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Unformatted text preview: EE103 Formulas Vectors and matrices • Relation between inner product and angle: x T y = bardbl x bardblbardbl y bardbl cos negationslash ( x, y ) • Flop counts for basic operations ( α is a scalar, x and y are n-vectors, A is an m × n-matrix, B is an n × p-matrix) Inner product x T y : 2 n − 1 flops ( ≈ 2 n flops for large n ) Vector addition x + y : n flops Scalar multiplication αx : n flops Matrix-vector product Ax : m (2 n − 1) flops ( ≈ 2 mn flops for large n ) Matrix-matrix product AB : mp (2 n − 1) flops ( ≈ 2 mpn flops for large n ) Solving linear equations • Cost of solving Ax = b when A is n × n and upper or lower triangular: n 2 flops • Cost of Cholesky factorization A = LL T : (1 / 3) n 3 flops if A is n × n • Cost of LU factorization A = PLU : (2 / 3) n 3 flops if A is n × n Matrix norm and condition number • Definition of matrix norm: bardbl A bardbl = max x negationslash =0 bardbl Ax bardbl bardbl x bardbl • Properties of the matrix norm bardbl...
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- Spring '09
- ax, Ri, flops Matrix-vector product, flops Scalar multiplication