{"id":478790,"date":"2023-08-09T09:38:12","date_gmt":"2023-08-09T09:38:12","guid":{"rendered":""},"modified":"2023-09-05T11:17:35","modified_gmt":"2023-09-05T11:17:35","slug":"round-off-error","status":"publish","type":"wiki","link":"https:\/\/oneproxy.pro\/vn\/wiki\/round-off-error\/","title":{"rendered":"L\u1ed7i l\u00e0m tr\u00f2n"},"content":{"rendered":"<h2>Gi\u1edbi thi\u1ec7u<\/h2>\n<p>Trong l\u0129nh v\u1ef1c t\u00ednh to\u00e1n s\u1ed1 v\u00e0 t\u00ednh to\u00e1n khoa h\u1ecdc, kh\u00e1i ni\u1ec7m sai s\u1ed1 l\u00e0m tr\u00f2n \u0111\u00f3ng vai tr\u00f2 quan tr\u1ecdng trong vi\u1ec7c hi\u1ec3u nh\u1eefng h\u1ea1n ch\u1ebf v\u00e0 th\u00e1ch th\u1ee9c li\u00ean quan \u0111\u1ebfn vi\u1ec7c bi\u1ec3u di\u1ec5n s\u1ed1 th\u1ef1c tr\u00ean h\u1ec7 th\u1ed1ng m\u00e1y t\u00ednh k\u1ef9 thu\u1eadt s\u1ed1. L\u1ed7i l\u00e0m tr\u00f2n xu\u1ea5t hi\u1ec7n do s\u1ef1 kh\u00e1c bi\u1ec7t c\u1ed1 h\u1eefu gi\u1eefa t\u00ednh ch\u1ea5t li\u00ean t\u1ee5c c\u1ee7a s\u1ed1 th\u1ef1c v\u00e0 t\u00ednh ch\u1ea5t r\u1eddi r\u1ea1c c\u1ee7a bi\u1ec3u di\u1ec5n k\u1ef9 thu\u1eadt s\u1ed1. B\u00e0i vi\u1ebft n\u00e0y \u0111i s\u00e2u v\u00e0o l\u1ecbch s\u1eed, s\u1ef1 ph\u1ee9c t\u1ea1p, c\u00e1c lo\u1ea1i v\u00e0 \u00fd ngh\u0129a c\u1ee7a c\u00e1c l\u1ed7i l\u00e0m tr\u00f2n trong t\u00ednh to\u00e1n s\u1ed1.<\/p>\n<h2>Ngu\u1ed3n g\u1ed1c v\u00e0 \u0111\u1ec1 c\u1eadp s\u1edbm<\/h2>\n<p>Kh\u00e1i ni\u1ec7m sai s\u1ed1 l\u00e0m tr\u00f2n b\u1eaft ngu\u1ed3n t\u1eeb bu\u1ed5i b\u00ecnh minh c\u1ee7a m\u00e1y t\u00ednh k\u1ef9 thu\u1eadt s\u1ed1. Ngay t\u1eeb gi\u1eefa th\u1ebf k\u1ef7 20, nh\u1eefng ng\u01b0\u1eddi ti\u00ean phong trong l\u0129nh v\u1ef1c khoa h\u1ecdc m\u00e1y t\u00ednh, nh\u01b0 John W. Mauchly v\u00e0 J. Presper Eckert, \u0111\u00e3 nh\u1eadn ra nh\u1eefng h\u1ea1n ch\u1ebf c\u1ee7a vi\u1ec7c bi\u1ec3u di\u1ec5n s\u1ed1 th\u1ef1c \u1edf \u0111\u1ecbnh d\u1ea1ng nh\u1ecb ph\u00e2n. Vi\u1ec7c nh\u1eadn ra r\u1eb1ng kh\u00f4ng ph\u1ea3i t\u1ea5t c\u1ea3 c\u00e1c s\u1ed1 th\u1ef1c \u0111\u1ec1u c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n ch\u00ednh x\u00e1c d\u01b0\u1edbi d\u1ea1ng nh\u1ecb ph\u00e2n \u0111\u00e3 d\u1eabn \u0111\u1ebfn kh\u00e1i ni\u1ec7m sai s\u1ed1 l\u00e0m tr\u00f2n. S\u1ef1 \u0111\u1ec1 c\u1eadp \u0111\u00e1ng ch\u00fa \u00fd \u0111\u1ea7u ti\u00ean v\u1ec1 thu\u1eadt ng\u1eef n\u00e0y xu\u1ea5t hi\u1ec7n trong c\u00e1c cu\u1ed9c th\u1ea3o lu\u1eadn xung quanh s\u1ef1 ph\u00e1t tri\u1ec3n c\u1ee7a c\u00e1c m\u00e1y t\u00ednh \u0111\u1eddi \u0111\u1ea7u nh\u01b0 ENIAC.