ๎Šฝ๎†ก๎Š‹๎กซ ๎ฑ๎†ก๎†ƒ๎‡ ๎Š‹๎กซ ๎‚ฅ๎…”๎Š‹๎…”๎‡‰๎กซ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ
โ—ฆ ๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎…น๎ˆ๎†ก ๎ข‚๎€ƒ๎ˆ ๎€ท๎…”๎…”๎‰ฝ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎ ๎ €๎ขƒ
โ—ฆ -hpddm_schwarz_method ๎ข‚๎…”๎‰ฝ๎ˆŸ๎ฅพ๎‰ญ๎…”๎‰ฝ๎ฅพ๎ˆถ๎‰ฝ๎ˆŸ๎ขƒ
โ—ฆ -hpddm_geneo_nu n
โ—ฆ -hpddm_level_2_p p
๎ ‚๎ ‡
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ ƒ๎กช ๎ฅ๎Œผ๎Ž™๎†๎‚ถ ๎Œฆ๎Žฉ๎‹ผ๎š๎ฃ๎Œผ ๎Žฉ๎“๎Œผ๎š๎‹ผ๎ญ๎Žฉ๎š ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ƒ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
โ—ฆ ๎‰ญ๎ˆถ๎…น๎Š ๎‰ฝ๎Š‹ ๎€ท๎€ท๎‚ก ๎ข‚๎ƒฒ๎‰ข๎‡ซ๎ˆ๎ˆ๎…”๎ˆฅ๎†ก ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ
โ—ฆ โˆผ ๎‡ซ๎ˆฅ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎‰ญ๎†ก๎‰ฝ๎‰ฝ๎‡ซ๎…น๎ˆ๎†ก ๎†ก๎ˆ๎…”๎‰ฝ๎Š‹๎‡ซ๎†ƒ๎‡ซ๎Š‹๎‹Š ๎ข‚๎ฑ๎…”๎‡‰๎†ก๎‰ญ๎‰ฝ๎‰ฝ๎…”๎‰ฝ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎ €๎ †๎ขƒ
โ—ฆ ๎‡ ๎‡ซ๎‡Œ๎‡ ๎ˆ๎‹Š ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎‹–๎…”๎…น๎ˆ๎†ก ๎ข‚๎‚๎ˆถ๎ˆ๎‡ซ๎Šฝ๎†ก๎Š‹๎กซ ๎ฑ๎†ก๎†ƒ๎‡ ๎Š‹๎กซ ๎‚ฅ๎…”๎Š‹๎…”๎‡‰๎กซ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ
โ—ฆ ๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎…น๎ˆ๎†ก ๎ข‚๎€ƒ๎ˆ ๎€ท๎…”๎…”๎‰ฝ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎ ๎ €๎ขƒ
โ—ฆ -hpddm_schwarz_method ๎ข‚๎…”๎‰ฝ๎ˆŸ๎ฅพ๎‰ญ๎…”๎‰ฝ๎ฅพ๎ˆถ๎‰ฝ๎ˆŸ๎ขƒ
โ—ฆ -hpddm_geneo_nu n
โ—ฆ -hpddm_level_2_p p
๎ ‚๎ ‡
๎ƒฒ๎ผ๎Œž๎ฃ๎ญ๎š๎ผ๎Œฆ๎ญ๎ผ๎š๎ฐ๎Ž™๎  ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
๎€ƒ๎Ž‹๎ ๎Œผ๎Œž๎š๎‹ผ๎ฐ๎Œฆ ๎Œฏ๎Œผ๎Œฆ๎Žฉ๎Ž•๎“๎Žฉ๎ฃ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™
A =
๏ฃฎ
๏ฃฏ
๏ฃฐ
A
๎ €๎ €
๎Ÿฟ A
๎ €ฮ“
๎Ÿฟ A
๎ ๎ 
A
๎ ฮ“
A
ฮ“๎ €
A
ฮ“๎ 
A
ฮ“ฮ“
๏ฃน
๏ฃบ
๏ฃป
b =
๏ฃฎ
๏ฃฏ
๏ฃฐ
b
๎ €
b
๎ 
b
ฮ“
๏ฃน
๏ฃบ
๏ฃป
=โ‡’
๎€€
A
ฮ“ฮ“
โˆ’ A
ฮ“๎ €
A
โˆ’๎ €
๎ €๎ €
A
๎ €ฮ“
โˆ’ A
ฮ“๎ 
A
โˆ’๎ €
๎ ๎ 
A
๎ ฮ“
๎€
x
ฮ“
= b
ฮ“
โˆ’ A
ฮ“๎ €
A
โˆ’๎ €
๎ €๎ €
b
๎ €
โˆ’ A
ฮ“๎ 
A
โˆ’๎ €
๎ ๎ 
b
๎ 
= g
ฮ“
= g
(๎ €)
ฮ“
+ g
(๎ )
ฮ“
๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎ขŽ๎‰ฝ๎ข
S
p
= A
(๎ €)
ฮ“ฮ“
โˆ’ A
ฮ“๎ €
A
โˆ’๎ €
๎ €๎ €
A
๎ €ฮ“
+ A
(๎ )
ฮ“ฮ“
โˆ’ A
ฮ“๎ 
A
โˆ’๎ €
๎ ๎ 
A
๎ ฮ“
= S
(๎ €)
p
+ S
(๎ )
p
๎ ‚๎ ˆ
๎€ƒ๎Ž‹๎ ๎Œผ๎Œž๎š๎‹ผ๎ฐ๎Œฆ ๎Œฏ๎Œผ๎Œฆ๎Žฉ๎Ž•๎“๎Žฉ๎ฃ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™
A =
๏ฃฎ
๏ฃฏ
๏ฃฐ
A
๎ €๎ €
๎Ÿฟ A
๎ €ฮ“
๎Ÿฟ A
๎ ๎ 
A
๎ ฮ“
A
ฮ“๎ €
A
ฮ“๎ 
A
ฮ“ฮ“
๏ฃน
๏ฃบ
๏ฃป
b =
๏ฃฎ
๏ฃฏ
๏ฃฐ
b
๎ €
b
๎ 
b
ฮ“
๏ฃน
๏ฃบ
๏ฃป
=โ‡’
๎€€
A
ฮ“ฮ“
โˆ’ A
ฮ“๎ €
A
โˆ’๎ €
๎ €๎ €
A
๎ €ฮ“
โˆ’ A
ฮ“๎ 
A
โˆ’๎ €
๎ ๎ 
A
๎ ฮ“
๎€
x
ฮ“
= b
ฮ“
โˆ’ A
ฮ“๎ €
A
โˆ’๎ €
๎ €๎ €
b
๎ €
โˆ’ A
ฮ“๎ 
A
โˆ’๎ €
๎ ๎ 
b
๎ 
= g
ฮ“
= g
(๎ €)
ฮ“
+ g
(๎ )
ฮ“
๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎ขŽ๎‰ฝ๎ข
S
p
= A
(๎ €)
ฮ“ฮ“
โˆ’ A
ฮ“๎ €
A
โˆ’๎ €
๎ €๎ €
A
๎ €ฮ“
+ A
(๎ )
ฮ“ฮ“
โˆ’ A
ฮ“๎ 
A
โˆ’๎ €
๎ ๎ 
A
๎ ฮ“
= S
(๎ €)
p
+ S
(๎ )
p
๎ ‚๎ ˆ
๎€ญ๎Žฉ๎Ž™๎Œฏ๎Œผ๎Ž™๎ฃ๎Œผ๎Œฏ ๎ฃ๎ฃ๎ฃ๎ญ๎Œผ๎Ž•
๎ƒก๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ S
p
x
ฮ“
= g
ฮ“
S
(๎ €)
p
= S
(๎ )
p
=โ‡’ M
โˆ’๎ €
=
๎ €
๎ ƒ
๎€
S
(๎ €)
p
โˆ’๎ €
+ S
(๎ )
p
โˆ’๎ €
๎€‘
A
(i)
=
๎€ข
A
ii
A
iฮ“
A
ฮ“i
A
(i)
ฮ“ฮ“
๎€ฃ
=
๎€ข
I ๎Ÿฟ
A
ฮ“i
A
ii
โˆ’๎ €
I
๎€ฃ๎€ข
I ๎Ÿฟ
๎Ÿฟ S
(i)
p
๎€ฃ๎€ข
I A
(i)
ฮ“ฮ“
โˆ’๎ €
A
iฮ“
๎Ÿฟ I
๎€ฃ
๎ ƒ๎Ÿฟ
๎€ญ๎Žฉ๎Ž™๎Œฏ๎Œผ๎Ž™๎ฃ๎Œผ๎Œฏ ๎ฃ๎ฃ๎ฃ๎ญ๎Œผ๎Ž•
๎ƒก๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ S
p
x
ฮ“
= g
ฮ“
S
(๎ €)
p
= S
(๎ )
p
=โ‡’ M
โˆ’๎ €
=
๎ €
๎ ƒ
๎€
S
(๎ €)
p
โˆ’๎ €
+ S
(๎ )
p
โˆ’๎ €
๎€‘
A
(i)
=
๎€ข
A
ii
A
iฮ“
A
ฮ“i
A
(i)
ฮ“ฮ“
๎€ฃ
=
๎€ข
I ๎Ÿฟ
A
ฮ“i
A
ii
โˆ’๎ €
I
๎€ฃ๎€ข
I ๎Ÿฟ
๎Ÿฟ S
(i)
p
๎€ฃ๎€ข
I A
(i)
ฮ“ฮ“
โˆ’๎ €
A
iฮ“
๎Ÿฟ I
๎€ฃ
๎ ƒ๎Ÿฟ
๎ฅ๎Œผ๎Ž™๎Œผ๎š๎‹ผ๎Ž‹ ๎Ž™๎Žฉ๎ญ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ
โ„ฆ
๎ €
โ„ฆ
๎ 
โ„ฆ
๎ ‚
๎ข‚๎ฅ๎ˆถ๎‰ฝ๎‰ฝ๎†ก๎ˆ๎†ก๎Š‹ ๎…”๎ˆฅ๎†Ž ๎ƒจ๎†ก๎‹Š ๎ ๎Ÿฟ๎Ÿฟ๎ …๎ขƒ
๎ƒฒ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎Š‹๎†ก๎…”๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
๎ ƒ๎ €
๎ฅ๎Œผ๎Ž™๎Œผ๎š๎‹ผ๎Ž‹ ๎Ž™๎Žฉ๎ญ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ
โ„ฆ
๎ €
โ„ฆ
๎ 
โ„ฆ
๎ ‚
๎ €
๎ขŽ๎ €๎ข
๎ 
๎ขŽ๎ €๎ข
๎ ‚
๎ขŽ๎ €๎ข
๎ ƒ
๎ขŽ๎ €๎ข
๎ „
๎ขŽ๎ €๎ข
๎ €
๎ขŽ๎ ๎ข
๎ 
๎ขŽ๎ ๎ข
๎ ‚
๎ขŽ๎ ๎ข
๎ ƒ
๎ขŽ๎ ๎ข
๎ „
๎ขŽ๎ ๎ข
๎ …
๎ขŽ๎ ๎ข
๎ €
๎ขŽ๎ ‚๎ข
๎ 
๎ขŽ๎ ‚๎ข
๎ ‚
๎ขŽ๎ ‚๎ข
๎ ƒ
๎ขŽ๎ ‚๎ข
๎ „
๎ขŽ๎ ‚๎ข
๎ …
๎ขŽ๎ ‚๎ข
๎ †
๎ขŽ๎ ‚๎ข
๎ข‚๎ฅ๎ˆถ๎‰ฝ๎‰ฝ๎†ก๎ˆ๎†ก๎Š‹ ๎…”๎ˆฅ๎†Ž ๎ƒจ๎†ก๎‹Š ๎ ๎Ÿฟ๎Ÿฟ๎ …๎ขƒ
๎‚•๎ˆถ๎†ƒ๎…”๎ˆ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
A
(i)
=
๎€ข
A
ii
A
iฮ“
A
ฮ“i
A
(i)
ฮ“ฮ“
๎€ฃ
๎ ƒ๎ €
๎ฅ๎Œผ๎Ž™๎Œผ๎š๎‹ผ๎Ž‹ ๎Ž™๎Žฉ๎ญ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ
โ„ฆ
๎ €
โ„ฆ
๎ 
โ„ฆ
๎ ‚
๎ €
๎ขŽ๎ €๎ข
ฮ“
๎ ‚
๎ขŽ๎ €๎ข
ฮ“
๎ 
๎ขŽ๎ €๎ข
ฮ“
๎ ƒ
๎ขŽ๎ €๎ข
ฮ“
๎ €
๎ขŽ๎ ๎ข
ฮ“
๎ ‚
๎ขŽ๎ ๎ข
ฮ“
๎ 
๎ขŽ๎ ๎ข
ฮ“
๎ ƒ
๎ขŽ๎ ๎ข
ฮ“
๎ ‚
๎ขŽ๎ ‚๎ข
ฮ“
๎ €
๎ขŽ๎ ‚๎ข
ฮ“
๎ 
๎ขŽ๎ ‚๎ข
ฮ“
๎ข‚๎ฅ๎ˆถ๎‰ฝ๎‰ฝ๎†ก๎ˆ๎†ก๎Š‹ ๎…”๎ˆฅ๎†Ž ๎ƒจ๎†ก๎‹Š ๎ ๎Ÿฟ๎Ÿฟ๎ …๎ขƒ
๎†๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‡‰ ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡ซ๎ˆถ๎‰ญ ๎†Ž๎กฒ๎ˆถ๎กฒ๎‡‰๎กฒ
S
(i)
p
= A
(i)
ฮ“ฮ“
โˆ’ A
ฮ“i
A
โˆ’๎ €
ii
A
iฮ“
๎ ƒ๎ €
๎ฅ๎Œผ๎Ž™๎Œผ๎š๎‹ผ๎Ž‹ ๎Ž™๎Žฉ๎ญ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ
๎ €
๎ขŽ๎ €๎ข
ฮ“
๎ ‚
๎ขŽ๎ €๎ข
ฮ“
๎ 
๎ขŽ๎ €๎ข
ฮ“
๎ ƒ
๎ขŽ๎ €๎ข
ฮ“
๎ €
๎ขŽ๎ ๎ข
ฮ“
๎ ‚
๎ขŽ๎ ๎ข
ฮ“
๎ 
๎ขŽ๎ ๎ข
ฮ“
๎ ƒ
๎ขŽ๎ ๎ข
ฮ“
๎ ‚
๎ขŽ๎ ‚๎ข
ฮ“
๎ €
๎ขŽ๎ ‚๎ข
ฮ“
๎ 
๎ขŽ๎ ‚๎ข
ฮ“
๎ €
๎ ‚
๎ ƒ
๎ „
๎ 
๎‚๎Š ๎ˆŸ๎‰ข ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ๎‰ฝ {B
(i)
}
๎ ‚
i=๎ €
๎ƒก๎‰ญ๎‡ซ๎ˆŸ๎…”๎ˆ ๎†ƒ๎ˆถ๎ˆฅ๎‰ฝ๎Š‹๎‰ญ๎…”๎‡ซ๎ˆฅ๎Š‹๎‰ฝ
๎ข‚๎‚ก๎…”๎ˆฅ๎†Ž๎†ก๎ˆ ๎ €๎ ˆ๎ ˆ๎ ‚๎ขƒ
๎ ƒ๎ €
๎ฅ๎Œผ๎Ž™๎Œผ๎š๎‹ผ๎Ž‹ ๎Ž™๎Žฉ๎ญ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ
๎ €
๎ขŽ๎ €๎ข
ฮ“
๎ ‚
๎ขŽ๎ €๎ข
ฮ“
๎ 
๎ขŽ๎ €๎ข
ฮ“
๎ ƒ
๎ขŽ๎ €๎ข
ฮ“
๎ €
๎ขŽ๎ ๎ข
ฮ“
๎ ‚
๎ขŽ๎ ๎ข
ฮ“
๎ 
๎ขŽ๎ ๎ข
ฮ“
๎ ƒ
๎ขŽ๎ ๎ข
ฮ“
๎ ‚
๎ขŽ๎ ‚๎ข
ฮ“
๎ €
๎ขŽ๎ ‚๎ข
ฮ“
๎ 
๎ขŽ๎ ‚๎ข
ฮ“
๎ €
๎ 
๎ ƒ
๎ „
๎ ‚
๎ …
๎ †
๎‚๎Š ๎ˆŸ๎‰ข ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ๎‰ฝ {B
(i)
}
๎ ‚
i=๎ €
๎€ท๎Š ๎…”๎ˆ ๎†ƒ๎ˆถ๎ˆฅ๎‰ฝ๎Š‹๎‰ญ๎…”๎‡ซ๎ˆฅ๎Š‹๎‰ฝ
๎ข‚๎ข๎…”๎‰ญ๎‡ ๎…”๎Š‹ ๎…”๎ˆฅ๎†Ž ๎ƒจ๎ˆถ๎Š ๎‹‰ ๎ €๎ ˆ๎ ˆ๎ €๎ขƒ
๎ ƒ๎ €
๎€ญ๎Žฉ๎Ž™๎Œฏ๎Œผ๎Ž™๎ฃ๎Œผ๎Œฏ ๎ฃ๎ฃ๎ฃ๎ญ๎Œผ๎Ž•
โˆ€i โˆˆ J๎ €; NK, S
(i)
p
x
(i)
ฮ“
= g
(i)
ฮ“
+ ฮป
(i)
ฮ“
R
(i)
ฮ“
T
ฮป
(i)
ฮ“
=
๎Ÿฟ
N
๎˜
i=๎ €
B
(i)
x
(i)
ฮ“
= ๎Ÿฟ
N
๎˜
i=๎ €
B
(i)
ฮป
(i)
ฮ“
= ๎Ÿฟ
๎ ƒ๎ 
๎€ญ๎Žฉ๎Ž™๎Œฏ๎Œผ๎Ž™๎ฃ๎Œผ๎Œฏ ๎ฃ๎ฃ๎ฃ๎ญ๎Œผ๎Ž•
โˆ€i โˆˆ J๎ €; NK, S
(i)
p
x
(i)
ฮ“
= g
(i)
ฮ“
+ ฮป
(i)
ฮ“
R
(i)
ฮ“
T
ฮป
(i)
ฮ“
=
๎Ÿฟ
N
๎˜
i=๎ €
B
(i)
x
(i)
ฮ“
= ๎Ÿฟ
N
๎˜
i=๎ €
B
(i)
ฮป
(i)
ฮ“
= ๎Ÿฟ
๎ ƒ๎ 
๎€ญ๎Žฉ๎Ž™๎Œฏ๎Œผ๎Ž™๎ฃ๎Œผ๎Œฏ ๎ฃ๎ฃ๎ฃ๎ญ๎Œผ๎Ž•
โˆ€i โˆˆ J๎ €; NK, S
(i)
p
x
(i)
ฮ“
= g
(i)
ฮ“
+ ฮป
(i)
ฮ“
R
(i)
ฮ“
T
ฮป
(i)
ฮ“
=
๎Ÿฟ
N
๎˜
i=๎ €
B
(i)
x
(i)
ฮ“
= ๎Ÿฟ
N
๎˜
i=๎ €
B
(i)
ฮป
(i)
ฮ“
= ๎Ÿฟ
๎ ƒ๎ 
๎€ญ๎Žฉ๎Ž™๎Œฏ๎Œผ๎Ž™๎ฃ๎Œผ๎Œฏ ๎ฃ๎ฃ๎ฃ๎ญ๎Œผ๎Ž•
โˆ€i โˆˆ J๎ €; NK, S
(i)
p
x
(i)
ฮ“
= g
(i)
ฮ“
+ ฮป
(i)
ฮ“
R
(i)
ฮ“
T
ฮป
(i)
ฮ“
=
๎Ÿฟ
N
๎˜
i=๎ €
B
(i)
x
(i)
ฮ“
= ๎Ÿฟ
N
๎˜
i=๎ €
B
(i)
ฮป
(i)
ฮ“
= ๎Ÿฟ
๎ ƒ๎ 
๎ƒก๎š๎ฐ๎Ž•๎‹ผ๎Ž‹ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
๎„๎ˆฅ๎‡ซ๎‰ฉ๎Š ๎†ก ๎†Ž๎‡ซ๎‰ฝ๎‰ข๎ˆ๎…”๎†ƒ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎กธ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎†ก๎†Ž ๎‰ญ๎†ก๎…”๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ x
ฮ“
=โ‡’ x
(i)
ฮ“
= B
(i)
T
x
ฮ“
โ—ฆ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ ๎ˆถ๎‡‰ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
N
๎˜
i=๎ €
B
(i)
S
(i)
p
B
(i)
T
x
ฮ“
=
N
๎˜
i=๎ €
B
(i)
g
(i)
ฮ“
โ—ฆ ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ
M
โˆ’๎ €
=
N
๎˜
i=๎ €
B
(i)
D
(i)
p
S
(i)
p
โ€ 
D
(i)
p
B
(i)
T
๎…”๎‰ข๎‰ข๎ˆ๎‡ซ๎†ก๎†Ž ๎Š‹๎ˆถ ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎‡ซ๎ˆฅ ๎บ๎ˆŸ(S
p
)
๎ ƒ๎ ‚
๎ƒก๎š๎ฐ๎Ž•๎‹ผ๎Ž‹ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
๎„๎ˆฅ๎‡ซ๎‰ฉ๎Š ๎†ก ๎†Ž๎‡ซ๎‰ฝ๎‰ข๎ˆ๎…”๎†ƒ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎กธ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎†ก๎†Ž ๎‰ญ๎†ก๎…”๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ x
ฮ“
=โ‡’ x
(i)
ฮ“
= B
(i)
T
x
ฮ“
โ—ฆ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ ๎ˆถ๎‡‰ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
N
๎˜
i=๎ €
B
(i)
S
(i)
p
B
(i)
T
x
ฮ“
=
N
๎˜
i=๎ €
B
(i)
g
(i)
ฮ“
โ—ฆ ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ
M
โˆ’๎ €
=
N
๎˜
i=๎ €
B
(i)
D
(i)
p
S
(i)
p
โ€ 
D
(i)
p
B
(i)
T
๎…”๎‰ข๎‰ข๎ˆ๎‡ซ๎†ก๎†Ž ๎Š‹๎ˆถ ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎‡ซ๎ˆฅ ๎บ๎ˆŸ(S
p
)
๎ ƒ๎ ‚
๎ƒก๎š๎ฐ๎Ž•๎‹ผ๎Ž‹ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
๎„๎ˆฅ๎‡ซ๎‰ฉ๎Š ๎†ก ๎†Ž๎‡ซ๎‰ฝ๎‰ข๎ˆ๎…”๎†ƒ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎กธ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎†ก๎†Ž ๎‰ญ๎†ก๎…”๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ x
ฮ“
=โ‡’ x
(i)
ฮ“
= B
(i)
T
x
ฮ“
โ—ฆ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ ๎ˆถ๎‡‰ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
N
๎˜
i=๎ €
B
(i)
S
(i)
p
B
(i)
T
x
ฮ“
=
N
๎˜
i=๎ €
B
(i)
g
(i)
ฮ“
โ—ฆ ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ
M
โˆ’๎ €
=
N
๎˜
i=๎ €
B
(i)
D
(i)
p
S
(i)
p
โ€ 
D
(i)
p
B
(i)
T
๎…”๎‰ข๎‰ข๎ˆ๎‡ซ๎†ก๎†Ž ๎Š‹๎ˆถ ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎‡ซ๎ˆฅ ๎บ๎ˆŸ(S
p
)
๎ ƒ๎ ‚
๎ƒก๎š๎ฐ๎Ž•๎‹ผ๎Ž‹ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
๎„๎ˆฅ๎‡ซ๎‰ฉ๎Š ๎†ก ๎†Ž๎‡ซ๎‰ฝ๎‰ข๎ˆ๎…”๎†ƒ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎กธ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎†ก๎†Ž ๎‰ญ๎†ก๎…”๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ x
ฮ“
=โ‡’ x
(i)
ฮ“
= B
(i)
T
x
ฮ“
โ—ฆ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ ๎ˆถ๎‡‰ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
N
๎˜
i=๎ €
B
(i)
S
(i)
p
B
(i)
T
x
ฮ“
=
N
๎˜
i=๎ €
B
(i)
g
(i)
ฮ“
โ—ฆ ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ
M
โˆ’๎ €
=
N
๎˜
i=๎ €
B
(i)
D
(i)
p
S
(i)
p
โ€ 
D
(i)
p
B
(i)
T
๎…”๎‰ข๎‰ข๎ˆ๎‡ซ๎†ก๎†Ž ๎Š‹๎ˆถ ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎‡ซ๎ˆฅ ๎บ๎ˆŸ(S
p
)
๎ ƒ๎ ‚
๎€ฅ๎‹ผ๎Ž‹๎‹ผ๎Ž™๎Œฆ๎ฐ๎Ž™๎  ๎€ท๎Žฉ๎Ž•๎‹ผ๎ฐ๎Ž™ ๎€ท๎Œผ๎Œฆ๎Žฉ๎Ž•๎“๎Žฉ๎ฃ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ฅ๎…”๎ˆ๎…”๎ˆฅ๎†ƒ๎†ก๎†Ž ๎‰ญ๎†ก๎‰ฝ๎‡ซ๎†Ž๎Š ๎…”๎ˆ
N
๎˜
i=๎ €
R
(i)
ฮ“
T
D
(i)
p
B
(i)
T
r
ฮ“
= ๎Ÿฟ
๎ƒก๎‰ญ๎ˆถ๎‡ฟ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ƒจ
ฮ“
=
๎จ
B
(๎ €)
D
(๎ €)
p
R
(๎ €)
ฮ“
ยท ยท ยท B
(N)
D
(N)
p
R
(N)
ฮ“
๎ฉ
๎ƒฒ
p
=
N
๎˜
i=๎ €
B
(i)
S
(i)
p
B
(i)
T
=โ‡’
๎ƒจ
T
ฮ“
๎ƒฒ
p
P = ๎Ÿฟ ๎‹๎‡ซ๎Š‹๎‡  P = I โˆ’ ๎ƒจ
ฮ“
๎€€
๎ƒจ
T
ฮ“
๎ƒฒ
p
๎ƒจ
ฮ“
๎€
โˆ’๎ €
๎ƒจ
T
ฮ“
๎ƒฒ
p
๎ ƒ๎ ƒ
๎€ฅ๎‹ผ๎Ž‹๎‹ผ๎Ž™๎Œฆ๎ฐ๎Ž™๎  ๎€ท๎Žฉ๎Ž•๎‹ผ๎ฐ๎Ž™ ๎€ท๎Œผ๎Œฆ๎Žฉ๎Ž•๎“๎Žฉ๎ฃ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ฅ๎…”๎ˆ๎…”๎ˆฅ๎†ƒ๎†ก๎†Ž ๎‰ญ๎†ก๎‰ฝ๎‡ซ๎†Ž๎Š ๎…”๎ˆ
N
๎˜
i=๎ €
R
(i)
ฮ“
T
D
(i)
p
B
(i)
T
r
ฮ“
= ๎Ÿฟ
๎ƒก๎‰ญ๎ˆถ๎‡ฟ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ƒจ
ฮ“
=
๎จ
B
(๎ €)
D
(๎ €)
p
R
(๎ €)
ฮ“
ยท ยท ยท B
(N)
D
(N)
p
R
(N)
ฮ“
๎ฉ
๎ƒฒ
p
=
N
๎˜
i=๎ €
B
(i)
S
(i)
p
B
(i)
T
=โ‡’
๎ƒจ
T
ฮ“
๎ƒฒ
p
P = ๎Ÿฟ ๎‹๎‡ซ๎Š‹๎‡  P = I โˆ’ ๎ƒจ
ฮ“
๎€€
๎ƒจ
T
ฮ“
๎ƒฒ
p
๎ƒจ
ฮ“
๎€
โˆ’๎ €
๎ƒจ
T
ฮ“
๎ƒฒ
p
๎ ƒ๎ ƒ
๎€ฅ๎‹ผ๎Ž‹๎‹ผ๎Ž™๎Œฆ๎ฐ๎Ž™๎  ๎€ท๎Žฉ๎Ž•๎‹ผ๎ฐ๎Ž™ ๎€ท๎Œผ๎Œฆ๎Žฉ๎Ž•๎“๎Žฉ๎ฃ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ฅ๎…”๎ˆ๎…”๎ˆฅ๎†ƒ๎†ก๎†Ž ๎‰ญ๎†ก๎‰ฝ๎‡ซ๎†Ž๎Š ๎…”๎ˆ
N
๎˜
i=๎ €
R
(i)
ฮ“
T
D
(i)
p
B
(i)
T
r
ฮ“
= ๎Ÿฟ
๎ƒก๎‰ญ๎ˆถ๎‡ฟ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ƒจ
ฮ“
=
๎จ
B
(๎ €)
D
(๎ €)
p
R
(๎ €)
ฮ“
ยท ยท ยท B
(N)
D
(N)
p
R
(N)
ฮ“
๎ฉ
๎ƒฒ
p
=
N
๎˜
i=๎ €
B
(i)
S
(i)
p
B
(i)
T
=โ‡’
๎ƒจ
T
ฮ“
๎ƒฒ
p
P = ๎Ÿฟ ๎‹๎‡ซ๎Š‹๎‡  P = I โˆ’ ๎ƒจ
ฮ“
๎€€
๎ƒจ
T
ฮ“
๎ƒฒ
p
๎ƒจ
ฮ“
๎€
โˆ’๎ €
๎ƒจ
T
ฮ“
๎ƒฒ
p
๎ ƒ๎ ƒ
๎€ท๎ผ๎‹ผ๎Ž‹ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎„๎ˆฅ๎‡ซ๎‰ฉ๎Š ๎†ก ๎‰ญ๎†ก๎…”๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กธ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎†ก๎†Ž ๎†Ž๎‡ซ๎‰ฝ๎‰ข๎ˆ๎…”๎†ƒ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ ฮป
ฮ“
=โ‡’ ฮป
(i)
ฮ“
= B
(i)
T
ฮป
ฮ“
โ—ฆ ๎†Ž๎Š ๎…”๎ˆ ๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ S
(i)
d
= S
(i)
p
โ€ 
โ—ฆ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ ๎ˆถ๎‡‰ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โˆ€i โˆˆ J๎ €; NK, x
(i)
ฮ“
= S
(i)
d
(g
(i)
ฮ“
+ B
(i)
T
ฮป
ฮ“
) + R
(i)
ฮ“
ฮฑ
(i)
๎Ÿฟ = R
(i)
ฮ“
T
(g
(i)
ฮ“
+ B
(i)
T
ฮป
ฮ“
)
โ—ฆ ๎‰ฝ๎…”๎†Ž๎†Ž๎ˆ๎†ก๎ขš๎‰ข๎ˆถ๎‡ซ๎ˆฅ๎Š‹ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎Š ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎€ข
๎ƒฒ
d
๎ƒจ
ฮ“
๎ƒจ
T
ฮ“
๎Ÿฟ
๎€ฃ๎€ข
ฮป
ฮ“
ฮฑ
๎€ฃ
=
๎€ข
โˆ’๎…น
d
โˆ’๎‡Œ
ฮ“
๎€ฃ
๎ ƒ๎ „
๎€ท๎ผ๎‹ผ๎Ž‹ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎„๎ˆฅ๎‡ซ๎‰ฉ๎Š ๎†ก ๎‰ญ๎†ก๎…”๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กธ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎†ก๎†Ž ๎†Ž๎‡ซ๎‰ฝ๎‰ข๎ˆ๎…”๎†ƒ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ ฮป
ฮ“
=โ‡’ ฮป
(i)
ฮ“
= B
(i)
T
ฮป
ฮ“
โ—ฆ ๎†Ž๎Š ๎…”๎ˆ ๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ S
(i)
d
= S
(i)
p
โ€ 
โ—ฆ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ ๎ˆถ๎‡‰ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โˆ€i โˆˆ J๎ €; NK, x
(i)
ฮ“
= S
(i)
d
(g
(i)
ฮ“
+ B
(i)
T
ฮป
ฮ“
) + R
(i)
ฮ“
ฮฑ
(i)
๎Ÿฟ = R
(i)
ฮ“
T
(g
(i)
ฮ“
+ B
(i)
T
ฮป
ฮ“
)
โ—ฆ ๎‰ฝ๎…”๎†Ž๎†Ž๎ˆ๎†ก๎ขš๎‰ข๎ˆถ๎‡ซ๎ˆฅ๎Š‹ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎Š ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎€ข
๎ƒฒ
d
๎ƒจ
ฮ“
๎ƒจ
T
ฮ“
๎Ÿฟ
๎€ฃ๎€ข
ฮป
ฮ“
ฮฑ
๎€ฃ
=
๎€ข
โˆ’๎…น
d
โˆ’๎‡Œ
ฮ“
๎€ฃ
๎ ƒ๎ „
๎€ท๎ผ๎‹ผ๎Ž‹ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎„๎ˆฅ๎‡ซ๎‰ฉ๎Š ๎†ก ๎‰ญ๎†ก๎…”๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กธ๎†ก๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎Š‹๎†ก๎†Ž ๎†Ž๎‡ซ๎‰ฝ๎‰ข๎ˆ๎…”๎†ƒ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ ฮป
ฮ“
=โ‡’ ฮป
(i)
ฮ“
= B
(i)
T
ฮป
ฮ“
โ—ฆ ๎†Ž๎Š ๎…”๎ˆ ๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ S
(i)
d
= S
(i)
p
โ€ 
โ—ฆ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ ๎ˆถ๎‡‰ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โˆ€i โˆˆ J๎ €; NK, x
(i)
ฮ“
= S
(i)
d
(g
(i)
ฮ“
+ B
(i)
T
ฮป
ฮ“
) + R
(i)
ฮ“
ฮฑ
(i)
๎Ÿฟ = R
(i)
ฮ“
T
(g
(i)
ฮ“
+ B
(i)
T
ฮป
ฮ“
)
โ—ฆ ๎‰ฝ๎…”๎†Ž๎†Ž๎ˆ๎†ก๎ขš๎‰ข๎ˆถ๎‡ซ๎ˆฅ๎Š‹ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎Š ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎€ข
๎ƒฒ
d
๎ƒจ
ฮ“
๎ƒจ
T
ฮ“
๎Ÿฟ
๎€ฃ๎€ข
ฮป
ฮ“
ฮฑ
๎€ฃ
=
๎€ข
โˆ’๎…น
d
โˆ’๎‡Œ
ฮ“
๎€ฃ
๎ ƒ๎ „
๎ƒฒ๎ฐ๎Ž•๎ฐ๎Ž‹๎‹ผ๎š ๎ผ๎ญ๎ฐ๎Ž‹๎ฐ๎ญ๎ฃ ๎š๎Žฉ๎ผ๎ญ๎ฐ๎Ž™๎Œผ๎ฃ
โ—ฆ exchange
โ—ฆ statistics
โ—ฆ renumber
๎ ƒ๎ …
๎ƒก๎†๎ƒพ๎ƒฒ๎Œฆ
๎บ๎Ž™๎ญ๎š๎Žฉ๎Œฏ๎ผ๎Œฆ๎ญ๎ฐ๎Žฉ๎Ž™
โ—ฆ ๎ƒก๎ˆถ๎‰ญ๎Š‹๎…”๎…น๎ˆ๎†ก๎กซ ๎†๎‹‰๎Š‹๎†ก๎ˆฅ๎‰ฝ๎‡ซ๎…น๎ˆ๎†ก ๎ƒพ๎ˆถ๎ˆถ๎ˆ๎ˆ‡๎‡ซ๎Š‹ ๎‡‰๎ˆถ๎‰ญ ๎ƒฒ๎†ƒ๎‡ซ๎†ก๎ˆฅ๎Š‹๎‡ซ๎‹ช๎†ƒ ๎€ญ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก ๎ˆถ๎‡‰ ๎†Ž๎…”๎Š‹๎…” ๎‰ฝ๎Š‹๎‰ญ๎Š ๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
โ€ข Vec ๎…”๎ˆฅ๎†Ž Mat
