Appendix A. Cubic Spline
|
|
|
- Laurence Simpson
- 5 years ago
- Views:
Transcription
1 Numerical Simulation of Space Plasmas (I [AP-4036] Appendix A by Ling-Hsiao Lyu April 017 Appendix A Cubic Spline We can write the piece-wise continuous function f cubic spline in the following fm: f (x f (x (x +1 (x +1 + f (x +1 (x (x +1 + [a (x (x +1 + b ](x (x +1 (x +1 (A1 The constants {a, b, f 1 n 1} are chosen such that the matching conditions f cubic spline can be fulfilled, ie, and d f (x 1 x x dx d f (x x x +1 dx x x x x d f (x 1 x x d f (x x x +1 d d x x x x One can obtain the following two types of recursion fmula: f 1 + f [ + 1 ]+ f (x f h 0 (x 1 + ( 1 (A f 1 + f [ + ]+ f (x +1 6 [ f 1 1 h 0 (x f 0 1 ] 1 (A3 where f 0 [ f (x +1 f (x ] / [x +1 ] and x +1 As we can see from equations (A and (A3, the coefficients on the left-hand side depend only on the distribution of grids F a given tabulate data set: { (x, f 1 n }, we can solve equation (A (A3, and evaluate the first and the second derivatives f and f at each grid x, as well as the constants {a, b, f 1 n 1}, based on given boundary conditions at and We can also evaluate the integration value, ie, the area under the cubic-spline curve, Area f (xdx This integration value is useful in inetic plasma simulation to determine mass density, charge density, current density, and thermal pressure at each grid point Exercise A1 Verify Equation (A and Equation (A3 A-1
2 Numerical Simulation of Space Plasmas (I [AP-4036] Appendix A by Ling-Hsiao Lyu April 017 Verifying Equation (A Let f ( x f 1 x + f x + (x (x (ax + b (1 where f 1 f ( and f f ( To express the undetermined constants, a and b, in terms of the first derivative of f (x at x and x, we tae derivative of Eq (1 It yields f f ( x f + [(x + (x ](ax + b + (x (x a ( Let f 1 f ( and f f ( It yields and f f 1 f + ( (a + b (3 f f f + ( (a + b (4 Solving Eqs (3 and (4 f the unnown constants, a and b, it yields f a 1 + f ( f f 1 ( (5 3 b 1 f 1 f 1 + f ( (x + x + f f 1 1 ( (x + x (6 3 1 Substituting Eqs (5 and (6 into Eqs (3 and (4 to eliminate a and b, it yields Thus, and f ( x f 1 x + f x + (x (x {(x x ( f 1 + (x f f [(x + (x ]} f ( x f + (x (x ( f ( 1 + f f + [(x x + (x x ] 1 {(x x ( f 1 + (x f f f 1 [(x + (x ]} f ( x [(x + (x ] ( f ( 1 + f f + ( {(x x f + (x x f f f [(x + (x ]} A- (7 (8 (9
3 Numerical Simulation of Space Plasmas (I [AP-4036] Appendix A by Ling-Hsiao Lyu April 017 Eq (9 can be written as f ( x ( { f [(x x + (x x ]+ f [(x x + (x x ] f [(x + (x ]} (10 Let f 1 f ( and f f ( Eq (10 yields Liewise, f f (x 0 x f 0 x x 0 + f 1 x f 1 4 f f f f 1 ( (11 f f + 4 f 1 6 f f 1 ( (1 + (x (x {(x x ( 1 f 0 + (x f 1 f 0 [(x + (x ]} it yields f 0 4 f f f 0 ( (13 Eqs (11 and (14 yields f 1 f + 4 f f 0 ( (14 f f + ( + f 1 + f 3[ 1 f 0 ( + f f 1 ( ] (15 Let x +1 and ( f 0 ( f +1 f / (x +1 Eq (15 yields 1 f 1 + ( f + 1 f +1 3[ ( f ( f 0 ] ( f 1 + ( 1 +1 f + 1 f +1 3[( f ( f 0 ] (17 A-3