<\/p>\n<h2>Hi\u1ec3u l\u1ed7i l\u00e0m tr\u00f2n<\/h2>\n<p>V\u1ec1 c\u1ed1t l\u00f5i, l\u1ed7i l\u00e0m tr\u00f2n xu\u1ea5t ph\u00e1t t\u1eeb \u0111\u1ed9 ch\u00ednh x\u00e1c h\u1eefu h\u1ea1n c\u1ee7a h\u1ec7 th\u1ed1ng k\u1ef9 thu\u1eadt s\u1ed1. M\u00e1y t\u00ednh s\u1eed d\u1ee5ng c\u00e1c bit h\u1eefu h\u1ea1n \u0111\u1ec3 bi\u1ec3u di\u1ec5n s\u1ed1 th\u1ef1c, d\u1eabn \u0111\u1ebfn kh\u00f4ng th\u1ec3 bi\u1ec3u di\u1ec5n ch\u00ednh x\u00e1c m\u1ecdi s\u1ed1 th\u1ef1c. S\u1ef1 kh\u00e1c bi\u1ec7t gi\u1eefa gi\u00e1 tr\u1ecb th\u1ef1c v\u00e0 bi\u1ec3u di\u1ec5n nh\u1ecb ph\u00e2n c\u1ee7a n\u00f3 g\u00e2y ra m\u1ed9t l\u1ed7i nh\u1ecf \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 l\u1ed7i l\u00e0m tr\u00f2n. L\u1ed7i n\u00e0y tr\u1edf n\u00ean nghi\u00eam tr\u1ecdng h\u01a1n khi t\u00ednh to\u00e1n bao g\u1ed3m c\u00e1c ph\u00e9p to\u00e1n nh\u01b0 c\u1ed9ng, tr\u1eeb, nh\u00e2n v\u00e0 chia, truy\u1ec1n v\u00e0 khu\u1ebfch \u0111\u1ea1i s\u1ef1 kh\u00e1c bi\u1ec7t ban \u0111\u1ea7u.<\/p>\n<h2>C\u01a1 ch\u1ebf n\u1ed9i b\u1ed9<\/h2>\n<p>C\u01a1 ch\u1ebf l\u00e0m tr\u00f2n sai s\u1ed1 xoay quanh c\u00e1ch bi\u1ec3u di\u1ec5n s\u1ed1 nh\u1ecb ph\u00e2n v\u00e0 \u0111\u1ed9 ch\u00ednh x\u00e1c h\u1eefu h\u1ea1n c\u1ee7a m\u00e1y t\u00ednh. Khi m\u1ed9t s\u1ed1 th\u1ef1c \u0111\u01b0\u1ee3c chuy\u1ec3n \u0111\u1ed5i th\u00e0nh s\u1ed1 nh\u1ecb ph\u00e2n, ph\u1ea7n ph\u00e2n s\u1ed1 c\u1ee7a n\u00f3 c\u00f3 th\u1ec3 c\u1ea7n ph\u1ea3i \u0111\u01b0\u1ee3c c\u1eaft ng\u1eafn ho\u1eb7c x\u1ea5p x\u1ec9. Vi\u1ec7c c\u1eaft b\u1edbt n\u00e0y d\u1eabn \u0111\u1ebfn sai l\u1ec7ch gi\u1eefa gi\u00e1 tr\u1ecb th\u1ef1c v\u00e0 gi\u00e1 tr\u1ecb \u0111\u01b0\u1ee3c l\u01b0u tr\u1eef. C\u00e1c ph\u00e9p to\u00e1n ti\u1ebfp theo li\u00ean quan \u0111\u1ebfn nh\u1eefng con s\u1ed1 g\u1ea7n \u0111\u00fang n\u00e0y s\u1ebd t\u1ea1o ra c\u00e1c l\u1ed7i, \u1ea3nh h\u01b0\u1edfng \u0111\u1ebfn k\u1ebft qu\u1ea3 t\u00ednh to\u00e1n cu\u1ed1i c\u00f9ng.