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Romberg Integration - USM
More precisely, suppose that we compute approximations T 1;1 and T 2;1 to the integral, using the Composite Trapezoidal Rule with one and two subintervals, respectively. That is, T 1;1 = b a 2 [f(a) + f(b)] T 2;1 = b a 4 f(a) + 2f a+ b 2 + f(b) : Suppose that f has continuous derivatives of all orders on [a;b]. Then, the Composite Trapezoidal
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Linear Interpolating Splines - USM
www.math.usm.eduLinear Interpolating Splines We have seen that high-degree polynomial interpolation can be problematic. However, if the tting function is only required to have a few continuous derivatives, then one can construct a piecewise polynomial to t the data. We now precisely de ne what we mean by a piecewise polynomial.
Orthogonality of Bessel Functions - USM
www.math.usm.eduNormalization Now that we have orthogonal Bessel functions, we seek orthonormal Bessel functions. From Z a 0 ˆ[J (kˆ)]2 dˆ= lim k0!k a[k 0J (ka)J (ka) kJ (ka)J (ka)]
Three-Dimensional Coordinate Systems
www.math.usm.eduJim Lambers MAT 169 Fall Semester 2009-10 Lecture 17 Notes These notes correspond to Section 10.1 in the text. Three-Dimensional Coordinate Systems
Properties of Sturm-Liouville Eigenfunctions and Eigenvalues
www.math.usm.eduReal Eigenvalues Just as a symmetric matrix has real eigenvalues, so does a (self-adjoint) Sturm-Liouville operator. Proposition 2 The eigenvalues of a regular or periodic Sturm-Liouville problem are real.
The Secant Method - USM
www.math.usm.eduJim Lambers MAT 772 Fall Semester 2010-11 Lecture 4 Notes These notes correspond to Sections 1.5 and 1.6 in the text. The Secant Method One drawback of Newton’s method is that it is necessary to evaluate f0(x) at various points, which may not be practical for some choices of f.
Gram-Schmidt Orthogonalization - USM
www.math.usm.edueach 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 ...
The Wronskian - math.usm.edu
www.math.usm.eduWe de ne a second-order linear di erential operator Lby L[y] = y00+ p(t)y0+ q(t)y: Then, a initial value problem with a second-order homogeneous linear ODE can be stated as L[y] = 0; y(t 0) = y 0; y0(t 0) = z 0: We state a result concerning existence and uniqueness of solutions to such ODE, analogous to the Existence-Uniqueness Theorem for rst ...
Gaussian Elimination and Back Substitution
www.math.usm.eduThe process of eliminating variables from the equations, or, equivalently, zeroing entries of the corresponding matrix, in order to reduce the system to upper …
Properties of Sturm-Liouville Eigenfunctions and …
www.math.usm.eduThe following essential result characterizes the behavior of the entire set of eigenvalues of Sturm-Liouville problems. Proposition 6 The set of eigenvalues of a regular Sturm-Liouville problem is countably in nite, and is a monotonically increasing sequence 0 < 1 < 2 < < n< n+1 < with lim n!1 n = 1. The same is true for a periodic Sturm ...
Orthogonality of Bessel Functions - USM
www.math.usm.eduOrthogonality of Bessel Functions Since Bessel functions often appear in solutions of PDE, it is necessary to be able to compute coe cients of series whose terms include Bessel functions. Therefore, we need to understand their orthogonality properties. Consider the Bessel equation ˆ2 d2J (kˆ) dˆ2 + ˆ dJ (kˆ) dˆ + (k2ˆ2 2)J (kˆ) = 0 ...
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