AIMS lecture notes on numerical analysis by Olver P.J.

By Olver P.J.

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Otherwise, there is no guarantee that a particular set of iterates will converge, although if they do, the limiting value is necessarily a root of our equation. 1, including period doubling bifurcations and chaotic behavior. The reader is invited to experiment with simple examples; further details can be found in [42]. 20. 39) can be viewed as a implicit equation defining the eccentric anomaly u as a function of the mean anomaly m. To solve Kepler’s equation by Newton’s Method, we introduce the iterative function u − sin u − m .

If they are both special lower triangular, so is their product. Similarly, if U, U are (special) upper triangular matrices, so is their product U U . The L U Factorization We have almost arrived at our first important result. 14). 13), and the basic property of the identity matrix I , we conclude that L U = (L1 L2 L3 )(E3 E2 E1 A) = L1 L2 (L3 E3 )E2 E1 A = L1 L2 I E2 E1 A = L1 (L2 E2 )E1 A = L1 I E1 A = (L1 E1 )A = I A = A. In other words, we have factored the coefficient matrix A = L U into a product of a special lower triangular matrix L and an upper triangular matrix U with the nonzero pivots on its main diagonal.

Olver AIMS Lecture Notes 2006 Peter J. Olver 3. Review of Matrix Algebra Vectors and matrices are essential for modern analysis of systems of equations — algebrai, differential, functional, etc. In this part, we will review the most basic facts of matrix arithmetic. See [38] for full details. 1. Matrices and Vectors. A matrix is a rectangular array of numbers. 32 ), 0 , 0 − 47 1 3 , −2 5 are all examples of matrices. We use the notation a 11 a  21 A=  .. am1 a12 a22 .. ... .. a1n a2n ..

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