# 1 Review of Newton Polynomials

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 cs: introduction to numerical analysis 0/0/0 Lecture 8: Polynomial Interpolation: Using Newton Polynomials and Error Analysis Instructor: Professor Amos Ron Scribes: Giordano Fusco, Mark Cowlishaw, Nathanael Fillmore Review of Newton Polynomials Recall the polynomial interpolation problem we have been studying for the last few lectures. Given X = (x 0,x,,x n ) and the value of f at the given points F = (f(x 0 ),f(x ),,f(x n )), find the polynomial p Π n so that p X = F. We showed in the previous lecture that the solution to this problem can be written as a linear combination of Newton polynomials. p n (t) = f[x 0 ] N 0 (t) + f[x 0,x ] N (t) + + f[x 0,x,...,x n ] N n (t) Where the polynomials N i (t) are the Newton polynomials. i N i (t) = (t x j ) j=0 Note that here, the empty product, corresponding to N 0 (t) is. Recall that the divided differences f[x 0,x,...,x i ] are defined using a table. x 0 D[0,0] D[0,]... D[0,N] x D[,0]... D[,N ] x N D[N,0]. Where the first row of the table corresponds to the divided differences f[x 0 ],f[x 0,x ],...,f[x 0,...,x n ], and each cell of the table is computed as follows. { xi if j = 0 D[i,j] = D[i,j ] D[i+,j ]] x i x i+j Otherwise = f[x i,x i+,...,x i+j ] () Note that the diagonal in this table, corresponding to the last entry of each row (D[i,n i] for i = 0,,...,n) computes the divided differences that we would calculate if the interpolation points X = (x 0,x,...,x n ) were given in reverse order (i.e. (x n,x n,...,x,x 0 )). As always, the final divided difference (the coefficient of the nth order Newton polynomial) is the same no matter the order of the points. f[x 0,x,...,x n,x n ] = f[x n,x,...,x n,x n ] = D[0,n] (In general, D[i,j] = f[x i,x i+,...,x i+j ] is the coefficient of the highest degree term of any jth degree polynomial that interpolates the points (x i,f(x i )),(x i+,f(x i+ )),...,(x i+j,f(x i+j )).) To illustrate, consider the following example.

2 Example.. Given the input points X = (0,,) and corresponding function values F = (,,0) find the polynomial interpolant p Π. To solve this problem using Newton polynomials, we build the following divided difference table The first row corresponds to the divided differences f[0],f[0,],f[0,,], which are the divided differences we would calculate when building the solution in the order X = (0,,). p (t) = (t 0) + (t 0)(t ) The diagonal (consisting of the last entry in each row) corresponds to the divided differences f[],f[,],f[0,,], which are the divided differences we would calculate when building the solution in the order X = (,,0). p (t) = 0 + (t ) + (t )(t ) It is easy to verify that these polynomials are equivalent. Access and Manipulation of Polynomials in Newton Form We would like to show that Newton polynomials have properties desirable for numerical analysis, particularly:. Newton polynomials are easy to evaluate. That is, the computational cost of evaluating the polynomial at a point does not grow too quickly with respect to the degree.. Newton Polynomials are easy to differentiate. That is, the computational cost of evaluating the derivative at a point does not grow too large with respect to the degree of the polynomial.. Evaluating Newton Polynomials Consider the following example. Example.. What is the value of the polynomial at the point t =? The parameters of this problem are: P(t) = + (t ) (t )(t + 7) centers:, 7 and coefficients:,, Note that the number of coefficients is always one more than the number of centers. The order in which centers and coefficients appear is important. We are going to show that it is inefficient to simply plug in the numbers and compute the operations. In particular we are going to see how the number of operations grows as size of the polynomial increases. The following table shows the number of multiplications required for each term.