<\/p>\n<h2>C\u00e1c t\u00ednh n\u0103ng ch\u00ednh c\u1ee7a L\u1ed7i l\u00e0m tr\u00f2n<\/h2>\n<ol>\n<li><strong>B\u1ea3n ch\u1ea5t t\u00edch l\u0169y<\/strong>: C\u00e1c l\u1ed7i l\u00e0m tr\u00f2n t\u00edch l\u0169y v\u1edbi m\u1ed7i ph\u00e9p t\u00ednh s\u1ed1 h\u1ecdc, c\u00f3 kh\u1ea3 n\u0103ng d\u1eabn \u0111\u1ebfn sai l\u1ec7ch \u0111\u00e1ng k\u1ec3 so v\u1edbi k\u1ebft qu\u1ea3 l\u00fd t\u01b0\u1edfng.<\/li>\n<li><strong>S\u1ef1 ph\u1ee5 thu\u1ed9c v\u00e0o \u0111\u1ed9 ch\u00ednh x\u00e1c<\/strong>: \u0110\u1ed9 l\u1edbn c\u1ee7a l\u1ed7i l\u00e0m tr\u00f2n ph\u1ee5 thu\u1ed9c v\u00e0o s\u1ed1 bit \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 bi\u1ec3u th\u1ecb m\u1ed9t s\u1ed1; \u0111\u1ed9 ch\u00ednh x\u00e1c cao h\u01a1n s\u1ebd gi\u1ea3m nh\u01b0ng kh\u00f4ng lo\u1ea1i b\u1ecf \u0111\u01b0\u1ee3c l\u1ed7i.<\/li>\n<li><strong>S\u1ef1 truy\u1ec1n l\u1ed7i<\/strong>: C\u00e1c l\u1ed7i xu\u1ea5t hi\u1ec7n trong m\u1ed9t b\u01b0\u1edbc t\u00ednh to\u00e1n c\u00f3 th\u1ec3 lan sang c\u00e1c b\u01b0\u1edbc ti\u1ebfp theo, c\u00f3 kh\u1ea3 n\u0103ng l\u00e0m t\u0103ng sai s\u1ed1 t\u1ed5ng th\u1ec3.<\/li>\n<li><strong>S\u1ef1 \u1ed5n \u0111\u1ecbnh v\u00e0 b\u1ea5t \u1ed5n<\/strong>: M\u1ed9t s\u1ed1 thu\u1eadt to\u00e1n nh\u1ea1y c\u1ea3m h\u01a1n v\u1edbi c\u00e1c l\u1ed7i l\u00e0m tr\u00f2n, d\u1eabn \u0111\u1ebfn t\u00ednh kh\u00f4ng \u1ed5n \u0111\u1ecbnh v\u1ec1 s\u1ed1 l\u01b0\u1ee3ng v\u00e0 k\u1ebft qu\u1ea3 kh\u00f4ng ch\u00ednh x\u00e1c.<\/li>\n<\/ol>\n<h2>C\u00e1c lo\u1ea1i l\u1ed7i l\u00e0m tr\u00f2n<\/h2>\n<table>\n<thead>\n<tr>\n<th>Ki\u1ec3u<\/th>\n<th>S\u1ef1 mi\u00eau t\u1ea3<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>L\u1ed7i l\u00e0m tr\u00f2n tuy\u1ec7t \u0111\u1ed1i<\/strong><\/td>\n<td>S\u1ef1 kh\u00e1c bi\u1ec7t tuy\u1ec7t \u0111\u1ed1i gi\u1eefa gi\u00e1 tr\u1ecb t\u00ednh to\u00e1n v\u00e0 gi\u00e1 tr\u1ecb th\u1ef1c.<\/td>\n<\/tr>\n<tr>\n<td><strong>L\u1ed7i l\u00e0m tr\u00f2n t\u01b0\u01a1ng \u0111\u1ed1i<\/strong><\/td>\n<td>T\u1ef7 l\u1ec7 gi\u1eefa sai s\u1ed1 l\u00e0m tr\u00f2n tuy\u1ec7t \u0111\u1ed1i v\u1edbi gi\u00e1 tr\u1ecb th\u1ef1c.<\/td>\n<\/tr>\n<tr>\n<td><strong>Gia\u0309m thi\u00ea\u0309u l\u00f4i<\/strong><\/td>\n<td>Ph\u00e1t sinh t\u1eeb s\u1ef1 g\u1ea7n \u0111\u00fang c\u1ee7a ph\u1ea7n ph\u00e2n s\u1ed1 c\u1ee7a s\u1ed1 th\u1ef1c trong qu\u00e1 tr\u00ecnh chuy\u1ec3n \u0111\u1ed5i sang nh\u1ecb ph\u00e2n.