4 Numerical Simulation of Space Plasmas (I [AP-4036] Appendix A by Ling-Hsiao Lyu April 017 Case 1: Step 1: Determine { f, f n 1} from a tabulate data set { (x, f 1 n } with fixed boundary conditions of f at x and x Given f ( and f (, equation (A can be rewritten as f B + f +1 C 1 + C f 1 f 1 + f B + f +1 C 1 + C f 1 + f B 1 + C f +1 C f f 3 n f n 1 (A4 B C 0! 0 1 B 3 C B n C n 0! 0 1 B n 1 f ( f (x 3 f ( f ( + ( C f ( ( + (x 3 C 3 ( 3 + ( C n ( + C n 1 f ( C n 1 (A4' where B ( + 1, C 1, f n 1 Equation (A4 (A4' is the governing equation of { f, f n 1} with a fixed boundary condition of f at x and x Step : Evaluate { f, f 1 n}, {a, b, f 1 n 1}, and Area f (xdx from the results obtained in step 1 Using the matching conditions f the cubic spline, we can show that f 1 n 1, we have a [ f +1 + f f 0 ] b [ f 0 f ] f ( / [ f f +1 ] (A5 (A6 (A7 f n, we have f ( ( / h n 1 [ f + f ( ] (A8 The area under the cubic spline curve between x, is given by A-4
5 Numerical Simulation of Space Plasmas (I [AP-4036] Appendix A by Ling-Hsiao Lyu April 017 x f (x dx (x +1 [ f (x + f (x +1 a 1 b +1 6 ] (A9 x The Area f (xdx can be obtained by summation of equation (A9 over all intervals Case : Step 1: Determine { f, f 1 n 1} from a tabulate data set { (x, f 1 n } with periodic boundary conditions: f ( f (, f ( f (, and f ( f ( Since f ( f (, we have f 0 (x 0 f 0 Given periodic boundary conditions: f ( f ( and f ( f (, equation (A can be rewritten as f + f B + f +1 C + C f 1 + f B + f +1 C 1 + C f 1 + f B + f ( C 1 + C f 1 f n f n 1 (A10 B 1 C B C! 0 0!!! 0 0! 1 B n C n C n B n 1 f ( f ( f ( f + ( C 1 ( + ( C ( 3 + ( C n ( + C n 1 (A10' where B ( + 1, C 1, f 1 n 1 Equation (A10 (A10' is the governing equation of { f, f 1 n 1} with periodic boundary conditions: f ( f ( and f ( f ( Step : Step in Case is the same as Step in Case 1 A-5
6 Numerical Simulation of Space Plasmas (I [AP-4036] Appendix A by Ling-Hsiao Lyu April 017 Case 3: Step 1: Determine { f, f n 1} from a tabulate data set { (x, f 1 n } with fixed boundary conditions of f at x and x Given f ( and f (, equation (A3 can be rewritten as f B + f +1 C (6 / 1 [ f 0 f 0 1 ] f ( f 1 + f B + f +1 C (6 / 1 [ f 0 f 0 1 ] f 1 + f B (6 / 1 [ f 0 f 0 1 ] f ( C f f 3 n f n 1 (A11 B C 0! 0 1 B 3 C B n C n 0! 0 1 B n 1 f ( f (x 3 f ( f (6 / h 1 [ f 0 ( f 0 ( ] f ( (6 / h [ f 0 (x 3 f 0 ( ] (6 / h n 3 [ f 0 ( f 0 ( 3 ] (6 / h n [ f 0 f 0 ( ] C n 1 f ( (A11' where B ( + 1, C 1, f n 1 Equation (A11 (A11' is the governing equation of { f, f n 1} with a fixed boundary condition of f at x and x Step : Evaluate { f, f 1 n}, {a, b, f 1 n 1}, and Area f (xdx from the results obtained in step 1 Using the matching conditions f the cubic spline, we can show that f 1 n 1, we have a (h / 6[ f +1 f ] b (h / 6[ f +1 + f ] f f 0 ( / 6[ f + f +1 ] (A1 (A13 (A14 f n, we have f ( f 0 + (h n 1 / 6[ f + f ( ] (A15 A-6
7 Numerical Simulation of Space Plasmas (I [AP-4036] Appendix A by Ling-Hsiao Lyu April 017 Substituting equations (A1 and (A13 into equation (A9, one can obtain the area under the cubic-spline curve between x The Area f (xdx can be obtained by summation of equation (A9 over all intervals Case 4: Step 1: Determine { f, f 1 n 1} from a tabulate data set { (x, f 1 n } with periodic boundary conditions: f ( f (, f ( f (, and f ( f ( Since f ( f (, we have f 0 (x 0 f 0 Given periodic boundary conditions: f ( f (, and f ( f (, equation (A3 can be rewritten as f + f B + f +1 C (6 / h n 1 [ f 0 f 0 ] f 1 + f B + f +1 C (6 / 1 [ f 0 f 0 1 ] f 1 + f B + f ( C (6 / 1 [ f 0 f 0 1 ] f 1 f n f n 1 (A16 B 1 C B C! 0 0!!! 0 0! 1 B n C n C n B n 1 f ( f ( f ( f (6 / h n 1 [ f 0 ( f 0 ] (6 / h 1 [ f 0 ( f 0 ( ] (6 / h n 3 [ f 0 ( f 0 ( 3 ] (6 / h n [ f 0 f 0 ( ] (A16' where B ( + 1, C 1, f 1 n 1 Equation (A16 (A16' is the governing equation of { f, f 1 n 1} with periodic boundary conditions: f ( f ( and f ( f ( Step : Step in Case 4 is the same as Step in Case 3 A-7