3 polynomial: P(t) = + (t ) (t )(t + 7) number of multiplications: 0 In general, when the degree of the polynomial is n we have n = n(n + ) multiplications. Note that we are counting only the number of multiplication because multiplications are far more computationally expensive than additions, so this gives a good estimate of the complexity of the evaluation. This brute force method is highly inefficient. In fact suppose that we add a new term P(t) = + (t ) (t )(t + 7) + 6(t )(t + 7)(t 8) then we would need three more multiplications. However, we can observe that (t ) appears as a factor in each term and (t + 7) is a factor in most of the terms. We can make use of this fact to rewrite the polynomial as P(t) = + (t )[ (t + 7)] Notice that P (t) = (t + 7) is a Newton polynomial itself. The parameters of P (t) are the same as before, but crossing out the leftmost ones, 7,, We can rewrite the original polynomial as P (t) = +(t )P (t). This did not happen by accident. In fact suppose we have one more term, 7, 8,,, 6 Now the whole polynomial would be P (t) = + (t )P (t) (i.e. only the index of the polynomial is shifted). This indicates that each additional term requires just one more multiplication. So the number of multiplications is linear in the number of terms (and, at worst, linear in the degree of the polynomial). Now we are ready to consider the general case. The Newton polynomial of degree n is and its parameters are P n (t) = c 0 + c (t x 0 ) c n (t x 0 ) (t x n ) n centers: x 0, x,..., x n and n + coefficients : c 0, c, c,..., c n If we delete the first two parameters we have that the parameters of P n (t) = c + c (t x ) c n (t x ) (t x n ) are x,..., x n c, c,..., c n

4 In general has parameters P n j (t) = c j + c j+ (t x j ) c n (t x j ) (t x n ) The algorithm to evaluate the polynomial is. P 0 (t) = c n. for j = n,...,0. P n j+ (t) = c j + (t x j )P n j (t) We call this technique nested multiplication. x j,..., x n c j, c j+,..., c n Example.. Consider the Newton polynomial P(t) = (t ) + 7(t )(t + ) whose parameters are x 0 =, x = c 0 =, c =, c = 7 We want to evaluate P(t) at t =, which means we want to compute P (). The steps are. P 0 () = 7. P () = + ( + )7 = 0. P () = + ( )0 = 00. Differentiating Newton Polynomials We are going to show how to evaluate the derivative of a Newton polynomial efficiently. Consider the following example. Example.. Find the derivative of the following cubic polynomial and evaluate it at a certain point. P(t) = (t ) + 7(t )(t + ) + 9(t )(t + )(t ) Just finding the derivative symbolically could be difficult if we use the product rule. In fact, for the last term we would have to take the derivative of (t ) and multiply by the other terms, then add the derivative of (t + ) times the other terms, and so on... You can imagine that as the degree of the polynomial grows, this task quickly becomes quite difficult (or at least time-consuming). Fortunately, there is a method that makes this task easier. The idea is to look at the recurrence relation defining the Newton representation and look at what happens when we differentiate it. Recall that P n (t) = c 0 + (t x 0 )P n (t) When c 0 is the first coefficient, and x 0 the first center. Differentiating on both sides we get P n (t) = P n (t) + (t x 0 )P n (t) We can easily compute P n(t) if we already know P n (t) and P n (t). The algorithm is

5 . P 0 (t) = c n. P 0 (t) = 0. for j = n,...,0. P n j+ (t) = c j + (t x j )P n j (t). P n j (t) = P n j (t) + (t x j )P n j (t) In other words, we are building the following two vectors starting from the top. The arrows indicate the dependences. P 0 (t) P 0 (t) = P n (t) P n (t) Note that the coefficients do not appear explicitly in the formula of the derivatives (vector on the right). However, the coefficients are used in the computation of the polynomials (vector on the left), and the polynomials are used in the computation of the derivatives. So the derivatives still depend on the coefficients. Note that this method is very efficient. Once we have calculated the values of P i (t) (at a cost of n multiplications), we need only one multiplication per term to evaluate the derivative, for a total of n multiplications. On the other hand, it is not easy to integrate Newton polynomials, but later in the course we will see some efficient methods for integrating functions and these methods will work for Newton polynomials. We end with another example. Example.. Evaluate p(t) = + (t-) + (t-)(t-7) and its derivative. Nodes: 7 Coefficients: Thus: p 0 (t) = p 0 (t) = 0 p (t) = + (t 7)p 0 (t) p (t) = p 0(t) + (t 7)p 0 (t) p (t) = + (t )p (t) p (t) = p (t) + (t )p (t) So if, say, t =, we just plug into the expressions above.