<\/td>\n<\/tr>\n<tr>\n<td><strong>L\u1ed7i h\u1ee7y<\/strong><\/td>\n<td>X\u1ea3y ra khi hai gi\u00e1 tr\u1ecb g\u1ea7n b\u1eb1ng nhau b\u1ecb tr\u1eeb \u0111i, d\u1eabn \u0111\u1ebfn m\u1ea5t \u0111\u1ed9 ch\u00ednh x\u00e1c \u0111\u00e1ng k\u1ec3.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>S\u1eed d\u1ee5ng v\u00e0 gi\u1ea3m thi\u1ec3u l\u1ed7i l\u00e0m tr\u00f2n<\/h2>\n<p>S\u1ef1 hi\u1ec3u bi\u1ebft v\u1ec1 sai s\u1ed1 l\u00e0m tr\u00f2n l\u00e0 \u0111i\u1ec1u c\u1ea7n thi\u1ebft trong nhi\u1ec1u l\u0129nh v\u1ef1c kh\u00e1c nhau nh\u01b0 m\u00f4 ph\u1ecfng khoa h\u1ecdc, m\u00f4 h\u00ecnh t\u00e0i ch\u00ednh v\u00e0 ph\u00e2n t\u00edch k\u1ef9 thu\u1eadt. M\u1eb7c d\u00f9 vi\u1ec7c lo\u1ea1i b\u1ecf ho\u00e0n to\u00e0n sai s\u1ed1 l\u00e0m tr\u00f2n l\u00e0 kh\u00f4ng th\u1ec3 nh\u01b0ng v\u1eabn c\u00f3 c\u00e1c chi\u1ebfn l\u01b0\u1ee3c \u0111\u1ec3 gi\u1ea3m thi\u1ec3u t\u00e1c \u0111\u1ed9ng c\u1ee7a n\u00f3:<\/p>\n<ol>\n<li><strong>Qu\u1ea3n l\u00fd ch\u00ednh x\u00e1c<\/strong>: S\u1eed d\u1ee5ng c\u00e1c lo\u1ea1i d\u1eef li\u1ec7u c\u00f3 \u0111\u1ed9 ch\u00ednh x\u00e1c cao h\u01a1n \u0111\u1ec3 gi\u1ea3m \u1ea3nh h\u01b0\u1edfng c\u1ee7a l\u1ed7i l\u00e0m tr\u00f2n.<\/li>\n<li><strong>L\u1ef1a ch\u1ecdn thu\u1eadt to\u00e1n<\/strong>: Ch\u1ecdn c\u00e1c thu\u1eadt to\u00e1n \u00edt b\u1ecb khu\u1ebfch \u0111\u1ea1i l\u1ed7i h\u01a1n.<\/li>\n<li><strong>Ph\u00e2n t\u00edch l\u1ed7i<\/strong>: Th\u01b0\u1eddng xuy\u00ean ph\u00e2n t\u00edch v\u00e0 theo d\u00f5i vi\u1ec7c lan truy\u1ec1n l\u1ed7i \u0111\u1ec3 x\u00e1c \u0111\u1ecbnh c\u00e1c \u0111i\u1ec3m quan tr\u1ecdng trong t\u00ednh to\u00e1n.<\/li>\n<li><strong>Gi\u1edbi h\u1ea1n l\u1ed7i<\/strong>: S\u1eed d\u1ee5ng c\u00e1c k\u1ef9 thu\u1eadt to\u00e1n h\u1ecdc \u0111\u1ec3 thi\u1ebft l\u1eadp gi\u1edbi h\u1ea1n tr\u00ean c\u1ee7a sai s\u1ed1 \u0111\u01b0\u1ee3c \u0111\u01b0a ra.<\/li>\n<\/ol>\n<h2>L\u1ed7i l\u00e0m tr\u00f2n trong ph\u1ed1i c\u1ea3nh<\/h2>\n<table>\n<thead>\n<tr>\n<th>\u0111\u1eb7c tr\u01b0ng<\/th>\n<th>L\u1ed7i l\u00e0m tr\u00f2n<\/th>\n<th>\u0110i\u1ec1u kho\u1ea3n t\u01b0\u01a1ng t\u1ef1<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Thi\u00ean nhi\u00ean<\/strong><\/td>\n<td>X\u1ea5p x\u1ec9 s\u1ed1<\/td>\n<td><strong>Gia\u0309m thi\u00ea\u0309u l\u00f4i<\/strong>: T\u01b0\u01a1ng t\u1ef1, nh\u01b0ng t\u1eadp trung v\u00e0o t\u00ednh g\u1ea7n \u0111\u00fang trong qu\u00e1 tr\u00ecnh chuy\u1ec3n \u0111\u1ed5i.