MA2501 Numerical Methods Spring 2015
Norwegian University of Science and Technology Department of Mathematics MA5 Numerical Methods Spring 5 Solutions to exercise set 9 Find approximate values of the following integrals using the adaptive
Numerical interpolation, extrapolation and fi tting of data
Chapter 6 Numerical interpolation, extrapolation and fi tting of data 6.1 Introduction Numerical interpolation and extrapolation is perhaps one of the most used tools in numerical applications to physics.
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
Chapter 4: Interpolation and Approximation. October 28, 2005
Chapter 4: Interpolation and Approximation October 28, 2005 Outline 1 2.4 Linear Interpolation 2 4.1 Lagrange Interpolation 3 4.2 Newton Interpolation and Divided Differences 4 4.3 Interpolation Error
Q 0 x if x 0 x x 1. S 1 x if x 1 x x 2. i 0,1,...,n 1, and L x L n 1 x if x n 1 x x n
. - Piecewise Linear-Quadratic Interpolation Piecewise-polynomial Approximation: Problem: Givenn pairs of data points x i, y i, i,,...,n, find a piecewise-polynomial Sx S x if x x x Sx S x if x x x 2 :
MATH 1014 Tutorial Notes 8
MATH4 Calculus II (8 Spring) Topics covered in tutorial 8:. Numerical integration. Approximation integration What you need to know: Midpoint rule & its error Trapezoid rule & its error Simpson s rule &
Data Analysis-I. Interpolation. Soon-Hyung Yook. December 4, Soon-Hyung Yook Data Analysis-I December 4, / 1
Data Analysis-I Interpolation Soon-Hyung Yook December 4, 2015 Soon-Hyung Yook Data Analysis-I December 4, 2015 1 / 1 Table of Contents Soon-Hyung Yook Data Analysis-I December 4, 2015 2 / 1 Introduction
x x2 2 + x3 3 x4 3. Use the divided-difference method to find a polynomial of least degree that fits the values shown: (b)
Numerical Methods - PROBLEMS. The Taylor series, about the origin, for log( + x) is x x2 2 + x3 3 x4 4 + Find an upper bound on the magnitude of the truncation error on the interval x.5 when log( + x)
Answers to Homework 9: Numerical Integration
Math 8A Spring Handout # Sergey Fomel April 3, Answers to Homework 9: Numerical Integration. a) Suppose that the function f x) = + x ) is known at three points: x =, x =, and x 3 =. Interpolate the function
Kevin James. MTHSC 206 Section 12.5 Equations of Lines and Planes
MTHSC 206 Section 12.5 Equations of Lines and Planes Definition A line in R 3 can be described by a point and a direction vector. Given the point r 0 and the direction vector v. Any point r on the line
COS 424: Interacting with Data
COS 424: Interacting with Data Lecturer: Rob Schapire Lecture #14 Scribe: Zia Khan April 3, 2007 Recall from previous lecture that in regression we are trying to predict a real value given our data. Specically,
Rewriting Absolute Value Functions as Piece-wise Defined Functions
Rewriting Absolute Value Functions as Piece-wise Defined Functions Consider the absolute value function f ( x) = 2x+ 4-3. Sketch the graph of f(x) using the strategies learned in Algebra II finding the
SPLINE INTERPOLATION
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 16 (x