โ€ข PC ๎…”๎ˆฅ๎†Ž KSP
โ€ข SNES๎กซ TS๎กซ ๎…”๎ˆฅ๎†Ž Tao
โ—ฆ ๎Š ๎‰ฝ๎†ก๎‡‰๎Š ๎ˆ ๎‡‰๎ˆถ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎‡ซ๎ˆ๎‡ซ๎ˆฅ๎‡Œ ๎ˆถ๎Š‹๎‡ ๎†ก๎‰ญ ๎ˆ๎‡ซ๎…น๎‰ญ๎…”๎‰ญ๎‡ซ๎†ก๎‰ฝ ๎ขŽ๎‚ก๎„๎‚ก๎ƒก๎ƒฒ๎กซ hypre๎ข
โ—ฆ https://petsc.org
โ—ฆ https://gitlab.com/petsc/petsc
โ—ฆ ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡‰๎…”๎†ƒ๎†ก๎†Ž ๎‹๎‡ซ๎Š‹๎‡  ๎ฑ๎ƒก๎€ท๎€ท๎‚ก
๎ ƒ๎ †
๎€ท๎‹ผ๎ญ๎‹ผ ๎Œฏ๎ฐ๎ฃ๎ญ๎š๎ฐ๎Œž๎ผ๎ญ๎ฐ๎Žฉ๎Ž™
๎‚ถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎‡‰๎ˆถ๎ˆ๎ˆ๎ˆถ๎‹ ๎…” ๎ €๎€ท ๎‰ญ๎ˆถ๎‹๎ขš๎‹๎‡ซ๎‰ฝ๎†ก ๎†ƒ๎ˆถ๎ˆฅ๎Š‹๎‡ซ๎‡Œ๎Š ๎ˆถ๎Š ๎‰ฝ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ ๎กฐ๎Ÿฟ
๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ ๎กฐ๎ €
๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ ๎กฐ๎ 
๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ ๎กฐ๎ ‚
=โ‡’ ๎…”๎†ƒ๎†ƒ๎†ก๎‰ฝ๎‰ฝ๎‡ซ๎…น๎ˆ๎†ก ๎Šฝ๎‡ซ๎…” GlobalNumbering
๎ ƒ๎ ˆ
๎€ท๎‹ผ๎ญ๎‹ผ ๎Œฏ๎ฐ๎ฃ๎ญ๎š๎ฐ๎Œž๎ผ๎ญ๎ฐ๎Žฉ๎Ž™
๎‚ถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎‡‰๎ˆถ๎ˆ๎ˆ๎ˆถ๎‹ ๎…” ๎ €๎€ท ๎‰ญ๎ˆถ๎‹๎ขš๎‹๎‡ซ๎‰ฝ๎†ก ๎†ƒ๎ˆถ๎ˆฅ๎Š‹๎‡ซ๎‡Œ๎Š ๎ˆถ๎Š ๎‰ฝ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ ๎กฐ๎Ÿฟ
๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ ๎กฐ๎ €
๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ ๎กฐ๎ 
๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ ๎กฐ๎ ‚
=โ‡’ ๎…”๎†ƒ๎†ƒ๎†ก๎‰ฝ๎‰ฝ๎‡ซ๎…น๎ˆ๎†ก ๎Šฝ๎‡ซ๎…” GlobalNumbering
๎ ƒ๎ ˆ
๎ƒก๎†๎ƒพ๎ƒฒ๎Œฆ ๎Ž•๎‹ผ๎ญ๎š๎ฐ๎ข ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎†ก๎Š‹๎Š‹๎‡ซ๎ˆฅ๎‡Œ ๎Š ๎‰ข ๎…” Mat
โ—ฆ ๎‰ฝ๎…”๎ˆŸ๎†ก ๎‡ซ๎ˆฅ๎‰ข๎Š ๎Š‹ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ ๎…”๎‰ฝ ๎ฑ๎ƒก๎€ท๎€ท๎‚ก ๎Š‹๎‹Š๎‰ข๎†ก๎‰ฝ
โ—ฆ ๎‰ฝ๎‡ซ๎ˆŸ๎‰ข๎ˆ๎‡ซ๎‹ช๎†ก๎†Ž ๎ˆŸ๎…”๎†ƒ๎‰ญ๎ˆถ MatCreate(Th,A,Pk)
โ—ฆ Mat<complex> ๎‡‰๎ˆถ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ
๎ƒฒ๎‹๎‡ซ๎Š‹๎†ƒ๎‡ ๎‡ซ๎ˆฅ๎‡Œ ๎…น๎†ก๎Š‹๎‹๎†ก๎†ก๎ˆฅ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ
โ—ฆ ChangeNumbering(Mat<K>,K[int])
โ—ฆ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ€ข inverse ๎Š‹๎ˆถ ๎‡Œ๎ˆถ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎Š‹๎ˆถ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก
โ€ข exchange ๎Š‹๎ˆถ ๎Š ๎‰ข๎†Ž๎…”๎Š‹๎†ก ๎‡Œ๎‡ ๎ˆถ๎‰ฝ๎Š‹ ๎Šฝ๎…”๎ˆ๎Š ๎†ก๎‰ฝ
๎ „๎Ÿฟ
๎ƒก๎†๎ƒพ๎ƒฒ๎Œฆ ๎Ž•๎‹ผ๎ญ๎š๎ฐ๎ข ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ƒฒ๎†ก๎Š‹๎Š‹๎‡ซ๎ˆฅ๎‡Œ ๎Š ๎‰ข ๎…” Mat
โ—ฆ ๎‰ฝ๎…”๎ˆŸ๎†ก ๎‡ซ๎ˆฅ๎‰ข๎Š ๎Š‹ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ ๎…”๎‰ฝ ๎ฑ๎ƒก๎€ท๎€ท๎‚ก ๎Š‹๎‹Š๎‰ข๎†ก๎‰ฝ
โ—ฆ ๎‰ฝ๎‡ซ๎ˆŸ๎‰ข๎ˆ๎‡ซ๎‹ช๎†ก๎†Ž ๎ˆŸ๎…”๎†ƒ๎‰ญ๎ˆถ MatCreate(Th,A,Pk)
โ—ฆ Mat<complex> ๎‡‰๎ˆถ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ
๎ƒฒ๎‹๎‡ซ๎Š‹๎†ƒ๎‡ ๎‡ซ๎ˆฅ๎‡Œ ๎…น๎†ก๎Š‹๎‹๎†ก๎†ก๎ˆฅ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ
โ—ฆ ChangeNumbering(Mat<K>,K[int])
โ—ฆ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ€ข inverse ๎Š‹๎ˆถ ๎‡Œ๎ˆถ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎Š‹๎ˆถ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก
โ€ข exchange ๎Š‹๎ˆถ ๎Š ๎‰ข๎†Ž๎…”๎Š‹๎†ก ๎‡Œ๎‡ ๎ˆถ๎‰ฝ๎Š‹ ๎Šฝ๎…”๎ˆ๎Š ๎†ก๎‰ฝ
๎ „๎Ÿฟ
Mat ๎Ž‹๎ฐ๎Ž™๎Œผ๎‹ผ๎š ๎Žฉ๎“๎Œผ๎š๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎‚๎Š ๎‰ฝ๎Š‹ ๎…”๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก matrix
โ—ฆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ A * x
โ—ฆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‰ข๎ˆถ๎‰ฝ๎†ก๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ A' * x
โ—ฆ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก Aห†
-1
* x
โ—ฆ ๎ฑ๎†ก๎‰ญ๎ˆŸ๎‡ซ๎Š‹๎‡ซ๎…”๎ˆฅ
๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‰ข๎ˆถ๎‰ฝ๎†ก๎†Ž ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก A'ห†
-1
* x
โ—ฆ ๎ˆฅ๎…”๎Š‹๎‡ซ๎Šฝ๎†ก ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎‡ซ๎ˆฅ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ€ข KSPSolve
โ€ข MatMatMult
โ€ข ๎ˆŸ๎ˆถ๎‰ญ๎†ก ๎…”๎Š‹ Mat ๎ˆŸ๎…”๎ˆฅ๎Š ๎…”๎ˆ ๎‰ข๎…”๎‡Œ๎†ก๎‰ฝ
๎ „๎ €
Mat ๎Ž‹๎ฐ๎Ž™๎Œผ๎‹ผ๎š ๎Žฉ๎“๎Œผ๎š๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎‚๎Š ๎‰ฝ๎Š‹ ๎…”๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก matrix
โ—ฆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ A * x
โ—ฆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‰ข๎ˆถ๎‰ฝ๎†ก๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹ A' * x
โ—ฆ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก Aห†
-1
* x
โ—ฆ ๎ฑ๎†ก๎‰ญ๎ˆŸ๎‡ซ๎Š‹๎‡ซ๎…”๎ˆฅ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‰ข๎ˆถ๎‰ฝ๎†ก๎†Ž ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก A'ห†
-1
* x
โ—ฆ ๎ˆฅ๎…”๎Š‹๎‡ซ๎Šฝ๎†ก ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎‡ซ๎ˆฅ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ€ข KSPSolve
โ€ข MatMatMult
โ€ข ๎ˆŸ๎ˆถ๎‰ญ๎†ก ๎…”๎Š‹ Mat ๎ˆŸ๎…”๎ˆฅ๎Š ๎…”๎ˆ ๎‰ข๎…”๎‡Œ๎†ก๎‰ฝ
๎ „๎ €
๎€ฅ๎‹ผ๎ฃ๎ฐ๎Œฆ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎ฐ๎Ž™๎  ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎†Ž๎†ก๎‡‰๎…”๎Š ๎ˆ๎Š‹ ๎Š‹๎ˆถ ๎€ฅ๎‚๎…”๎†ƒ๎ˆถ๎…น๎‡ซ ๎‹๎‡ซ๎Š‹๎‡  ILU๎ขŽ๎Ÿฟ๎ข ๎…”๎‰ฝ ๎…” ๎‰ฝ๎Š ๎…น๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ
โ—ฆ ๎…”๎Š‹๎Š‹๎…”๎†ƒ๎‡  ๎…” KSP ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ set ๎…”๎ˆฅ๎†Ž sparams
โ—ฆ -ksp_type๎กซ -pc_type
โ—ฆ -help ๎‡Œ๎†ก๎ˆฅ๎†ก๎‰ญ๎…”๎Š‹๎†ก๎†Ž ๎†Ž๎‹Š๎ˆฅ๎…”๎ˆŸ๎‡ซ๎†ƒ๎…”๎ˆ๎ˆ๎‹Š
๎„๎‰ข๎†Ž๎…”๎Š‹๎‡ซ๎ˆฅ๎‡Œ ๎…” Mat
โ—ฆ Mat = matrix ๎‡ซ๎‡‰ ๎‰ฝ๎…”๎ˆŸ๎†ก ๎‰ข๎…”๎Š‹๎Š‹๎†ก๎‰ญ๎ˆฅ ๎ˆถ๎‰ญ ๎‹ช๎‰ญ๎‰ฝ๎Š‹ ๎Š ๎‰ข๎†Ž๎…”๎Š‹๎†ก
โ—ฆ Mat = Mat ๎‡ซ๎‡‰ ๎†Ž๎‡ซ๎‡‰๎‡‰๎†ก๎‰ญ๎†ก๎ˆฅ๎Š‹ ๎‰ข๎…”๎Š‹๎Š‹๎†ก๎‰ญ๎ˆฅ
๎ „๎ 
๎€ท๎ฐ๎š๎Œผ๎Œฆ๎ญ ๎š๎‹ผ๎Œฆ๎ญ๎Žฉ๎š๎ฐ๎ฎ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ
โ—ฆ -pc_type ๎ข‚๎ˆ๎Š ๎ฅพ๎†ƒ๎‡ ๎ˆถ๎ˆ๎†ก๎‰ฝ๎ˆ‡๎‹Š๎ขƒ
โ—ฆ -pc_factor_mat_solver_type ๎ข‚๎ˆŸ๎Š ๎ˆŸ๎‰ข๎‰ฝ๎ฅพ๎‰ฝ๎Š ๎‰ข๎†ก๎‰ญ๎ˆ๎Š ๎ขƒ
โ—ฆ -help ๎‡‰๎ˆถ๎‰ญ ๎‹ช๎ˆฅ๎†ก๎ขš๎Š‹๎Š ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ ๎…” ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ
โ—ฆ ๎†ก๎กฒ๎‡Œ๎กฒ๎กซ -mat_mumps_icntl_4 ๎ 
๎ „๎ ‚
๎ƒฒ๎Œฆ๎ซ๎›๎‹ผ๎š๎ฎ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
โ—ฆ -pc_type ๎ข‚๎…”๎‰ฝ๎ˆŸ๎ฅพ๎‡Œ๎…”๎‰ฝ๎ˆŸ๎ขƒ
โ—ฆ -pc_asm_overlap n
โ—ฆ -pc_asm_type ๎ข‚๎…น๎…”๎‰ฝ๎‡ซ๎†ƒ๎ฅพ๎‰ญ๎†ก๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎†ƒ๎Š‹๎ฅพ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‰ข๎ˆถ๎ˆ๎…”๎Š‹๎†ก๎ฅพ๎ˆฅ๎ˆถ๎ˆฅ๎†ก๎ขƒ
โ—ฆ -sub_pc_type๎กซ -sub_ksp_type
โ—ฆ ๎ˆŸ๎ˆถ๎‰ญ๎†ก ๎…”๎Š‹ PCASM ๎ˆŸ๎…”๎ˆฅ๎Š ๎…”๎ˆ ๎‰ข๎…”๎‡Œ๎†ก
๎ „๎ ƒ
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ ƒ๎กช ๎‚•๎ฐ๎Žฉ๎ผ๎˜๎ฐ๎Ž‹๎Ž‹๎Œผ๎ข—๎€ฅ๎š๎‹ผ๎ญ๎ผ๎ข—๎ฅ๎Œผ๎Ž‹๎š๎‹ผ๎Ž™๎Œฏ ๎Œผ๎˜๎ผ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎Š‹๎‡ ๎†ก ๎ˆฅ๎ˆถ๎ˆฅ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎ƒก๎€ท๎†
โˆ†u + ฮป exp
u
= ๎Ÿฟ ๎‡ซ๎ˆฅ โ„ฆ
u
(
x
,
y
) = cos
ฯ€
x
cos
ฯ€
y
๎ˆถ๎ˆฅ
ฮ“
๎‚ฅ๎†ก๎‹๎Š‹๎ˆถ๎ˆฅ ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž
u
n+๎ €
= u
n
+ w ๎‰ฝ๎Š ๎†ƒ๎‡  ๎Š‹๎‡ ๎…”๎Š‹ F(u
n
) + ๎†Ž
w
F(u
n
) = ๎Ÿฟ
๎‹๎‡ซ๎Š‹๎‡  ๎†Ž
w
F(u
n
) = โˆ†w + ฮป exp
u
n
w
๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ƒ๎กผ๎ฃ๎Œผ๎˜๎กฒ๎Œผ๎Œฏ๎“๎ขƒ ๎ค‹ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ƒ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ „๎ „
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ ƒ๎กช ๎‚•๎ฐ๎Žฉ๎ผ๎˜๎ฐ๎Ž‹๎Ž‹๎Œผ๎ข—๎€ฅ๎š๎‹ผ๎ญ๎ผ๎ข—๎ฅ๎Œผ๎Ž‹๎š๎‹ผ๎Ž™๎Œฏ ๎Œผ๎˜๎ผ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎Š‹๎‡ ๎†ก ๎ˆฅ๎ˆถ๎ˆฅ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎ƒก๎€ท๎†
โˆ†u + ฮป exp
u
= ๎Ÿฟ ๎‡ซ๎ˆฅ โ„ฆ
u
(
x
,
y
) = cos
ฯ€
x
cos
ฯ€
y
๎ˆถ๎ˆฅ
ฮ“
๎‚ฅ๎†ก๎‹๎Š‹๎ˆถ๎ˆฅ ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž
u
n+๎ €
= u
n
+ w ๎‰ฝ๎Š ๎†ƒ๎‡  ๎Š‹๎‡ ๎…”๎Š‹ F(u
n
) + ๎†Ž
w
F(u
n
) = ๎Ÿฟ
๎‹๎‡ซ๎Š‹๎‡  ๎†Ž
w
F(u
n
) = โˆ†w + ฮป exp
u
n
w
๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ƒ๎กผ๎ฃ๎Œผ๎˜๎กฒ๎Œผ๎Œฏ๎“๎ขƒ ๎ค‹ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ƒ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎ „๎ „
๎€ฅ๎Ž‹๎Žฉ๎Œฆ๎Ž† ๎Ž•๎‹ผ๎ญ๎š๎ฐ๎Œฆ๎Œผ๎ฃ
๎‚•๎‡ซ๎ˆŸ๎‡ซ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ˆถ๎‡‰ ๎ˆŸ๎ˆถ๎ˆฅ๎ˆถ๎ˆ๎‡ซ๎Š‹๎‡ ๎‡ซ๎†ƒ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎Š ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ๎ˆ๎†ก varf
โ—ฆ ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ๎ˆ๎†ก fespace
๎€ƒ๎ˆ๎Š‹๎†ก๎‰ญ๎ˆฅ๎…”๎Š‹๎‡ซ๎Šฝ๎†ก
โ—ฆ ๎ขจ๎†Ž๎†ก๎†ƒ๎ˆถ๎Š ๎‰ข๎ˆ๎†ก๎†Ž๎ขฉ fespace ๎…”๎ˆฅ๎†Ž ๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎‰ข๎ˆ๎†ก varf
โ—ฆ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ MatNest ๎Š‹๎ˆถ ๎ˆŸ๎…”๎Š‹๎†ƒ๎‡  ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎…น๎ˆ๎ˆถ๎†ƒ๎ˆ‡ ๎‰ฝ๎‹Š๎ˆฅ๎Š‹๎…”๎‹‰
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎‡ซ๎ˆฅ๎‡Œ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  MatNest
โ—ฆ ๎ˆฅ๎ˆถ ๎†ก๎‹‰๎‰ข๎ˆ๎‡ซ๎†ƒ๎‡ซ๎Š‹ ๎‰ญ๎†ก๎‰ข๎‰ญ๎†ก๎‰ฝ๎†ก๎ˆฅ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ โ‡’ ๎ˆฅ๎ˆถ LU๎กซ ๎€ฅ๎‚๎…”๎†ƒ๎ˆถ๎…น๎‡ซ
โ—ฆ ๎…”๎Š ๎Š‹๎ˆถ๎ˆŸ๎…”๎Š‹๎‡ซ๎†ƒ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‰ญ MatConvert
๎ „๎ …
๎€ฅ๎Ž‹๎Žฉ๎Œฆ๎Ž† ๎Ž•๎‹ผ๎ญ๎š๎ฐ๎Œฆ๎Œผ๎ฃ
๎‚•๎‡ซ๎ˆŸ๎‡ซ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ˆถ๎‡‰ ๎ˆŸ๎ˆถ๎ˆฅ๎ˆถ๎ˆ๎‡ซ๎Š‹๎‡ ๎‡ซ๎†ƒ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎Š ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ๎ˆ๎†ก varf
โ—ฆ ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ๎ˆ๎†ก fespace
๎€ƒ๎ˆ๎Š‹๎†ก๎‰ญ๎ˆฅ๎…”๎Š‹๎‡ซ๎Šฝ๎†ก
โ—ฆ ๎ขจ๎†Ž๎†ก๎†ƒ๎ˆถ๎Š ๎‰ข๎ˆ๎†ก๎†Ž๎ขฉ fespace ๎…”๎ˆฅ๎†Ž ๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎‰ข๎ˆ๎†ก varf
โ—ฆ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ MatNest ๎Š‹๎ˆถ ๎ˆŸ๎…”๎Š‹๎†ƒ๎‡  ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎…น๎ˆ๎ˆถ๎†ƒ๎ˆ‡ ๎‰ฝ๎‹Š๎ˆฅ๎Š‹๎…”๎‹‰
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎‡ซ๎ˆฅ๎‡Œ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  MatNest
โ—ฆ ๎ˆฅ๎ˆถ ๎†ก๎‹‰๎‰ข๎ˆ๎‡ซ๎†ƒ๎‡ซ๎Š‹ ๎‰ญ๎†ก๎‰ข๎‰ญ๎†ก๎‰ฝ๎†ก๎ˆฅ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ โ‡’ ๎ˆฅ๎ˆถ LU๎กซ ๎€ฅ๎‚๎…”๎†ƒ๎ˆถ๎…น๎‡ซ
โ—ฆ ๎…”๎Š ๎Š‹๎ˆถ๎ˆŸ๎…”๎Š‹๎‡ซ๎†ƒ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‰ญ MatConvert
๎ „๎ …
๎€ฅ๎Ž‹๎Žฉ๎Œฆ๎Ž† ๎Ž•๎‹ผ๎ญ๎š๎ฐ๎Œฆ๎Œผ๎ฃ
๎‚•๎‡ซ๎ˆŸ๎‡ซ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ˆถ๎‡‰ ๎ˆŸ๎ˆถ๎ˆฅ๎ˆถ๎ˆ๎‡ซ๎Š‹๎‡ ๎‡ซ๎†ƒ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎Š ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ๎ˆ๎†ก varf
โ—ฆ ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ๎ˆ๎†ก fespace
๎€ƒ๎ˆ๎Š‹๎†ก๎‰ญ๎ˆฅ๎…”๎Š‹๎‡ซ๎Šฝ๎†ก
โ—ฆ ๎ขจ๎†Ž๎†ก๎†ƒ๎ˆถ๎Š ๎‰ข๎ˆ๎†ก๎†Ž๎ขฉ fespace ๎…”๎ˆฅ๎†Ž ๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎‰ข๎ˆ๎†ก varf
โ—ฆ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ MatNest ๎Š‹๎ˆถ ๎ˆŸ๎…”๎Š‹๎†ƒ๎‡  ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎…น๎ˆ๎ˆถ๎†ƒ๎ˆ‡ ๎‰ฝ๎‹Š๎ˆฅ๎Š‹๎…”๎‹‰
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎‡ซ๎ˆฅ๎‡Œ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  MatNest
โ—ฆ ๎ˆฅ๎ˆถ ๎†ก๎‹‰๎‰ข๎ˆ๎‡ซ๎†ƒ๎‡ซ๎Š‹ ๎‰ญ๎†ก๎‰ข๎‰ญ๎†ก๎‰ฝ๎†ก๎ˆฅ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ โ‡’ ๎ˆฅ๎ˆถ LU๎กซ ๎€ฅ๎‚๎…”๎†ƒ๎ˆถ๎…น๎‡ซ
โ—ฆ ๎…”๎Š ๎Š‹๎ˆถ๎ˆŸ๎…”๎Š‹๎‡ซ๎†ƒ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‰ญ MatConvert
๎ „๎ …
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ „๎กช ๎ƒก๎Žฉ๎ฐ๎ฃ๎ฃ๎Žฉ๎Ž™ ๎Œผ๎˜๎กฒ ๎›๎ฐ๎ญ๎ซ ๎‚ฅ๎Œผ๎ผ๎Ž•๎‹ผ๎Ž™๎Ž™ ๎€ฅ๎€ญ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ „๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
โ—ฆ ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‹ช๎ˆฅ๎‡ซ๎Š‹๎†ก ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ ๎‹๎‡ซ๎Š‹๎‡  ๎ˆฅ๎ˆถ ๎†ก๎‰ฝ๎‰ฝ๎†ก๎ˆฅ๎Š‹๎‡ซ๎…”๎ˆ ๎€ฅ๎€ญ
โ—ฆ ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎†ƒ๎ˆถ๎ˆฅ๎‰ฝ๎Š‹๎‰ญ๎…”๎‡ซ๎ˆฅ๎Š‹
๎š
โ„ฆ
u = ๎Ÿฟ
๎€ข
A c
c
T
๎Ÿฟ
๎€ฃ๎€ข
u
ฮป
๎€ฃ
=
๎€ข
b
๎Ÿฟ
๎€ฃ
โ—ฆ ๎‹๎‰ญ๎‡ซ๎Š‹๎Š‹๎†ก๎ˆฅ ๎…”๎‰ฝ Mat N = [[A,c],[c',0]]
โ—ฆ A ๎‡ซ๎‰ฝ ๎…” Mat ๎…”๎ˆฅ๎†Ž c ๎‡‰๎ˆถ๎ˆ๎ˆ๎ˆถ๎‹๎‰ฝ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎Š ๎‰ฝ๎†ก ๎†ก๎‡ซ๎Š‹๎‡ ๎†ก๎‰ญ ห†
-1
๎ˆถ๎‰ญ MatConvert ๎ค‹ KSPSolve
โ—ฆ ฮป ๎‰ฝ๎Š‹๎ˆถ๎‰ญ๎†ก๎†Ž ๎ˆถ๎ˆฅ๎ˆ๎‹Š ๎ˆถ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  ๎Š‹๎‡ ๎†ก ๎ˆ๎ˆถ๎‹๎†ก๎‰ฝ๎Š‹ ๎‰ญ๎…”๎ˆฅ๎ˆ‡
๎ „๎ †
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ „๎กช ๎ƒก๎Žฉ๎ฐ๎ฃ๎ฃ๎Žฉ๎Ž™ ๎Œผ๎˜๎กฒ ๎›๎ฐ๎ญ๎ซ ๎‚ฅ๎Œผ๎ผ๎Ž•๎‹ผ๎Ž™๎Ž™ ๎€ฅ๎€ญ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ „๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
โ—ฆ ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‹ช๎ˆฅ๎‡ซ๎Š‹๎†ก ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ ๎‹๎‡ซ๎Š‹๎‡  ๎ˆฅ๎ˆถ ๎†ก๎‰ฝ๎‰ฝ๎†ก๎ˆฅ๎Š‹๎‡ซ๎…”๎ˆ ๎€ฅ๎€ญ
โ—ฆ ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎†ƒ๎ˆถ๎ˆฅ๎‰ฝ๎Š‹๎‰ญ๎…”๎‡ซ๎ˆฅ๎Š‹
๎š
โ„ฆ
u = ๎Ÿฟ
๎€ข
A c
c
T
๎Ÿฟ
๎€ฃ๎€ข
u
ฮป
๎€ฃ
=
๎€ข
b
๎Ÿฟ
๎€ฃ
โ—ฆ ๎‹๎‰ญ๎‡ซ๎Š‹๎Š‹๎†ก๎ˆฅ ๎…”๎‰ฝ Mat N = [[A,c],[c',0]]
โ—ฆ A ๎‡ซ๎‰ฝ ๎…” Mat ๎…”๎ˆฅ๎†Ž c ๎‡‰๎ˆถ๎ˆ๎ˆ๎ˆถ๎‹๎‰ฝ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎Š ๎‰ฝ๎†ก ๎†ก๎‡ซ๎Š‹๎‡ ๎†ก๎‰ญ ห†
-1
๎ˆถ๎‰ญ MatConvert ๎ค‹ KSPSolve
โ—ฆ ฮป ๎‰ฝ๎Š‹๎ˆถ๎‰ญ๎†ก๎†Ž ๎ˆถ๎ˆฅ๎ˆ๎‹Š ๎ˆถ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  ๎Š‹๎‡ ๎†ก ๎ˆ๎ˆถ๎‹๎†ก๎‰ฝ๎Š‹ ๎‰ญ๎…”๎ˆฅ๎ˆ‡
๎ „๎ †
๎ƒจ๎Œผ๎Œฆ๎ญ๎‹ผ๎Ž™๎ ๎ผ๎Ž‹๎‹ผ๎š ๎Ž•๎‹ผ๎ญ๎š๎ฐ๎Œฆ๎Œผ๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ …๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก matrix ๎…”๎‰ญ๎†ก ๎ˆฅ๎ˆถ๎Š‹ ๎‰ฝ๎‰ฉ๎Š ๎…”๎‰ญ๎†ก
โ—ฆ matrix Loc = varf(Ph, Vh) ๎กธ๎กธ ๎‚•๎ˆถ๎†ƒ๎กฒ๎ˆฅ ๎ค ๎„ช๎‡ ๎กฒ๎ˆฅ๎†Ž๎ˆถ๎‡‰
โ—ฆ ๎†ƒ๎ˆถ๎Š ๎‰ข๎ˆ๎‡ซ๎ˆฅ๎‡Œ ๎ ๎€ท ๎…”๎ˆฅ๎†Ž ๎ ‚๎€ท ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ
fespace๎‰ฝ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก๎†Ž ๎‹๎‡ซ๎Š‹๎‡  ๎Š‹๎‡ ๎†ก ๎‰ฝ๎…”๎ˆŸ๎†ก ๎‰ข๎…”๎‰ญ๎Š‹๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎ˆถ๎ˆฅ๎†ก ๎‰ฝ๎‰ฉ๎Š ๎…”๎‰ญ๎†ก Mat A ๎‡‰๎ˆถ๎‰ญ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆฅ๎‡Œ Vh
โ—ฆ ๎ˆถ๎ˆฅ๎†ก ๎‰ฝ๎‰ฉ๎Š ๎…”๎‰ญ๎†ก Mat B ๎‡‰๎ˆถ๎‰ญ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆฅ๎‡Œ Ph
โ—ฆ ๎ˆถ๎ˆฅ๎†ก Mat C(A,B,Loc)
=โ‡’ DmeshCreate(Th) ๎ค‹ MatCreate(A|B)
๎ „๎ ‡
๎ƒจ๎Œผ๎Œฆ๎ญ๎‹ผ๎Ž™๎ ๎ผ๎Ž‹๎‹ผ๎š ๎Ž•๎‹ผ๎ญ๎š๎ฐ๎Œฆ๎Œผ๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ …๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก matrix ๎…”๎‰ญ๎†ก ๎ˆฅ๎ˆถ๎Š‹ ๎‰ฝ๎‰ฉ๎Š ๎…”๎‰ญ๎†ก
โ—ฆ matrix Loc = varf(Ph, Vh) ๎กธ๎กธ ๎‚•๎ˆถ๎†ƒ๎กฒ๎ˆฅ ๎ค ๎„ช๎‡ ๎กฒ๎ˆฅ๎†Ž๎ˆถ๎‡‰
โ—ฆ ๎†ƒ๎ˆถ๎Š ๎‰ข๎ˆ๎‡ซ๎ˆฅ๎‡Œ ๎ ๎€ท ๎…”๎ˆฅ๎†Ž ๎ ‚๎€ท ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ
fespace๎‰ฝ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก๎†Ž ๎‹๎‡ซ๎Š‹๎‡  ๎Š‹๎‡ ๎†ก ๎‰ฝ๎…”๎ˆŸ๎†ก ๎‰ข๎…”๎‰ญ๎Š‹๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎ˆถ๎ˆฅ๎†ก ๎‰ฝ๎‰ฉ๎Š ๎…”๎‰ญ๎†ก Mat A ๎‡‰๎ˆถ๎‰ญ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆฅ๎‡Œ Vh