6 Error Analysis of Polynomial Interpolation It is very important to understand the error in polynomial interpolation. Given a function f : [a,b] R and a set of points X := {x 0,...,x n } [a,b], a polynomial interpolant has the property that the value of the polynomial at the points X is the same as the function. We denote this as P X = f X Note that the error of our polynomial P with respect to the true function f is defined as e(t) := f(t) P(t) for all t [a,b]. Ultimately, we are going to give an upper bound for the error. Of course, it is easy to bound the error very loosely, for example it is easy to see that e(t) max x [a,b] f(x) +max x [a,b] P(x) holds for any t [a,b]. However, this bound may be much larger than the true error - we want a tighter estimate of the error, and one that gives us more insight into the sources of error. Figure : Interpolating one point What do we expect, therefore, of a good error formula?. First, we would expect that e(x i ) = 0 for all our interpolation points x i X.. Second, consider interpolating a function f based on one point (x 0,f(x 0 )), as in Figure. Of course, there will always be some error in this case, unless f is itself constant. If the function f moves very fast, then the error will be big. If the function f moves slowly, then there will be less error. The derivative is what controls this. So we require that f not be too large. Figure : Interpolating two points Consider next interpolating a function f based on two points, as in Figure. Here the error doesn t look too bad, even though f is rather large, because the slope of our interpolating polynomial p is also large. So here we need to look at the second derivative, f. 6

7 So we can guess that a good error formula if we interpolate n + points will somehow involve () (t x i ) for all points of interpolation x i, and () the (n + )st derivative. We can see this in another way from the examples below. Example.. The main problem is that we can always cheat. Suppose that we are given the points of Figure a. The points are aligned and the solution is the linear polynomial of Figure b (note that for aligned points the polynomial is linear regardless of the number of points). However, knowing the points, we could decide to choose a very bad function for which polynomial interpolation behaves extremely poorly, as in Figure c (a) (b) (c) Figure : Example of bad polynomial interpolation The way to recognize this kind of function is to look at the derivative. The function of the previous example has a derivative that changes frequently. Now, consider another example Example.. The first derivative of the function in Figure changes frequently, but the interpolation works pretty well in this case Figure : Example of good polynomial interpolation From example. we can see that looking only at the first derivative we might not get enough information. In fact, the general formula for calculating the error of a polynomial interpolant P is given by f(t) P(t) = f(n+) (c) (n + )! (t x 0)(t x ) (t x n ) where c lies somewhere in the smallest interval containing X {t}. 7

8 Note that to do the error analysis we do not need to know the interpolating polynomial. We just need to know the derivative of the function and the position of the points. Also note that we are not going to compute the exact value of the error, but we are looking for a good bound. Example.. Suppose that we are given the function f(t) = sin t and the points X = (.,.,.). We want to estimate f() p(), i.e., we want to estimate the error at the point t =. Using the error formula, we calculate that f() P() = f() (c t ) (t.)(t.)(t.)! where the point c t depends on the value of t. In other words we don t know exactly where c t is, but we know that it lies in the interval containing X. In this case we consider c t [.,.]. The rd derivative of sin t is cos t, which is bounded in absolute value by, so f () (c t ) So we can estimate that f() P() (.)(.)(.)! = 6 Note that, since we do not know the precise value of c t, this is almost certainly an overestimate of the exact error. However, without more knowledge, this is the best we can do. 8

### 1 Cubic Hermite Spline Interpolation

cs412: introduction to numerical analysis 10/26/10 Lecture 13: Cubic Hermite Spline Interpolation II Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mark Cowlishaw, Nathanael Fillmore 1 Cubic Hermite

### 1 Review of Least Squares Solutions to Overdetermined Systems

cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares

### Lecture 3: Solving Equations Using Fixed Point Iterations

cs41: introduction to numerical analysis 09/14/10 Lecture 3: Solving Equations Using Fixed Point Iterations Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mark Cowlishaw, Nathanael Fillmore Our problem,

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Sect 3.2 Synthetic Division

94 Objective 1: Sect 3.2 Synthetic Division Division Algorithm Recall that when dividing two numbers, we can check our answer by the whole number (quotient) times the divisor plus the remainder. This should