<\/td>\n<\/tr>\n<tr>\n<td><strong>Hi\u1ec7u \u1ee9ng tr\u00ean \u0111\u1ed9 ch\u00ednh x\u00e1c<\/strong><\/td>\n<td>L\u00e0m suy gi\u1ea3m \u0111\u1ed9 ch\u00ednh x\u00e1c<\/td>\n<td><strong>L\u1ed7i d\u1ea5u ph\u1ea9y \u0111\u1ed9ng<\/strong>: Thu\u1eadt ng\u1eef t\u1ed5ng qu\u00e1t h\u01a1n bao g\u1ed3m nh\u1eefng \u0111i\u1ec3m kh\u00f4ng ch\u00ednh x\u00e1c trong s\u1ed1 h\u1ecdc d\u1ea5u ph\u1ea9y \u0111\u1ed9ng.<\/td>\n<\/tr>\n<tr>\n<td><strong>S\u1ef1 ph\u1ee5 thu\u1ed9c v\u00e0o ho\u1ea1t \u0111\u1ed9ng<\/strong><\/td>\n<td>T\u0103ng theo ho\u1ea1t \u0111\u1ed9ng<\/td>\n<td><strong>L\u1ed7i l\u00e0m tr\u00f2n<\/strong>: Th\u01b0\u1eddng \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng thay th\u1ebf cho nhau nh\u01b0ng c\u00f3 th\u1ec3 \u0111\u1ec1 c\u1eadp c\u1ee5 th\u1ec3 \u0111\u1ebfn c\u00e1c ph\u00e9p to\u00e1n l\u00e0m tr\u00f2n.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Quan \u0111i\u1ec3m v\u00e0 c\u00f4ng ngh\u1ec7 t\u01b0\u01a1ng lai<\/h2>\n<p>S\u1ef1 ti\u1ebfn b\u1ed9 kh\u00f4ng ng\u1eebng c\u1ee7a ph\u1ea7n c\u1ee9ng v\u00e0 ph\u1ea7n m\u1ec1m m\u00e1y t\u00ednh m\u1edf ra c\u01a1 h\u1ed9i gi\u1ea3m thi\u1ec3u l\u1ed7i l\u00e0m tr\u00f2n s\u1ed1. C\u00e1c c\u00f4ng ngh\u1ec7 m\u1edbi n\u1ed5i nh\u01b0 \u0111i\u1ec7n to\u00e1n l\u01b0\u1ee3ng t\u1eed v\u00e0 c\u00e1c thu\u1eadt to\u00e1n s\u1ed1 c\u1ea3i ti\u1ebfn h\u1ee9a h\u1eb9n s\u1ebd n\u00e2ng cao \u0111\u1ed9 ch\u00ednh x\u00e1c v\u00e0 gi\u1ea3m s\u1ef1 lan truy\u1ec1n l\u1ed7i. C\u00e1c nh\u00e0 nghi\u00ean c\u1ee9u \u0111ang kh\u00e1m ph\u00e1 nh\u1eefng c\u00e1ch m\u1edbi \u0111\u1ec3 c\u00e2n b\u1eb1ng hi\u1ec7u qu\u1ea3 t\u00ednh to\u00e1n v\u1edbi \u0111\u1ed9 ch\u00ednh x\u00e1c, m\u1edf ra k\u1ef7 nguy\u00ean t\u00ednh to\u00e1n s\u1ed1 ch\u00ednh x\u00e1c h\u01a1n.<\/p>\n<h2>M\u00e1y ch\u1ee7 proxy v\u00e0 l\u1ed7i l\u00e0m tr\u00f2n<\/h2>\n<p>M\u1eb7c d\u00f9 d\u01b0\u1eddng nh\u01b0 kh\u00f4ng li\u00ean quan nh\u01b0ng m\u00e1y ch\u1ee7 proxy v\u00e0 l\u1ed7i l\u00e0m tr\u00f2n giao nhau trong c\u00e1c t\u00ecnh hu\u1ed1ng li\u00ean quan \u0111\u1ebfn truy\u1ec1n d\u1eef li\u1ec7u v\u00e0 t\u00ednh to\u00e1n t\u1eeb xa. C\u00e1c m\u00e1y ch\u1ee7 proxy c\u00f3 th\u1ec3 \u0111\u01b0a ra c\u00e1c d\u1ea1ng x\u1ea5p x\u1ec9 v\u00e0 sai s\u1ed1 c\u1ee7a ri\u00eang ch\u00fang, t\u01b0\u01a1ng t\u1ef1 nh\u01b0 