โ—ฆ ๎ˆถ๎ˆฅ๎†ก ๎‰ฝ๎‰ฉ๎Š ๎…”๎‰ญ๎†ก Mat B ๎‡‰๎ˆถ๎‰ญ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆฅ๎‡Œ Ph
โ—ฆ ๎ˆถ๎ˆฅ๎†ก Mat C(A,B,Loc)
=โ‡’ DmeshCreate(Th) ๎ค‹ MatCreate(A|B)
๎ „๎ ‡
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ †๎กช ๎ƒฒ๎ญ๎Žฉ๎Ž†๎Œผ๎ฃ ๎Œผ๎˜๎ผ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ †๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎…” ๎ƒก๎ˆถ๎‡ซ๎‰ฝ๎†ก๎Š ๎‡ซ๎ˆ๎ˆ๎†ก ๎‡ซ๎ˆฅ๎‹ซ๎ˆถ๎‹ ๎Š‹๎‡ ๎†ก ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ
โˆ’โˆ†u + โˆ‡p = ๎Ÿฟ ๎‡ซ๎ˆฅ โ„ฆ
โˆ‡ ยท u = ๎Ÿฟ
โ—ฆ ๎Š‹๎‹๎ˆถ varf๎‰ฝ
โ—ฆ ๎Š‹๎‹๎ˆถ fespace ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎Š‹๎‹๎ˆถ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎†ก๎†Ž Mat๎‰ฝ A ๎…”๎ˆฅ๎†Ž B
โ—ฆ ๎†ƒ๎ˆถ๎Š ๎‰ข๎ˆ๎†ก๎†Ž ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ Mat N = [[A,B],[B',0]]
โ—ฆ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‰ข๎ˆถ๎‰ฝ๎†ก๎†Ž ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎ˆฅ๎ˆถ๎Š‹ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎†ก๎†Ž ๎†ก๎‹‰๎‰ข๎ˆ๎‡ซ๎†ƒ๎‡ซ๎Š‹๎ˆ๎‹Š
๎ „๎ ˆ
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ †๎กช ๎ƒฒ๎ญ๎Žฉ๎Ž†๎Œผ๎ฃ ๎Œผ๎˜๎ผ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ †๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎…” ๎ƒก๎ˆถ๎‡ซ๎‰ฝ๎†ก๎Š ๎‡ซ๎ˆ๎ˆ๎†ก ๎‡ซ๎ˆฅ๎‹ซ๎ˆถ๎‹ ๎Š‹๎‡ ๎†ก ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ
โˆ’โˆ†u + โˆ‡p = ๎Ÿฟ ๎‡ซ๎ˆฅ โ„ฆ
โˆ‡ ยท u = ๎Ÿฟ
โ—ฆ ๎Š‹๎‹๎ˆถ varf๎‰ฝ
โ—ฆ ๎Š‹๎‹๎ˆถ fespace ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎Š‹๎‹๎ˆถ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎†ก๎†Ž Mat๎‰ฝ A ๎…”๎ˆฅ๎†Ž B
โ—ฆ ๎†ƒ๎ˆถ๎Š ๎‰ข๎ˆ๎†ก๎†Ž ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ Mat N = [[A,B],[B',0]]
โ—ฆ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‰ข๎ˆถ๎‰ฝ๎†ก๎†Ž ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎ˆฅ๎ˆถ๎Š‹ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎†ก๎†Ž ๎†ก๎‹‰๎‰ข๎ˆ๎‡ซ๎†ƒ๎‡ซ๎Š‹๎ˆ๎‹Š
๎ „๎ ˆ
๎‚ก๎ผ๎Ž‹๎ญ๎ฐ๎ ๎š๎ฐ๎Œฏ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ๎ฃ
โ—ฆ ๎‰ข๎ˆถ๎‡ซ๎ˆฅ๎Š‹๎ขš๎‹๎‡ซ๎‰ฝ๎†ก ๎…”๎‡Œ๎‡Œ๎‰ญ๎†ก๎‡Œ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ€ข hypre ๎ขŽ๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‡ ๎‹Š๎‰ข๎‰ญ๎†ก๎ข ๎ข‚๎ข๎…”๎ˆ๎‡Œ๎ˆถ๎Š ๎Š‹ ๎…”๎ˆฅ๎†Ž ๎„ต๎…”๎ˆฅ๎‡Œ ๎ ๎Ÿฟ๎Ÿฟ๎ ๎ขƒ
โ€ข ๎€ƒ๎‚ก๎ƒฒ ๎ข‚๎ฑ๎‡ซ๎‰ข๎Š‹๎ˆŸ๎…”๎‡ซ๎‰ญ ๎…”๎ˆฅ๎†Ž ๎„ด๎Š  ๎ ๎Ÿฟ๎Ÿฟ๎ †๎ขƒ
โ—ฆ ๎‰ฝ๎ˆŸ๎ˆถ๎ˆถ๎Š‹๎‡ ๎†ก๎†Ž ๎…”๎‡Œ๎‡Œ๎‰ญ๎†ก๎‡Œ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ€ข ๎ฅ๎€ƒ๎‚ก๎ฅ ๎ข‚๎€ƒ๎†Ž๎…”๎ˆŸ๎‰ฝ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ ƒ๎ขƒ
โ€ข ๎‚ก๎‚• ๎ขŽ๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎ˆŸ๎ˆ๎ข ๎ข‚๎ฅ๎†ก๎†ก ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎Ÿฟ๎ …๎ขƒ
๎ƒฒ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ๎‰ฝ ๎ˆถ๎‡‰ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎…น๎ˆ๎ˆถ๎†ƒ๎ˆ‡ ๎‰ฝ๎‡ซ๎‹–๎†ก
โ—ฆ MatNullSpace๎กซ ๎†ก๎กฒ๎‡Œ๎กฒ๎กซ ๎‰ญ๎‡ซ๎‡Œ๎‡ซ๎†Ž ๎…น๎ˆถ๎†Ž๎‹Š ๎ˆŸ๎ˆถ๎†Ž๎†ก๎‰ฝ
๎ …๎Ÿฟ
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ ‡๎กช ๎ฃ๎ฃ๎ฃ๎ญ๎Œผ๎Ž• ๎Žฉ๎š ๎Ž‹๎ฐ๎Ž™๎Œผ๎‹ผ๎š ๎Œผ๎Ž‹๎‹ผ๎ฃ๎ญ๎ฐ๎Œฆ๎ฐ๎ญ๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‡๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎Š‹๎‡ ๎†ก ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ
โˆ’๎†Ž๎‡ซ๎Šฝ ฯƒ = f ๎‡ซ๎ˆฅ โ„ฆ
u = ๎Ÿฟ ๎ˆถ๎ˆฅ ฮ“
D
๎ƒพ๎ˆถ๎ˆถ๎ˆ๎‰ฝ
โ—ฆ ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎…”๎ˆ fespace
โ—ฆ ๎…น๎ˆ๎ˆถ๎†ƒ๎ˆ‡ ๎‰ฝ๎‡ซ๎‹–๎†ก ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎‰ญ๎†ก๎†ก
โ—ฆ MatNullSpace ๎‹๎‡ซ๎Š‹๎‡  ๎Š‹๎‡ ๎†ก ๎‰ญ๎‡ซ๎‡Œ๎‡ซ๎†Ž ๎…น๎ˆถ๎†Ž๎‹Š ๎ˆŸ๎ˆถ๎†Ž๎†ก๎‰ฝ
โ€ข ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ (๎ €, ๎Ÿฟ, ๎Ÿฟ) (๎Ÿฟ, ๎ €, ๎Ÿฟ) (๎Ÿฟ, ๎Ÿฟ, ๎ €)
โ€ข ๎‰ญ๎ˆถ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ (y, โˆ’x, ๎Ÿฟ) (โˆ’z, ๎Ÿฟ, x) (๎Ÿฟ, z, โˆ’y)
๎ …๎ €
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ ‡๎กช ๎ฃ๎ฃ๎ฃ๎ญ๎Œผ๎Ž• ๎Žฉ๎š ๎Ž‹๎ฐ๎Ž™๎Œผ๎‹ผ๎š ๎Œผ๎Ž‹๎‹ผ๎ฃ๎ญ๎ฐ๎Œฆ๎ฐ๎ญ๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ‡๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎Š‹๎‡ ๎†ก ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ
โˆ’๎†Ž๎‡ซ๎Šฝ ฯƒ = f ๎‡ซ๎ˆฅ โ„ฆ
u = ๎Ÿฟ ๎ˆถ๎ˆฅ ฮ“
D
๎ƒพ๎ˆถ๎ˆถ๎ˆ๎‰ฝ
โ—ฆ ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎…”๎ˆ fespace
โ—ฆ ๎…น๎ˆ๎ˆถ๎†ƒ๎ˆ‡ ๎‰ฝ๎‡ซ๎‹–๎†ก ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎‰ญ๎†ก๎†ก
โ—ฆ MatNullSpace ๎‹๎‡ซ๎Š‹๎‡  ๎Š‹๎‡ ๎†ก ๎‰ญ๎‡ซ๎‡Œ๎‡ซ๎†Ž ๎…น๎ˆถ๎†Ž๎‹Š ๎ˆŸ๎ˆถ๎†Ž๎†ก๎‰ฝ
โ€ข ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ (๎ €, ๎Ÿฟ, ๎Ÿฟ) (๎Ÿฟ, ๎ €, ๎Ÿฟ) (๎Ÿฟ, ๎Ÿฟ, ๎ €)
โ€ข ๎‰ญ๎ˆถ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ (y, โˆ’x, ๎Ÿฟ) (โˆ’z, ๎Ÿฟ, x) (๎Ÿฟ, z, โˆ’y)
๎ …๎ €
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ ˆ๎กช ๎ ๎Œผ๎Žฉ๎Ž•๎Œผ๎ญ๎š๎ฐ๎Œฆ ๎Ž•๎ผ๎Ž‹๎ญ๎ฐ๎ ๎š๎ฐ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ˆ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ
๎†ƒ๎Š ๎‰ญ๎ˆ ๎†ƒ๎Š ๎‰ญ๎ˆ E โˆ’ k
๎ 
E = f ๎‡ซ๎ˆฅ โ„ฆ
(๎†ƒ๎Š ๎‰ญ๎ˆ E) ร— n โˆ’ ik(E ร— n) ร— n = ๎Ÿฟ ๎ˆถ๎ˆฅ ฮ“
R
๎ƒพ๎ˆถ๎ˆถ๎ˆ๎‰ฝ
โ—ฆ ๎‚ฅ๎†ข๎†Ž๎†ข๎ˆ๎†ก๎†ƒ ๎†ก๎†Ž๎‡Œ๎†ก ๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ
โ—ฆ ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎‰ฝ๎ˆŸ๎ˆถ๎ˆถ๎Š‹๎‡ ๎†ก๎‰ญ๎‰ฝ
โ—ฆ buildMatEdgeRecursive
โ—ฆ ๎ˆŸ๎ˆถ๎‰ญ๎†ก ๎…”๎Š‹ PCMG ๎ˆŸ๎…”๎ˆฅ๎Š ๎…”๎ˆ ๎‰ข๎…”๎‡Œ๎†ก
๎ …๎ 
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ ˆ๎กช ๎ ๎Œผ๎Žฉ๎Ž•๎Œผ๎ญ๎š๎ฐ๎Œฆ ๎Ž•๎ผ๎Ž‹๎ญ๎ฐ๎ ๎š๎ฐ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ ˆ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ
๎†ƒ๎Š ๎‰ญ๎ˆ ๎†ƒ๎Š ๎‰ญ๎ˆ E โˆ’ k
๎ 
E = f ๎‡ซ๎ˆฅ โ„ฆ
(๎†ƒ๎Š ๎‰ญ๎ˆ E) ร— n โˆ’ ik(E ร— n) ร— n = ๎Ÿฟ ๎ˆถ๎ˆฅ ฮ“
R
๎ƒพ๎ˆถ๎ˆถ๎ˆ๎‰ฝ
โ—ฆ ๎‚ฅ๎†ข๎†Ž๎†ข๎ˆ๎†ก๎†ƒ ๎†ก๎†Ž๎‡Œ๎†ก ๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎‰ฝ
โ—ฆ ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎‰ฝ๎ˆŸ๎ˆถ๎ˆถ๎Š‹๎‡ ๎†ก๎‰ญ๎‰ฝ
โ—ฆ buildMatEdgeRecursive
โ—ฆ ๎ˆŸ๎ˆถ๎‰ญ๎†ก ๎…”๎Š‹ PCMG ๎ˆŸ๎…”๎ˆฅ๎Š ๎…”๎ˆ ๎‰ข๎…”๎‡Œ๎†ก
๎ …๎ 
๎ข๎ฐ๎Œผ๎Ž‹๎Œฏ๎ฃ๎“๎Ž‹๎ฐ๎ญ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎Ÿฟ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎‰ฝ๎†ก๎‰ข๎…”๎‰ญ๎…”๎Š‹๎†ก ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎†ก๎…”๎†ƒ๎‡  ๎‹ช๎†ก๎ˆ๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ ๎…”๎‰ญ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎ˆ๎†ก๎…”๎Šฝ๎†ก๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ IS
โ—ฆ fespace Vh(Pk,Pq,Pt) โ‡’ Vh [u,v,w] = [1,2,3]
โ—ฆ ๎ˆฅ๎†ก๎‹ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ -fieldsplit_%d_
๎‚ถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ—ฆ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ ๎†ƒ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ๎‡ซ๎‹–๎…”๎…น๎ˆ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎…” string[int]
โ—ฆ ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎‹‰๎‡ซ๎ˆŸ๎…”๎Š‹๎†ก ๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹
๎†๎‹‰๎…”๎ˆŸ๎‰ข๎ˆ๎†ก๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎ƒฒ๎Š‹๎ˆถ๎ˆ‡๎†ก๎‰ฝ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ข‚๎‚‘๎ˆฅ๎†ก๎‰ข๎ˆ๎†ก๎‹Š ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ
A =
๎€ข
C B
B
T
๎Ÿฟ
๎€ฃ
โ‰ˆ ๎‰ข๎†ƒ
A
=
๎€ข ๎€ฃ
๎ขš๎ˆ‡๎‰ฝ๎‰ข๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡Œ๎ˆŸ๎‰ญ๎†ก๎‰ฝ
๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Šฝ๎†ก๎ˆ๎ˆถ๎†ƒ๎‡ซ๎Š‹๎‹Š๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก
๎ข๎ฐ๎Œผ๎Ž‹๎Œฏ๎ฃ๎“๎Ž‹๎ฐ๎ญ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎Ÿฟ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎‰ฝ๎†ก๎‰ข๎…”๎‰ญ๎…”๎Š‹๎†ก ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎†ก๎…”๎†ƒ๎‡  ๎‹ช๎†ก๎ˆ๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ ๎…”๎‰ญ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎ˆ๎†ก๎…”๎Šฝ๎†ก๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ IS
โ—ฆ fespace Vh(Pk,Pk,Pq) โ‡’ Vh [u,v,p] = [1,1,2]
โ—ฆ ๎ˆฅ๎†ก๎‹ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ -fieldsplit_%d_
๎‚ถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ—ฆ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ ๎†ƒ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ๎‡ซ๎‹–๎…”๎…น๎ˆ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎…” string[int]
โ—ฆ ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎‹‰๎‡ซ๎ˆŸ๎…”๎Š‹๎†ก ๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹
๎†๎‹‰๎…”๎ˆŸ๎‰ข๎ˆ๎†ก๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎ƒฒ๎Š‹๎ˆถ๎ˆ‡๎†ก๎‰ฝ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ข‚๎‚‘๎ˆฅ๎†ก๎‰ข๎ˆ๎†ก๎‹Š ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ
A =
๎€ข
C B
B
T
๎Ÿฟ
๎€ฃ
โ‰ˆ ๎‰ข๎†ƒ
A
=
๎€ข
C
โˆ’
๎ €
๎Ÿฟ
๎Ÿฟ I
๎€ฃ
๎ขš๎ˆ‡๎‰ฝ๎‰ข๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡Œ๎ˆŸ๎‰ญ๎†ก๎‰ฝ ๎ขš๎‰ข๎†ƒ๎กผ๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก
๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Šฝ๎†ก๎ˆ๎ˆถ๎†ƒ๎‡ซ๎Š‹๎‹Š๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎ˆ๎Š 
๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎‰ข๎‰ญ๎†ก๎‰ฝ๎‰ฝ๎Š ๎‰ญ๎†ก๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡ฟ๎…”๎†ƒ๎ˆถ๎…น๎‡ซ
๎ข๎ฐ๎Œผ๎Ž‹๎Œฏ๎ฃ๎“๎Ž‹๎ฐ๎ญ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎Ÿฟ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎‰ฝ๎†ก๎‰ข๎…”๎‰ญ๎…”๎Š‹๎†ก ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎†ก๎…”๎†ƒ๎‡  ๎‹ช๎†ก๎ˆ๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ ๎…”๎‰ญ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎ˆ๎†ก๎…”๎Šฝ๎†ก๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ IS
โ—ฆ fespace Vh(Pk,Pk,Pq) โ‡’ Vh [u,v,p] = [1,1,2]
โ—ฆ ๎ˆฅ๎†ก๎‹ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ -fieldsplit_%d_
๎‚ถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ—ฆ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ ๎†ƒ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ๎‡ซ๎‹–๎…”๎…น๎ˆ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎…” string[int]
โ—ฆ ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎‹‰๎‡ซ๎ˆŸ๎…”๎Š‹๎†ก ๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹
๎†๎‹‰๎…”๎ˆŸ๎‰ข๎ˆ๎†ก๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎ƒฒ๎Š‹๎ˆถ๎ˆ‡๎†ก๎‰ฝ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ข‚๎‚‘๎ˆฅ๎†ก๎‰ข๎ˆ๎†ก๎‹Š ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ
A =
๎€ข
C B
B
T
๎Ÿฟ
๎€ฃ
โ‰ˆ ๎‰ข๎†ƒ
A
=
๎€ข
๎ˆ‡๎‰ฝ๎‰ข
C
๎Ÿฟ
๎Ÿฟ I
๎€ฃ
๎ขš๎ˆ‡๎‰ฝ๎‰ข๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡‰๎‡Œ๎ˆŸ๎‰ญ๎†ก๎‰ฝ ๎ขš๎‰ข๎†ƒ๎กผ๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก
๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Šฝ๎†ก๎ˆ๎ˆถ๎†ƒ๎‡ซ๎Š‹๎‹Š๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡Œ๎…”๎ˆŸ๎‡Œ
๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎‰ข๎‰ญ๎†ก๎‰ฝ๎‰ฝ๎Š ๎‰ญ๎†ก๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡ฟ๎…”๎†ƒ๎ˆถ๎…น๎‡ซ
๎ข๎ฐ๎Œผ๎Ž‹๎Œฏ๎ฃ๎“๎Ž‹๎ฐ๎ญ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎Ÿฟ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎‰ฝ๎†ก๎‰ข๎…”๎‰ญ๎…”๎Š‹๎†ก ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎†ก๎…”๎†ƒ๎‡  ๎‹ช๎†ก๎ˆ๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ ๎…”๎‰ญ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎ˆ๎†ก๎…”๎Šฝ๎†ก๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ IS
โ—ฆ fespace Vh(Pk,Pk,Pq) โ‡’ Vh [u,v,p] = [1,1,2]
โ—ฆ ๎ˆฅ๎†ก๎‹ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ -fieldsplit_%d_
๎‚ถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ—ฆ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ ๎†ƒ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ๎‡ซ๎‹–๎…”๎…น๎ˆ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎…” string[int]
โ—ฆ ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎‹‰๎‡ซ๎ˆŸ๎…”๎Š‹๎†ก ๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹
๎†๎‹‰๎…”๎ˆŸ๎‰ข๎ˆ๎†ก๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎ƒฒ๎Š‹๎ˆถ๎ˆ‡๎†ก๎‰ฝ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ข‚๎‚‘๎ˆฅ๎†ก๎‰ข๎ˆ๎†ก๎‹Š ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ
A =
๎€ข
C B
B
T
๎Ÿฟ
๎€ฃ
โ‰ˆ ๎‰ข๎†ƒ
A
=
๎€ข
๎ˆ‡๎‰ฝ๎‰ข
C
B
๎Ÿฟ I
๎€ฃ
๎ขš๎ˆ‡๎‰ฝ๎‰ข๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡‰๎‡Œ๎ˆŸ๎‰ญ๎†ก๎‰ฝ ๎ขš๎‰ข๎†ƒ๎กผ๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎‰ข๎ˆ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎Šฝ๎†ก
๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Šฝ๎†ก๎ˆ๎ˆถ๎†ƒ๎‡ซ๎Š‹๎‹Š๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡Œ๎…”๎ˆŸ๎‡Œ
๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎‰ข๎‰ญ๎†ก๎‰ฝ๎‰ฝ๎Š ๎‰ญ๎†ก๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡ฟ๎…”๎†ƒ๎ˆถ๎…น๎‡ซ
๎ข๎ฐ๎Œผ๎Ž‹๎Œฏ๎ฃ๎“๎Ž‹๎ฐ๎ญ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎Ÿฟ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎‰ฝ๎†ก๎‰ข๎…”๎‰ญ๎…”๎Š‹๎†ก ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎†ก๎…”๎†ƒ๎‡  ๎‹ช๎†ก๎ˆ๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ ๎…”๎‰ญ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎ˆ๎†ก๎…”๎Šฝ๎†ก๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ IS
โ—ฆ fespace Vh(Pk,Pk,Pq) โ‡’ Vh [u,v,p] = [1,1,2]
โ—ฆ ๎ˆฅ๎†ก๎‹ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ -fieldsplit_%d_
๎‚ถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ—ฆ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ ๎†ƒ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ๎‡ซ๎‹–๎…”๎…น๎ˆ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎…” string[int]
โ—ฆ ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎‹‰๎‡ซ๎ˆŸ๎…”๎Š‹๎†ก ๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹
๎†๎‹‰๎…”๎ˆŸ๎‰ข๎ˆ๎†ก๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎ƒฒ๎Š‹๎ˆถ๎ˆ‡๎†ก๎‰ฝ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ข‚๎‚‘๎ˆฅ๎†ก๎‰ข๎ˆ๎†ก๎‹Š ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ
A =
๎€ข
C B
B
T
๎Ÿฟ
๎€ฃ
โ‰ˆ ๎‰ข๎†ƒ
A
=
๎€ข
๎ˆ‡๎‰ฝ๎‰ข
C
๎Ÿฟ
๎Ÿฟ โˆ’๎ˆ‡๎‰ฝ๎‰ข
S
๎€ฃ
๎ขš๎ˆ‡๎‰ฝ๎‰ข๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡‰๎‡Œ๎ˆŸ๎‰ญ๎†ก๎‰ฝ ๎ขš๎‰ข๎†ƒ๎กผ๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‰ฝ๎†ƒ๎‡ ๎Š ๎‰ญ
๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Šฝ๎†ก๎ˆ๎ˆถ๎†ƒ๎‡ซ๎Š‹๎‹Š๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡Œ๎…”๎ˆŸ๎‡Œ
๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎‰ข๎‰ญ๎†ก๎‰ฝ๎‰ฝ๎Š ๎‰ญ๎†ก๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡ฟ๎…”๎†ƒ๎ˆถ๎…น๎‡ซ
๎ขš๎‰ข๎†ƒ๎กผ๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎‰ฝ๎†ƒ๎‡ ๎Š ๎‰ญ๎กผ๎‡‰๎…”๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎†Ž๎‡ซ๎…”๎‡Œ