Taylor Series and Numerical Approximations
Taylor Series and Numerical Approximations Hilary Weller h.weller@reading.ac.uk August 7, 05 An introduction to the concept of a Taylor series and how these are used in numerical analysis to find numerical
CONTROL POLYGONS FOR CUBIC CURVES
On-Line Geometric Modeling Notes CONTROL POLYGONS FOR CUBIC CURVES Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis Overview B-Spline
Lagrange Multipliers Method
APPENDIX N Lagrange Multipliers Method Imagine a Cartesian coordinate system of n + m dimension with the axes labeled x 1, x 2,...,x and a function 1 E(x), wherex = (x 1, x 2,...,x ). Suppose that we are
In practice one often meets a situation where the function of interest, f(x), is only represented by a discrete set of tabulated points,
1 Interpolation 11 Introduction In practice one often meets a situation where the function of interest, f(x), is only represented by a discrete set of tabulated points, {x i, y i = f(x i ) i = 1 n, obtained,
Section 5.5 More Integration Formula (The Substitution Method) 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 5.5 More Integration Formula (The Substitution Method) 2 Lectures College of Science MATHS : Calculus I (University of Bahrain) Integrals / 7 The Substitution Method Idea: To replace a relatively
Preface. 2 Linear Equations and Eigenvalue Problem 22
Contents Preface xv 1 Errors in Computation 1 1.1 Introduction 1 1.2 Floating Point Representation of Number 1 1.3 Binary Numbers 2 1.3.1 Binary number representation in computer 3 1.4 Significant Digits
Introduction Linear system Nonlinear equation Interpolation
Interpolation Interpolation is the process of estimating an intermediate value from a set of discrete or tabulated values. Suppose we have the following tabulated values: y y 0 y 1 y 2?? y 3 y 4 y 5 x
Chapter 2. Deriving the Vlasov Equation From the Klimontovich Equation 19. Deriving the Vlasov Equation From the Klimontovich Equation
Chapter 2. Deriving the Vlasov Equation From the Klimontovich Equation 19 Chapter 2. Deriving the Vlasov Equation From the Klimontovich Equation Topics or concepts to learn in Chapter 2: 1. The microscopic
16.4. Power Series. Introduction. Prerequisites. Learning Outcomes
Power Series 6.4 Introduction In this Section we consider power series. These are examples of infinite series where each term contains a variable, x, raised to a positive integer power. We use the ratio
MATH3283W LECTURE NOTES: WEEK 6 = 5 13, = 2 5, 1 13
MATH383W LECTURE NOTES: WEEK 6 //00 Recursive sequences (cont.) Examples: () a =, a n+ = 3 a n. The first few terms are,,, 5 = 5, 3 5 = 5 3, Since 5
CURVE FITTING LEAST SQUARE LINE. Consider the class of linear function of the form. = Ax+ B...(1)
CURVE FITTIG LEAST SQUARE LIE Consider the class of linear function of the form y = f( x) = B...() In previous chapter we saw how to construct a polynomial that passes through a set of points. If all numerical
Chapter 5 HIGH ACCURACY CUBIC SPLINE APPROXIMATION FOR TWO DIMENSIONAL QUASI-LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS
Chapter 5 HIGH ACCURACY CUBIC SPLINE APPROXIMATION FOR TWO DIMENSIONAL QUASI-LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS 5.1 Introduction When a physical system depends on more than one variable a general
