Climate Modeling In-Class Discussion: Legendre Polynomials.

Slides:

Advertisements

Similar presentations

Arc Length and Curvature

Advertisements

Spherical Harmonic Lighting Jaroslav Křivánek. Overview Function approximation Function approximation Spherical harmonics Spherical harmonics Some other.

Session 4: the polynomial class Constructing and manipulating objects.

Copyright © Cengage Learning. All rights reserved. 13 Vector Functions.

1 Chapter 4 Interpolation and Approximation Lagrange Interpolation The basic interpolation problem can be posed in one of two ways: The basic interpolation.

Vector Products (Dot Product). Vector Algebra The Three Products.

Signal , Weight Vector Spaces and Linear Transformations

Fundamentals of Applied Electromagnetics

The Dot Product Sections 6.7. Objectives Calculate the dot product of two vectors. Calculate the angle between two vectors. Use the dot product to determine.

Approximation on Finite Elements Bruce A. Finlayson Rehnberg Professor of Chemical Engineering.

The Cross Product of Two Vectors In Space Section

6.4 Vectors and Dot Products The Definition of the Dot Product of Two Vectors The dot product of u = and v = is Ex.’s Find each dot product.

The Cross Product Third Type of Multiplying Vectors.

Chapter 9 Function Approximation

MAT 4725 Numerical Analysis Section 8.2 Orthogonal Polynomials and Least Squares Approximations (Part II)

AP Calculus Chapter 1, Section 3

Vector Functions 10. Derivatives and Integrals of Vector Functions 10.2.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 3.3 Introduction to Polynomials.

2-1 Operations on Polynomials. Refer to the algebraic expression above to complete the following: 1)How many terms are there? 2)Give an example of like.

TODAY IN CALCULUS…  Warm Up: Review simplifying radicals  Learning Targets :  You will use special products and factorization techniques to factor polynomials.

2.1 Rates of Change and Limits. What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided.

1 Spring 2003 Prof. Tim Warburton MA557/MA578/CS557 Lecture 8.

6. Introduction to Spectral method. Finite difference method – approximate a function locally using lower order interpolating polynomials. Spectral method.

4.4 Legendre Functions Legendre polynomials. The differential equation: = -1 < x < 1 around x = 0.

F UNDAMENTAL T HEOREM OF CALCULUS 4-B. Fundamental Theorem of Calculus If f(x) is continuous at every point [a, b] And F(x) is the antiderivative of f(x)

Advanced Higher Mathematics Methods in Algebra and Calculus Geometry, Proof and Systems of Equations Applications of Algebra and Calculus AH.

Chapter 5 Polynomials: An Introduction to Algebra.

AP CALCULUS 1009 : Product and Quotient Rules. PRODUCT RULE FOR DERIVATIVES Product Rule: (In Words) ________________________________________________.

Section 4.2 Definite Integral Math 1231: Single-Variable Calculus.

Section 5.1 Length and Dot Product in ℝ n. Let v = ‹v 1, v 2, v 3,..., v n › and w = ‹w 1, w 2, w 3,..., w n › be vectors in ℝ n. The dot product.

4.4 The Fundamental Theorem of Calculus. Essential Question: How are the integral & the derivative related?

Solving Equations Binomials Simplifying Polynomials

12 Vector-Valued Functions

Section 12.3 The Dot Product

ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Eigenfunctions Fourier Series of CT Signals Trigonometric Fourier.

Simplifying Algebraic Expressions. 1. Evaluate each expression using the given values of the variables (similar to p.72 #37-49)

SEC 2: EVALUATING ABSOLUTE VALUES & SIMPLIFYING EXPRESSIONS Learning Target: 1.I will simplify complex algebraic expressions by combining like terms, adding.

1 Chapter 4 Interpolation and Approximation Lagrange Interpolation The basic interpolation problem can be posed in one of two ways: The basic interpolation.

Section 4.2 – The Dot Product. The Dot Product (inner product) where is the angle between the two vectors we refer to the vectors as ORTHOGONAL.