๎ข๎ฐ๎Œผ๎Ž‹๎Œฏ๎ฃ๎“๎Ž‹๎ฐ๎ญ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎Ÿฟ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎‰ฝ๎†ก๎‰ข๎…”๎‰ญ๎…”๎Š‹๎†ก ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎†ก๎…”๎†ƒ๎‡  ๎‹ช๎†ก๎ˆ๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ ๎…”๎‰ญ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎ˆ๎†ก๎…”๎Šฝ๎†ก๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ IS
โ—ฆ fespace Vh(Pk,Pk,Pq) โ‡’ Vh [u,v,p] = [1,1,2]
โ—ฆ ๎ˆฅ๎†ก๎‹ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ -fieldsplit_%d_
๎‚ถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ—ฆ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ ๎†ƒ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ๎‡ซ๎‹–๎…”๎…น๎ˆ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎…” string[int]
โ—ฆ ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎‹‰๎‡ซ๎ˆŸ๎…”๎Š‹๎†ก ๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹
๎†๎‹‰๎…”๎ˆŸ๎‰ข๎ˆ๎†ก๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎ƒฒ๎Š‹๎ˆถ๎ˆ‡๎†ก๎‰ฝ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ข‚๎‚‘๎ˆฅ๎†ก๎‰ข๎ˆ๎†ก๎‹Š ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ
A =
๎€ข
C B
B
T
๎Ÿฟ
๎€ฃ
โ‰ˆ ๎‰ข๎†ƒ
A
=
๎€ข
๎ˆ‡๎‰ฝ๎‰ข
C
๎Ÿฟ
B
T
๎ˆ‡๎‰ฝ๎‰ข
S
๎€ฃ
๎ขš๎ˆ‡๎‰ฝ๎‰ข๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡‰๎‡Œ๎ˆŸ๎‰ญ๎†ก๎‰ฝ ๎ขš๎‰ข๎†ƒ๎กผ๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‰ฝ๎†ƒ๎‡ ๎Š ๎‰ญ
๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Šฝ๎†ก๎ˆ๎ˆถ๎†ƒ๎‡ซ๎Š‹๎‹Š๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡Œ๎…”๎ˆŸ๎‡Œ
๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎‰ข๎‰ญ๎†ก๎‰ฝ๎‰ฝ๎Š ๎‰ญ๎†ก๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡ฟ๎…”๎†ƒ๎ˆถ๎…น๎‡ซ
๎ขš๎‰ข๎†ƒ๎กผ๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎‰ฝ๎†ƒ๎‡ ๎Š ๎‰ญ๎กผ๎‡‰๎…”๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎ˆ๎ˆถ๎‹๎†ก๎‰ญ
๎ข๎ฐ๎Œผ๎Ž‹๎Œฏ๎ฃ๎“๎Ž‹๎ฐ๎ญ ๎“๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎Œฏ๎ฐ๎ญ๎ฐ๎Žฉ๎Ž™๎Œผ๎š๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎Ÿฟ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
โ—ฆ ๎‰ฝ๎†ก๎‰ข๎…”๎‰ญ๎…”๎Š‹๎†ก ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎†ก๎…”๎†ƒ๎‡  ๎‹ช๎†ก๎ˆ๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ ๎…”๎‰ญ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎ˆ๎†ก๎…”๎Šฝ๎†ก๎†Ž
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ IS
โ—ฆ fespace Vh(Pk,Pk,Pq) โ‡’ Vh [u,v,p] = [1,1,2]
โ—ฆ ๎ˆฅ๎†ก๎‹ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ -fieldsplit_%d_
๎‚ถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ—ฆ ๎‰ข๎‰ญ๎†ก๎‹ช๎‹‰๎†ก๎‰ฝ ๎†ƒ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ๎‡ซ๎‹–๎…”๎…น๎ˆ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎…” string[int]
โ—ฆ ๎…”๎‰ข๎‰ข๎‰ญ๎ˆถ๎‹‰๎‡ซ๎ˆŸ๎…”๎Š‹๎†ก ๎ƒฒ๎†ƒ๎‡ ๎Š ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹
๎†๎‹‰๎…”๎ˆŸ๎‰ข๎ˆ๎†ก๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎ƒฒ๎Š‹๎ˆถ๎ˆ‡๎†ก๎‰ฝ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ข‚๎‚‘๎ˆฅ๎†ก๎‰ข๎ˆ๎†ก๎‹Š ๎ ๎Ÿฟ๎ €๎ ‚๎ขƒ
A =
๎€ข
C B
B
T
๎Ÿฟ
๎€ฃ
โ‰ˆ ๎‰ข๎†ƒ
A
=
๎€ข
C
โˆ’๎ €
B
๎Ÿฟ ๎ˆ‡๎‰ฝ๎‰ข
S
๎€ฃ
๎ขš๎ˆ‡๎‰ฝ๎‰ข๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‡Œ๎ˆŸ๎‰ญ๎†ก๎‰ฝ ๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎‰ข๎‰ญ๎†ก๎‰ฝ๎‰ฝ๎Š ๎‰ญ๎†ก๎กผ๎ˆ‡๎‰ฝ๎‰ข๎กผ๎ˆŸ๎…”๎‹‰๎กผ๎‡ซ๎Š‹๎‰ฝ ๎ €
๎ขš๎‰ข๎†ƒ๎กผ๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‰ฝ๎†ƒ๎‡ ๎Š ๎‰ญ ๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎Šฝ๎†ก๎ˆ๎ˆถ๎†ƒ๎‡ซ๎Š‹๎‹Š๎กผ๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก
๎ˆ๎Š  ๎ขš๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎‰ข๎‰ญ๎†ก๎‰ฝ๎‰ฝ๎Š ๎‰ญ๎†ก๎กผ๎ˆ‡๎‰ฝ๎‰ข๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎‰ญ๎‡ซ๎†ƒ๎‡ ๎…”๎‰ญ๎†Ž๎‰ฝ๎ˆถ๎ˆฅ
๎ขš๎‰ข๎†ƒ๎กผ๎‹ช๎†ก๎ˆ๎†Ž๎‰ฝ๎‰ข๎ˆ๎‡ซ๎Š‹๎กผ๎‰ฝ๎†ƒ๎‡ ๎Š ๎‰ญ๎กผ๎‡‰๎…”๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎Š ๎‰ข๎‰ข๎†ก๎‰ญ
๎‚ก๎‹ผ๎ญ๎š๎ฐ๎ข๎ขš๎š๎š๎Œผ๎Œผ ๎Žฉ๎“๎Œผ๎š๎‹ผ๎ญ๎Žฉ๎š๎ฃ
MatShell
โ—ฆ ๎Š ๎‰ฝ๎†ก๎‰ญ๎ขš๎‰ข๎‰ญ๎ˆถ๎Šฝ๎‡ซ๎†Ž๎†ก๎†Ž ๎‰ญ๎ˆถ๎Š ๎Š‹๎‡ซ๎ˆฅ๎†ก๎‰ฝ ๎ขŽMatMult๎กซ MatMultTranspose๎ข
โ—ฆ ๎ขŽ๎Š ๎‰ฝ๎†ก๎‰ญ๎ขš๎‰ข๎‰ญ๎ˆถ๎Šฝ๎‡ซ๎†Ž๎†ก๎†Ž๎ข ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ ๎ˆ๎†ก๎‰ฝ๎‰ฝ ๎‡ซ๎ˆฅ๎Š‹๎Š ๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก
โ—ฆ func ๎‰ข๎…”๎‰ฝ๎‰ฝ๎†ก๎†Ž ๎Š‹๎ˆถ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ
โ—ฆ ๎ˆŸ๎Š ๎‰ฝ๎Š‹ ๎Š ๎‰ฝ๎†ก ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ KSP ๎‡ซ๎‰ฝ ๎ˆถ๎…น๎ˆ๎‡ซ๎Šฝ๎‡ซ๎ˆถ๎Š ๎‰ฝ ๎Š‹๎ˆถ ๎…” Mat ๎Š‹๎‹Š๎‰ข๎†ก
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ PCSHELL
๎ …๎ ƒ
๎‚ก๎‹ผ๎ญ๎š๎ฐ๎ข๎ขš๎š๎š๎Œผ๎Œผ ๎Žฉ๎“๎Œผ๎š๎‹ผ๎ญ๎Žฉ๎š๎ฃ
MatShell
โ—ฆ ๎Š ๎‰ฝ๎†ก๎‰ญ๎ขš๎‰ข๎‰ญ๎ˆถ๎Šฝ๎‡ซ๎†Ž๎†ก๎†Ž ๎‰ญ๎ˆถ๎Š ๎Š‹๎‡ซ๎ˆฅ๎†ก๎‰ฝ ๎ขŽMatMult๎กซ MatMultTranspose๎ข
โ—ฆ ๎ขŽ๎Š ๎‰ฝ๎†ก๎‰ญ๎ขš๎‰ข๎‰ญ๎ˆถ๎Šฝ๎‡ซ๎†Ž๎†ก๎†Ž๎ข ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ ๎ˆ๎†ก๎‰ฝ๎‰ฝ ๎‡ซ๎ˆฅ๎Š‹๎Š ๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก
โ—ฆ func ๎‰ข๎…”๎‰ฝ๎‰ฝ๎†ก๎†Ž ๎Š‹๎ˆถ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ
โ—ฆ ๎ˆŸ๎Š ๎‰ฝ๎Š‹ ๎Š ๎‰ฝ๎†ก ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ KSP ๎‡ซ๎‰ฝ ๎ˆถ๎…น๎ˆ๎‡ซ๎Šฝ๎‡ซ๎ˆถ๎Š ๎‰ฝ ๎Š‹๎ˆถ ๎…” Mat ๎Š‹๎‹Š๎‰ข๎†ก
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ PCSHELL
๎ …๎ ƒ
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ €๎ €๎กช ๎ €๎€ท ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œฏ๎ฐ๎š๎š๎Œผ๎š๎Œผ๎Ž™๎Œฆ๎Œผ๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎ขš๎ˆถ๎‰ญ๎†Ž๎†ก๎‰ญ ๎†ƒ๎†ก๎ˆฅ๎Š‹๎†ก๎‰ญ๎†ก๎†Ž ๎‰ฝ๎†ƒ๎‡ ๎†ก๎ˆŸ๎†ก
n
๎ 
๏ฃฎ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฐ
๎  โˆ’๎ €
โˆ’๎ € ๎  โˆ’๎ €
๎กฒ
๎กฒ
๎กฒ
๎กฒ
๎กฒ
๎กฒ
๎กฒ
๎กฒ
๎กฒ
โˆ’๎ € ๎ 
๏ฃน
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃป
x =
๏ฃฎ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฐ
๎ €
๎ €
๎กฒ
๎กฒ
๎กฒ
๎ €
๏ฃน
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃป
โ—ฆ ๎Š ๎‰ฝ๎†ก๎‡‰๎Š ๎ˆ ๎ˆŸ๎ˆถ๎ˆฅ๎‡ซ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎กซ ๎†ก๎กฒ๎‡Œ๎กฒ๎กซ -ksp_view_singularvalues
โ—ฆ ๎Šฝ๎…”๎‰ญ๎‡ซ๎…”๎…น๎ˆ๎†ก precon ๎Š‹๎ˆถ ๎‰ฝ๎Š ๎‰ข๎‰ข๎ˆ๎‹Š ๎…” PCSHELL
โ—ฆ ๎ €๎€ท meshL
๎ …๎ „
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ €๎ €๎กช ๎ €๎€ท ๎š๎ฐ๎Ž™๎ฐ๎ญ๎Œผ ๎Œฏ๎ฐ๎š๎š๎Œผ๎š๎Œผ๎Ž™๎Œฆ๎Œผ๎ฃ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎ขš๎ˆถ๎‰ญ๎†Ž๎†ก๎‰ญ ๎†ƒ๎†ก๎ˆฅ๎Š‹๎†ก๎‰ญ๎†ก๎†Ž ๎‰ฝ๎†ƒ๎‡ ๎†ก๎ˆŸ๎†ก
n
๎ 
๏ฃฎ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฐ
๎  โˆ’๎ €
โˆ’๎ € ๎  โˆ’๎ €
๎กฒ
๎กฒ
๎กฒ
๎กฒ
๎กฒ
๎กฒ
๎กฒ
๎กฒ
๎กฒ
โˆ’๎ € ๎ 
๏ฃน
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃป
x =
๏ฃฎ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฐ
๎ €
๎ €
๎กฒ
๎กฒ
๎กฒ
๎ €
๏ฃน
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃป
โ—ฆ ๎Š ๎‰ฝ๎†ก๎‡‰๎Š ๎ˆ ๎ˆŸ๎ˆถ๎ˆฅ๎‡ซ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎กซ ๎†ก๎กฒ๎‡Œ๎กฒ๎กซ -ksp_view_singularvalues
โ—ฆ ๎Šฝ๎…”๎‰ญ๎‡ซ๎…”๎…น๎ˆ๎†ก precon ๎Š‹๎ˆถ ๎‰ฝ๎Š ๎‰ข๎‰ข๎ˆ๎‹Š ๎…” PCSHELL
โ—ฆ ๎ €๎€ท meshL
๎ …๎ „
๎‚ฅ๎Žฉ๎Ž™๎Ž‹๎ฐ๎Ž™๎Œผ๎‹ผ๎š ๎ฃ๎Žฉ๎Ž‹๎˜๎Œผ๎š๎ฃ SNES
โ—ฆ ๎†ก๎…”๎‰ฝ๎‹Š๎ขš๎Š‹๎ˆถ๎ขš๎Š ๎‰ฝ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡‰๎…”๎†ƒ๎†ก ๎Š‹๎ˆถ ๎ขŽ๎‰ฉ๎Š ๎…”๎‰ฝ๎‡ซ๎ขš๎ข๎‚ฅ๎†ก๎‹๎Š‹๎ˆถ๎ˆฅ ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ
โ—ฆ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก F(u) = b
โ—ฆ ๎‡ ๎…”๎ˆฅ๎†Ž๎ˆ๎†ก ๎Šฝ๎…”๎‰ญ๎‡ซ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‡ซ๎ˆฅ๎†ก๎‰ฉ๎Š ๎…”๎ˆ๎‡ซ๎Š‹๎‡ซ๎†ก๎‰ฝ ๎ˆถ๎ˆฅ u
โ—ฆ ๎ˆŸ๎Š ๎‰ฝ๎Š‹ ๎‰ข๎‰ญ๎ˆถ๎Šฝ๎‡ซ๎†Ž๎†ก ๎Š‹๎‹๎ˆถ func๎‰ฝ
โ€ข ๎Š‹๎ˆถ ๎†ก๎Šฝ๎…”๎ˆ๎Š ๎…”๎Š‹๎†ก ๎‰ญ๎†ก๎‰ฝ๎‡ซ๎†Ž๎Š ๎…”๎ˆ๎‰ฝ ๎‰ฝ๎Š‹๎ˆถ๎‰ญ๎†ก๎†Ž ๎…”๎‰ฝ K[int]
โ€ข ๎Š‹๎ˆถ ๎Š ๎‰ข๎†Ž๎…”๎Š‹๎†ก ๎…” ๎‚๎…”๎†ƒ๎ˆถ๎…น๎‡ซ๎…”๎ˆฅ ๎‰ฝ๎Š‹๎ˆถ๎‰ญ๎†ก๎†Ž ๎…”๎‰ฝ Mat
โ—ฆ ChangeNumbering ๎Š‹๎ˆถ ๎‡Œ๎ˆถ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎Š‹๎ˆถ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ
โ‡’ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ ๎‡‰๎ˆถ๎‰ญ ๎Š‹๎‡ ๎†ก ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ๎‡ซ๎‹–๎†ก๎†Ž ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ๎‰ฝ ๎†ƒ๎ˆถ๎ˆฅ๎‹ช๎‡Œ๎Š ๎‰ญ๎†ก๎†Ž ๎Šฝ๎‡ซ๎…” set
๎ …๎ …
๎‚ฅ๎Žฉ๎Ž™๎Ž‹๎ฐ๎Ž™๎Œผ๎‹ผ๎š ๎ฃ๎Žฉ๎Ž‹๎˜๎Œผ๎š๎ฃ SNES
โ—ฆ ๎†ก๎…”๎‰ฝ๎‹Š๎ขš๎Š‹๎ˆถ๎ขš๎Š ๎‰ฝ๎†ก ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡‰๎…”๎†ƒ๎†ก ๎Š‹๎ˆถ ๎ขŽ๎‰ฉ๎Š ๎…”๎‰ฝ๎‡ซ๎ขš๎ข๎‚ฅ๎†ก๎‹๎Š‹๎ˆถ๎ˆฅ ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ
โ—ฆ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก F(u) = b
โ—ฆ ๎‡ ๎…”๎ˆฅ๎†Ž๎ˆ๎†ก ๎Šฝ๎…”๎‰ญ๎‡ซ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‡ซ๎ˆฅ๎†ก๎‰ฉ๎Š ๎…”๎ˆ๎‡ซ๎Š‹๎‡ซ๎†ก๎‰ฝ ๎ˆถ๎ˆฅ u
โ—ฆ ๎ˆŸ๎Š ๎‰ฝ๎Š‹ ๎‰ข๎‰ญ๎ˆถ๎Šฝ๎‡ซ๎†Ž๎†ก ๎Š‹๎‹๎ˆถ func๎‰ฝ
โ€ข ๎Š‹๎ˆถ ๎†ก๎Šฝ๎…”๎ˆ๎Š ๎…”๎Š‹๎†ก ๎‰ญ๎†ก๎‰ฝ๎‡ซ๎†Ž๎Š ๎…”๎ˆ๎‰ฝ ๎‰ฝ๎Š‹๎ˆถ๎‰ญ๎†ก๎†Ž ๎…”๎‰ฝ K[int]
โ€ข ๎Š‹๎ˆถ ๎Š ๎‰ข๎†Ž๎…”๎Š‹๎†ก ๎…” ๎‚๎…”๎†ƒ๎ˆถ๎…น๎‡ซ๎…”๎ˆฅ ๎‰ฝ๎Š‹๎ˆถ๎‰ญ๎†ก๎†Ž ๎…”๎‰ฝ Mat
โ—ฆ ChangeNumbering ๎Š‹๎ˆถ ๎‡Œ๎ˆถ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎Š‹๎ˆถ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ
โ‡’ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ ๎‡‰๎ˆถ๎‰ญ ๎Š‹๎‡ ๎†ก ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ๎‡ซ๎‹–๎†ก๎†Ž ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ๎‰ฝ ๎†ƒ๎ˆถ๎ˆฅ๎‹ช๎‡Œ๎Š ๎‰ญ๎†ก๎†Ž ๎Šฝ๎‡ซ๎…” set
๎ …๎ …
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ €๎ ๎กช ๎‚ฅ๎Œผ๎›๎ญ๎Žฉ๎Ž™ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก
โˆ‡J(u) = ๎Ÿฟ ๎‡ซ๎ˆฅ โ„ฆ
๎‹๎‡ซ๎Š‹๎‡  J(u) =
๎š
โ„ฆ
๎ €
๎ 
f
๎€€
||โˆ‡u||
๎ 
๎€
โˆ’ cu
๎‰ฝ๎Š ๎…น๎‡ฟ๎†ก๎†ƒ๎Š‹ ๎Š‹๎ˆถ u
๎ˆ๎ˆถ๎‹๎†ก๎‰ญ
โฉฝ u โฉฝ u
๎Š ๎‰ข๎‰ข๎†ก๎‰ญ
๎‡ซ๎ˆฅ C
๎…”๎ˆฅ๎†Ž f : x 7โ†’ (๎ € + a)x โˆ’ ๎ˆ๎ˆถ๎‡Œ(๎ € + x)
โ—ฆ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎†ก ๎†ƒ๎ˆถ๎ˆฅ๎‰ฝ๎Š‹๎…”๎ˆฅ๎Š‹ ๎Š‹๎†ก๎‰ญ๎ˆŸ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ F(u) = b
โ—ฆ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎†ก ๎‰ญ๎†ก๎‰ฝ๎‡ซ๎†Ž๎Š ๎…”๎ˆ๎‰ฝ ๎…”๎ˆฅ๎†Ž ๎Š ๎‰ข๎†Ž๎…”๎Š‹๎†ก ๎Š‹๎‡ ๎†ก ๎‚๎…”๎†ƒ๎ˆถ๎…น๎‡ซ๎…”๎ˆฅ
โ—ฆ ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ ๎‡ซ๎ˆฅ SNESSolve ๎‡‰๎ˆถ๎‰ญ ๎Š‹๎‡ ๎†ก ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎‰ฝ
โ—ฆ ๎ˆฅ๎ˆถ ๎…”๎Š ๎Š‹๎ˆถ๎ˆŸ๎…”๎Š‹๎‡ซ๎†ƒ ๎†Ž๎‡ซ๎‡‰๎‡‰๎†ก๎‰ญ๎†ก๎ˆฅ๎Š‹๎‡ซ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ …๎ †
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ €๎ ๎กช ๎‚ฅ๎Œผ๎›๎ญ๎Žฉ๎Ž™ ๎Ž•๎Œผ๎ญ๎ซ๎Žฉ๎Œฏ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎ ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก
โˆ‡J(u) = ๎Ÿฟ ๎‡ซ๎ˆฅ โ„ฆ
๎‹๎‡ซ๎Š‹๎‡  J(u) =
๎š
โ„ฆ
๎ €
๎ 
f
๎€€
||โˆ‡u||
๎ 
๎€
โˆ’ cu
๎‰ฝ๎Š ๎…น๎‡ฟ๎†ก๎†ƒ๎Š‹ ๎Š‹๎ˆถ u
๎ˆ๎ˆถ๎‹๎†ก๎‰ญ
โฉฝ u โฉฝ u
๎Š ๎‰ข๎‰ข๎†ก๎‰ญ
๎‡ซ๎ˆฅ C
๎…”๎ˆฅ๎†Ž f : x 7โ†’ (๎ € + a)x โˆ’ ๎ˆ๎ˆถ๎‡Œ(๎ € + x)
โ—ฆ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎†ก ๎†ƒ๎ˆถ๎ˆฅ๎‰ฝ๎Š‹๎…”๎ˆฅ๎Š‹ ๎Š‹๎†ก๎‰ญ๎ˆŸ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ F(u) = b
โ—ฆ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎†ก ๎‰ญ๎†ก๎‰ฝ๎‡ซ๎†Ž๎Š ๎…”๎ˆ๎‰ฝ ๎…”๎ˆฅ๎†Ž ๎Š ๎‰ข๎†Ž๎…”๎Š‹๎†ก ๎Š‹๎‡ ๎†ก ๎‚๎…”๎†ƒ๎ˆถ๎…น๎‡ซ๎…”๎ˆฅ
โ—ฆ ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ ๎‡ซ๎ˆฅ SNESSolve ๎‡‰๎ˆถ๎‰ญ ๎Š‹๎‡ ๎†ก ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎‰ฝ
โ—ฆ ๎ˆฅ๎ˆถ ๎…”๎Š ๎Š‹๎ˆถ๎ˆŸ๎…”๎Š‹๎‡ซ๎†ƒ ๎†Ž๎‡ซ๎‡‰๎‡‰๎†ก๎‰ญ๎†ก๎ˆฅ๎Š‹๎‡ซ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ …๎ †
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ €๎ ‚๎กช ๎ฃ๎Žฉ๎Ž‹๎ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎ฃ๎ญ๎š๎ผ๎Œฆ๎ญ๎ฐ๎Žฉ๎Ž™ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎€ท๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎Š‹๎ˆถ ๎…” ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎ˆŸ๎†ก๎‰ฝ๎‡  ๎…”๎†Ž๎…”๎‰ข๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎†ƒ๎†ก๎ˆฅ๎Š‹๎‰ญ๎…”๎ˆ๎‡ซ๎‹–๎†ก๎†Ž ๎‰ข๎ˆถ๎‰ฝ๎Š‹๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ DmeshCreate ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎ˆŸ๎…”๎†ƒ๎‰ญ๎ˆถ N2O
โ—ฆ ๎ค‹ restrict ๎Š‹๎ˆถ ๎‡Œ๎ˆถ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎Š‹๎ˆถ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎‰ญ๎†ก๎†Ž๎Š ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆฅ๎†ก๎†ก๎†Ž๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎Š ๎ˆŸ ๎†ƒ๎ˆถ๎ˆฅ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎ …๎ ‡
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ €๎ ‚๎กช ๎ฃ๎Žฉ๎Ž‹๎ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎š๎Œผ๎Œฆ๎Žฉ๎Ž™๎ฃ๎ญ๎š๎ผ๎Œฆ๎ญ๎ฐ๎Žฉ๎Ž™ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎ ‚๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎€ท๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎Š‹๎ˆถ ๎…” ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎ˆŸ๎†ก๎‰ฝ๎‡  ๎…”๎†Ž๎…”๎‰ข๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎†ƒ๎†ก๎ˆฅ๎Š‹๎‰ญ๎…”๎ˆ๎‡ซ๎‹–๎†ก๎†Ž ๎‰ข๎ˆถ๎‰ฝ๎Š‹๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ DmeshCreate ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎ˆŸ๎…”๎†ƒ๎‰ญ๎ˆถ N2O
โ—ฆ ๎ค‹ restrict ๎Š‹๎ˆถ ๎‡Œ๎ˆถ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎ˆ๎ˆถ๎†ƒ๎…”๎ˆ ๎Š‹๎ˆถ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ
โ—ฆ ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎‰ญ๎†ก๎†Ž๎Š ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆฅ๎†ก๎†ก๎†Ž๎†ก๎†Ž ๎Š‹๎ˆถ ๎‰ฝ๎Š ๎ˆŸ ๎†ƒ๎ˆถ๎ˆฅ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎ …๎ ‡
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ €๎ ƒ๎กช ๎ฃ๎ฃ๎ฃ๎ญ๎Œผ๎Ž•๎ฃ ๎›๎ฐ๎ญ๎ซ ๎Ž•๎ผ๎Ž‹๎ญ๎ฐ๎“๎Ž‹๎Œผ ๎ƒจ๎ฑ๎ƒฒ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎ ƒ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ
โˆ†u + k
๎ 
u = f
i
๎‡ซ๎ˆฅ โ„ฆ
u ยท n = ๎Ÿฟ ๎ˆถ๎ˆฅ ฮ“
๎‹๎‡ซ๎Š‹๎‡  ๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎‰ข๎ˆ๎†ก ๎‰ข๎ˆถ๎‡ซ๎ˆฅ๎Š‹ ๎‰ฝ๎ˆถ๎Š ๎‰ญ๎†ƒ๎†ก๎‰ฝ {f
i
}
i=๎ €,๎ ,...
โ—ฆ KSPSolve ๎ค‹ complex[int,int]
โ—ฆ ๎Š ๎‰ฝ๎†ก ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ๎ˆ๎†ก ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎…”๎ˆฅ ๎†ก๎‹‰๎…”๎†ƒ๎Š‹ ๎‡‰๎…”๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎…น๎ˆ๎ˆถ๎†ƒ๎ˆ‡ ๎ฅ๎‚ก๎ƒจ๎†๎ƒฒ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎ฑ๎ƒก๎€ท๎€ท๎‚ก
๎ …๎ ˆ
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ €๎ ƒ๎กช ๎ฃ๎ฃ๎ฃ๎ญ๎Œผ๎Ž•๎ฃ ๎›๎ฐ๎ญ๎ซ ๎Ž•๎ผ๎Ž‹๎ญ๎ฐ๎“๎Ž‹๎Œผ ๎ƒจ๎ฑ๎ƒฒ ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎ ƒ๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎Š‹๎‡ ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ
โˆ†u + k
๎ 
u = f
i
๎‡ซ๎ˆฅ โ„ฆ
u ยท n = ๎Ÿฟ ๎ˆถ๎ˆฅ ฮ“
๎‹๎‡ซ๎Š‹๎‡  ๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎‰ข๎ˆ๎†ก ๎‰ข๎ˆถ๎‡ซ๎ˆฅ๎Š‹ ๎‰ฝ๎ˆถ๎Š ๎‰ญ๎†ƒ๎†ก๎‰ฝ {f
i
}
i=๎ €,๎ ,...