1 3 µ2 p ν (τ ν ) = I ν. s ν (τ ν ) + (1) µ dq ν. dτ ν. and. = Q ν {( 1 µ 2) p ν (τ ν ) }, (2)
1. Plane-parallel atmospheres with polarized scattering. In the treatment by Chandrasehar (196, pp 38-5), the transfer equations are expressed in terms of I l and I r, the intensities in directions parallel
ECONOMICS 207 SPRING 2006 LABORATORY EXERCISE 5 KEY. 8 = 10(5x 2) = 9(3x + 8), x 50x 20 = 27x x = 92 x = 4. 8x 2 22x + 15 = 0 (2x 3)(4x 5) = 0
ECONOMICS 07 SPRING 006 LABORATORY EXERCISE 5 KEY Problem. Solve the following equations for x. a 5x 3x + 8 = 9 0 5x 3x + 8 9 8 = 0(5x ) = 9(3x + 8), x 0 3 50x 0 = 7x + 7 3x = 9 x = 4 b 8x x + 5 = 0 8x
MATHEMATICS Lecture. 4 Chapter.8 TECHNIQUES OF INTEGRATION By Dr. Mohammed Ramidh
MATHEMATICS Lecture. 4 Chapter.8 TECHNIQUES OF INTEGRATION By TECHNIQUES OF INTEGRATION OVERVIEW The Fundamental Theorem connects antiderivatives and the definite integral. Evaluating the indefinite integral,
Calculus I Announcements
Slide 1 Calculus I Announcements Read sections 4.2,4.3,4.4,4.1 and 5.3 Do the homework from sections 4.2,4.3,4.4,4.1 and 5.3 Exam 3 is Thursday, November 12th See inside for a possible exam question. Slide
Stress analysis of a stepped bar
Stress analysis of a stepped bar Problem Find the stresses induced in the axially loaded stepped bar shown in Figure. The bar has cross-sectional areas of A ) and A ) over the lengths l ) and l ), respectively.
h p://edugen.wileyplus.com/edugen/courses/crs1404/pc/b03/c2hlch...
n this appendix we discuss... 1 of 4 16-Sep-12 19:35 APPENDIX C In this appendix we discuss how to calculate the moment of inertia of an area. 1 The moment of inertia of an area was first introduced in
Lecture 7: Interpolation
Lecture 7: Interpolation ECE 401: Signal and Image Analysis University of Illinois 2/9/2017 1 Sampling Review 2 Interpolation and Upsampling 3 Spectrum of Interpolated Signals Outline 1 Sampling Review
Block Circulant and Toeplitz Structures in the Linearized Hartree Fock Equation on Finite Lattices: Tensor Approach
Comput. Methods Appl. Math. 2017; 17 (3):431 455 Research Article Venera Khoromsaia and Boris N. Khoromsij* Bloc Circulant and Toeplitz Structures in the Linearized Hartree Foc Equation on Finite Lattices:
Edexcel GCE Further Pure Mathematics (FP1) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Further Pure Mathematics (FP1) Required Knowledge Information Sheet FP1 Formulae Given in Mathematical Formulae and Statistical Tables Booklet Summations o =1 2 = 1 + 12 + 1 6 o =1 3 = 1 64
Fourier series
11.1-11.2. Fourier series Yurii Lyubarskii, NTNU September 5, 2016 Periodic functions Function f defined on the whole real axis has period p if Properties f (t) = f (t + p) for all t R If f and g have
Cubic Splines; Bézier Curves
Cubic Splines; Bézier Curves 1 Cubic Splines piecewise approximation with cubic polynomials conditions on the coefficients of the splines 2 Bézier Curves computer-aided design and manufacturing MCS 471
Quartic Equation. By CH vd Westhuizen A unique Solution assuming Complex roots. Ax^4 + Bx^3 + Cx^2 + Dx + E = 0
Quartic Equation By CH vd Westhuizen A unique Solution assuming Complex roots The general Quartic is given by Ax^4 + Bx^3 + Cx^ + Dx + E = 0 As in the third order polynomial we are first going to reduce
Lecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.
