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11 - Solving linear systems Naive Gaussian elimination...

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Solving linear systems Naive Gaussian elimination Pivoting Sparse/banded matrices Performance Gaussian elimination Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Gaussian elimination MATH 2070U 1 / 33
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Solving linear systems Naive Gaussian elimination Pivoting Sparse/banded matrices Performance Gaussian elimination 1 Solving linear systems of equations Easy-to-solve systems 2 Naive Gaussian elimination 3 Pivoting 4 Sparse and banded matrices 5 Performance of algorithms Using cputime to time programs Timed comparison of backslash vs. invert-and-multiply Operation counts c D. Aruliah (UOIT) Gaussian elimination MATH 2070U 2 / 33
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Solving linear systems Naive Gaussian elimination Pivoting Sparse/banded matrices Performance Linear systems of equations Rule for matrix multiplication permits representation of linear systems of equations using matrices and vectors e.g., linear system of equations 2 x 1 + x 2 + x 3 = 4 4 x 1 + 3 x 2 + 3 x 3 + x 4 = 11 8 x 1 + 7 x 2 + 9 x 3 + 5 x 4 = 29 6 x 1 + 7 x 2 + 9 x 3 + 8 x 4 = 30 can be written as [ A ] { x } = { b } with 2 1 1 0 4 3 3 1 8 7 9 5 6 7 9 8 | {z } [ A ] x 1 x 2 x 3 x 4 | {z } { x } = 4 11 29 30 | {z } { b } c D. Aruliah (UOIT) Gaussian elimination MATH 2070U 4 / 33
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Solving linear systems Naive Gaussian elimination Pivoting Sparse/banded matrices Performance Solving linear systems of equations in M ATLAB Want to solve n × n system of linear equations [ A ] { x } = { b } Backslash operator does this in M ATLAB : >> A = [ 1 3 ; 4 7 ] >> b = [ 4; 11 ] >> x = A \ b Read x = A \ b as { x } = [ A ] - 1 { b } x = A \ b = x = mldivide(A,b) “matrix left divide” c D. Aruliah (UOIT) Gaussian elimination MATH 2070U 5 / 33
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Solving linear systems Naive Gaussian elimination Pivoting Sparse/banded matrices Performance Solving linear systems in M ATLAB Present goal: to understand what \ does I Gaussian elimination I Pivoting See doc mldivide “matrix left division” Solution of [ A ] { x } = { b } Never solve linear systems by computing [ A ] - 1 and { x } = [ A ] - 1 { b } ! Use backslash ( mldivide ) or some solver enclosing sensible algorithms c D. Aruliah (UOIT) Gaussian elimination MATH 2070U 6 / 33
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Solving linear systems Naive Gaussian elimination Pivoting Sparse/banded matrices Performance Easy systems Diagonal system example Solve the linear system of equations 2 3 - 4 x 1 x 2 x 3 = 5 9 1 2 x 1 = 5 x 1 = 5 2 3 x 2 = 9 x 2 = 9 3 = 3 - 4 x 3 = 1 x 3 = - 1 4 Equations are completely decoupled c D. Aruliah (UOIT) Gaussian elimination MATH 2070U 7 / 33
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Solving linear systems Naive Gaussian elimination Pivoting Sparse/banded matrices Performance Easy systems Diagonal systems Given vector { b } = [ b 1 , . . . , b n ] T R n and diagonal matrix [ D ] = diag ( { d } ) with { d } = [ d 1 , . . . , d n ] T R n , wish to solve linear system of equations [ D ] { x } = { b } , i.e., d 1 d 2 .
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