### since by using a computer we are limited to the use of elementary arithmetic operations

> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations

### Newton Divided-Difference Interpolating Polynomials

Section 3 Newton Divided-Difference Interpolating Polynomials The standard form for representing an nth order interpolating polynomial is straightforward. There are however, other representations of nth

### Numerical methods for finding the roots of a function

Numerical methods for finding the roots of a function The roots of a function f (x) are defined as the values for which the value of the function becomes equal to zero. So, finding the roots of f (x) means

### INTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y).

INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 =0, x 1 = π 4, x

### Computational Geometry Lab: FEM BASIS FUNCTIONS FOR A TETRAHEDRON

Computational Geometry Lab: FEM BASIS FUNCTIONS FOR A TETRAHEDRON John Burkardt Information Technology Department Virginia Tech http://people.sc.fsu.edu/ jburkardt/presentations/cg lab fem basis tetrahedron.pdf

### Lectures 5-6: Taylor Series

Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

### What s Ahead. Interpolating Methods: Lagrange, Newton, Hermite, Monomial, and Chebyshev

Chapter 6 Interpolation ***** Updated 5/21/12 HG What s Ahead Polynomial Interpolation: Existence and Uniqueness Interpolation Error Interpolating Methods: Lagrange, Newton, Hermite, Monomial, and Chebyshev

### November 16, 2015. Interpolation, Extrapolation & Polynomial Approximation

Interpolation, Extrapolation & Polynomial Approximation November 16, 2015 Introduction In many cases we know the values of a function f (x) at a set of points x 1, x 2,..., x N, but we don t have the analytic

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### MATH 2300 review problems for Exam 3 ANSWERS

MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

### The finite field with 2 elements The simplest finite field is

The finite field with 2 elements The simplest finite field is GF (2) = F 2 = {0, 1} = Z/2 It has addition and multiplication + and defined to be 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 0 0 = 0 0 1 = 0

### Inverses. Stephen Boyd. EE103 Stanford University. October 27, 2015

Inverses Stephen Boyd EE103 Stanford University October 27, 2015 Outline Left and right inverses Inverse Solving linear equations Examples Pseudo-inverse Left and right inverses 2 Left inverses a number

### 4.3 Lagrange Approximation

206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

### Rational Polynomial Functions

Rational Polynomial Functions Rational Polynomial Functions and Their Domains Today we discuss rational polynomial functions. A function f(x) is a rational polynomial function if it is the quotient of

### Lecture 2: Universality

CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal

### Higher Order Linear Differential Equations with Constant Coefficients

Higher Order Linear Differential Equations with Constant Coefficients Part I. Homogeneous Equations: Characteristic Roots Objectives: Solve n-th order homogeneous linear equations where a n,, a 1, a 0

### Polynomials Classwork

Polynomials Classwork What Is a Polynomial Function? Numerical, Analytical and Graphical Approaches Anatomy of an n th -degree polynomial function Def.: A polynomial function of degree n in the vaiable

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### The Epsilon-Delta Limit Definition:

The Epsilon-Delta Limit Definition: A Few Examples Nick Rauh 1. Prove that lim x a x 2 = a 2. (Since we leave a arbitrary, this is the same as showing x 2 is continuous.) Proof: Let > 0. We wish to find

### Practical Numerical Training UKNum

Practical Numerical Training UKNum 7: Systems of linear equations C. Mordasini Max Planck Institute for Astronomy, Heidelberg Program: 1) Introduction 2) Gauss Elimination 3) Gauss with Pivoting 4) Determinants

### Lecture 10: Distinct Degree Factoring

CS681 Computational Number Theory Lecture 10: Distinct Degree Factoring Instructor: Piyush P Kurur Scribe: Ramprasad Saptharishi Overview Last class we left of with a glimpse into distant degree factorization.