Lecture 9 Approximations of Laplace s Equation, Finite Element Method Mathématiques appliquées (MATH54-1) B. Dewals, C. Geuzaine V1.2 23/11/218 1 Learning objectives of this lecture Apply the finite difference
Differential equations
Differential equations Math 7 Spring Practice problems for April Exam Problem Use the method of elimination to find the x-component of the general solution of x y = 6x 9x + y = x 6y 9y Soln: The system
Examination paper for TMA4110 Matematikk 3
Department of Mathematical Sciences Examination paper for TMA11 Matematikk 3 Academic contact during examination: Eugenia Malinnikova Phone: 735557 Examination date: 6th May, 15 Examination time (from
1. Solve the boundary-value problems or else show that no solutions exist. y (x) = c 1 e 2x + c 2 e 3x. (3)
Diff. Eqns. Problem Set 6 Solutions. Solve the boundary-value problems or else show that no solutions exist. a y + y 6y, y, y 4 b y + 9y x + e x, y, yπ a Assuming y e rx is a solution, we get the characteristic
6.1. Linearized Wave Equations in a Uniform Isotropic MHD Plasma. = 0 into Ohm s law yields E 0
Chapter 6. Linear Waves in the MHD Plasma 85 Chapter 6. Linear Waves in the MHD Plasma Topics or concepts to learn in Chapter 6:. Linearize the MHD equations. The eigen-mode solutions of the MHD waves
Math 578: Assignment 2
Math 578: Assignment 2 13. Determine whether the natural cubic spline that interpolates the table is or is not the x 0 1 2 3 y 1 1 0 10 function 1 + x x 3 x [0, 1] f(x) = 1 2(x 1) 3(x 1) 2 + 4(x 1) 3 x
How to write proofs - II
How to write proofs - II Problem (Ullman and Hopcroft, Page 104, Exercise 4.8). Let G be the grammar Prove that S ab ba A a as baa B b bs abb L(G) = { x {a, b} x contains equal number of a s and b s }.
