x 2 b 1 b 2 b 3. x 1 x 3 c n 2 d n 1 e n 1 x n 1 b n 1 5 c n 1 d n x n b n
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- William Harrison
- 5 years ago
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Transcription
1 Tridiagonal linear systems <latexit sha1_base"y/w9be0+f+voao/bayzqn8cqme">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</latexit> <latexit 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sha1_base"orobbvdflaozwqrnvx0nyisye">aaacmxicbvbns8naenbfxq9hlyhe8luqu9vbwrgcvbepzbozpiubtdidqcx0xjvx+gf0zn9u+/tiodwdg8wagee+furqgpe/nmztfwfxaxll119ynrcq1dqvyqrnoc0zmembkbmqqkebbuqytwwnjrwhddjebx9cnynqldnlopixrihacoavugrhpzlcnir1lgnyc/wpqjnptxpvpxzegs9sumglmbjezlszrcaldnygmizfsqqfiqwgumwy8ldumezimaztqqjtmffyvljrmkod0cstr/zypyv1mnwpikwwqvfwiktxfhasy0zf/ggknhoxaasa1sfopzpnonqumcajvalc8dhed8ejh9uxj9cvpxawgcuinnasssdsiumjtdh/m9ksab80thv+xwg9zzguujyczjzjk1ytlqktthrik8kxfn1xlppzpyeqcmzjr/k+fogdrwqtg</latexit> <latexit sha1_base"orobbvdflaozwqrnvx0nyisye">aaacmxicbvbns8naenbfxq9hlyhe8luqu9vbwrgcvbepzbozpiubtdidqcx0xjvx+gf0zn9u+/tiodwdg8wagee+furqgpe/nmztfwfxaxll119ynrcq1dqvyqrnoc0zmembkbmqqkebbuqytwwnjrwhddjebx9cnynqldnlopixrihacoavugrhpzlcnir1lgnyc/wpqjnptxpvpxzegs9sumglmbjezlszrcaldnygmizfsqqfiqwgumwy8ldumezimaztqqjtmffyvljrmkod0cstr/zypyv1mnwpikwwqvfwiktxfhasy0zf/ggknhoxaasa1sfopzpnonqumcajvalc8dhed8ejh9uxj9cvpxawgcuinnasssdsiumjtdh/m9ksab80thv+xwg9zzguujyczjzjk1ytlqktthrik8kxfn1xlppzpyeqcmzjr/k+fogdrwqtg</latexit> A tridiagonal linear system of equations takes the form: d 1 e 1 c 1 d e c d e c n d n 1 e n 1 x x x n 1 b 1 b b b n 1 c n 1 d n x n b n The name tridiagonal comes from the fact that there are at most three non-zero entries per row of the coefficient matrix, and these entries can only appear on the main diagonal, sub-diagonal, and super-diagonal These systems appear in many applications
2 Example: multistage countercurrent extractor (W + Sm) Sm 0 0 W (W + Sm) Sm 0 0 W (W + Sm) Sm 0 0 W (W + Sm) x x x Wx in 0 0 Sy in Example: n, W 1, S, m, x in 01, y in 0 Model problem from previous lecture
3 Example: multistage countercurrent extractor <latexit sha1_base"y/w9be0+f+voao/bayzqn8cqme">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</latexit> <latexit 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sha1_base"yi/dijygcvvsdigmmv0dajm">aaacinicbvhbthsxehwwlskqkbhllajihahi0gihbsjdj0ciiqpgwuebtjchruxzmiv+s9cnba91sjmuweiwnt/e8xkufopgv81bevn+dkqv/+w9rrn1z8mhehqznmzhmleihoymcjdznbpikjxtjxupn0cy0wmbgcq1+xviuhawdnagzpnigo+e0zomnixkkqftgmhiz+p1ipdvnlv0nkmzqgwua/cmnuh9ubqcmzfxjwdhpkxledzdpwlgs8ukcrsztlwxyjfmooasjnuwmgzfqp9bzutihtlmsjvszyxiziwgumm/b+jzmrasyqduzecuvtsrm9qocfuxqucfgubvogehkwz0gixnhagocuwa0at1i+yozxdphkqxnzrfurkh/mdvinfzymohn+1tbygxnpg+jqxlu0imsefougwxmsjol9apw+qjrbs/jxchbbpzjseiwyjvyrduhkj/ln/pc/ppj1zxvq81ppjnv+a0rlwea</latexit> Model problem from previous lecture Example: System for a general n <latexit 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4 Solving tridiagonal linear systems <latexit