โ—ฆ KSPSolve ๎ค‹ complex[int,int]
โ—ฆ ๎Š ๎‰ฝ๎†ก ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ๎ˆ๎†ก ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก ๎‹๎‡ซ๎Š‹๎‡  ๎…”๎ˆฅ ๎†ก๎‹‰๎…”๎†ƒ๎Š‹ ๎‡‰๎…”๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎…น๎ˆ๎ˆถ๎†ƒ๎ˆ‡ ๎ฅ๎‚ก๎ƒจ๎†๎ƒฒ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎ฑ๎ƒก๎€ท๎€ท๎‚ก
๎ …๎ ˆ
๎†๎Ž™๎Œฏ๎ขš๎ผ๎ฃ๎Œผ๎š ๎“๎Œผ๎š๎ฃ๎“๎Œผ๎Œฆ๎ญ๎ฐ๎˜๎Œผ๎ฃ
โ—ฆ ๎Š‹๎‡ซ๎ˆŸ๎†ก๎‰ฝ๎Š‹๎†ก๎‰ข๎‰ข๎†ก๎‰ญ๎‰ฝ TS ๎ˆŸ๎…”๎ˆฅ๎Š ๎…”๎ˆ ๎‰ข๎…”๎‡Œ๎†ก๎‰ฝ
โ—ฆ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก๎‰ญ๎‰ฝ Tao ๎ˆŸ๎…”๎ˆฅ๎Š ๎…”๎ˆ ๎‰ข๎…”๎‡Œ๎†ก๎‰ฝ
โ—ฆ ๎†ก๎…”๎‰ฝ๎‹Š ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎…”๎…น๎‡ซ๎ˆ๎‡ซ๎Š‹๎‹Š ๎…น๎†ก๎Š‹๎‹๎†ก๎†ก๎ˆฅ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆถ๎…น๎‡ฟ๎†ก๎†ƒ๎Š‹๎‰ฝ
๎ †๎Ÿฟ
๎ƒฒ๎‚•๎†๎ƒก๎Œฆ
๎บ๎Ž™๎ญ๎š๎Žฉ๎Œฏ๎ผ๎Œฆ๎ญ๎ฐ๎Žฉ๎Ž™
โ—ฆ ๎ƒฒ๎†ƒ๎…”๎ˆ๎…”๎…น๎ˆ๎†ก ๎‚•๎‡ซ๎…น๎‰ญ๎…”๎‰ญ๎‹Š ๎‡‰๎ˆถ๎‰ญ ๎†๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎…”๎ˆ๎Š ๎†ก ๎ƒก๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ ๎€ญ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎…น๎Š ๎‡ซ๎ˆ๎Š‹ ๎ˆถ๎ˆฅ ๎Š‹๎ˆถ๎‰ข ๎ˆถ๎‡‰ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ
โ—ฆ ๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก ๎ˆถ๎‡‰ ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎†Ž๎…”๎Š‹๎…” ๎‰ฝ๎Š‹๎‰ญ๎Š ๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎‰ฝ
โ€ข EPS
โ€ข ST
โ—ฆ ๎Š ๎‰ฝ๎†ก๎‡‰๎Š ๎ˆ ๎‡‰๎ˆถ๎‰ญ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎‡ซ๎ˆ๎‡ซ๎ˆฅ๎‡Œ ๎ˆถ๎Š‹๎‡ ๎†ก๎‰ญ ๎ˆ๎‡ซ๎…น๎‰ญ๎…”๎‰ญ๎‡ซ๎†ก๎‰ฝ ๎ขŽ๎€ƒ๎ƒจ๎ƒก๎€ƒ๎€ญ๎‚‘๎ข
โ—ฆ http://slepc.upv.es
โ—ฆ https://gitlab.com/slepc/slepc
๎ †๎ €
๎บ๎Ž™๎ญ๎Œผ๎š๎š๎‹ผ๎Œฆ๎Œผ
๎ƒก๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ—ฆ ๎ˆถ๎ˆฅ๎†ก Mat ๎ˆถ๎‰ญ ๎Š‹๎‹๎ˆถ ๎‡‰๎ˆถ๎‰ญ ๎‡Œ๎†ก๎ˆฅ๎†ก๎‰ญ๎…”๎ˆ๎‡ซ๎‹–๎†ก๎†Ž ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ
โ—ฆ ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎…”๎ˆ๎Š ๎†ก๎‰ฝ๎กธ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ
โ€ข ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ€ข ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎‡‰๎ˆถ๎‰ญ MatNest
โ—ฆ ๎†ƒ๎…”๎‰ญ๎†ก๎‡‰๎Š ๎ˆ ๎‹๎‡ซ๎Š‹๎‡  ๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎‰ญ๎ขš๎Š‹๎‹Š๎‰ข๎†ก
๎ƒฒ๎‰ข๎†ก๎†ƒ๎Š‹๎‰ญ๎…”๎ˆ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎‡ซ๎ˆŸ๎‰ข๎‰ญ๎ˆถ๎Šฝ๎†ก ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡‰๎ˆถ๎‰ญ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ KSP ๎†ƒ๎ˆถ๎ˆฅ๎‹ช๎‡Œ๎Š ๎‰ญ๎†ก๎†Ž ๎Šฝ๎‡ซ๎…” set
๎ †๎ 
๎บ๎Ž™๎ญ๎Œผ๎š๎š๎‹ผ๎Œฆ๎Œผ
๎ƒก๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ—ฆ ๎ˆถ๎ˆฅ๎†ก Mat ๎ˆถ๎‰ญ ๎Š‹๎‹๎ˆถ ๎‡‰๎ˆถ๎‰ญ ๎‡Œ๎†ก๎ˆฅ๎†ก๎‰ญ๎…”๎ˆ๎‡ซ๎‹–๎†ก๎†Ž ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ
โ—ฆ ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎…”๎ˆ๎Š ๎†ก๎‰ฝ๎กธ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ
โ€ข ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ€ข ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎‡‰๎ˆถ๎‰ญ MatNest
โ—ฆ ๎†ƒ๎…”๎‰ญ๎†ก๎‡‰๎Š ๎ˆ ๎‹๎‡ซ๎Š‹๎‡  ๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎‰ญ๎ขš๎Š‹๎‹Š๎‰ข๎†ก
๎ƒฒ๎‰ข๎†ก๎†ƒ๎Š‹๎‰ญ๎…”๎ˆ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎‡ซ๎ˆŸ๎‰ข๎‰ญ๎ˆถ๎Šฝ๎†ก ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡‰๎ˆถ๎‰ญ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ KSP ๎†ƒ๎ˆถ๎ˆฅ๎‹ช๎‡Œ๎Š ๎‰ญ๎†ก๎†Ž ๎Šฝ๎‡ซ๎…” set
๎ †๎ 
๎บ๎Ž™๎ญ๎Œผ๎š๎š๎‹ผ๎Œฆ๎Œผ
๎ƒก๎…”๎‰ญ๎…”๎ˆŸ๎†ก๎Š‹๎†ก๎‰ญ๎‰ฝ
โ—ฆ ๎ˆถ๎ˆฅ๎†ก Mat ๎ˆถ๎‰ญ ๎Š‹๎‹๎ˆถ ๎‡‰๎ˆถ๎‰ญ ๎‡Œ๎†ก๎ˆฅ๎†ก๎‰ญ๎…”๎ˆ๎‡ซ๎‹–๎†ก๎†Ž ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ
โ—ฆ ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎…”๎ˆ๎Š ๎†ก๎‰ฝ๎กธ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ
โ€ข ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ€ข ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎‡‰๎ˆถ๎‰ญ MatNest
โ—ฆ ๎†ƒ๎…”๎‰ญ๎†ก๎‡‰๎Š ๎ˆ ๎‹๎‡ซ๎Š‹๎‡  ๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎‰ญ๎ขš๎Š‹๎‹Š๎‰ข๎†ก
๎ƒฒ๎‰ข๎†ก๎†ƒ๎Š‹๎‰ญ๎…”๎ˆ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
โ—ฆ ๎‡ซ๎ˆŸ๎‰ข๎‰ญ๎ˆถ๎Šฝ๎†ก ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎‡Œ๎†ก๎ˆฅ๎†ƒ๎†ก ๎‡‰๎ˆถ๎‰ญ ๎‰ฝ๎ˆถ๎ˆŸ๎†ก ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ
โ—ฆ ๎Š ๎ˆฅ๎†Ž๎†ก๎‰ญ๎ˆ๎‹Š๎‡ซ๎ˆฅ๎‡Œ KSP ๎†ƒ๎ˆถ๎ˆฅ๎‹ช๎‡Œ๎Š ๎‰ญ๎†ก๎†Ž ๎Šฝ๎‡ซ๎…” set
๎ †๎ 
๎‚ก๎ฐ๎ฃ๎Œฆ๎Œผ๎Ž‹๎Ž‹๎‹ผ๎Ž™๎Œผ๎Žฉ๎ผ๎ฃ
๎‚ก๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ขš๎‡‰๎‰ญ๎†ก๎†ก ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ๎‰ฝ
โ—ฆ ๎‡ฟ๎Š ๎‰ฝ๎Š‹ ๎…”๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  KSPSolve ๎‹๎‡ซ๎Š‹๎‡  ๎…” MatShell
โ—ฆ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎Š‹๎†ก๎†Ž ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ ๎ˆถ๎‡‰ ST
๎ƒก๎†ก๎‰ญ๎‡ซ๎ˆถ๎†Ž๎‡ซ๎†ƒ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎€ญ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ ๎‰ข๎…”๎‰ญ๎Š‹๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ ๎Š‹๎‡ ๎…”๎Š‹ ๎ˆŸ๎…”๎‹Š ๎…น๎†ก ๎‡ซ๎ˆŸ๎…น๎…”๎ˆ๎…”๎ˆฅ๎†ƒ๎†ก๎†Ž
๎ฑ๎ƒก๎€ท๎€ท๎‚ก ๎‚‘๎‰ญ๎‹Š๎ˆ๎ˆถ๎Šฝ ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ
โ—ฆ -ksp_type ๎‡ ๎‰ข๎†Ž๎†Ž๎ˆŸ
โ—ฆ -st_ksp_type ๎‡ ๎‰ข๎†Ž๎†Ž๎ˆŸ
๎ †๎ ‚
๎‚ก๎ฐ๎ฃ๎Œฆ๎Œผ๎Ž‹๎Ž‹๎‹ผ๎Ž™๎Œผ๎Žฉ๎ผ๎ฃ
๎‚ก๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ขš๎‡‰๎‰ญ๎†ก๎†ก ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ๎‰ฝ
โ—ฆ ๎‡ฟ๎Š ๎‰ฝ๎Š‹ ๎…”๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  KSPSolve ๎‹๎‡ซ๎Š‹๎‡  ๎…” MatShell
โ—ฆ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎Š‹๎†ก๎†Ž ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ ๎ˆถ๎‡‰ ST
๎ƒก๎†ก๎‰ญ๎‡ซ๎ˆถ๎†Ž๎‡ซ๎†ƒ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎€ญ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ ๎‰ข๎…”๎‰ญ๎Š‹๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ ๎Š‹๎‡ ๎…”๎Š‹ ๎ˆŸ๎…”๎‹Š ๎…น๎†ก ๎‡ซ๎ˆŸ๎…น๎…”๎ˆ๎…”๎ˆฅ๎†ƒ๎†ก๎†Ž
๎ฑ๎ƒก๎€ท๎€ท๎‚ก ๎‚‘๎‰ญ๎‹Š๎ˆ๎ˆถ๎Šฝ ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ
โ—ฆ -ksp_type ๎‡ ๎‰ข๎†Ž๎†Ž๎ˆŸ
โ—ฆ -st_ksp_type ๎‡ ๎‰ข๎†Ž๎†Ž๎ˆŸ
๎ †๎ ‚
๎‚ก๎ฐ๎ฃ๎Œฆ๎Œผ๎Ž‹๎Ž‹๎‹ผ๎Ž™๎Œผ๎Žฉ๎ผ๎ฃ
๎‚ก๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ขš๎‡‰๎‰ญ๎†ก๎†ก ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ๎‰ฝ
โ—ฆ ๎‡ฟ๎Š ๎‰ฝ๎Š‹ ๎…”๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  KSPSolve ๎‹๎‡ซ๎Š‹๎‡  ๎…” MatShell
โ—ฆ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ
โ—ฆ ๎ˆ๎‡ซ๎ˆŸ๎‡ซ๎Š‹๎†ก๎†Ž ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ ๎ˆถ๎‡‰ ST
๎ƒก๎†ก๎‰ญ๎‡ซ๎ˆถ๎†Ž๎‡ซ๎†ƒ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ
๎€ญ๎Š ๎‰ฝ๎Š‹๎ˆถ๎ˆŸ ๎‰ข๎…”๎‰ญ๎Š‹๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ ๎Š‹๎‡ ๎…”๎Š‹ ๎ˆŸ๎…”๎‹Š ๎…น๎†ก ๎‡ซ๎ˆŸ๎…น๎…”๎ˆ๎…”๎ˆฅ๎†ƒ๎†ก๎†Ž
๎ฑ๎ƒก๎€ท๎€ท๎‚ก ๎‚‘๎‰ญ๎‹Š๎ˆ๎ˆถ๎Šฝ ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ
โ—ฆ -ksp_type ๎‡ ๎‰ข๎†Ž๎†Ž๎ˆŸ
โ—ฆ -st_ksp_type ๎‡ ๎‰ข๎†Ž๎†Ž๎ˆŸ
๎ †๎ ‚
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ €๎กช ๎ƒฒ๎ญ๎Œผ๎Ž†๎Ž‹๎Žฉ๎˜๎ข—๎ƒก๎Žฉ๎ฐ๎Ž™๎Œฆ๎‹ผ๎š๎Œฝ ๎Žฉ๎“๎Œผ๎š๎‹ผ๎ญ๎Žฉ๎š ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ข๎‡ซ๎ˆฅ๎†Ž ๎Š‹๎‡ ๎†ก ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ
๎€ท๎Š‹๎‚ฅ: ฮ“
N
โ†’ R
g 7โ†’ โˆ‚
n
v
๎‹๎‡ ๎†ก๎‰ญ๎†ก v ๎‰ฝ๎…”๎Š‹๎‡ซ๎‰ฝ๎‹ช๎†ก๎‰ฝ
โˆ’โˆ†v = ๎Ÿฟ ๎‡ซ๎ˆฅ โ„ฆ
v = ๎Ÿฟ ๎ˆถ๎ˆฅ ฮ“
D
v = g ๎ˆถ๎ˆฅ ฮ“
N
โ—ฆ ๎†ก๎‰ฉ๎Š ๎‡ซ๎Šฝ๎…”๎ˆ๎†ก๎ˆฅ๎Š‹ ๎Š‹๎ˆถ ๎‹ช๎ˆฅ๎†Ž๎‡ซ๎ˆฅ๎‡Œ (ฮป, u) ๎‰ฝ๎กฒ๎Š‹๎กฒ Au = ฮปBu
โ—ฆ ๎‰ฝ๎‡ ๎‡ซ๎‡‰๎Š‹๎ขš๎…”๎ˆฅ๎†Ž๎ขš๎‡ซ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎Š‹ ๎‰ฝ๎‰ข๎†ก๎†ƒ๎Š‹๎‰ญ๎…”๎ˆ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎‰ฝ๎…”๎ˆŸ๎†ก Mat ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‡‰๎ˆถ๎‰ญ A ๎…”๎ˆฅ๎†Ž B ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ Mat B(A,Loc)
๎ †๎ ƒ
๎†๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ ๎ €๎กช ๎ƒฒ๎ญ๎Œผ๎Ž†๎Ž‹๎Žฉ๎˜๎ข—๎ƒก๎Žฉ๎ฐ๎Ž™๎Œฆ๎‹ผ๎š๎Œฝ ๎Žฉ๎“๎Œผ๎š๎‹ผ๎ญ๎Žฉ๎š ๎ข‚๎Œผ๎ข๎‹ผ๎Ž•๎“๎Ž‹๎Œผ๎ €๎กฒ๎Œผ๎Œฏ๎“๎ขƒ
๎€ƒ๎‡ซ๎ˆŸ
๎ข๎‡ซ๎ˆฅ๎†Ž ๎Š‹๎‡ ๎†ก ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‰ฝ ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ
๎€ท๎Š‹๎‚ฅ: ฮ“
N
โ†’ R
g 7โ†’ โˆ‚
n
v
๎‹๎‡ ๎†ก๎‰ญ๎†ก v ๎‰ฝ๎…”๎Š‹๎‡ซ๎‰ฝ๎‹ช๎†ก๎‰ฝ
โˆ’โˆ†v = ๎Ÿฟ ๎‡ซ๎ˆฅ โ„ฆ
v = ๎Ÿฟ ๎ˆถ๎ˆฅ ฮ“
D
v = g ๎ˆถ๎ˆฅ ฮ“
N
โ—ฆ ๎†ก๎‰ฉ๎Š ๎‡ซ๎Šฝ๎…”๎ˆ๎†ก๎ˆฅ๎Š‹ ๎Š‹๎ˆถ ๎‹ช๎ˆฅ๎†Ž๎‡ซ๎ˆฅ๎‡Œ (ฮป, u) ๎‰ฝ๎กฒ๎Š‹๎กฒ Au = ฮปBu
โ—ฆ ๎‰ฝ๎‡ ๎‡ซ๎‡‰๎Š‹๎ขš๎…”๎ˆฅ๎†Ž๎ขš๎‡ซ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎Š‹ ๎‰ฝ๎‰ข๎†ก๎†ƒ๎Š‹๎‰ญ๎…”๎ˆ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
โ—ฆ ๎‰ฝ๎…”๎ˆŸ๎†ก Mat ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‡‰๎ˆถ๎‰ญ A ๎…”๎ˆฅ๎†Ž B ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ Mat B(A,Loc)
๎ †๎ ƒ
๎€ƒ๎“๎“๎Ž‹๎ฐ๎Œฆ๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ
๎€ƒ๎ผ๎ ๎Ž•๎Œผ๎Ž™๎ญ๎Œผ๎Œฏ ๎‚•๎‹ผ๎ ๎š๎‹ผ๎Ž™๎ ๎ฐ๎‹ผ๎Ž™ ๎š๎Žฉ๎š ๎ฃ๎ญ๎‹ผ๎Œž๎ฐ๎Ž‹๎ฐ๎ญ๎ฃ ๎‹ผ๎Ž™๎‹ผ๎Ž‹๎ฃ๎ฃ๎ฐ๎ฃ
๎ข‚๎‚ก๎ˆถ๎Š ๎ˆ๎‡ซ๎ˆฅ ๎†ก๎Š‹ ๎…”๎ˆ๎กฒ ๎ ๎Ÿฟ๎ €๎ ˆ๎ขƒ
โ—ฆ https://github.com/prj-/moulin2019al
โ—ฆ ๎‚ฅ๎ˆถ๎ˆฅ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ๎ขš๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ๎กฒ๎†ก๎†Ž๎‰ข
โ€ข -gamma ๎ขŽ๎Ÿฟ.๎ €๎ข
โ€ข -mesh ๎ขŽ๎ข๎ˆ๎…”๎Š‹๎ƒก๎ˆ๎…”๎Š‹๎†ก๎ ‚๎€ท๎กฒ๎ˆŸ๎†ก๎‰ฝ๎‡ ๎ข
โ€ข -Re ๎ขŽ๎ „๎Ÿฟ๎ข
โ—ฆ ๎†๎‡ซ๎‡Œ๎†ก๎ˆฅ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ๎กฒ๎†ก๎†Ž๎‰ข
โ€ข -shift_real ๎ขŽ๎ €๎Ÿฟ
โˆ’๎ …
๎ข
โ€ข -shift_imag ๎ขŽ๎Ÿฟ.๎ …๎ข
โ€ข -nev ๎ขŽ๎ „๎ข
โ€ข -recycle ๎ขŽ๎Ÿฟ๎ข
๎ †๎ „
๎€ฅ๎‹ผ๎ฃ๎Œผ๎š๎Ž‹๎Žฉ๎›
๎ †๎ …
๎‚•๎Œผ๎‹ผ๎Œฏ๎ฐ๎Ž™๎  ๎ผ๎Ž™๎ฃ๎ญ๎‹ผ๎Œž๎Ž‹๎Œผ ๎Œผ๎ฐ๎ ๎Œผ๎Ž™๎˜๎Œผ๎Œฆ๎ญ๎Žฉ๎š
๎ †๎ †
๎ƒพ๎ซ๎‹ผ๎Ž™๎Ž† ๎ฃ๎Žฉ๎ผ๎กฎ
๎ƒฆ๎ผ๎Œผ๎ฃ๎ญ๎ฐ๎Žฉ๎Ž™๎ฃ๎กณ
๎ †๎ †
๎‚ถ๎“๎ญ๎ฐ๎Ž•๎‹ผ๎Ž‹ ๎Œฆ๎Žฉ๎Ž•๎“๎ฐ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎“๎š๎Žฉ๎Œฆ๎Œผ๎ฃ๎ฃ ๎บ ๎ ๎Œผ๎Ž™๎Œผ๎š๎‹ผ๎Ž‹ ๎˜๎‹ผ๎š๎ฐ๎‹ผ๎Œž๎Ž‹๎Œผ๎ฃ
โ—ฆ ๎‰ญ๎†ก๎ˆŸ๎ˆถ๎Šฝ๎†ก ๎…”๎ˆฅ๎‹Š ๎‰ข๎‰ญ๎†ก๎Šฝ๎‡ซ๎ˆถ๎Š ๎‰ฝ ๎‡ซ๎ˆฅ๎‰ฝ๎Š‹๎…”๎ˆฅ๎†ƒ๎†ก ๎ˆถ๎‡‰ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก
โ—ฆ ๎‡ซ๎ˆฅ๎‰ฝ๎Š‹๎…”๎ˆ๎ˆ ๎‡Œ๎†ƒ๎†ƒ๎กธ๎†ƒ๎ˆ๎…”๎ˆฅ๎‡Œ ๎…”๎ˆฅ๎†Ž ๎‡Œ๎‡‰๎ˆถ๎‰ญ๎Š‹๎‰ญ๎…”๎ˆฅ
โ—ฆ ๎ˆŸ๎…”๎ˆ‡๎†ก ๎‰ฝ๎Š ๎‰ญ๎†ก ๎‹Š๎ˆถ๎Š  ๎‡ ๎…”๎Šฝ๎†ก ๎…” ๎‹๎ˆถ๎‰ญ๎ˆ‡๎‡ซ๎ˆฅ๎‡Œ ๎‚ก๎ƒก๎บ ๎‡ซ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ
๎ค‘ ๎†ก๎‹‰๎‰ข๎ˆถ๎‰ญ๎Š‹ ๎ข๎ข๎กผ๎€ท๎บ๎ƒจ๎ค๎ขผ๎ข€๎ƒก๎„ญ๎€ท๎ข๎กธ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ขš๎‰ฝ๎ˆถ๎Š ๎‰ญ๎†ƒ๎†ก๎‰ฝ
๎ค‘ ๎†ก๎‹‰๎‰ข๎ˆถ๎‰ญ๎Š‹ ๎ƒก๎†๎ƒพ๎ƒฒ๎€ญ๎กผ๎€ท๎บ๎ƒจ๎ค๎ขผ๎ข€๎ƒก๎„ญ๎€ท๎ข๎กธ๎‰ข๎†ก๎Š‹๎‰ฝ๎†ƒ
๎ค‘ ๎†ก๎‹‰๎‰ข๎ˆถ๎‰ญ๎Š‹ ๎ƒก๎†๎ƒพ๎ƒฒ๎€ญ๎กผ๎€ƒ๎ƒจ๎€ญ๎ฑ๎ค๎…”๎‰ญ๎†ƒ๎‡ ๎ขš๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ
๎ค‘ ๎†ก๎‹‰๎‰ข๎ˆถ๎‰ญ๎Š‹ ๎ƒก๎†๎ƒพ๎ƒฒ๎€ญ๎กผ๎„ช๎€ƒ๎ƒจ๎ค๎ขผ๎ข€๎ƒก๎†๎ƒพ๎ƒฒ๎€ญ๎กผ๎€ท๎บ๎ƒจ๎ข๎กธ๎ขผ๎ข€๎ƒก๎†๎ƒพ๎ƒฒ๎€ญ๎กผ๎€ƒ๎ƒจ๎€ญ๎ฑ๎ข
๎‚ถ๎“๎ญ๎ฐ๎Ž•๎‹ผ๎Ž‹ ๎Œฆ๎Žฉ๎Ž•๎“๎ฐ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎“๎š๎Žฉ๎Œฆ๎Œผ๎ฃ๎ฃ ๎บ๎บ ๎Œฆ๎Ž‹๎Žฉ๎Ž™๎ฐ๎Ž™๎  ๎ญ๎ซ๎Œผ ๎š๎Œผ๎“๎Žฉ๎ฃ๎ฐ๎ญ๎Žฉ๎š๎ฐ๎Œผ๎ฃ
๎ค‘ ๎‡Œ๎‡ซ๎Š‹ ๎†ƒ๎ˆ๎ˆถ๎ˆฅ๎†ก ๎‡ ๎Š‹๎Š‹๎‰ข๎‰ฝ๎กช๎กธ๎กธ๎‡Œ๎‡ซ๎Š‹๎‡ ๎Š ๎…น๎กฒ๎†ƒ๎ˆถ๎ˆŸ๎กธ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎กธ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ขš๎‰ฝ๎ˆถ๎Š ๎‰ญ๎†ƒ๎†ก๎‰ฝ
๎ค‘ ๎‡Œ๎‡ซ๎Š‹ ๎†ƒ๎ˆ๎ˆถ๎ˆฅ๎†ก ๎‡ ๎Š‹๎Š‹๎‰ข๎‰ฝ๎กช๎กธ๎กธ๎‡Œ๎‡ซ๎Š‹๎ˆ๎…”๎…น๎กฒ๎†ƒ๎ˆถ๎ˆŸ๎กธ๎‰ข๎†ก๎Š‹๎‰ฝ๎†ƒ๎กธ๎‰ข๎†ก๎Š‹๎‰ฝ๎†ƒ
๎‚ถ๎“๎ญ๎ฐ๎Ž•๎‹ผ๎Ž‹ ๎Œฆ๎Žฉ๎Ž•๎“๎ฐ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎“๎š๎Žฉ๎Œฆ๎Œผ๎ฃ๎ฃ ๎บ๎บ๎บ ๎Œฆ๎Žฉ๎Ž•๎“๎ฐ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎š๎Žฉ๎š ๎š๎Œผ๎‹ผ๎Ž‹ ๎ฃ๎Œฆ๎‹ผ๎Ž‹๎‹ผ๎š๎ฃ
๎ค‘ ๎†ƒ๎†Ž ๎ขผ๎ข€๎ƒก๎†๎ƒพ๎ƒฒ๎€ญ๎กผ๎€ท๎บ๎ƒจ๎ข ๎ฅท๎ฅท ๎กฒ๎กธ๎†ƒ๎ˆถ๎ˆฅ๎‹ช๎‡Œ๎Š ๎‰ญ๎†ก ๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎ˆŸ๎Š ๎ˆŸ๎‰ข๎‰ฝ
๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‰ข๎…”๎‰ญ๎ˆŸ๎†ก๎Š‹๎‡ซ๎‰ฝ ๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎ˆŸ๎†ก๎Š‹๎‡ซ๎‰ฝ
๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‡ ๎‹Š๎‰ข๎‰ญ๎†ก ๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‰ฝ๎Š ๎‰ข๎†ก๎‰ญ๎ˆ๎Š 
๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‰ฝ๎ˆ๎†ก๎‰ข๎†ƒ ๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‡ ๎‰ข๎†Ž๎†Ž๎ˆŸ
๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‰ข๎Š‹๎‰ฝ๎†ƒ๎ˆถ๎Š‹๎†ƒ๎‡  ๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎‰ฝ๎‰ข๎…”๎‰ญ๎‰ฝ๎†ก
๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎‰ข๎…”๎†ƒ๎ˆ‡ ๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎Š‹๎†ก๎Š‹๎‡Œ๎†ก๎ˆฅ
๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎ˆŸ๎ˆŸ๎‡Œ ๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‰ข๎…”๎‰ญ๎ˆŸ๎ˆŸ๎‡Œ
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‡‰๎ˆถ๎‰ญ๎Š‹๎‰ญ๎…”๎ˆฅ๎ขš๎…น๎‡ซ๎ˆฅ๎†Ž๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ๎ค๎ˆฅ๎ˆถ ๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎‰ญ๎ขš๎Š‹๎‹Š๎‰ข๎†ก๎ค๎‰ญ๎†ก๎…”๎ˆ
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎†Ž๎†ก๎…น๎Š ๎‡Œ๎‡Œ๎‡ซ๎ˆฅ๎‡Œ๎ค๎ˆฅ๎ˆถ
๎ค‘ ๎ˆŸ๎…”๎ˆ‡๎†ก
๎‚ถ๎“๎ญ๎ฐ๎Ž•๎‹ผ๎Ž‹ ๎Œฆ๎Žฉ๎Ž•๎“๎ฐ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎“๎š๎Žฉ๎Œฆ๎Œผ๎ฃ๎ฃ ๎บ๎„ช ๎Œฆ๎Žฉ๎Ž•๎“๎ฐ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎š๎Žฉ๎š ๎Œฆ๎Žฉ๎Ž•๎“๎Ž‹๎Œผ๎ข ๎ฃ๎Œฆ๎‹ผ๎Ž‹๎‹ผ๎š๎ฃ๎กซ ๎Žฉ๎“๎ญ๎ฐ๎Žฉ๎Ž™๎‹ผ๎Ž‹
๎ค‘ ๎†ก๎‹‰๎‰ข๎ˆถ๎‰ญ๎Š‹ ๎ƒก๎†๎ƒพ๎ƒฒ๎€ญ๎กผ๎€ƒ๎ƒจ๎€ญ๎ฑ๎ค๎…”๎‰ญ๎†ƒ๎‡ ๎ขš๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ขš๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰
๎ค‘ ๎กฒ๎กธ๎†ƒ๎ˆถ๎ˆฅ๎‹ช๎‡Œ๎Š ๎‰ญ๎†ก ๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎ˆŸ๎Š ๎ˆŸ๎‰ข๎‰ฝ๎ขš๎†Ž๎‡ซ๎‰ญ๎ค๎…”๎‰ญ๎†ƒ๎‡ ๎ขš๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ข๎…”๎‰ญ๎ˆŸ๎†ก๎Š‹๎‡ซ๎‰ฝ๎ขš๎†Ž๎‡ซ๎‰ญ๎ค๎…”๎‰ญ๎†ƒ๎‡ ๎ขš๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎ˆŸ๎†ก๎Š‹๎‡ซ๎‰ฝ๎ขš๎†Ž๎‡ซ๎‰ญ๎ค๎…”๎‰ญ๎†ƒ๎‡ ๎ขš๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ฝ๎Š ๎‰ข๎†ก๎‰ญ๎ˆ๎Š ๎ขš๎†Ž๎‡ซ๎‰ญ๎ค๎…”๎‰ญ๎†ƒ๎‡ ๎ขš๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ ๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‰ฝ๎ˆ๎†ก๎‰ข๎†ƒ
๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‡ ๎‰ข๎†Ž๎†Ž๎ˆŸ ๎ขš๎ขš๎†Ž๎ˆถ๎‹๎ˆฅ๎ˆ๎ˆถ๎…”๎†Ž๎ขš๎‡ ๎Š‹๎ˆถ๎ˆถ๎ˆ
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ข๎Š‹๎‰ฝ๎†ƒ๎ˆถ๎Š‹๎†ƒ๎‡ ๎ขš๎†Ž๎‡ซ๎‰ญ๎ค๎…”๎‰ญ๎†ƒ๎‡ ๎ขš๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ฝ๎Š ๎‡ซ๎Š‹๎†ก๎‰ฝ๎‰ข๎…”๎‰ญ๎‰ฝ๎†ก๎ขš๎†Ž๎‡ซ๎‰ญ๎ค๎…”๎‰ญ๎†ƒ๎‡ ๎ขš๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎‰ข๎…”๎†ƒ๎ˆ‡๎ขš๎†Ž๎‡ซ๎‰ญ๎ค๎…”๎‰ญ๎†ƒ๎‡ ๎ขš๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎Š‹๎†ก๎Š‹๎‡Œ๎†ก๎ˆฅ๎ขš๎†Ž๎‡ซ๎‰ญ๎ค๎…”๎‰ญ๎†ƒ๎‡ ๎ขš๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‡‰๎ˆถ๎‰ญ๎Š‹๎‰ญ๎…”๎ˆฅ๎ขš๎…น๎‡ซ๎ˆฅ๎†Ž๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ๎ค๎ˆฅ๎ˆถ
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎‰ญ๎ขš๎Š‹๎‹Š๎‰ข๎†ก๎ค๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰ ๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎†Ž๎†ก๎…น๎Š ๎‡Œ๎‡Œ๎‡ซ๎ˆฅ๎‡Œ๎ค๎ˆฅ๎ˆถ
๎ค‘ ๎ˆŸ๎…”๎ˆ‡๎†ก
๎‚ถ๎“๎ญ๎ฐ๎Ž•๎‹ผ๎Ž‹ ๎Œฆ๎Žฉ๎Ž•๎“๎ฐ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™ ๎“๎š๎Žฉ๎Œฆ๎Œผ๎ฃ๎ฃ ๎„ช ๎š๎ฐ๎Ž™๎‹ผ๎Ž‹ ๎Œฆ๎Žฉ๎Ž•๎“๎ฐ๎Ž‹๎‹ผ๎ญ๎ฐ๎Žฉ๎Ž™
๎ค‘ ๎†ƒ๎†Ž ๎ขผ๎ข€๎ข๎ข๎กผ๎€ท๎บ๎ƒจ๎ข
๎ค‘ ๎‡Œ๎‡ซ๎Š‹ ๎†ƒ๎‡ ๎†ก๎†ƒ๎ˆ‡๎ˆถ๎Š ๎Š‹ ๎†Ž๎†ก๎Šฝ๎†ก๎ˆ๎ˆถ๎‰ข
๎ค‘ ๎…”๎Š ๎Š‹๎ˆถ๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎‡‰ ๎ขš๎‡ซ
๎ค‘ ๎กฒ๎กธ๎†ƒ๎ˆถ๎ˆฅ๎‹ช๎‡Œ๎Š ๎‰ญ๎†ก ๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ˆถ๎Š ๎Š‹๎ขš๎‡ ๎†Ž๎‡‰๎ „
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ข๎†ก๎Š‹๎‰ฝ๎†ƒ๎ค๎ขผ๎ข€๎ƒก๎†๎ƒพ๎ƒฒ๎€ญ๎กผ๎„ช๎€ƒ๎ƒจ๎ข๎กธ๎ˆ๎‡ซ๎…น
๎ขš๎ขš๎‹๎‡ซ๎Š‹๎‡ ๎ขš๎‰ข๎†ก๎Š‹๎‰ฝ๎†ƒ๎กผ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ค๎ขผ๎ข€๎ƒก๎†๎ƒพ๎ƒฒ๎€ญ๎กผ๎„ช๎€ƒ๎ƒจ๎ข๎ขš๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎กธ๎ˆ๎‡ซ๎…น
โ—ฆ ๎‡ซ๎‡‰ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎‡ซ๎‰ฝ ๎ˆฅ๎ˆถ๎Š‹ ๎†Ž๎†ก๎Š‹๎†ก๎†ƒ๎Š‹๎†ก๎†Ž๎กซ ๎ˆถ๎Šฝ๎†ก๎‰ญ๎‹๎‰ญ๎‡ซ๎Š‹๎†ก ๎‚ก๎ƒก๎บ๎ƒจ๎„๎‚ฅ
๎ค‘ ๎ˆŸ๎…”๎ˆ‡๎†ก ๎ขš๎‡ฟ๎ ƒ
๎ค‘ ๎†ก๎‹‰๎‰ข๎ˆถ๎‰ญ๎Š‹ ๎ƒก๎€ƒ๎ƒพ๎ฑ๎ค๎ขผ๎ข€๎ƒก๎€ƒ๎ƒพ๎ฑ๎ข๎กช๎ขผ๎ข€๎ข๎ข๎กผ๎€ท๎บ๎ƒจ๎ข๎กธ๎‰ฝ๎‰ญ๎†ƒ๎กธ๎ˆŸ๎‰ข๎‡ซ
๎ค‘ ๎†ก๎‹‰๎‰ข๎ˆถ๎‰ญ๎Š‹ ๎ƒก๎€ƒ๎ƒพ๎ฑ๎ค๎ขผ๎ข€๎ƒก๎€ƒ๎ƒพ๎ฑ๎ข๎กช๎ขผ๎ข€๎ข๎ข๎กผ๎€ท๎บ๎ƒจ๎ข๎กธ๎‰ฝ๎‰ญ๎†ƒ๎กธ๎ˆฅ๎‹
โ—ฆ ๎‰ฝ๎†ก๎Š‹๎Š ๎‰ข ๎ค—๎กธ๎กฒ๎‡‰๎‰ญ๎†ก๎†ก๎‡‰๎†ก๎ˆŸ๎ค‹๎ค‹๎กฒ๎‰ข๎‰ญ๎†ก๎‡‰ ๎ˆถ๎‰ญ ๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก ๎ข๎ข๎กผ๎บ๎‚ฅ๎€ญ๎‚•๎„๎€ท๎†๎ƒก๎€ƒ๎ƒพ๎ฑ
๎…”๎ˆฅ๎†Ž ๎ข๎ข๎กผ๎‚•๎‚ถ๎€ƒ๎€ท๎ƒก๎€ƒ๎ƒพ๎ฑ
๎ƒจ๎Œผ๎š๎Œผ๎š๎Œผ๎Ž™๎Œฆ๎Œผ๎ฃ ๎ฐ
๎€ƒ๎†Ž๎…”๎ˆŸ๎‰ฝ๎กซ ๎‚ก๎…”๎‰ญ๎ˆ‡๎กซ ๎ฑ๎…”๎‰ญ๎Š ๎ˆฅ ๎ฑ๎กฒ ๎€ฅ๎…”๎‹Š๎‰ญ๎…”๎ˆ‡๎Š‹๎…”๎‰ญ๎กซ ๎ƒพ๎ˆถ๎ˆฅ๎‹Š ๎‚ก๎กฒ ๎‚‘๎†ก๎…”๎Šฝ๎†ก๎ˆฅ๎‹Š๎กซ ๎…”๎ˆฅ๎†Ž ๎ƒก๎…”๎ˆฅ๎…”๎‹Š๎‡ซ๎ˆถ๎Š‹๎‡ซ๎‰ฝ ๎ƒก๎…”๎‰ข๎…”๎†Ž๎ˆถ๎‰ข๎ˆถ๎Š ๎ˆ๎ˆถ๎‰ฝ
๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ ƒ๎ข๎กฒ ๎ขจ๎„๎ˆ๎Š‹๎‰ญ๎…”๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎…น๎ˆ๎†ก ๎บ๎ˆŸ๎‰ข๎ˆ๎‡ซ๎†ƒ๎‡ซ๎Š‹ ๎ข๎‡ซ๎ˆฅ๎‡ซ๎Š‹๎†ก ๎†๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹ ๎€ƒ๎ˆฅ๎…”๎ˆ๎‹Š๎‰ฝ๎†ก๎‰ฝ ๎‡ซ๎ˆฅ ๎ƒฒ๎ˆถ๎ˆ๎‡ซ๎†Ž ๎‚ก๎†ก๎†ƒ๎‡ ๎…”๎ˆฅ๎‡ซ๎†ƒ๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡ 
๎ˆถ๎Šฝ๎†ก๎‰ญ ๎…” ๎ฑ๎…”๎ˆ๎‡‰ ๎…” ๎€ฅ๎‡ซ๎ˆ๎ˆ๎‡ซ๎ˆถ๎ˆฅ ๎€ท๎†ก๎‡Œ๎‰ญ๎†ก๎†ก๎‰ฝ ๎ˆถ๎‡‰ ๎ข๎‰ญ๎†ก๎†ก๎†Ž๎ˆถ๎ˆŸ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช Proceedings of the 2004 ACM/IEEE
Conference on Supercomputing๎กฒ ๎ƒฒ๎€ญ๎Ÿฟ๎ ƒ๎กฒ ๎บ๎†๎†๎† ๎€ญ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎†ก๎‰ญ ๎ƒฒ๎ˆถ๎†ƒ๎‡ซ๎†ก๎Š‹๎‹Š๎กซ ๎ ‚๎ ƒ๎กช๎ €๎ข—๎ ‚๎ ƒ๎กช๎ €๎ „๎กฒ
๎€ƒ๎ˆ ๎€ท๎…”๎…”๎‰ฝ๎กซ ๎ฑ๎Š ๎‰ฝ๎‰ฝ๎…”๎ˆŸ๎กซ ๎‚•๎…”๎Š ๎‰ญ๎…” ๎ฅ๎‰ญ๎‡ซ๎‡Œ๎ˆถ๎‰ญ๎‡ซ๎กซ ๎ƒก๎‡ซ๎†ก๎‰ญ๎‰ญ๎†ก ๎‚๎ˆถ๎ˆ๎‡ซ๎Šฝ๎†ก๎Š‹๎กซ ๎…”๎ˆฅ๎†Ž ๎ƒก๎‡ซ๎†ก๎‰ญ๎‰ญ๎†ก๎ขš๎ฑ๎†ก๎ˆฅ๎‰ญ๎‡ซ ๎ƒพ๎ˆถ๎Š ๎‰ญ๎ˆฅ๎‡ซ๎†ก๎‰ญ ๎ขŽ๎ ๎Ÿฟ๎ ๎ €๎ข๎กฒ ๎ขจ๎€ƒ
๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎ˆ๎†ก๎Šฝ๎†ก๎ˆ ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ ๎…น๎…”๎‰ฝ๎†ก๎†Ž ๎ˆถ๎ˆฅ ๎…” ๎‡ ๎‡ซ๎†ก๎‰ญ๎…”๎‰ญ๎†ƒ๎‡ ๎‹Š ๎ˆถ๎‡‰ ๎‰ญ๎ˆถ๎…น๎Š ๎‰ฝ๎Š‹ ๎†ƒ๎ˆถ๎…”๎‰ญ๎‰ฝ๎†ก
๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช SIAM Journal on Scienti๎‹ฉc Computing ๎ ƒ๎ ‚๎กฒ๎ ‚๎กซ ๎€ƒ๎ €๎ ˆ๎Ÿฟ๎ †๎ข—๎€ƒ๎ €๎ ˆ๎ ๎ ‡๎กฒ ๎ผ๎š๎Ž‹๎กช
https://github.com/prj-/aldaas2019multi๎กฒ
๎€ƒ๎ˆŸ๎†ก๎‰ฝ๎Š‹๎ˆถ๎‹Š๎กซ ๎ƒก๎…”๎Š‹๎‰ญ๎‡ซ๎†ƒ๎ˆ‡๎กซ ๎บ๎…”๎‡ซ๎ˆฅ ๎€ท๎Š ๎‡‰๎‡‰๎กซ ๎‚๎†ก๎…”๎ˆฅ๎ขš๎„ต๎Šฝ๎†ก๎‰ฝ ๎‚•๎ขซ๎†๎‹‰๎†ƒ๎†ก๎ˆ๎ˆ๎†ก๎ˆฅ๎Š‹๎กซ ๎…”๎ˆฅ๎†Ž ๎‚๎…”๎†ƒ๎ˆ‡๎ˆถ ๎‚‘๎ˆถ๎‰ฝ๎Š‹๎†ก๎‰ญ ๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ €๎ข๎กฒ ๎ขจ๎€ƒ ๎‡‰๎Š ๎ˆ๎ˆ๎‹Š
๎…”๎‰ฝ๎‹Š๎ˆฅ๎†ƒ๎‡ ๎‰ญ๎ˆถ๎ˆฅ๎ˆถ๎Š ๎‰ฝ ๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎‡‰๎‰ญ๎ˆถ๎ˆฅ๎Š‹๎…”๎ˆ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž ๎†Ž๎‹Š๎ˆฅ๎…”๎ˆŸ๎‡ซ๎†ƒ ๎‰ฝ๎†ƒ๎‡ ๎†ก๎†Ž๎Š ๎ˆ๎‡ซ๎ˆฅ๎‡Œ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช
SIAM Journal on Matrix Analysis and Applications ๎ ๎ ‚๎กฒ๎ €๎กซ ๎‰ข๎‰ข๎กฒ ๎ €๎ „๎ข—๎ ƒ๎ €๎กฒ ๎ผ๎š๎Ž‹๎กช
https://mumps-solver.org๎กฒ
๎€ฅ๎…”๎ˆ๎…”๎‹Š๎กซ ๎ƒฒ๎…”๎Š‹๎‡ซ๎‰ฝ๎‡ ๎กซ ๎„ญ๎‡ซ๎ˆ๎ˆ๎‡ซ๎…”๎ˆŸ ๎€ท๎กฒ ๎ฅ๎‰ญ๎ˆถ๎‰ข๎‰ข๎กซ ๎‚•๎ˆถ๎‡ซ๎‰ฝ ๎€ญ๎Š ๎‰ญ๎‡‰๎ˆŸ๎…”๎ˆฅ ๎‚ก๎†ƒ๎บ๎ˆฅ๎ˆฅ๎†ก๎‰ฝ๎กซ ๎…”๎ˆฅ๎†Ž ๎€ฅ๎…”๎‰ญ๎‰ญ๎‹Š ๎ข๎กฒ ๎ƒฒ๎ˆŸ๎‡ซ๎Š‹๎‡  ๎ขŽ๎ €๎ ˆ๎ ˆ๎ †๎ข๎กฒ
๎ขจ๎†๎‡‰๎‹ช๎†ƒ๎‡ซ๎†ก๎ˆฅ๎Š‹ ๎ˆŸ๎…”๎ˆฅ๎…”๎‡Œ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹ ๎ˆถ๎‡‰ ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ๎‡ซ๎‰ฝ๎ˆŸ ๎‡ซ๎ˆฅ ๎ˆถ๎…น๎‡ฟ๎†ก๎†ƒ๎Š‹๎ขš๎ˆถ๎‰ญ๎‡ซ๎†ก๎ˆฅ๎Š‹๎†ก๎†Ž ๎ˆฅ๎Š ๎ˆŸ๎†ก๎‰ญ๎‡ซ๎†ƒ๎…”๎ˆ ๎‰ฝ๎ˆถ๎‡‰๎Š‹๎‹๎…”๎‰ญ๎†ก
๎ˆ๎‡ซ๎…น๎‰ญ๎…”๎‰ญ๎‡ซ๎†ก๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช Modern Software Tools in Scienti๎‹ฉc Computing๎กซ ๎‰ข๎‰ข๎กฒ ๎ €๎ …๎ ‚๎ข—๎ ๎Ÿฟ๎ ๎กฒ
๎ƒจ๎Œผ๎š๎Œผ๎š๎Œผ๎Ž™๎Œฆ๎Œผ๎ฃ ๎ฐ๎ฐ
๎€ญ๎…”๎‡ซ๎กซ ๎„ด๎‡ซ๎…”๎ˆถ๎ขš๎€ญ๎‡ ๎Š ๎…”๎ˆฅ๎กซ ๎‚ก๎…”๎ˆ‡๎‰ฝ๎‹Š๎ˆŸ๎‡ซ๎ˆ๎‡ซ๎…”๎ˆฅ ๎€ท๎‰ญ๎‹Š๎‡ฟ๎…”๎กซ ๎…”๎ˆฅ๎†Ž ๎‚ก๎…”๎‰ญ๎†ƒ๎Š ๎‰ฝ ๎ƒฒ๎…”๎‰ญ๎ˆ‡๎‡ซ๎‰ฝ ๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ ‚๎ข๎กฒ ๎ขจ๎ƒจ๎†ก๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎†ƒ๎Š‹๎†ก๎†Ž ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก
๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  ๎‡ ๎…”๎‰ญ๎ˆŸ๎ˆถ๎ˆฅ๎‡ซ๎†ƒ ๎ˆถ๎Šฝ๎†ก๎‰ญ๎ˆ๎…”๎‰ข ๎‡‰๎ˆถ๎‰ญ ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎†Ž๎†ก๎‹ช๎ˆฅ๎‡ซ๎Š‹๎†ก
๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช SIAM Journal on Numerical Analysis ๎ ƒ๎ €๎กฒ๎ ƒ๎กซ ๎‰ข๎‰ข๎กฒ ๎ €๎ ๎Ÿฟ๎ ˆ๎ข—๎ €๎ ๎ ‚๎ €๎กฒ
๎ƒฒ๎Š‹๎ขš๎€ญ๎‹Š๎‰ญ๎กซ ๎€ƒ๎ˆŸ๎‡ซ๎ˆ‡๎กซ ๎‚ก๎…”๎‰ญ๎Š‹๎‡ซ๎ˆฅ ๎‚๎กฒ ๎ฅ๎…”๎ˆฅ๎†Ž๎†ก๎‰ญ๎กซ ๎…”๎ˆฅ๎†Ž ๎ƒฒ๎Š‹๎†ก๎‰ข๎‡ ๎†ก๎ˆฅ ๎ƒพ๎‡ ๎ˆถ๎ˆŸ๎…”๎‰ฝ ๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ †๎ข๎กฒ ๎ขจ๎‚ถ๎‰ข๎Š‹๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก๎†Ž
๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎‰ข๎ˆ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎Šฝ๎†ก๎กซ ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก๎กซ ๎…”๎ˆฅ๎†Ž ๎‰ญ๎†ก๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎†ƒ๎Š‹๎†ก๎†Ž ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช
SIAM Journal on Scienti๎‹ฉc Computing ๎ ๎ ˆ๎กฒ๎ …๎กซ ๎‰ข๎‰ข๎กฒ ๎ ๎ ƒ๎Ÿฟ๎ ๎ข—๎ ๎ ƒ๎ ๎ „๎กฒ
๎ข๎…”๎ˆ๎‡Œ๎ˆถ๎Š ๎Š‹๎กซ ๎ƒจ๎ˆถ๎…น๎†ก๎‰ญ๎Š‹ ๎€ท๎กฒ ๎…”๎ˆฅ๎†Ž ๎„๎ˆ๎‰ญ๎‡ซ๎ˆ‡๎†ก ๎‚ก๎†ก๎‡ซ๎†ก๎‰ญ ๎„ต๎…”๎ˆฅ๎‡Œ ๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ ๎ข๎กฒ ๎ขจhypre๎กช ๎€ƒ ๎ˆ๎‡ซ๎…น๎‰ญ๎…”๎‰ญ๎‹Š ๎ˆถ๎‡‰ ๎‡ ๎‡ซ๎‡Œ๎‡ 
๎‰ข๎†ก๎‰ญ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎ˆฅ๎†ƒ๎†ก ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช Computational Scienceโ€”ICCS 2002๎กซ ๎‰ข๎‰ข๎กฒ ๎ …๎ ‚๎ ๎ข—๎ …๎ ƒ๎ €๎กฒ
๎ข๎…”๎‰ญ๎‡ ๎…”๎Š‹๎กซ ๎€ญ๎‡ ๎…”๎‰ญ๎…น๎†ก๎ˆ ๎…”๎ˆฅ๎†Ž ๎ข๎‰ญ๎…”๎ˆฅ๎†‡๎ˆถ๎‡ซ๎‰ฝ๎ขš๎„ด๎…”๎Šฝ๎‡ซ๎†ก๎‰ญ ๎ƒจ๎ˆถ๎Š ๎‹‰ ๎ขŽ๎ €๎ ˆ๎ ˆ๎ €๎ข๎กฒ ๎ขจ๎€ƒ ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž ๎ˆถ๎‡‰ ๎‹ช๎ˆฅ๎‡ซ๎Š‹๎†ก ๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹
๎Š‹๎†ก๎…”๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎…”๎ˆฅ๎†Ž ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎†ƒ๎ˆถ๎ˆฅ๎ˆฅ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆฅ๎‡Œ ๎…”๎ˆฅ๎†Ž ๎‡ซ๎Š‹๎‰ฝ ๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ ๎‰ฝ๎ˆถ๎ˆ๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎…”๎ˆ๎‡Œ๎ˆถ๎‰ญ๎‡ซ๎Š‹๎‡ ๎ˆŸ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช International
Journal for Numerical Methods in Engineering ๎ ‚๎ ๎กฒ๎ …๎กซ ๎‰ข๎‰ข๎กฒ ๎ €๎ ๎Ÿฟ๎ „๎ข—๎ €๎ ๎ ๎ †๎กฒ
๎ฅ๎†ก๎†ก๎กซ ๎‚ก๎‡ซ๎†ƒ๎‡ ๎…”๎†ก๎ˆ ๎„ญ๎กฒ๎กซ ๎€ญ๎‡ ๎‰ญ๎‡ซ๎‰ฝ๎Š‹๎ˆถ๎‰ข๎‡ ๎†ก๎‰ญ ๎‚ก๎กฒ ๎ƒฒ๎‡ซ๎†ก๎‡‰๎†ก๎‰ญ๎Š‹๎กซ ๎‚๎ˆถ๎ˆฅ๎…”๎Š‹๎‡ ๎…”๎ˆฅ ๎‚๎กฒ ๎ฑ๎Š ๎กซ ๎ƒจ๎…”๎‹Š ๎ƒฒ๎กฒ ๎ƒพ๎Š ๎ˆŸ๎‡ซ๎ˆฅ๎…”๎‰ญ๎ˆถ๎กซ ๎…”๎ˆฅ๎†Ž
๎‚ก๎…”๎‰ญ๎‹–๎‡ซ๎ˆถ ๎ฅ๎กฒ ๎ƒฒ๎…”๎ˆ๎…” ๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ …๎ข๎กฒ ML 5.0 Smoothed Aggregation Userโ€™s Guide๎กฒ ๎ƒพ๎†ก๎†ƒ๎‡ ๎กฒ ๎‰ญ๎†ก๎‰ข๎กฒ
๎ƒฒ๎€ƒ๎‚ฅ๎€ท๎ ๎Ÿฟ๎Ÿฟ๎ …๎ขš๎ ๎ …๎ ƒ๎ ˆ๎กฒ ๎ƒฒ๎…”๎ˆฅ๎†Ž๎‡ซ๎…” ๎‚ฅ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‚•๎…”๎…น๎ˆถ๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ๎‡ซ๎†ก๎‰ฝ๎กฒ ๎ผ๎š๎Ž‹๎กช
https://trilinos.github.io/ml.html๎กฒ
๎ƒจ๎Œผ๎š๎Œผ๎š๎Œผ๎Ž™๎Œฆ๎Œผ๎ฃ ๎ฐ๎ฐ๎ฐ
๎ฅ๎†ก๎Š ๎‹–๎…”๎‡ซ๎ˆฅ๎†ก๎กซ ๎€ญ๎‡ ๎‰ญ๎‡ซ๎‰ฝ๎Š‹๎ˆถ๎‰ข๎‡ ๎†ก ๎…”๎ˆฅ๎†Ž ๎‚๎†ก๎…”๎ˆฅ๎ขš๎ข๎‰ญ๎…”๎ˆฅ๎†‡๎ˆถ๎‡ซ๎‰ฝ ๎ƒจ๎†ก๎ˆŸ๎…”๎†ƒ๎ˆ๎†ก ๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ ˆ๎ข๎กฒ ๎ขจ๎ฅ๎ˆŸ๎‰ฝ๎‡ ๎กช ๎€ƒ ๎ ‚๎ขš๎€ท ๎‹ช๎ˆฅ๎‡ซ๎Š‹๎†ก
๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹ ๎ˆŸ๎†ก๎‰ฝ๎‡  ๎‡Œ๎†ก๎ˆฅ๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ ๎‹๎‡ซ๎Š‹๎‡  ๎…น๎Š ๎‡ซ๎ˆ๎Š‹๎ขš๎‡ซ๎ˆฅ ๎‰ข๎‰ญ๎†ก๎ขš ๎…”๎ˆฅ๎†Ž ๎‰ข๎ˆถ๎‰ฝ๎Š‹๎ขš๎‰ข๎‰ญ๎ˆถ๎†ƒ๎†ก๎‰ฝ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎‡‰๎…”๎†ƒ๎‡ซ๎ˆ๎‡ซ๎Š‹๎‡ซ๎†ก๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช
International Journal for Numerical Methods in Engineering ๎ †๎ ˆ๎กฒ๎ €๎ €๎กซ ๎‰ข๎‰ข๎กฒ ๎ €๎ ‚๎Ÿฟ๎ ˆ๎ข—๎ €๎ ‚๎ ‚๎ €๎กฒ
๎ผ๎š๎Ž‹๎กช http://geuz.org/gmsh๎กฒ
๎ฅ๎ˆถ๎‰ฝ๎‰ฝ๎†ก๎ˆ๎†ก๎Š‹๎กซ ๎ƒก๎‡ซ๎†ก๎‰ญ๎‰ญ๎†ก ๎…”๎ˆฅ๎†Ž ๎€ญ๎‡ ๎‰ญ๎‡ซ๎‰ฝ๎Š‹๎‡ซ๎…”๎ˆฅ ๎ƒจ๎†ก๎‹Š ๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ …๎ข๎กฒ ๎ขจ๎‚ฅ๎ˆถ๎ˆฅ๎ขš๎ˆถ๎Šฝ๎†ก๎‰ญ๎ˆ๎…”๎‰ข๎‰ข๎‡ซ๎ˆฅ๎‡Œ ๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ
๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ ๎‡ซ๎ˆฅ ๎‰ฝ๎Š‹๎‰ญ๎Š ๎†ƒ๎Š‹๎Š ๎‰ญ๎…”๎ˆ ๎ˆŸ๎†ก๎†ƒ๎‡ ๎…”๎ˆฅ๎‡ซ๎†ƒ๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช Archives of Computational
Methods in Engineering ๎ €๎ ‚๎กฒ๎ ƒ๎กซ ๎‰ข๎‰ข๎กฒ ๎ „๎ €๎ „๎ข—๎ „๎ †๎ ๎กฒ
๎ฑ๎…”๎‡‰๎†ก๎‰ญ๎‰ฝ๎‰ฝ๎…”๎‰ฝ๎กซ ๎ƒจ๎‹Š๎…”๎†Ž๎‡ ๎กซ ๎ƒก๎‡ซ๎†ก๎‰ญ๎‰ญ๎†ก ๎‚๎ˆถ๎ˆ๎‡ซ๎Šฝ๎†ก๎Š‹๎กซ ๎…”๎ˆฅ๎†Ž ๎ข๎‰ญ๎†ข๎†Ž๎†ข๎‰ญ๎‡ซ๎†ƒ ๎‚ฅ๎…”๎Š‹๎…”๎‡‰ ๎ขŽ๎ ๎Ÿฟ๎ €๎ †๎ข๎กฒ ๎ขจ๎€ƒ๎ˆฅ ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹–
๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž ๎Š‹๎‹Š๎‰ข๎†ก ๎Š‹๎‡ ๎†ก๎ˆถ๎‰ญ๎‹Š ๎‡‰๎ˆถ๎‰ญ ๎‚•๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ๎ขซ๎‰ฝ ๎…”๎ˆ๎‡Œ๎ˆถ๎‰ญ๎‡ซ๎Š‹๎‡ ๎ˆŸ ๎…”๎ˆฅ๎†Ž ๎…” ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎‹–๎†ก๎†Ž ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก๎†Ž
๎‰ญ๎†ก๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎†ƒ๎Š‹๎†ก๎†Ž ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช SIAM Journal on Scienti๎‹ฉc Computing
๎ ‚๎ ˆ๎กฒ๎ ƒ๎กซ ๎€ƒ๎ €๎ ‚๎ ƒ๎ „๎ข—๎€ƒ๎ €๎ ‚๎ …๎ „๎กฒ
๎ฑ๎†ก๎†ƒ๎‡ ๎Š‹๎กซ ๎ข๎‰ญ๎†ข๎†Ž๎†ข๎‰ญ๎‡ซ๎†ƒ ๎ขŽ๎ ๎Ÿฟ๎ €๎ ๎ข๎กฒ ๎ขจ๎‚ฅ๎†ก๎‹ ๎†Ž๎†ก๎Šฝ๎†ก๎ˆ๎ˆถ๎‰ข๎ˆŸ๎†ก๎ˆฅ๎Š‹ ๎‡ซ๎ˆฅ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†ก๎ˆŸ๎ค‹๎ค‹๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช Journal of Numerical
Mathematics ๎ ๎Ÿฟ๎กฒ๎ ‚๎ขš๎ ƒ๎กซ ๎‰ข๎‰ข๎กฒ ๎ ๎ „๎ €๎ข—๎ ๎ …๎ …๎กฒ
๎ฑ๎†ข๎ˆฅ๎ˆถ๎ˆฅ๎กซ ๎ƒก๎…”๎‰ฝ๎†ƒ๎…”๎ˆ๎กซ ๎ƒก๎‡ซ๎†ก๎‰ญ๎‰ญ๎†ก ๎ƒจ๎…”๎ˆŸ๎†ก๎Š‹๎กซ ๎…”๎ˆฅ๎†Ž ๎‚๎†ก๎…”๎ˆฅ ๎ƒจ๎ˆถ๎ˆŸ๎…”๎ˆฅ ๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ ๎ข๎กฒ ๎ขจ๎ƒก๎…”๎ƒฒ๎Š‹๎‡ซ๎„ด๎กช ๎…” ๎‡ ๎‡ซ๎‡Œ๎‡ ๎ขš๎‰ข๎†ก๎‰ญ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎ˆฅ๎†ƒ๎†ก
๎‰ข๎…”๎‰ญ๎…”๎ˆ๎ˆ๎†ก๎ˆ ๎†Ž๎‡ซ๎‰ญ๎†ก๎†ƒ๎Š‹ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ ๎‡‰๎ˆถ๎‰ญ ๎‰ฝ๎‰ข๎…”๎‰ญ๎‰ฝ๎†ก ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎Šฝ๎†ก ๎†Ž๎†ก๎‹ช๎ˆฅ๎‡ซ๎Š‹๎†ก ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช Parallel
Computing ๎ ๎ ‡๎กฒ๎ ๎กซ ๎‰ข๎‰ข๎กฒ ๎ ‚๎Ÿฟ๎ €๎ข—๎ ‚๎ ๎ €๎กฒ ๎ผ๎š๎Ž‹๎กช http://pastix.gforge.inria.fr๎กฒ
๎ƒจ๎Œผ๎š๎Œผ๎š๎Œผ๎Ž™๎Œฆ๎Œผ๎ฃ ๎ฐ๎˜
๎ฑ๎†ก๎‰ญ๎ˆฅ๎…”๎ˆฅ๎†Ž๎†ก๎‹–๎กซ ๎„ช๎‡ซ๎†ƒ๎†ก๎ˆฅ๎Š‹๎†ก๎กซ ๎‚๎ˆถ๎‰ฝ๎†ก ๎†๎กฒ ๎ƒจ๎ˆถ๎ˆŸ๎…”๎ˆฅ๎กซ ๎…”๎ˆฅ๎†Ž ๎„ช๎‡ซ๎†ƒ๎†ก๎ˆฅ๎Š‹๎†ก ๎„ช๎‡ซ๎†Ž๎…”๎ˆ ๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ „๎ข๎กฒ ๎ขจ๎ƒฒ๎‚•๎†๎ƒก๎†ƒ๎กช ๎…” ๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎…น๎ˆ๎†ก
๎…”๎ˆฅ๎†Ž ๎‹ซ๎†ก๎‹‰๎‡ซ๎…น๎ˆ๎†ก ๎Š‹๎ˆถ๎ˆถ๎ˆ๎ˆ‡๎‡ซ๎Š‹ ๎‡‰๎ˆถ๎‰ญ ๎Š‹๎‡ ๎†ก ๎‰ฝ๎ˆถ๎ˆ๎Š ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‡‰ ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎…”๎ˆ๎Š ๎†ก ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช ACM
Transactions on Mathematical Software ๎ ‚๎ €๎กฒ๎ ‚๎กซ ๎‰ข๎‰ข๎กฒ ๎ ‚๎ „๎ €๎ข—๎ ‚๎ …๎ ๎กฒ ๎ผ๎š๎Ž‹๎กช
https://slepc.upv.es๎กฒ
๎ฑ๎‡ซ๎‰ข๎Š‹๎ˆŸ๎…”๎‡ซ๎‰ญ๎กซ ๎ƒจ๎…”๎ˆ๎‡‰ ๎…”๎ˆฅ๎†Ž ๎‚๎‡ซ๎ˆฅ๎†ƒ๎‡ ๎…”๎ˆถ ๎„ด๎Š  ๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ †๎ข๎กฒ ๎ขจ๎‚ฅ๎ˆถ๎†Ž๎…”๎ˆ ๎…”๎Š ๎‹‰๎‡ซ๎ˆ๎‡ซ๎…”๎‰ญ๎‹Š ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‡ซ๎ˆฅ๎‡Œ ๎‡ซ๎ˆฅ
H(๎†ƒ๎Š ๎‰ญ๎ˆ) ๎…”๎ˆฅ๎†Ž H(๎†Ž๎‡ซ๎Šฝ) ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช SIAM Journal on Numerical Analysis ๎ ƒ๎ „๎กฒ๎ …๎กซ
๎‰ข๎‰ข๎กฒ ๎ ๎ ƒ๎ ‡๎ ‚๎ข—๎ ๎ „๎Ÿฟ๎ ˆ๎กฒ
๎‚๎ˆถ๎ˆ๎‡ซ๎Šฝ๎†ก๎Š‹๎กซ ๎ƒก๎‡ซ๎†ก๎‰ญ๎‰ญ๎†ก๎กซ ๎ข๎‰ญ๎†ข๎†Ž๎†ข๎‰ญ๎‡ซ๎†ƒ ๎ฑ๎†ก๎†ƒ๎‡ ๎Š‹๎กซ ๎ข๎‰ญ๎†ข๎†Ž๎†ข๎‰ญ๎‡ซ๎†ƒ ๎‚ฅ๎…”๎Š‹๎…”๎‡‰๎กซ ๎…”๎ˆฅ๎†Ž ๎€ญ๎‡ ๎‰ญ๎‡ซ๎‰ฝ๎Š‹๎ˆถ๎‰ข๎‡ ๎†ก ๎ƒก๎‰ญ๎Š ๎†Ž๎ขซ๎‡ ๎ˆถ๎ˆŸ๎ˆŸ๎†ก ๎ขŽ๎ ๎Ÿฟ๎ €๎ ‚๎ข๎กฒ
๎ขจ๎ƒฒ๎†ƒ๎…”๎ˆ๎…”๎…น๎ˆ๎†ก ๎€ท๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎€ท๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ƒก๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎‰ฝ ๎‡‰๎ˆถ๎‰ญ ๎ฑ๎†ก๎Š‹๎†ก๎‰ญ๎ˆถ๎‡Œ๎†ก๎ˆฅ๎†ก๎ˆถ๎Š ๎‰ฝ ๎†๎ˆ๎ˆ๎‡ซ๎‰ข๎Š‹๎‡ซ๎†ƒ
๎ƒก๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช Proceedings of the International Conference on High Performance
Computing, Networking, Storage and Analysis, SC13๎กฒ ๎€ƒ๎€ญ๎‚ก๎กฒ
๎‚๎ˆถ๎ˆ๎‡ซ๎Šฝ๎†ก๎Š‹๎กซ ๎ƒก๎‡ซ๎†ก๎‰ญ๎‰ญ๎†ก ๎…”๎ˆฅ๎†Ž ๎ƒก๎‡ซ๎†ก๎‰ญ๎‰ญ๎†ก๎ขš๎ฑ๎†ก๎ˆฅ๎‰ญ๎‡ซ ๎ƒพ๎ˆถ๎Š ๎‰ญ๎ˆฅ๎‡ซ๎†ก๎‰ญ ๎ขŽ๎ ๎Ÿฟ๎ €๎ …๎ข๎กฒ ๎ขจ๎€ฅ๎ˆ๎ˆถ๎†ƒ๎ˆ‡ ๎‡ซ๎Š‹๎†ก๎‰ญ๎…”๎Š‹๎‡ซ๎Šฝ๎†ก ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ ๎…”๎ˆฅ๎†Ž
๎‰ญ๎†ก๎†ƒ๎‹Š๎†ƒ๎ˆ๎‡ซ๎ˆฅ๎‡Œ ๎‡‰๎ˆถ๎‰ญ ๎‡ซ๎ˆŸ๎‰ข๎‰ญ๎ˆถ๎Šฝ๎†ก๎†Ž ๎‰ฝ๎†ƒ๎…”๎ˆ๎…”๎…น๎‡ซ๎ˆ๎‡ซ๎Š‹๎‹Š ๎ˆถ๎‡‰ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช Proceedings of the 2016
International Conference for High Performance Computing, Networking, Storage
and Analysis๎กฒ ๎ƒฒ๎€ญ๎ €๎ …๎กฒ ๎บ๎†๎†๎†๎กฒ
๎‚‘๎ˆฅ๎†ก๎‰ข๎ˆ๎†ก๎‹Š๎กซ ๎‚ก๎…”๎Š‹๎Š‹๎‡ ๎†ก๎‹ ๎ขŽ๎ ๎Ÿฟ๎ €๎ ‚๎ข๎กฒ Nested and Hierarchical Solvers in PETSc๎กฒ ๎ผ๎š๎Ž‹๎กช
https://www.caam.rice.edu/~mk51/presentations/SIAMCSE13.pdf๎กฒ
๎ƒจ๎Œผ๎š๎Œผ๎š๎Œผ๎Ž™๎Œฆ๎Œผ๎ฃ ๎˜
๎‚•๎‡ซ๎กซ ๎„ด๎‡ซ๎…”๎ˆถ๎‹Š๎†ก ๎ขŽ๎ ๎Ÿฟ๎Ÿฟ๎ „๎ข๎กฒ ๎ขจ๎€ƒ๎ˆฅ ๎‚ถ๎Šฝ๎†ก๎‰ญ๎Šฝ๎‡ซ๎†ก๎‹ ๎ˆถ๎‡‰ ๎ƒฒ๎Š ๎‰ข๎†ก๎‰ญ๎‚•๎„๎กช ๎€ƒ๎ˆ๎‡Œ๎ˆถ๎‰ญ๎‡ซ๎Š‹๎‡ ๎ˆŸ๎‰ฝ๎กซ ๎บ๎ˆŸ๎‰ข๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กซ ๎…”๎ˆฅ๎†Ž ๎„๎‰ฝ๎†ก๎‰ญ
๎บ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡‰๎…”๎†ƒ๎†ก๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช ACM Transactions on Mathematical Software ๎ ‚๎ €๎กฒ๎ ‚๎กซ ๎‰ข๎‰ข๎กฒ ๎ ‚๎Ÿฟ๎ ๎ข—๎ ‚๎ ๎ „๎กฒ ๎ผ๎š๎Ž‹๎กช
http://crd-legacy.lbl.gov/~xiaoye/SuperLU๎กฒ
๎‚ก๎…”๎ˆฅ๎†Ž๎†ก๎ˆ๎กซ ๎‚๎…”๎ˆฅ ๎ขŽ๎ €๎ ˆ๎ ˆ๎ ‚๎ข๎กฒ ๎ขจ๎€ฅ๎…”๎ˆ๎…”๎ˆฅ๎†ƒ๎‡ซ๎ˆฅ๎‡Œ ๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช Communications in
Numerical Methods in Engineering ๎ ˆ๎กฒ๎ ‚๎กซ ๎‰ข๎‰ข๎กฒ ๎ ๎ ‚๎ ‚๎ข—๎ ๎ ƒ๎ €๎กฒ
๎‚ก๎ˆถ๎Š ๎ˆ๎‡ซ๎ˆฅ๎กซ ๎‚๎ˆถ๎‡ ๎…”๎ˆฅ๎ˆฅ๎กซ ๎ƒก๎‡ซ๎†ก๎‰ญ๎‰ญ๎†ก ๎‚๎ˆถ๎ˆ๎‡ซ๎Šฝ๎†ก๎Š‹๎กซ ๎…”๎ˆฅ๎†Ž ๎‚ถ๎ˆ๎‡ซ๎Šฝ๎‡ซ๎†ก๎‰ญ ๎‚ก๎…”๎‰ญ๎‰ฉ๎Š ๎†ก๎Š‹ ๎ขŽ๎ ๎Ÿฟ๎ €๎ ˆ๎ข๎กฒ ๎ขจ๎€ƒ๎Š ๎‡Œ๎ˆŸ๎†ก๎ˆฅ๎Š‹๎†ก๎†Ž ๎‚•๎…”๎‡Œ๎‰ญ๎…”๎ˆฅ๎‡Œ๎‡ซ๎…”๎ˆฅ
๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ ๎‡‰๎ˆถ๎‰ญ ๎ˆ๎…”๎‰ญ๎‡Œ๎†ก๎ขš๎‰ฝ๎†ƒ๎…”๎ˆ๎†ก ๎‡ ๎‹Š๎†Ž๎‰ญ๎ˆถ๎†Ž๎‹Š๎ˆฅ๎…”๎ˆŸ๎‡ซ๎†ƒ ๎‰ฝ๎Š‹๎…”๎…น๎‡ซ๎ˆ๎‡ซ๎Š‹๎‹Š ๎…”๎ˆฅ๎…”๎ˆ๎‹Š๎‰ฝ๎‡ซ๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช Computer
Methods in Applied Mechanics and Engineering ๎ ‚๎ „๎ €๎กซ ๎‰ข๎‰ข๎กฒ ๎ †๎ €๎ ‡๎ข—๎ †๎ ƒ๎ ‚๎กฒ ๎ผ๎š๎Ž‹๎กช
https://github.com/prj-/moulin2019al๎กฒ
๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹–๎กซ ๎ฑ๎†ก๎‰ญ๎ˆŸ๎…”๎ˆฅ๎ˆฅ ๎ขŽ๎ €๎ ‡๎ †๎Ÿฟ๎ข๎กฒ ๎ขจ๎„”๎…น๎†ก๎‰ญ ๎†ก๎‡ซ๎ˆฅ๎†ก๎ˆฅ ๎ฅ๎‰ญ๎†ก๎ˆฅ๎‹–๎Šง๎…น๎†ก๎‰ญ๎‡Œ๎…”๎ˆฅ๎‡Œ ๎†Ž๎Š ๎‰ญ๎†ƒ๎‡  ๎…”๎ˆ๎Š‹๎†ก๎‰ญ๎ˆฅ๎‡ซ๎†ก๎‰ญ๎†ก๎ˆฅ๎†Ž๎†ก๎‰ฝ
๎„ช๎†ก๎‰ญ๎‡‰๎…”๎‡ ๎‰ญ๎†ก๎ˆฅ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช Vierteljahrsschrift der Naturforschenden Gesellschaft in Zรผrich ๎ €๎ „๎กซ
๎‰ข๎‰ข๎กฒ ๎ ๎ †๎ ๎ข—๎ ๎ ‡๎ …๎กฒ
๎ƒฒ๎‡ซ๎กซ ๎ฑ๎…”๎ˆฅ๎‡Œ ๎ขŽ๎ ๎Ÿฟ๎ €๎ ‚๎ข๎กฒ TetGen: A Quality Tetrahedral Mesh Generator and 3D Delaunay
Triangulator๎กฒ ๎ƒพ๎†ก๎†ƒ๎‡ ๎กฒ ๎‰ญ๎†ก๎‰ข๎กฒ ๎ €๎ ‚๎กฒ ๎ผ๎š๎Ž‹๎กช
http://wias-berlin.de/software/tetgen๎กฒ
๎ƒจ๎Œผ๎š๎Œผ๎š๎Œผ๎Ž™๎Œฆ๎Œผ๎ฃ ๎˜๎ฐ
๎ƒฒ๎‰ข๎‡ซ๎ˆ๎ˆ๎…”๎ˆฅ๎†ก๎กซ ๎‚ฅ๎‡ซ๎†ƒ๎ˆถ๎ˆ๎†ก๎กซ ๎„ช๎‡ซ๎†ƒ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎Š‹๎…” ๎€ท๎ˆถ๎ˆ๎†ก๎…”๎ˆฅ๎กซ ๎ƒก๎…”๎Š‹๎‰ญ๎‡ซ๎†ƒ๎†ก ๎ฑ๎…”๎Š ๎‰ญ๎†ก๎Š‹๎กซ ๎ข๎‰ญ๎†ข๎†Ž๎†ข๎‰ญ๎‡ซ๎†ƒ ๎‚ฅ๎…”๎Š‹๎…”๎‡‰๎กซ
๎€ญ๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎‰ฝ ๎ƒก๎†ก๎†ƒ๎‡ ๎‰ฝ๎Š‹๎†ก๎‡ซ๎ˆฅ๎กซ ๎…”๎ˆฅ๎†Ž ๎ƒจ๎ˆถ๎…น๎†ก๎‰ญ๎Š‹ ๎ƒฒ๎†ƒ๎‡ ๎†ก๎‡ซ๎†ƒ๎‡ ๎ˆ ๎ขŽ๎ ๎Ÿฟ๎ €๎ ‚๎ข๎กฒ ๎ขจ๎€ƒ๎…น๎‰ฝ๎Š‹๎‰ญ๎…”๎†ƒ๎Š‹ ๎‰ญ๎ˆถ๎…น๎Š ๎‰ฝ๎Š‹ ๎†ƒ๎ˆถ๎…”๎‰ญ๎‰ฝ๎†ก ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก๎‰ฝ
๎‡‰๎ˆถ๎‰ญ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ๎‰ฝ ๎ˆถ๎‡‰ ๎ƒก๎€ท๎†๎‰ฝ ๎Šฝ๎‡ซ๎…” ๎‡Œ๎†ก๎ˆฅ๎†ก๎‰ญ๎…”๎ˆ๎‡ซ๎‹–๎†ก๎†Ž ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ๎‰ฝ ๎‡ซ๎ˆฅ ๎Š‹๎‡ ๎†ก ๎ˆถ๎Šฝ๎†ก๎‰ญ๎ˆ๎…”๎‰ข๎‰ฝ๎ขฉ๎กฒ ๎บ๎ˆฅ๎กช
Numerische Mathematik ๎ €๎ ๎ …๎กฒ๎ ƒ๎กซ ๎‰ข๎‰ข๎กฒ ๎ †๎ ƒ๎ €๎ข—๎ †๎ †๎Ÿฟ๎กฒ
๎ƒฒ๎Š ๎‹–๎Š ๎ˆ‡๎‡ซ๎กซ ๎€ƒ๎Š‹๎‰ฝ๎Š ๎‰ฝ๎‡ ๎‡ซ ๎…”๎ˆฅ๎†Ž ๎ข๎‰ญ๎…”๎ˆฅ๎†‡๎ˆถ๎‡ซ๎‰ฝ๎ขš๎„ด๎…”๎Šฝ๎‡ซ๎†ก๎‰ญ ๎ƒจ๎ˆถ๎Š ๎‹‰ ๎ขŽ๎ ๎Ÿฟ๎ €๎ ƒ๎ข๎กฒ ๎ขจ๎€ƒ ๎†Ž๎‡ซ๎‰ฝ๎‰ฝ๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ ๎‹๎‡ซ๎Š‹๎‡  ๎ˆ‡๎†ก๎‰ญ๎ˆฅ๎†ก๎ˆ
๎†Ž๎†ก๎Š‹๎†ก๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‡‰๎ˆถ๎‰ญ ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ ๎‹ช๎ˆฅ๎‡ซ๎Š‹๎†ก ๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎†ƒ๎†ก๎‰ฝ ๎ˆถ๎ˆฅ ๎‰ฝ๎‡ ๎…”๎‰ญ๎†ก๎†Ž ๎ˆŸ๎†ก๎ˆŸ๎ˆถ๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎Š ๎Š‹๎†ก๎‰ญ๎‰ฝ๎ขฉ๎กฒ
๎บ๎ˆฅ๎กช International Journal for Numerical Methods in Engineering ๎ €๎Ÿฟ๎Ÿฟ๎กฒ๎ ๎กซ ๎‰ข๎‰ข๎กฒ ๎ €๎ ‚๎ …๎ข—๎ €๎ …๎ ƒ๎กฒ
๎บ๎Ž™๎Œฏ๎Œผ๎ข ๎ฐ
๎ขš๎€ท ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎Š‹๎ˆถ ๎‰ข๎‰ญ๎†ก๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก ๎…” macro๎กฒ ๎ ‚๎ †
๎ขš๎‡ ๎†ก๎ˆ๎‰ข ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎‡ ๎†ก๎ˆ๎‰ข๎กฒ ๎ €๎ ƒ๎ ƒ๎กซ ๎ €๎ ƒ๎ „
๎ขš๎ˆ‡๎‰ฝ๎‰ข๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ KSP ๎Š‹๎‹Š๎‰ข๎†ก๎กฒ ๎ €๎ ƒ๎ ƒ๎กซ ๎ €๎ ‡๎ ‡๎ข—๎ €๎ ˆ๎Ÿฟ
๎ขš๎ˆฅ๎‰ฝ ๎ˆฅ๎ˆถ ๎‰ฝ๎†ƒ๎‰ญ๎‡ซ๎‰ข๎Š‹๎กฒ ๎ ๎ …
๎ขš๎ˆฅ๎‹ ๎ˆฅ๎ˆถ ๎‹๎‡ซ๎ˆฅ๎†Ž๎ˆถ๎‹๎กฒ ๎ ๎ …
๎ขš๎‰ข๎†ƒ๎กผ๎Š‹๎‹Š๎‰ข๎†ก ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ PC ๎Š‹๎‹Š๎‰ข๎†ก๎กฒ ๎ €๎ ƒ๎ ƒ๎ข—๎ €๎ ƒ๎ …
๎ขš๎Šฝ ๎Šฝ๎†ก๎‰ญ๎…น๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‹Š๎กฒ ๎ ๎ …
๎ขš๎‹๎‡Œ ๎‹๎‡ซ๎Š‹๎‡  ๎‡Œ๎‰ญ๎…”๎‰ข๎‡ ๎‡ซ๎†ƒ๎‰ฝ๎กฒ ๎ ๎ …
๎กฒ๎ˆŸ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ ๎ˆถ๎‡‰ ๎†ƒ๎ˆถ๎ˆ๎Š ๎ˆŸ๎ˆฅ๎‰ฝ ๎ˆถ๎‡‰ ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎‰ฝ๎กฒ ๎ ‚๎ ƒ
๎กฒ๎ˆฅ ๎ˆ๎†ก๎…”๎†Ž๎‡ซ๎ˆฅ๎‡Œ ๎†Ž๎‡ซ๎ˆŸ๎†ก๎ˆฅ๎‰ฝ๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ˆถ๎‡‰ ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎‰ฝ๎กฒ ๎ ‚๎ ƒ
๎กฒ๎ˆฅ๎†Ž๎ˆถ๎‡‰ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ ๎ˆถ๎‡‰ ๎†Ž๎†ก๎‡Œ๎‰ญ๎†ก๎†ก๎‰ฝ ๎ˆถ๎‡‰ ๎‡‰๎‰ญ๎†ก๎†ก๎†Ž๎ˆถ๎ˆŸ ๎ˆถ๎‡‰ ๎…” fespace๎กฒ ๎ ๎ ‡๎กซ ๎ ๎ ˆ
๎กฒ๎‰ญ๎†ก๎‰ฝ๎‡ซ๎‹–๎†ก ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž ๎Š‹๎ˆถ ๎‰ญ๎†ก๎‰ฝ๎‡ซ๎‹–๎†ก ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎‰ฝ๎กฒ ๎ ‚๎ ƒ
๎€ƒ๎Š‹๎Š‹๎…”๎†ƒ๎‡ ๎€ญ๎ˆถ๎…”๎‰ญ๎‰ฝ๎†ก๎‚ถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ ๎‰ฝ๎†ก๎Š‹๎Š ๎‰ข ๎…” ๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎ˆ๎†ก๎Šฝ๎†ก๎ˆ ๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎กฒ ๎ ˆ๎ ๎ข—๎ €๎Ÿฟ๎ †
๎…น๎Š ๎‡ซ๎ˆ๎†Ž๎‚ก๎…”๎Š‹๎†๎†Ž๎‡Œ๎†ก๎ƒจ๎†ก๎†ƒ๎Š ๎‰ญ๎‰ฝ๎‡ซ๎Šฝ๎†ก ๎†ƒ๎‰ญ๎†ก๎…”๎Š‹๎†ก ๎…” ๎‡ ๎‡ซ๎†ก๎‰ญ๎…”๎‰ญ๎†ƒ๎‡ ๎‹Š ๎ˆถ๎‡‰ Mat๎‰ฝ ๎‹๎‡ซ๎Š‹๎‡  ๎‰ญ๎†ก๎‰ฝ๎‰ข๎†ก๎†ƒ๎Š‹๎‡ซ๎Šฝ๎†ก ๎‰ข๎‰ญ๎ˆถ๎ˆ๎ˆถ๎ˆฅ๎‡Œ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎ˆถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ๎‰ฝ๎กฒ
๎ €๎ …๎ €๎กซ ๎ €๎ …๎ 
๎…น๎Š ๎‡ซ๎ˆ๎†Ž๎ˆŸ๎†ก๎‰ฝ๎‡  ๎Š ๎ˆฅ๎‰ฝ๎Š‹๎‰ญ๎Š ๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎†Ž ๎ ๎€ท ๎ˆŸ๎†ก๎‰ฝ๎‡ ๎กฒ ๎ ๎ †
๎บ๎Ž™๎Œฏ๎Œผ๎ข ๎ฐ๎ฐ
๎€ญ๎‡ ๎…”๎ˆฅ๎‡Œ๎†ก๎‚ฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎‰ฝ๎‹๎‡ซ๎Š‹๎†ƒ๎‡  ๎…น๎†ก๎Š‹๎‹๎†ก๎†ก๎ˆฅ ๎ข๎‰ญ๎†ก๎†ก๎ข๎†๎‚ก ๎…”๎ˆฅ๎†Ž ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ๎กฒ ๎ €๎ ƒ๎Ÿฟ๎กซ ๎ €๎ ƒ๎ €๎กซ ๎ €๎ †๎ ƒ๎กซ ๎ €๎ †๎ „
๎€ญ๎‡ ๎…”๎ˆฅ๎‡Œ๎†ก๎‚ถ๎‰ข๎†ก๎‰ญ๎…”๎Š‹๎ˆถ๎‰ญ ๎†ƒ๎‡ ๎…”๎ˆฅ๎‡Œ๎†ก ๎Š‹๎‡ ๎†ก ๎ˆฅ๎Š ๎ˆŸ๎†ก๎‰ญ๎‡ซ๎†ƒ๎…”๎ˆ ๎Šฝ๎…”๎ˆ๎Š ๎†ก๎‰ฝ ๎ˆถ๎‡‰ ๎…” Mat๎กฒ ๎ ˆ๎ €
๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ข‚๎‡ซ๎ˆฅ๎Š‹๎กซ๎‡ซ๎ˆฅ๎Š‹๎ขƒ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎ ๎€ท ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎กฒ ๎ €๎ ‡๎Ÿฟ๎กซ ๎ €๎ ‡๎ €
๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ข‚๎‡ซ๎ˆฅ๎Š‹๎ขƒ ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎ €๎€ท ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎กฒ ๎ ‚๎ ƒ
๎†ƒ๎Š ๎…น๎†ก ๎‰ฝ๎Š‹๎‰ญ๎Š ๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎†Ž ๎ ‚๎€ท ๎†ƒ๎Š ๎…น๎†ก๎กฒ ๎ ๎ †
