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numan_c1 - MT 3802 NUMERICAL ANALYSIS 2008/2009 Dr Clare E...

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MT 3802 NUMERICAL ANALYSIS 2008/2009 Dr Clare E Parnell and Dr St´ ephane R´ egnier October 15, 2008
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Contents 0 Handout 1 0.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.3 Books and Website . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 Vector and Matrix Norms 4 1.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 The Basis and Dimension of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Vector Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2 Inner Product Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.3 Commonly Used Norms and Normed Linear Spaces . . . . . . . . . . . . . . . 8 1.4 Sub-ordinate Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.1 Commonly Used Sub-ordinate Matrix Norms . . . . . . . . . . . . . . . . . . 10 1.5 Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6 The Condition Number and Ill-Conditioned Matrices . . . . . . . . . . . . . . . . . . 14 1.6.1 An Example of Using Norms to Solve a Linear System of Equations . . . . . 15 1.7 Some Useful Results for Determining the Condition Number . . . . . . . . . . . . . . 17 1.7.1 Finding Norms of Inverse Matrices . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7.2 Limits on the Condition Number . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.8 Examples of Ill-Conditioned Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Iterative Methods 21 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 Single Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Sequences of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 i
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2.2.1 The Limit of a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.2 Convergence of a Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.3 Spectral radius and rate of convergence . . . . . . . . . . . . . . . . . . . . . 25 2.2.4 Gerschgorin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 The Jacobi Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 Convergence of the Jacobi Iteration Method . . . . . . . . . . . . . . . . . . . 29 2.4 The Gauss-Seidel Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 The Successive Over Relaxation Iterative Method . . . . . . . . . . . . . . . . . . . . 32 2.6 Consistently Ordered Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6.2 Tri-diagonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.3 Convergence for the SOR method . . . . . . . . . . . . . . . . . . . . . . . . . 36 3 An Introduction to the Approximation of Functions 39 3.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Reducing the Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3 Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 Minimising the Error Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.5 Piecewise Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.6 Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.7 The Basis Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4 Best Approximation 56 4.1 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2 Least Squares Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3 Orthogonal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4 Minimax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.5 Equi-oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.6 Chebyshev Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.7 Economisation of a Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 ii
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Chapter 0 Handout 0.1 Notation Throughout this course we will be using scalars, vectors and matrices. It is essential that you know what they are and can tell the difference between them!! Scalar: e.g. α , β or γ . Scalars belong to a field F such as R or C . Vectors: e.g. x or f ( x ). The first vector belongs to a Vector Space (defined later) such as R n , C n and is of the form x = ( x i , x 2 , . . . , x n ). The second vector is a continuous function such as the polynomial x 2 + x and belongs to a vector space such as C ( −∞ , ). In the lectures, vectors will be denoted by: x i.e. a small letter with a wiggly line under and in the online lecture notes by x i.e. a bold small letter . Matrices: e.g. A or B . An n × n matrix has the form A = a 11 a 12 . . . a 1 n a 21 a 22 . . . . . . . . . . . . . . . . . . a n 1 . . . . . . a nn . In the lectures, a matrix will be denoted by: A a capital letter with a straight line under 1
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and in the online lecture notes by A i.e. a bold capital letter .
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