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Unformatted text preview: UCLA Fall Quarter 2009-10 Electrical Engineering 103 Applied Numerical Computing Professor L. Vandenberghe Notes written in collaboration with S. Boyd (Stanford Univ.) Contents 1 Vectors 1 1.1 Definitions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Zero and unit vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Scalar-vector multiplication . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.7 Euclidean norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.8 Angle between vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.9 Vector inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Matrices 23 2.1 Definitions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Zero and identity matrices . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Matrix transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Matrix addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5 Scalar-matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6 Matrix-matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . 27 2.7 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.8 Matrix inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.9 Matrix norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 Complexity of matrix algorithms 47 3.1 Complexity analysis via flop count . . . . . . . . . . . . . . . . . . . . 47 3.2 Basic matrix and vector operations . . . . . . . . . . . . . . . . . . . . 48 3.3 Iterative algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4 Linear and R-linear convergence . . . . . . . . . . . . . . . . . . . . . 51 3.5 Quadratic convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.6 Superlinear convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 iv Contents 4 Linear equations 57 4.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Linear equations and matrix inverse . . . . . . . . . . . . . . . . . . . . 58 4.3 Range and nullspace of a matrix . . . . . . . . . . . . . . . . . . . . . 59 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5 Triangular matrices 65 5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Forward substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
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