sai s\u1ed1 l\u00e0m tr\u00f2n trong t\u00ednh to\u00e1n s\u1ed1. Hi\u1ec3u c\u1ea3 l\u1ed7i l\u00e0m tr\u00f2n v\u00e0 h\u00e0nh vi c\u1ee7a m\u00e1y ch\u1ee7 proxy l\u00e0 r\u1ea5t quan tr\u1ecdng khi x\u1eed l\u00fd c\u00e1c \u1ee9ng d\u1ee5ng s\u1eed d\u1ee5ng nhi\u1ec1u d\u1eef li\u1ec7u, \u0111\u1ea3m b\u1ea3o truy\u1ec1n v\u00e0 t\u00ednh to\u00e1n th\u00f4ng tin ch\u00ednh x\u00e1c.<\/p>\n<h2>Li\u00ean k\u1ebft li\u00ean quan<\/h2>\n<p>\u0110\u1ec3 bi\u1ebft th\u00eam th\u00f4ng tin chuy\u00ean s\u00e2u v\u1ec1 l\u1ed7i l\u00e0m tr\u00f2n, \u0111\u1ed9 \u1ed5n \u0111\u1ecbnh s\u1ed1 v\u00e0 c\u00e1c kh\u00e1i ni\u1ec7m li\u00ean quan, b\u1ea1n c\u00f3 th\u1ec3 kh\u00e1m ph\u00e1 c\u00e1c t\u00e0i nguy\u00ean sau:<\/p>\n<ul>\n<li><a href=\"https:\/\/www.computer.org\/csdl\/home\" target=\"_new\" rel=\"noopener nofollow\">Hi\u1ec7p h\u1ed9i m\u00e1y t\u00ednh IEEE<\/a><\/li>\n<li><a href=\"http:\/\/www2.math.uu.se\/~svante\/papers\/sjN15.pdf\" target=\"_new\" rel=\"noopener nofollow\">Ph\u00e2n t\u00edch s\u1ed1: To\u00e1n h\u1ecdc t\u00ednh to\u00e1n khoa h\u1ecdc<\/a><\/li>\n<li><a href=\"https:\/\/www.nist.gov\/\" target=\"_new\" rel=\"noopener nofollow\">S\u1ed5 tay h\u00e0m s\u1ed1 to\u00e1n h\u1ecdc c\u1ee7a NIST<\/a><\/li>\n<\/ul>\n<p>T\u00f3m l\u1ea1i, l\u1ed7i l\u00e0m tr\u00f2n l\u00e0 m\u1ed9t th\u00e1ch th\u1ee9c c\u01a1 b\u1ea3n trong t\u00ednh to\u00e1n s\u1ed1, \u1ea3nh h\u01b0\u1edfng \u0111\u1ebfn nhi\u1ec1u l\u0129nh v\u1ef1c v\u00e0 \u1ee9ng d\u1ee5ng kh\u00e1c nhau. B\u1eb1ng c\u00e1ch hi\u1ec3u r\u00f5 ngu\u1ed3n g\u1ed1c, c\u01a1 ch\u1ebf, lo\u1ea1i v\u00e0 chi\u1ebfn l\u01b0\u1ee3c gi\u1ea3m nh\u1eb9, c\u00e1c c\u00e1 nh\u00e2n v\u00e0 ng\u00e0nh c\u00f3 th\u1ec3 x\u1eed l\u00fd s\u1ef1 ph\u1ee9c t\u1ea1p c\u1ee7a c\u00e1c ph\u00e9p t\u00ednh s\u1ed1, \u0111\u01b0a ra quy\u1ebft \u0111\u1ecbnh s\u00e1ng su\u1ed1t \u0111\u1ec3 \u0111\u1ea1t \u0111\u01b0\u1ee3c k\u1ebft qu\u1ea3 ch\u00ednh x\u00e1c h\u01a1n.<\/p>","protected":false},"featured_media":470389,"menu_order":0,"template":"","meta":{"_acf_changed":false,"content-type":"","inline_featured_image":false,"footnotes":""},"class_list":["post-478790","wiki","type-wiki","status-publish","has-post-thumbnail","hentry"],"acf":{"faq_title":"Frequently Asked Questions about <mark>Round-off Error: Exploring Precision in Numerical Computations<\/mark>","faq_items":[{"question":"What is round-off error?","answer":"<p>Round-off error refers to the small discrepancies that arise when representing real numbers in digital computing systems. Due to the finite precision of computers, not all real numbers can be precisely represented in binary format, leading to tiny errors in calculations.