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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 .Next, to obtain 2, we need to ensure that it is orthogonal to 1, and then normalize an intermediate step, we seek a function 2of the form 2= 2+c12 1such that 1| 2 = 0. Then, we can set 2= 2/ 2 . Taking the scalar product of both sides ofthe above equation with 1, we obtain0 = 1| 2 = 1| 2 +c12 1| 1.

That is, any family of orthogonal polynomials satis es a three-term recurrence relation, in which 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 ...

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