sha1_base"bekrhueniaoqsftloik+wewkw">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</latexit> <latexit 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sha1_base"mcx9ch0nnsgcdwqp0nxtqeo">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</latexit> <latexit sha1_base"mcx9ch0nnsgcdwqp0nxtqeo">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</latexit> Tridiagonal linear systems can be solved very efficiently by using a special version of the Gaussian elimination algorithm This algorithm is called the Thomas algorithm (or sometime Crout) Key ideas of the algorithm: 1 Reduction to upper triangular form only requires eliminating c j terms Resulting upper triangular matrix only has two non-zeros per row Augmented system d 1 e 1 b 1 c 1 d e b c d e b c n d n 1 e n 1 b n 1 c n 1 d n b n Step 1: upper triangular form for i 1ton µ c i /d i d i+1 d i+1 b i+1 b i+1 end 1do µe i µb i
5 Solving tridiagonal linear systems <latexit sha1_base"awdxcd0djfmr9n1glmmnmw0">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</latexit> <latexit 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sha1_base"0gjakywbbwwrvf+itfdbvsqziw">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</latexit> Tridiagonal linear systems can be solved very efficiently by using a special version of the Gaussian elimination algorithm This algorithm is called the Thomas algorithm (or sometime Crout) Key ideas of the algorithm: 1 Reduction to upper triangular form only requires eliminating c j terms Resulting upper triangular matrix only has two non-zeros per row Upper triangular system (row echelon form) d 1 e 1 b 1 d e b d e b d n 1 e n 1 b n 1 d n b n Step : Back substitution x n b n /d n for i n 1to1do x i (b i e i x i+1 )/d i end
6 Solving tridiagonal linear systems: recap <latexit sha1_base"bekrhueniaoqsftloik+wewkw">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</latexit> <latexit 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sha1_base"mcx9ch0nnsgcdwqp0nxtqeo">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</latexit> Augmented system d 1 e 1 b 1 c 1 d e b c d e b c n d n 1 e n 1 b n 1 c n 1 d n b n Upper triangular system (row echelon form) d 1 e 1 b 1 d e b d e b d n 1 e n 1 b n 1 d n b n Step 1: Upper triangular form for i 1ton µ c i /d i d i+1 d i+1 b i+1 b i+1 end 1do µe i µb i Step : Back substitution How many floating point operations are required? x n b n /d n for i n 1to1do x i (b i e i x i+1 )/d i end Note: For numerical stability, the algorithm should only be used when d 1 e 1, d n c n 1, d j c j 1 + e j,j,,n 1 A tridiagonal matrix that satisfies this property is called diagonally dominant