<\/p>"},{"question":"How did the concept of round-off error originate?","answer":"<p>The concept of round-off error can be traced back to the early days of digital computing in the mid-20th century. Early computer pioneers like John W. Mauchly and J. Presper Eckert recognized the limitations of representing real numbers in binary, giving rise to the term \"round-off error.\"<\/p>"},{"question":"How does round-off error work?","answer":"<p>Round-off error occurs due to the finite number of bits used to represent real numbers in binary form. When converting a real number to binary, its fractional part might be truncated or approximated, leading to deviations from the actual value. Subsequent calculations then compound these errors, affecting the final results.<\/p>"},{"question":"What are the key features of round-off error?","answer":"<p>Round-off error exhibits several important characteristics:<\/p><ul><li><strong>Accumulative Nature<\/strong>: Errors accumulate with each arithmetic operation.<\/li><li><strong>Dependence on Precision<\/strong>: The number of bits used affects the error magnitude.<\/li><li><strong>Error Propagation<\/strong>: Errors from one step can affect subsequent steps.<\/li><li><strong>Stability and Instability<\/strong>: Some algorithms are more sensitive to errors, leading to instability.<\/li><\/ul>"},{"question":"What are the types of round-off error?","answer":"<p>There are different types of round-off error:<\/p><ul><li><strong>Absolute Round-off Error<\/strong>: The absolute difference between computed and true values.<\/li><li><strong>Relative Round-off Error<\/strong>: The ratio of absolute error to true value.<\/li><li><strong>Truncation Error<\/strong>: Arises from approximating a real number's fractional part.<\/li><li><strong>Cancellation Error<\/strong>: Occurs when subtracting nearly equal values, leading to precision loss.