ECE 8443 – Pattern Recognition ECE 3163 – Signals and Systems Objectives: Eigenfunctions Fourier Series of CT Signals Trigonometric Fourier Series Dirichlet.

6.4 Vector and Dot Products. Dot Product  This vector product results in a scalar  Example 1: Find the dot product.

Function Representation & Spherical Harmonics. Function approximation G(x)... function to represent B 1 (x), B 2 (x), … B n (x) … basis functions G(x)

Sec Math II 1.3.

An inner product on a vector space V is a function that, to each pair of vectors u and v in V, associates a real number and satisfies the following.

6.4 Vectors and Dot Products Objectives: Students will find the dot product of two vectors and use properties of the dot product. Students will find angles.

The Cross Product. We have two ways to multiply two vectors. One way is the scalar or dot product. The other way is called the vector product or cross.

MA4229 Lectures 13, 14 Week 12 Nov 1, Chapter 13 Approximation to periodic functions.

The following table contains the evaluation of the Taylor polynomial centered at a = 1 for f(x) = 1/x. What is the degree of this polynomial? x T(x) 0.5.

Quantum Two 1. 2 Angular Momentum and Rotations 3.

Notes Over 1.2.

Dot Product and Angle Between Two Vectors

Sullivan Algebra and Trigonometry: Section 10.5

Polynomial and Rational Inequalities

Quantum Mechanics of Angular Momentum

12.2 Finding Limits Algebraically

Lecture 3 0f 8 Topic 5: VECTORS 5.3 Scalar Product.

Law of sines Law of cosines Page 326, Textbook section 6.1

By the end of Week 2: You would learn how to plot equations in 2 variables in 3-space and how to describe and manipulate with vectors. These are just.

Section 3.2 – The Dot Product

8-2 Multiplying and Factoring

2.7 Two-variable inequalities (linear) 3.3 Systems of Inequalities

More Multiplication Properties of Exponents

Evaluating expressions and Properties of operations

Multiplying Powers with the Same Base

Differentiate the function: {image} .

Vectors and Dot Products

ECE 6382 Fall 2016 David R. Jackson Notes 24 Legendre Functions.

15. Legendre Functions Legendre Polynomials Orthogonality

ALGEBRA I - REVIEW FOR TEST 3-3

Warm Up Simplify the expression by using distributive property and then combining like terms. x(x + 5) + 4(x + 5)

Presentation transcript:

Climate Modeling In-Class Discussion: Legendre Polynomials

P 0 (x) = 1 P 1 (x) = x P 2 (x) = (3x 2 - 1)/2 P 3 (x) = (5x 3 - 3x)/2 P 4 (x) = (35x x 2 + 3)/8 P 5 (x) = (63x x x)/8 P 6 (x) = (231x x x 2 - 5)/16 Legendre Polynomials 0 - 6

P 0 (x) = 1 P 2 (x) = (3x 2 - 1)/2 P 4 (x) = (35x x 2 + 3)/8 P 6 (x) = (231x x x 2 - 5)/16 Plots: Even Polynomials

P 1 (x) = x P 3 (x) = (5x 3 - 3x)/2 P 5 (x) = (63x x x)/8 Plots: Odd Polynomials

Why? Convenient properties on the sphere when using x = sin(lat) Some examples: (a) Even P n (e.g., above) satisfy boundary conditions 1 & 2 All = 0 at x = 0. All are finite at x = 1. Basis Functions: Legendre Polynomials (1)

Why? Convenient properties on the sphere when using x = sin(lat) Eigenfunctions of this operator on the sphere. Simplifies evaluation of the derivatives (calculus becomes algebra). (b) Basis Functions: Legendre Polynomials (2)

Why? Convenient properties on the sphere when using x = sin(lat) Polynomials of different degrees are orthogonal. (c) Basis Functions: Legendre Polynomials (3) NOTE: The integral above is like taking the dot product with vectors: (A 1,B 1 ). (A 2,B 2 ) = A 1 A 2 + B 1 B 2 = 0 if the vectors are orthogonal The “components” of P n are its values at each x.

In-Class Discussion Legendre Polynomials ~ End ~