There is a unique function s(x) that has the required properties. It turns out to also satisfy
Numerical Analysis Grinshpan Natural Cubic Spline Let,, n be given nodes (strictly increasing) and let y,, y n be given values (arbitrary) Our goal is to produce a function s() with the following properties:
AP Calculus BC Syllabus Course Overview
AP Calculus BC Syllabus Course Overview Textbook Anton, Bivens, and Davis. Calculus: Early Transcendentals, Combined version with Wiley PLUS. 9 th edition. Hoboken, NJ: John Wiley & Sons, Inc. 2009. Course
A = 3 1. We conclude that the algebraic multiplicity of the eigenvalues are both one, that is,
65 Diagonalizable Matrices It is useful to introduce few more concepts, that are common in the literature Definition 65 The characteristic polynomial of an n n matrix A is the function p(λ) det(a λi) Example
2326 Problem Sheet 1. Solutions 1. First Order Differential Equations. Question 1: Solve the following, given an explicit solution if possible:
2326 Problem Sheet. Solutions First Order Differential Equations Question : Solve the following, given an explicit solution if possible:. 2 x = 4x3, () = 3, ( ) The given equation is a first order linear
Preliminary Examination in Numerical Analysis
Department of Applied Mathematics Preliminary Examination in Numerical Analysis August 7, 06, 0 am pm. Submit solutions to four (and no more) of the following six problems. Show all your work, and justify
14.9 Worked Examples. Example 14.2 Escape Velocity of Toro
14.9 Wored Examples Example 14. Escape Velocity of Toro The asteroid Toro, discovered in 1964, has a radius of about R = 5.0m and a mass of about m t =.0 10 15 g. Let s assume that Toro is a perfectly
Polynomial Functions. x n 2 a n. x n a 1. f x = a o. x n 1 a 2. x 0, , a 1
Polynomial Functions A polynomial function is a sum of multiples of an independent variable raised to various integer powers. The general form of a polynomial function is f x = a o x n a 1 x n 1 a 2 x
CURVE FITTING LEAST SQUARE LINE. Consider the class of linear function of the form
CURVE FITTIG LEAST SQUARE LIE Consider the class of linear function of the form f ( x) B...() In previous chapter we saw how to construct a polnomial that passes through a set of points. If all numerical
Bayes Rule for Minimizing Risk
Bayes Rule for Minimizing Risk Dennis Lee April 1, 014 Introduction In class we discussed Bayes rule for minimizing the probability of error. Our goal is to generalize this rule to minimize risk instead
(2) Dividing both sides of the equation in (1) by the divisor, 3, gives: =
Dividing Polynomials Prepared by: Sa diyya Hendrickson Name: Date: Let s begin by recalling the process of long division for numbers. Consider the following fraction: Recall that fractions are just division
Fixed point iteration and root finding
Fixed point iteration and root finding The sign function is defined as x > 0 sign(x) = 0 x = 0 x < 0. It can be evaluated via an iteration which is useful for some problems. One such iteration is given
We consider the problem of finding a polynomial that interpolates a given set of values:
Chapter 5 Interpolation 5. Polynomial Interpolation We consider the problem of finding a polynomial that interpolates a given set of values: x x 0 x... x n y y 0 y... y n where the x i are all distinct.
Lecture 10 Polynomial interpolation
Lecture 10 Polynomial interpolation Weinan E 1,2 and Tiejun Li 2 1 Department of Mathematics, Princeton University, weinan@princeton.edu 2 School of Mathematical Sciences, Peking University, tieli@pku.edu.cn
Roots and Coefficients Polynomials Preliminary Maths Extension 1
Preliminary Maths Extension Question If, and are the roots of x 5x x 0, find the following. (d) (e) Question If p, q and r are the roots of x x x 4 0, evaluate the following. pq r pq qr rp p q q r r p
Physics 6303 Lecture 2 August 22, 2018
Physics 6303 Lecture 2 August 22, 2018 LAST TIME: Coordinate system construction, covariant and contravariant vector components, basics vector review, gradient, divergence, curl, and Laplacian operators
3.1 Interpolation and the Lagrange Polynomial
MATH 4073 Chapter 3 Interpolation and Polynomial Approximation Fall 2003 1 Consider a sample x x 0 x 1 x n y y 0 y 1 y n. Can we get a function out of discrete data above that gives a reasonable estimate
In the derivation of Optimal Interpolation, we found the optimal weight matrix W that minimizes the total analysis error variance.