### UNIT - I LESSON - 1 The Solution of Numerical Algebraic and Transcendental Equations

UNIT - I LESSON - 1 The Solution of Numerical Algebraic and Transcendental Equations Contents: 1.0 Aims and Objectives 1.1 Introduction 1.2 Bisection Method 1.2.1 Definition 1.2.2 Computation of real root

### Zeros of Polynomial Functions

Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

### i = 0, 1,, N p(x i ) = f i,

Chapter 4 Curve Fitting We consider two commonly used methods for curve fitting, namely interpolation and least squares For interpolation, we use first polynomials then splines Then, we introduce least

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### Zeros of Polynomial Functions

Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### 3.2 Inequalities involving polynomials and rational functions

56 CHAPTER 3. RATIONAL FUNCTIONS 3.2 Inequalities involving polynomials and rational functions Textbook section 3.6 Now that we know a fairly extensive range of basic functions, we have the tools to study

### MINI LESSON. Lesson 5b Solving Quadratic Equations

MINI LESSON Lesson 5b Solving Quadratic Equations Lesson Objectives By the end of this lesson, you should be able to: 1. Determine the number and type of solutions to a QUADRATIC EQUATION by graphing 2.

### Differentiability and some of its consequences Definition: A function f : (a, b) R is differentiable at a point x 0 (a, b) if

Differentiability and some of its consequences Definition: A function f : (a, b) R is differentiable at a point x 0 (a, b) if f(x)) f(x 0 ) x x 0 exists. If the it exists for all x 0 (a, b) then f is said

### Solution to Homework 2

Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

### 4.5 Chebyshev Polynomials

230 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION 4.5 Chebyshev Polynomials We now turn our attention to polynomial interpolation for f (x) over [ 1, 1] based on the nodes 1 x 0 < x 1 < < x N 1. Both

### Continuous Functions, Smooth Functions and the Derivative

UCSC AMS/ECON 11A Supplemental Notes # 4 Continuous Functions, Smooth Functions and the Derivative c 2004 Yonatan Katznelson 1. Continuous functions One of the things that economists like to do with mathematical

### the recursion-tree method

the recursion- method recurrence into a 1 recurrence into a 2 MCS 360 Lecture 39 Introduction to Data Structures Jan Verschelde, 22 November 2010 recurrence into a The for consists of two steps: 1 Guess

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

### 3 Polynomial Interpolation

3 Polynomial Interpolation Read sections 7., 7., 7.3. 7.3.3 (up to p. 39), 7.3.5. Review questions 7. 7.4, 7.8 7.0, 7. 7.4, 7.7, 7.8. All methods for solving ordinary differential equations which we considered

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

### Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way

Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### Memoization/Dynamic Programming. The String reconstruction problem. CS125 Lecture 5 Fall 2016

CS125 Lecture 5 Fall 2016 Memoization/Dynamic Programming Today s lecture discusses memoization, which is a method for speeding up algorithms based on recursion, by using additional memory to remember

### Linear and quadratic Taylor polynomials for functions of several variables.

ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

### 1 Prior Probability and Posterior Probability

Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which

### Lecture 5 - Triangular Factorizations & Operation Counts

LU Factorization Lecture 5 - Triangular Factorizations & Operation Counts We have seen that the process of GE essentially factors a matrix A into LU Now we want to see how this factorization allows us

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### MCSE-004. Dec 2013 Solutions Manual IGNOUUSER

MCSE-004 Dec 2013 Solutions Manual IGNOUUSER 1 1. (a) Verify the distributive property of floating point numbers i.e. prove : a(b-c) ab ac a=.5555e1, b=.4545e1, c=.4535e1 Define : Truncation error, Absolute

### 3.5 Spline interpolation

3.5 Spline interpolation Given a tabulated function f k = f(x k ), k = 0,... N, a spline is a polynomial between each pair of tabulated points, but one whose coefficients are determined slightly non-locally.

### Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 01

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 01 Welcome to the series of lectures, on finite element analysis. Before I start,

### The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

### Examples of Functions

Examples of Functions In this document is provided examples of a variety of functions. The purpose is to convince the beginning student that functions are something quite different than polynomial equations.