<\/li><\/ul>"},{"question":"How can round-off error be managed?","answer":"<p>While eliminating round-off error entirely is impossible, you can reduce its impact:<\/p><ul><li><strong>Precision Management<\/strong>: Use higher precision data types.<\/li><li><strong>Algorithm Choice<\/strong>: Opt for algorithms less sensitive to error amplification.<\/li><li><strong>Error Analysis<\/strong>: Regularly analyze error propagation to identify critical points.<\/li><li><strong>Error Bounds<\/strong>: Establish upper bounds on introduced errors mathematically.<\/li><\/ul>"},{"question":"How does round-off error relate to proxy servers?","answer":"<p>Although seemingly unrelated, proxy servers and round-off errors intersect in scenarios involving data transmission. Proxy servers can introduce their own forms of approximation and error, akin to round-off error in numerical computations. Understanding both concepts is essential for accurate data transfer and computation.<\/p>"},{"question":"What does the future hold for round-off error?","answer":"<p>Advancements in hardware and software, such as quantum computing and improved algorithms, offer opportunities to mitigate round-off error. These technologies promise enhanced precision and reduced error propagation, leading to more accurate numerical computations.<\/p>"},{"question":"Where can I find more information about round-off error?","answer":"<p>For a deeper understanding of round-off error, numerical stability, and related concepts, you can explore the following resources:<\/p><ul><li><a href=\"https:\/\/www.computer.org\/csdl\/home\" target=\"_new\">IEEE Computer Society<\/a><\/li><li><a href=\"http:\/\/www2.math.uu.se\/~svante\/papers\/sjN15.pdf\" target=\"_new\">Numerical Analysis: Mathematics of Scientific Computing<\/a><\/li><li><a href=\"https:\/\/www.nist.gov\/\" target=\"_new\">NIST Handbook of Mathematical Functions<\/a><\/li><\/ul>"}]},"_links":{"self":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/wiki\/478790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/wiki"}],"about":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/types\/wiki"}],"version-history":[{"count":0,"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/wiki\/478790\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/media\/470389"}],"wp:attachment":[{"href":"https:\/\/oneproxy.pro\/vn\/wp-json\/wp\/v2\/media?parent=478790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}