๎€ท๎ˆŸ๎†ก๎‰ฝ๎‡ ๎€ญ๎‰ญ๎†ก๎…”๎Š‹๎†ก ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก ๎…” ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎ˆŸ๎†ก๎‰ฝ๎‡ ๎กฒ ๎ €๎ „๎ ƒ๎กซ ๎ €๎ „๎ „๎กซ ๎ €๎ †๎ ‡๎กซ ๎ €๎ †๎ ˆ
๎†๎‡ซ๎‡Œ๎†ก๎ˆฅ๎„ช๎…”๎ˆ๎Š ๎†ก ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎…”๎ˆ๎Š ๎†ก ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎…” ๎‰ญ๎†ก๎Šฝ๎†ก๎‰ญ๎‰ฝ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎Š ๎ˆฅ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎ขš
๎‡‰๎…”๎†ƒ๎†ก๎กฒ ๎ ‚๎ …
๎†๎ƒก๎ƒฒ ๎ƒฒ๎‚•๎†๎ƒก๎†ƒ ๎†ก๎‡ซ๎‡Œ๎†ก๎ˆฅ๎Šฝ๎…”๎ˆ๎Š ๎†ก ๎‰ข๎‰ญ๎ˆถ๎…น๎ˆ๎†ก๎ˆŸ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ๎กฒ ๎ €๎ ‡๎ ƒ
๎†ก๎‹‰๎†ƒ๎‡ ๎…”๎ˆฅ๎‡Œ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎Š ๎ˆฅ๎‡ซ๎†ƒ๎…”๎Š‹๎†ก ๎Šฝ๎…”๎ˆ๎Š ๎†ก๎‰ฝ ๎…”๎‰ฝ๎‰ฝ๎ˆถ๎†ƒ๎‡ซ๎…”๎Š‹๎†ก๎†Ž ๎Š‹๎ˆถ ๎†Ž๎Š ๎‰ข๎ˆ๎‡ซ๎†ƒ๎…”๎Š‹๎†ก๎†Ž ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ๎กฒ ๎ ˆ๎ €๎กซ ๎ €๎ ‚๎ ƒ
๎‡‰๎†ก๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก ๎‹ช๎ˆฅ๎‡ซ๎Š‹๎†ก ๎†ก๎ˆ๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก๎กฒ ๎ ๎ ‡๎กซ ๎ ๎ ˆ๎กซ ๎ ‚๎ ๎กซ ๎ ‚๎ ‚๎กซ ๎ ‚๎ ‡๎กซ ๎ €๎ ƒ๎ ˆ๎ข—๎ €๎ „๎ €๎กซ ๎ €๎ „๎ ƒ๎ข—๎ €๎ „๎ †๎กซ ๎ €๎ „๎ ˆ๎กซ ๎ €๎ …๎Ÿฟ๎กซ
๎ €๎ …๎ ‚๎ข—๎ €๎ …๎ ˆ๎กซ ๎ ๎Ÿฟ๎ ˆ๎กซ ๎ ๎ €๎ ‚
๎‡‰๎Š ๎ˆฅ๎†ƒ ๎‡‰๎Š ๎ˆฅ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กฒ ๎ ‚๎ …๎กซ ๎ €๎ †๎Ÿฟ๎กซ ๎ €๎ †๎ €๎กซ ๎ €๎ †๎ ƒ๎กซ ๎ €๎ †๎ „
๎บ๎Ž™๎Œฏ๎Œผ๎ข ๎ฐ๎ฐ๎ฐ
๎ฅ๎ˆ๎ˆถ๎…น๎…”๎ˆ๎‚ฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎‡Œ๎†ก๎ˆฅ๎†ก๎‰ญ๎…”๎Š‹๎†ก ๎…” ๎‡Œ๎ˆ๎ˆถ๎…น๎…”๎ˆ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎ˆถ๎‡‰ ๎Š‹๎‡ ๎†ก ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ๎กฒ ๎ €๎ ‚๎ ‡๎กซ ๎ €๎ ‚๎ ˆ
๎บ๎ข๎‚ก๎€ƒ๎€ญ๎ƒจ๎‚ถ ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ฝ๎Š‹๎…”๎Š‹๎†ก๎ˆŸ๎†ก๎ˆฅ๎Š‹ ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎…” ๎ˆŸ๎…”๎†ƒ๎‰ญ๎ˆถ๎กฒ ๎ ‚๎ †
๎บ๎ƒฒ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‹‰ ๎‰ฝ๎†ก๎Š‹๎กฒ ๎ €๎ …๎ ‚๎ข—๎ €๎ …๎ ˆ
๎‚‘๎ƒฒ๎ƒก ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎‚‘๎‰ญ๎‹Š๎ˆ๎ˆถ๎Šฝ ๎‰ฝ๎Š ๎…น๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎กฒ ๎ €๎ ‚๎ …๎กซ ๎ €๎ ƒ๎ ƒ๎กซ ๎ €๎ †๎Ÿฟ๎กซ ๎ €๎ †๎ €๎กซ ๎ €๎ ‡๎ „๎ข—๎ €๎ ‡๎ †๎กซ ๎ ๎Ÿฟ๎ ˆ
๎‚‘๎ƒฒ๎ƒก๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎กฒ ๎ €๎ ƒ๎ ๎กซ ๎ €๎ ƒ๎ ‚๎กซ ๎ €๎ „๎ ๎กซ ๎ €๎ „๎ ‚๎กซ ๎ €๎ ‡๎Ÿฟ๎กซ ๎ €๎ ‡๎ €๎กซ ๎ €๎ ‡๎ ‡๎ข—๎ €๎ ˆ๎Ÿฟ
๎‚•๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ๎€ญ๎ฅ ๎†ƒ๎ˆถ๎ˆฅ๎‡ฟ๎Š ๎‡Œ๎…”๎Š‹๎†ก ๎‡Œ๎‰ญ๎…”๎†Ž๎‡ซ๎†ก๎ˆฅ๎Š‹ ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎…” ๎‰ญ๎†ก๎Šฝ๎†ก๎‰ญ๎‰ฝ๎†ก ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎Š ๎ˆฅ๎‡ซ๎†ƒ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡‰๎…”๎†ƒ๎†ก๎กฒ ๎ ‚๎ …
๎ˆ๎ˆถ๎…”๎†Ž ๎‡ซ๎ˆฅ๎‰ฝ๎Š‹๎‰ญ๎Š ๎†ƒ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎Š‹๎ˆถ ๎ˆ๎ˆถ๎…”๎†Ž ๎…”๎†Ž๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‰ข๎ˆ๎Š ๎‡Œ๎‡ซ๎ˆฅ๎‰ฝ๎กฒ ๎ ‚๎ „
๎ˆŸ๎…”๎†ƒ๎‰ญ๎ˆถ ๎†Ž๎†ก๎‹ช๎ˆฅ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎ˆถ๎‡‰ ๎ˆŸ๎…”๎†ƒ๎‰ญ๎ˆถ๎กฒ ๎ ‚๎ †๎กซ ๎ ๎Ÿฟ๎ ˆ
๎‚ก๎…”๎Š‹ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎กฒ ๎ €๎ ‚๎ …๎กซ ๎ €๎ ƒ๎ ๎กซ ๎ €๎ ƒ๎ ‚๎กซ ๎ €๎ †๎Ÿฟ๎กซ ๎ €๎ †๎ €๎กซ ๎ ๎Ÿฟ๎ ˆ๎กซ ๎ ๎ €๎ 
๎‚ก๎…”๎Š‹ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž ๎‰ญ๎†ก๎…”๎ˆ๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎‹๎‰ญ๎…”๎‰ข๎‰ข๎‡ซ๎ˆฅ๎‡Œ ๎…” ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎กฒ ๎ €๎ ƒ๎Ÿฟ๎ข—๎ €๎ ƒ๎ ƒ๎กซ
๎ €๎ „๎ ƒ๎ข—๎ €๎ „๎ †๎กซ ๎ €๎ †๎ ƒ๎กซ ๎ €๎ †๎ „๎กซ ๎ €๎ ‡๎ „๎ข—๎ €๎ ‡๎ †๎กซ ๎ €๎ ˆ๎ €๎กซ ๎ €๎ ˆ๎ ๎กซ ๎ ๎ €๎Ÿฟ๎กซ ๎ ๎ €๎ ƒ
๎‚ก๎…”๎Š‹๎ค’๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ค‘ ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž ๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆ๎†ก๎‹‰๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎‹๎‰ญ๎…”๎‰ข๎‰ข๎‡ซ๎ˆฅ๎‡Œ ๎…” ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎กฒ ๎ €๎ ƒ๎Ÿฟ๎กซ
๎ €๎ ƒ๎ €
๎‚ก๎…”๎Š‹๎€ญ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎Š‹ ๎†ƒ๎ˆถ๎ˆฅ๎Šฝ๎†ก๎‰ญ๎Š‹ ๎…” MatNest ๎‡ซ๎ˆฅ๎Š‹๎ˆถ ๎…” ๎ˆŸ๎ˆถ๎‰ญ๎†ก ๎‰ฝ๎Š‹๎…”๎ˆฅ๎†Ž๎…”๎‰ญ๎†Ž ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎Š‹๎กฒ ๎ €๎ ƒ๎ ˆ๎ข—๎ €๎ „๎ ‚
๎บ๎Ž™๎Œฏ๎Œผ๎ข ๎ฐ๎˜
๎‚ก๎…”๎Š‹๎€ญ๎‰ญ๎†ก๎…”๎Š‹๎†ก ๎†ƒ๎‰ญ๎†ก๎…”๎Š‹๎†ก ๎…” ๎†Ž๎‡ซ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎…น๎Š ๎Š‹๎†ก๎†Ž Mat๎กฒ ๎ €๎ ƒ๎Ÿฟ๎กซ ๎ €๎ ƒ๎ €๎กซ ๎ €๎ „๎ ƒ๎กซ ๎ €๎ „๎ „
๎‚ก๎…”๎Š‹๎‚ก๎…”๎Š‹๎‚ก๎Š ๎ˆ๎Š‹ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ข—๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹๎กฒ ๎ €๎ ƒ๎ ๎กซ ๎ €๎ ƒ๎ ‚
๎‚ก๎…”๎Š‹๎‚ก๎Š ๎ˆ๎Š‹ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹๎กฒ ๎ €๎ †๎Ÿฟ๎กซ ๎ €๎ †๎ €
๎‚ก๎…”๎Š‹๎‚ก๎Š ๎ˆ๎Š‹๎ƒพ๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‰ข๎ˆถ๎‰ฝ๎†ก ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‰ข๎ˆถ๎‰ฝ๎†ก๎ข—๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ ๎‰ข๎‰ญ๎ˆถ๎†Ž๎Š ๎†ƒ๎Š‹๎กฒ ๎ €๎ †๎Ÿฟ๎กซ ๎ €๎ †๎ €
๎‚ก๎…”๎Š‹๎‚ฅ๎†ก๎‰ฝ๎Š‹ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎Š‹ ๎‡‰๎ˆถ๎‰ญ ๎‰ฝ๎Š‹๎ˆถ๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎ˆฅ๎†ก๎‰ฝ๎Š‹๎†ก๎†Ž ๎‰ฝ๎Š ๎…น๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎†ƒ๎†ก๎‰ฝ ๎‰ฝ๎Š‹๎ˆถ๎‰ญ๎†ก๎†Ž ๎‡ซ๎ˆฅ๎†Ž๎†ก๎‰ข๎†ก๎ˆฅ๎ขš
๎†Ž๎†ก๎ˆฅ๎Š‹๎ˆ๎‹Š๎กฒ ๎ €๎ ƒ๎ ˆ๎ข—๎ €๎ „๎ €๎กซ ๎ €๎ ‡๎ „๎ข—๎ €๎ ‡๎ †๎กซ ๎ ๎ €๎ €
๎‚ก๎…”๎Š‹๎‚ฅ๎Š ๎ˆ๎ˆ๎ƒฒ๎‰ข๎…”๎†ƒ๎†ก Mat ๎Š‹๎‡ ๎…”๎Š‹ ๎‰ญ๎†ก๎ˆŸ๎ˆถ๎Šฝ๎†ก๎‰ฝ ๎…” ๎ˆฅ๎Š ๎ˆ๎ˆ ๎‰ฝ๎‰ข๎…”๎†ƒ๎†ก ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎…” ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎กฒ ๎ €๎ „๎ ‡๎ข—๎ €๎ …๎Ÿฟ
๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎‰ญ๎†ก๎…”๎ˆ๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎‰ฝ๎‰ข๎…”๎‰ญ๎‰ฝ๎†ก ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎กฒ ๎ ‚๎ ๎กซ ๎ ‚๎ ‚๎กซ ๎ …๎ „๎กซ ๎ €๎ ƒ๎ ๎ข—๎ €๎ ƒ๎ ƒ๎กซ ๎ €๎ „๎ ƒ๎กซ ๎ €๎ „๎ „๎กซ ๎ ๎ €๎ ƒ
๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰๎ข‚๎‡ซ๎ˆฅ๎Š‹๎ขƒ ๎…”๎‰ญ๎‰ญ๎…”๎‹Š ๎ˆถ๎‡‰ ๎‰ฝ๎‰ข๎…”๎‰ญ๎‰ฝ๎†ก ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎†ƒ๎†ก๎‰ฝ๎กฒ ๎ ‚๎ ƒ
๎‚ก๎…”๎Š‹๎ƒฒ๎‡ ๎†ก๎ˆ๎ˆ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎Š‹ ๎‡‰๎ˆถ๎‰ญ ๎Š ๎‰ฝ๎†ก๎‰ญ๎ขš๎†Ž๎†ก๎‹ช๎ˆฅ๎†ก๎†Ž ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎Š‹๎‹Š๎‰ข๎†ก๎‰ฝ๎กฒ ๎ €๎ †๎Ÿฟ๎กซ ๎ €๎ †๎ €๎กซ ๎ €๎ ‡๎ ‡๎ข—๎ €๎ ˆ๎Ÿฟ
๎ˆŸ๎†ก๎†Ž๎‡ซ๎Š‹ ๎ ‚๎€ท ๎Šฝ๎‡ซ๎‰ฝ๎Š ๎…”๎ˆ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กฒ ๎ ๎ †
๎ˆŸ๎†ก๎‰ฝ๎‡  ๎ ๎€ท ๎ˆŸ๎†ก๎‰ฝ๎‡ ๎กฒ ๎ ๎ †๎ข—๎ ๎ ˆ๎กซ ๎ …๎ „
๎ˆŸ๎†ก๎‰ฝ๎‡ ๎ ‚ ๎ ‚๎€ท ๎ˆŸ๎†ก๎‰ฝ๎‡ ๎กฒ ๎ ๎ †
๎ˆŸ๎†ก๎‰ฝ๎‡ ๎‚• ๎ €๎€ท ๎ˆŸ๎†ก๎‰ฝ๎‡ ๎กฒ ๎ €๎ †๎ ๎กซ ๎ €๎ †๎ ‚
๎ˆŸ๎‰ข๎‡ซ๎€ญ๎ˆถ๎ˆŸ๎ˆŸ ๎‚ก๎ƒก๎บ ๎†ƒ๎ˆถ๎ˆŸ๎ˆŸ๎Š ๎ˆฅ๎‡ซ๎†ƒ๎…”๎Š‹๎ˆถ๎‰ญ๎กฒ ๎ …๎ „
๎ˆŸ๎‰ข๎‡ซ๎ƒจ๎†ก๎‰ฉ๎Š ๎†ก๎‰ฝ๎Š‹ ๎‚ก๎ƒก๎บ ๎‰ญ๎†ก๎‰ฉ๎Š ๎†ก๎‰ฝ๎Š‹๎กฒ ๎ …๎ „
๎ˆฅ๎†ก๎‹๎ ๎ˆถ๎ˆ๎†Ž ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎Š‹๎ˆถ ๎‡Œ๎ˆถ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎…” ๎ˆฅ๎†ก๎‹ ๎Š‹๎ˆถ ๎…”๎ˆฅ ๎ˆถ๎ˆ๎†Ž ๎ˆŸ๎†ก๎‰ฝ๎‰ฝ ๎‹๎‡ ๎†ก๎ˆฅ ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ trunc๎กฒ
๎ ‚๎ ‡๎กซ ๎ ๎ €๎ ‚
๎บ๎Ž™๎Œฏ๎Œผ๎ข ๎˜
๎‚ถ๎‚ก๎ƒก๎กผ๎‚ฅ๎„๎‚ก๎กผ๎ƒพ๎ฑ๎ƒจ๎†๎€ƒ๎€ท๎ƒฒ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ ๎ˆถ๎‡‰ ๎‚ถ๎‰ข๎†ก๎ˆฅ๎‚ก๎ƒก ๎Š‹๎‡ ๎‰ญ๎†ก๎…”๎†Ž๎‰ฝ๎กฒ ๎ „๎ …
๎ˆถ๎ˆฅ ๎‡ซ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎†ก ๎€ท๎‡ซ๎‰ญ๎‡ซ๎†ƒ๎‡ ๎ˆ๎†ก๎Š‹ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎ˆถ๎ˆฅ ๎‡Œ๎‡ซ๎Šฝ๎†ก๎ˆฅ ๎ˆ๎…”๎…น๎†ก๎ˆ๎‰ฝ๎กฒ ๎ ‚๎Ÿฟ๎กซ ๎ ‚๎ €
๎ƒก๎€ญ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎กฒ ๎ €๎ ‚๎ …๎กซ ๎ ๎Ÿฟ๎ ˆ
๎ƒก๎€ญ๎€ƒ๎ƒฒ๎‚ก ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ƒฒ๎†ƒ๎‡ ๎‹๎…”๎‰ญ๎‹– ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ๎กฒ ๎ €๎ ƒ๎ …
๎ƒก๎€ญ๎‚ก๎ฅ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆŸ๎Š ๎ˆ๎Š‹๎‡ซ๎‡Œ๎‰ญ๎‡ซ๎†Ž ๎ˆŸ๎†ก๎Š‹๎‡ ๎ˆถ๎†Ž๎‰ฝ๎กฒ ๎ €๎ …๎ €๎กซ ๎ €๎ …๎ 
๎ƒก๎€ญ๎ƒฒ๎ฑ๎†๎‚•๎‚• ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎…”๎…น๎‰ฝ๎Š‹๎‰ญ๎…”๎†ƒ๎Š‹ ๎‰ข๎‰ญ๎†ก๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎†ก๎‰ญ๎กฒ ๎ €๎ †๎Ÿฟ๎ข—๎ €๎ †๎ ‚
๎‰ข๎ˆ๎ˆถ๎Š‹ ๎…น๎…”๎‰ฝ๎‡ซ๎†ƒ ๎Šฝ๎‡ซ๎‰ฝ๎Š ๎…”๎ˆ๎‡ซ๎‹–๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กฒ ๎ ๎ †
๎‰ฉ๎‡‰๎ˆถ๎‰ญ๎†Ž๎†ก๎‰ญ ๎ˆถ๎‰ญ๎†Ž๎†ก๎‰ญ ๎ˆถ๎‡‰ ๎ˆฅ๎Š ๎ˆŸ๎†ก๎‰ญ๎‡ซ๎†ƒ๎…”๎ˆ ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‡Œ๎‰ญ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กฒ ๎ ‚๎Ÿฟ๎กซ ๎ ‚๎ €
๎‰ญ๎†ก๎…”๎ˆ๎ข‚๎‡ซ๎ˆฅ๎Š‹๎กซ๎‡ซ๎ˆฅ๎Š‹๎ขƒ ๎‰ญ๎†ก๎…”๎ˆ๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎ ๎€ท ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎กฒ ๎ ‚๎ ƒ
๎‰ญ๎†ก๎…”๎ˆ๎ข‚๎‡ซ๎ˆฅ๎Š‹๎ขƒ ๎‰ญ๎†ก๎…”๎ˆ๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎ €๎€ท ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎กฒ ๎ ‚๎ ƒ
๎‰ญ๎†ก๎…”๎ˆ๎ข‚๎‡ซ๎ˆฅ๎Š‹๎ขƒ๎ข‚๎‡ซ๎ˆฅ๎Š‹๎ขƒ ๎‰ญ๎†ก๎…”๎ˆ๎ขš๎Šฝ๎…”๎ˆ๎Š ๎†ก๎†Ž ๎…”๎‰ญ๎‰ญ๎…”๎‹Š ๎ˆถ๎‡‰ ๎…”๎‰ญ๎‰ญ๎…”๎‹Š๎‰ฝ๎กฒ ๎ ‚๎ ƒ
๎‰ญ๎†ก๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ ๎‰ข๎†ก๎‰ญ๎ˆŸ๎Š ๎Š‹๎†ก ๎…” ๎ˆŸ๎…”๎Š‹๎‰ญ๎‡ซ๎‹‰ ๎‰ฝ๎ˆถ ๎Š‹๎‡ ๎…”๎Š‹ ๎‡ซ๎Š‹๎‰ฝ ๎‡ซ๎ˆฅ๎Š‹๎†ก๎‰ญ๎‡ซ๎ˆถ๎‰ญ ๎Š ๎ˆฅ๎ˆ‡๎ˆฅ๎ˆถ๎‹๎ˆฅ๎‰ฝ ๎…”๎‰ญ๎†ก ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎†ก๎†Ž ๎‹ช๎‰ญ๎‰ฝ๎Š‹๎กฒ
๎ €๎ ‚๎ ƒ
๎‰ญ๎†ก๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎†ƒ๎Š‹ ๎ˆฅ๎Š ๎ˆŸ๎…น๎†ก๎‰ญ๎‡ซ๎ˆฅ๎‡Œ ๎Š‹๎ˆถ ๎‡Œ๎ˆถ ๎‡‰๎‰ญ๎ˆถ๎ˆŸ ๎…” ๎ˆฅ๎†ก๎‹ ๎Š‹๎ˆถ ๎…”๎ˆฅ ๎ˆถ๎ˆ๎†Ž fespace ๎‹๎‡ ๎†ก๎ˆฅ ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ
new2old๎กฒ ๎ ‚๎ ‡๎กซ ๎ €๎ †๎ ‡๎กซ ๎ €๎ †๎ ˆ
๎บ๎Ž™๎Œฏ๎Œผ๎ข ๎˜๎ฐ
๎‰ฝ๎…”๎Šฝ๎†ก๎Šฝ๎Š‹๎ˆ‡ ๎†ก๎‹‰๎‰ข๎ˆถ๎‰ญ๎Š‹ ๎†Ž๎…”๎Š‹๎…” ๎Š‹๎ˆถ ๎ƒก๎…”๎‰ญ๎…”๎„ช๎‡ซ๎†ก๎‹๎กฒ ๎ †๎Ÿฟ
๎‰ฝ๎†ก๎Š‹ ๎‰ฝ๎Š ๎‰ข๎‰ข๎ˆ๎‹Š ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ ๎Š‹๎ˆถ ๎…” matrix ๎ˆถ๎‰ญ Mat๎กฒ ๎ ‚๎ ๎กซ ๎ ‚๎ ‚๎กซ ๎ €๎ ƒ๎ ƒ๎กซ ๎ €๎ †๎ ƒ๎กซ ๎ €๎ †๎ „๎กซ ๎ €๎ ‡๎ „๎ข—๎ €๎ ‡๎ †
๎ƒฒ๎‚ฅ๎†๎ƒฒ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆฅ๎ˆถ๎ˆฅ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก๎‰ญ๎กฒ ๎ €๎ ‚๎ …๎กซ ๎ €๎ †๎ ƒ๎กซ ๎ €๎ †๎ „
๎ƒฒ๎‚ฅ๎†๎ƒฒ๎ƒฒ๎ˆถ๎ˆ๎Šฝ๎†ก ๎‰ฝ๎ˆถ๎ˆ๎Šฝ๎†ก ๎…” ๎ˆฅ๎ˆถ๎ˆฅ๎ˆ๎‡ซ๎ˆฅ๎†ก๎…”๎‰ญ ๎‰ฝ๎‹Š๎‰ฝ๎Š‹๎†ก๎ˆŸ ๎ˆถ๎‡‰ ๎†ก๎‰ฉ๎Š ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ๎กฒ ๎ €๎ †๎ …๎กซ ๎ €๎ †๎ †
๎‰ฝ๎‰ฉ๎Š ๎…”๎‰ญ๎†ก ๎‰ฝ๎Š‹๎‰ญ๎Š ๎†ƒ๎Š‹๎Š ๎‰ญ๎†ก๎†Ž ๎ ๎€ท ๎‰ฝ๎‰ฉ๎Š ๎…”๎‰ญ๎†ก๎กฒ ๎ ๎ †
๎ƒฒ๎ƒพ ๎ƒฒ๎‚•๎†๎ƒก๎†ƒ ๎‰ฝ๎‰ข๎†ก๎†ƒ๎Š‹๎‰ญ๎…”๎ˆ ๎Š‹๎‰ญ๎…”๎ˆฅ๎‰ฝ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กฒ ๎ €๎ ‡๎ ƒ๎กซ ๎ €๎ ‡๎ ‡๎ข—๎ €๎ ˆ๎Ÿฟ
๎‰ฝ๎Š‹๎…”๎Š‹๎‡ซ๎‰ฝ๎Š‹๎‡ซ๎†ƒ๎‰ฝ ๎‰ข๎‰ญ๎‡ซ๎ˆฅ๎Š‹ ๎†Ž๎ˆถ๎ˆŸ๎…”๎‡ซ๎ˆฅ ๎†Ž๎†ก๎†ƒ๎ˆถ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ ๎‰ฝ๎Š‹๎…”๎Š‹๎‡ซ๎‰ฝ๎Š‹๎‡ซ๎†ƒ๎‰ฝ๎กฒ ๎ ˆ๎ €๎กซ ๎ €๎ ‚๎ ƒ
๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎ข‚๎‡ซ๎ˆฅ๎Š‹๎ขƒ ๎…”๎‰ญ๎‰ญ๎…”๎‹Š ๎ˆถ๎‡‰ ๎‰ฝ๎Š‹๎‰ญ๎‡ซ๎ˆฅ๎‡Œ๎‰ฝ๎กฒ ๎ ‚๎ ƒ๎กซ ๎ €๎ …๎ ‚๎ข—๎ €๎ …๎ ˆ
๎‰ฝ๎‹Š๎ˆŸ ๎€ฅ๎ˆถ๎ˆถ๎ˆ๎†ก๎…”๎ˆฅ ๎‹ซ๎…”๎‡Œ ๎‡‰๎ˆถ๎‰ญ ๎…”๎‰ฝ๎‰ฝ๎†ก๎ˆŸ๎…น๎ˆ๎‡ซ๎ˆฅ๎‡Œ ๎…” ๎‰ฝ๎‹Š๎ˆŸ๎ˆŸ๎†ก๎Š‹๎‰ญ๎‡ซ๎†ƒ matrix๎กฒ ๎ ‚๎ ๎กซ ๎ ‚๎ ‚
๎ƒพ๎…”๎ˆถ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎ˆถ๎‰ข๎Š‹๎‡ซ๎ˆŸ๎‡ซ๎‹–๎†ก๎‰ญ๎กฒ ๎ €๎ ‚๎ …๎กซ ๎ €๎ ‡๎ 
๎Š‹๎‡Œ๎Šฝ ๎Šฝ๎…”๎ˆ๎Š ๎†ก ๎‡‰๎ˆถ๎‰ญ ๎‡ซ๎ˆŸ๎‰ข๎ˆถ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎†ก๎‰ฝ๎‰ฝ๎†ก๎ˆฅ๎Š‹๎‡ซ๎…”๎ˆ ๎…น๎ˆถ๎Š ๎ˆฅ๎†Ž๎…”๎‰ญ๎‹Š ๎†ƒ๎ˆถ๎ˆฅ๎†Ž๎‡ซ๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎‰ฝ๎กฒ ๎ ‚๎ ๎กซ ๎ ‚๎ ‚
๎Š‹๎‰ญ๎Š ๎ˆฅ๎†ƒ ๎‹ช๎ˆ๎Š‹๎†ก๎‰ญ ๎…” ๎ˆŸ๎†ก๎‰ฝ๎‡  ๎Š ๎‰ฝ๎‡ซ๎ˆฅ๎‡Œ ๎…” ๎€ฅ๎ˆถ๎ˆถ๎ˆ๎†ก๎…”๎ˆฅ ๎†ก๎‹‰๎‰ข๎‰ญ๎†ก๎‰ฝ๎‰ฝ๎‡ซ๎ˆถ๎ˆฅ๎กฒ ๎ ‚๎ ‡๎กซ ๎ ๎ €๎ 
๎ƒพ๎ƒฒ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎Š‹๎‡ซ๎ˆŸ๎†ก๎‰ฝ๎Š‹๎†ก๎‰ข๎‰ข๎†ก๎‰ญ๎กฒ ๎ €๎ ‚๎ …๎กซ ๎ €๎ ‡๎ 
๎Šฝ๎…”๎‰ญ๎‡‰ ๎Šฝ๎…”๎‰ญ๎‡ซ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎…”๎ˆ ๎‡‰๎ˆถ๎‰ญ๎ˆŸ๎Š ๎ˆ๎…”๎Š‹๎‡ซ๎ˆถ๎ˆฅ๎กฒ ๎ ‚๎Ÿฟ๎ข—๎ ‚๎ ‚๎กซ ๎ €๎ ƒ๎ ˆ๎ข—๎ €๎ „๎ €๎กซ ๎ €๎ „๎ …๎กซ ๎ €๎ „๎ †
๎„ช๎†ก๎†ƒ ๎ƒก๎†๎ƒพ๎ƒฒ๎†ƒ ๎Šฝ๎†ก๎†ƒ๎Š‹๎ˆถ๎‰ญ๎กฒ ๎ €๎ ‚๎