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Gram-Schmidt Orthogonalization - USM

Jim LambersMAT 415/515 Fall Semester 2013-14 Lecture 3 NotesThese notes correspond to Section in the OrthogonalizationWe have seen that it can be very convenient to have an orthonormal basis for a given vectorspace, in order to compute expansions of arbitrary vectors within that space. Therefore, given anon-orthonormal basis, it is desirable to have a process for obtaining an orthonormal basis from , we have such a process, known asGram-Schmidt Orthogonalization . Suppose thatwe have a linearly independent, but not orthonormal, set of functions{ 1, 2,..}that span a givenvector spaceV. To construct an orthonormal set{ 1, 2,..}from this set, we proceed as , to obtain 1, we simply normalize 1: 1= 1 1.

each polynomial depends on the previous two. Table lists several families of orthogonal polynomials that can be generated from such a recurrence relation; we will see some of these families later in the course. Polynomials Scalar Product Legendre R 1 1 P n(x)P m(x)dx= 2 mn=(2n+ 1) Shifted Legendre R 1 0 P n(x)P m (x)dx= mn=(2n+ 1) Chebyshev ...

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