7 Exercise <latexit sha1_base"y/w9be0+f+voao/bayzqn8cqme">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</latexit> <latexit 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sha1_base"yi/dijygcvvsdigmmv0dajm">aaacinicbvhbthsxehwwlskqkbhllajihahi0gihbsjdj0ciiqpgwuebtjchruxzmiv+s9cnba91sjmuweiwnt/e8xkufopgv81bevn+dkqv/+w9rrn1z8mhehqznmzhmleihoymcjdznbpikjxtjxupn0cy0wmbgcq1+xviuhawdnagzpnigo+e0zomnixkkqftgmhiz+p1ipdvnlv0nkmzqgwua/cmnuh9ubqcmzfxjwdhpkxledzdpwlgs8ukcrsztlwxyjfmooasjnuwmgzfqp9bzutihtlmsjvszyxiziwgumm/b+jzmrasyqduzecuvtsrm9qocfuxqucfgubvogehkwz0gixnhagocuwa0at1i+yozxdphkqxnzrfurkh/mdvinfzymohn+1tbygxnpg+jqxlu0imsefougwxmsjol9apw+qjrbs/jxchbbpzjseiwyjvyrduhkj/ln/pc/ppj1zxvq81ppjnv+a0rlwea</latexit> <latexit sha1_base"yi/dijygcvvsdigmmv0dajm">aaacinicbvhbthsxehwwlskqkbhllajihahi0gihbsjdj0ciiqpgwuebtjchruxzmiv+s9cnba91sjmuweiwnt/e8xkufopgv81bevn+dkqv/+w9rrn1z8mhehqznmzhmleihoymcjdznbpikjxtjxupn0cy0wmbgcq1+xviuhawdnagzpnigo+e0zomnixkkqftgmhiz+p1ipdvnlv0nkmzqgwua/cmnuh9ubqcmzfxjwdhpkxledzdpwlgs8ukcrsztlwxyjfmooasjnuwmgzfqp9bzutihtlmsjvszyxiziwgumm/b+jzmrasyqduzecuvtsrm9qocfuxqucfgubvogehkwz0gixnhagocuwa0at1i+yozxdphkqxnzrfurkh/mdvinfzymohn+1tbygxnpg+jqxlu0imsefougwxmsjol9apw+qjrbs/jxchbbpzjseiwyjvyrduhkj/ln/pc/ppj1zxvq81ppjnv+a0rlwea</latexit> The Thomas algorithm is implemented in the tridisolve MATLAB function on the course webpage Exercise 1 Use the tridisolve function to solve the Countercurrent exchanger problem Compare the time it takes to solve this system with the time it takes standard Gaussian elimination applied to the full matrix d 1 e 1 c 1 d e c d e c n d n 1 e n 1 x x x n 1 b 1 b b b n 1 c j W, j 1,,n 1, d j (W + Sm), j1,,n, e j Sm, j 1,,n 1, b 1 Wx in, b j 0,j,,n 1, b n Sy in c n 1 d n x n b n
8 Sparse linear systems: spdiags <latexit sha1_base"y/w9be0+f+voao/bayzqn8cqme">aaadicjvjdb9mwfhubpkbsyjpfhutlxqne0mafpghckebgszzoaqrkd9raktzdyqyjtpifd+cd+gfpjflz+igep9dhx8tm+tfnitty/1qtb0bn/drnj1/8hdb9lyyrtichyjjmnfmijekhigxgbnca0taqf8/f19fnobshmfczfduouxupopwboqcleeoqcglynkuqagmmdzsyvadillvbtjyg8xfkvpgwobzyo0ladyizd0hqprpo/ifkmeuofgnbkevchtlsuzwig0bxfoolqv+pwlssioxfsdrsmwwqarkuqhycv9w8exnbj09t+9dewgsxvlmv1e1rnz8w88azn8888yplvcfb9ywn1+rtdbsaqdslwn9o/ednmfckofakzzht0chyxtkmucdiewkdoxdmlywshyimycekgrklplbpraabtpa9vqnkqxglfjulfanmn+vppbzkyftl+nsnyakgjjmhajbqzwk8rjaqggcncaia0tg+lysy0ehng8nifxmckeggolzkyoewrvcv1nfugtznakhzheokztxd8vgvwebynjvvugnwxflduqx+qpeuc8pick/fkhayjad9pv1/ah/0upfv++z9btt1vloikyvb+/auzkk9c</latexit> <latexit sha1_base"y/w9be0+f+voao/bayzqn8cqme">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</latexit> <latexit sha1_base"y/w9be0+f+voao/bayzqn8cqme">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</latexit> <latexit sha1_base"y/w9be0+f+voao/bayzqn8cqme">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</latexit> Tridiagonal systems are special examples of sparse linear systems MATLAB provides a convenient way to create tridiagonal systems (and other sparse systems) with their spdiags function For example, we can create the tridiagonal matrix in the linear system d 1 e 1 c 1 d e c d e c n d n 1 e n 1 x x x n 1 b 1 b b b n 1 c n 1 d n x n b n using A spdiags([[c; 0] d [0; e]],[-1 0 1],n,n) The system can then be solved using backslash \ x A\b If A is diagonally dominant then MATLAB will automatically use the Thomas algorithm to solve this system
Solving Linear Systems Using Gaussian Elimination. How can we solve
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