### Roots of Polynomials

Roots of Polynomials (Com S 477/577 Notes) Yan-Bin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### Power Series. We already know how to express one function as a series. Take a look at following equation: 1 1 r = r n. n=0

Math -0 Calc Power, Taylor, and Maclaurin Series Survival Guide One of the harder concepts that we have to become comfortable with during this semester is that of sequences and series We are working with

### Draft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then

CHAPTER : DERIVATIVES Specific Expectations Addressed in the Chapter Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function,

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Lecture 21 Integration: Left, Right and Trapezoid Rules

Lecture 1 Integration: Left, Right and Trapezoid Rules The Left and Right point rules In this section, we wish to approximate a definite integral b a f(x)dx, where f(x) is a continuous function. In calculus

### The Method of Lagrange Multipliers

The Method of Lagrange Multipliers S. Sawyer October 25, 2002 1. Lagrange s Theorem. Suppose that we want to maximize (or imize a function of n variables f(x = f(x 1, x 2,..., x n for x = (x 1, x 2,...,

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### Chapter 2. Vectors and m-files

Chapter 2 Vectors and m-files To gain a better appreciation for the way Matlab works, it may be helpful to recall two very different methods of integration from calculus. Most of the time students focus

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### To formulate more complex mathematical statements, we use the quantifiers there exists, written, and for all, written. If P(x) is a predicate, then

Discrete Math in CS CS 280 Fall 2005 (Kleinberg) Quantifiers 1 Quantifiers To formulate more complex mathematical statements, we use the quantifiers there exists, written, and for all, written. If P(x)

### In this lesson you will learn to find zeros of polynomial functions that are not factorable.

2.6. Rational zeros of polynomial functions. In this lesson you will learn to find zeros of polynomial functions that are not factorable. REVIEW OF PREREQUISITE CONCEPTS: A polynomial of n th degree has

### ROOTFINDING : A PRINCIPLE

ROOTFINDING : A PRINCIPLE We want to find the root α of a given function f(x). Thus we want to find the point x at which the graph of y = f(x) intersects the x-axis. One of the principles of numerical

### Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

### max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

### ( % . This matrix consists of \$ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&

Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important

### 1 Mathematical Induction

Extra Credit Homework Problems Note: these problems are of varying difficulty, so you might want to assign different point values for the different problems. I have suggested the point values each problem

### Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series

1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,

### Chapter 7. Induction and Recursion. Part 1. Mathematical Induction

Chapter 7. Induction and Recursion Part 1. Mathematical Induction The principle of mathematical induction is this: to establish an infinite sequence of propositions P 1, P 2, P 3,..., P n,... (or, simply

### Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

### Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

### > 2. Error and Computer Arithmetic

> 2. Error and Computer Arithmetic Numerical analysis is concerned with how to solve a problem numerically, i.e., how to develop a sequence of numerical calculations to get a satisfactory answer. Part

### LEAST SQUARES APPROXIMATION

LEAST SQUARES APPROXIMATION Another approach to approximating a function f(x) on an interval a x b is to seek an approximation p(x) with a small average error over the interval of approximation. A convenient

### Direct Methods for Solving Linear Systems. Linear Systems of Equations

Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### a = x 1 < x 2 < < x n = b, and a low degree polynomial is used to approximate f(x) on each subinterval. Example: piecewise linear approximation S(x)

Spline Background SPLINE INTERPOLATION Problem: high degree interpolating polynomials often have extra oscillations. Example: Runge function f(x) = 1 1+4x 2, x [ 1, 1]. 1 1/(1+4x 2 ) and P 8 (x) and P

### Sequences and Series

Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite

### Iterative Methods for Computing Eigenvalues and Eigenvectors

The Waterloo Mathematics Review 9 Iterative Methods for Computing Eigenvalues and Eigenvectors Maysum Panju University of Waterloo mhpanju@math.uwaterloo.ca Abstract: We examine some numerical iterative

### General Framework for an Iterative Solution of Ax b. Jacobi s Method

2.6 Iterative Solutions of Linear Systems 143 2.6 Iterative Solutions of Linear Systems Consistent linear systems in real life are solved in one of two ways: by direct calculation (using a matrix factorization,

### 8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

### Lecture 2: The Complexity of Some Problems

IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 2: The Complexity of Some Problems David Mix Barrington and Alexis Maciel July 18, 2000

### We shall turn our attention to solving linear systems of equations. Ax = b

59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### CHAPTER 5 Round-off errors

CHAPTER 5 Round-off errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any

### 5 Numerical Differentiation

D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### 1. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither.

Math Exam - Practice Problem Solutions. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither. (a) 5 (c) Since each row has a leading that

### 1 The Brownian bridge construction

The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof

### 4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

### Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f