Math 255 Honors: Gram-Schmidt Orthogonalization on the Space of Polynomials

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1 Math 55 Honors: Gram-Schmidt Orthogonalization on the Space of Polynomials David Moore May, 03 Abstract Gram-Schmidt Orthogonalization is a process to construct orthogonal vectors from some basis for a vector space. In this paper we will discuss the Gram-Schmidt process on the set of all polynomials with degree N, use the Gram-Schmidt process to generate the Legendre Polynomials, using Mathematica code, in their normalized and unnormalized forms. Space Axioms. Vector Space Axioms Any polynomial must be in the form N n=0 c nx n, for constants c 0,..., c n, where it is assumed that the polynomial is a function of some real paramater x. It is clear that the following axioms are satisfied, due to the properties and definitions of addition and multiplication in R. For any polynomials px), qx), rx), and scalars c, d R.. Addition Axioms a) p + q = q + p. b) p + q) + r = p + q + r). c) 0 + p = p + 0 = p. d) p) + p = 0.. Scalar Multiplication Axioms a) 0 p = 0 b) p = p c) cd)p = cdp) 3. Distributive Axioms a) cp + q) = cp + cq b) c + d)p = cp + dp Since the axioms are satisfied, the space of such polynomials is a vector space.

2 . Inner Product Space Axioms We can define the inner product between two polynomials, p, q in terms of a real integral: p, q = px)qx)dx The following axioms are satisfied, again with polynomials p, q, r, and real values c, d.. p, q = q, p, since px)qx) = qx)px) within the integral.. cp, q = cpx)qx)dx = c p, q. 3. p + r, q = px)qx) + rx)qx))dx = p, q + r, q. 4. p, p = px) dx > 0. So, the given definition satisfies all the properties of an inner product. We will also make use of the definition: A = A, A Now that we ve established the inner product and vector space, we can define Gram-Schmidt Orthogonalization. Gram-Schmidt Process. Definition The Gram-Schmidt process is defined as follows. Assume that S n is a basis with N elements for the set of polynomials of degree less than or equal to N. The Gram-Schmidt process creates one list of orthogonal vectors, w n.. let w 0 = S 0, e 0 = w 0 / w 0, w 0.. Then, define w = S w 0 S,w 0 w 0,w And in general, let w n = S n n i=0 w n Sn,wi w i,w i.. Properties Theorem. A nonzero vector A is orthogonal to a finite set of orthogonal vectors S if and only if it can be obtained from the Gram-Schmidt Process, from some vector C. ie, A = C C, S / S S... C, S n / S n S n, for some C in the vector space).

3 Proof. ) If A is obtained from the Gram-Schmidt process from some vector C, and is nonzero, then for any vector S n S, where S has N elements, A, S n = C S i C, S i / S i, S n ) i= = C, S n = C, S n S i C, S i / S i, S n i= ) C, S i / S i S i, S n 3) i= = C, S n C, S n / S n S n, S n 4) = C, S n C, S n / S n S n 5) = C, S n C, S n 6) = 0 7) Where 3) follows from the definition of a set of orthogonal vectors; S i, S j = 0 whenever i j. Since A, S n = 0, and since n was arbitrary, this means A is orthogonal to every vector in S. ) If A is nonzero vector orthogonal to every S i, then letting C = A, applying the Gram-Schmidt process to C yields: C = C S i S i, C / S i 8) i= S i 0/ S i 9) i= = C 0) = A ) So, since a vector produced from one step of the Gram-Schmidt orthogonalization process will be orthogonal to all vectors previous to it, using the process with {S n } as an input gives you the orthogonal basis {w n } as an output. The above theorem also implies that if you let S be a completely arbitrary basis for the space, then applying the Gram-Schmidt process gives you the general form of all possible orthogonal basis vectors. 3 Legendre Polynomials The most straightforward basis for the space of polynomials degree less than or equal to N is S = {, x, x,..., x N }, but, for example,, x = x dx = ) 3 ) 3 )/ = /3, so S is not an orthogonal basis. However, if we apply the Gram-Schmidt orthogonalization process to the set, letting N = for example, we wind up with: 3

4 . w 0 =.. w = x, x /, = x 0 = x. 3. w = x, x /, x, x / x, x. Since the rightmost term is the integral from to of an odd function, it s zero. The middle term evaluates to /3, and so w = x /3 = /33x ). Continuing to higher N, we end up with multiples of a family called the Legendre polynomials. 4 Legendre Polynomials Uniqueness A simple orthogonal basis could be in the form S = {c 0,0), c 0,) +c,) x,..., N i=0 c i,n)x i }, for some real constants {c i,j) } If we orthogonalize this basis, then as above, up to N = :. w 0 = c 0,0).. w = c 0,) + c,) x c 0,0) c0,0), c 0,) + c,) x /c 0,0) ) = c 0,) + c,) x /c 0,0) ) c 0,0)c 0,) + c 0,0) c,) x)dx = c 0,) + c,) x c 0,) = c,) x The integrals can take long to evaluate, but we find that each w i is just some multiple of the Legendre polynomials found in the last section, and that the constants c 0,), c 0,), c,) etc. disappear from the final result. The following is a mathematica-generated table continuing the process up to N = Table Table of Legendre Polynomials, generated by the code in appendix B: N Gram-Shmidt Generated Standard Normalized 0 c 0,0) xc,) x 3 3x ) c,) 3x ) x 5x 3 ) c 3,3) x 5x 3 ) 35x 4 30x + 3 ) c 4,4) 8 35x 4 30x + 3 ) 3 x x ) x 5x 3 ) 35x 4 30x + 3 ) This shows that the Legendre polynomials are unique in that any orthogonal basis consisting of polynomials of increasing degree, starting at a 0th degree polynomial, will just be multiples of the Legendre polynomials. To uniquely determine each polynomial, we can add an extra N equations. The standard Legendre Polynomials are constructed as in the table above, with the added condition that P n, P n = n + 4

5 The normalized Legendre Polynomials are constructed by specifying that P n, P n =. 4. Figure Plot of the first 3 normalized Legendre Polynomials 5 Appendix A Orthogonalization Code 5

6 B Legendre Polynomial Generation 6

7 C Plot Generation 7

APPENDIX B GRAM-SCHMIDT PROCEDURE OF ORTHOGONALIZATION. Let V be a finite dimensional inner product space spanned by basis vector functions

301 APPENDIX B GRAM-SCHMIDT PROCEDURE OF ORTHOGONALIZATION Let V be a finite dimensional inner product space spanned by basis vector functions {w 1, w 2,, w n }. According to the Gram-Schmidt Process an