hree-dimensional variational assimilation (3D-Var) In the derivation of Optimal Interpolation, we found the optimal weight matrix W that minimizes the total analysis error variance. Lorenc (1986) showed
2017 HSC Mathematics Extension 1 Marking Guidelines
07 HSC Mathematics Extension Marking Guidelines Section I Multiple-choice Answer Key Question Answer A B 3 B 4 C 5 D 6 D 7 A 8 C 9 C 0 B NESA 07 HSC Mathematics Extension Marking Guidelines Section II
2 Two-Point Boundary Value Problems
2 Two-Point Boundary Value Problems Another fundamental equation, in addition to the heat eq. and the wave eq., is Poisson s equation: n j=1 2 u x 2 j The unknown is the function u = u(x 1, x 2,..., x
1 Piecewise Cubic Interpolation
Piecewise Cubic Interpolation Typically the problem with piecewise linear interpolation is the interpolant is not differentiable as the interpolation points (it has a kinks at every interpolation point)
17.2 Nonhomogeneous Linear Equations. 27 September 2007
17.2 Nonhomogeneous Linear Equations 27 September 2007 Nonhomogeneous Linear Equations The differential equation to be studied is of the form ay (x) + by (x) + cy(x) = G(x) (1) where a 0, b, c are given
Appendix A Numerical Tables
Appendix A Numerical Tables A.1 The Gaussian Distribution The Gaussian distribution (Eq. 2.31)isdefinedas 1 f (x)= 2πσ 2 e (x μ) 2 2σ 2. (A.1) The maximum value is obtained at x = μ, and the value of (x
Dirichlet in the Upper-Half Plane
Dirichlet in the Upper-Half Plane Jenna Nelson and Gloria Mayorga Occidental College April 22, 2016 Jenna Nelson and Gloria Mayorga (Oxy) Dirichlet April 22, 2016 1 / 17 Overview 1 Evaluating real improper
Ten years analysis of Tropospheric refractivity variations
ANNALS OF GEOPHYSICS, VOL. 47, N. 4, August 2004 Ten years analysis of Tropospheric refractivity variations Stergios A. Isaakidis and Thomas D. Xenos Department of Electrical and Computer Engineering,
Logarithmic Functions
Metropolitan Community College The Natural Logarithmic Function The natural logarithmic function is defined on (0, ) as ln x = x 1 1 t dt. Example 1. Evaluate ln 1. Example 1. Evaluate ln 1. Solution.
Newton s Method and Linear Approximations
Newton s Method and Linear Approximations Curves are tricky. Lines aren t. Newton s Method and Linear Approximations Newton s Method for finding roots Goal: Where is f (x) = 0? f (x) = x 7 + 3x 3 + 7x
Matrix assembly by low rank tensor approximation
Matrix assembly by low rank tensor approximation Felix Scholz 13.02.2017 References Angelos Mantzaflaris, Bert Juettler, Boris Khoromskij, and Ulrich Langer. Matrix generation in isogeometric analysis
MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018)
MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018) COURSEWORK 3 SOLUTIONS Exercise ( ) 1. (a) Write A = (a ij ) n n and B = (b ij ) n n. Since A and B are diagonal, we have a ij = 0 and
The coordinates of the vertex of the corresponding parabola are p, q. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward.
Mathematics 10 Page 1 of 8 Quadratic Relations in Vertex Form The expression y ax p q defines a quadratic relation in form. The coordinates of the of the corresponding parabola are p, q. If a > 0, the
WORD: EXAMPLE(S): COUNTEREXAMPLE(S): EXAMPLE(S): COUNTEREXAMPLE(S): WORD: EXAMPLE(S): COUNTEREXAMPLE(S): EXAMPLE(S): COUNTEREXAMPLE(S): WORD:
Bivariate Data DEFINITION: In statistics, data sets using two variables. Scatter Plot DEFINITION: a bivariate graph with points plotted to show a possible relationship between the two sets of data. Positive
Lesson 5: The Graph of the Equation y = f(x)
Lesson 5: The Graph of the Equation y = f(x) Learning targets: I can identify when a function is increasing, decreasing, positive and negative and use interval notation to describe intervals where the
Finding an upper limit in the presence of an unknown background
PHYSICAL REVIEW D 66, 032005 2002 Finding an upper limit in the presence of an unnown bacground S. Yellin* Department of Physics, University of California, Santa Barbara, Santa Barbara, California 93106
Applied Numerical Analysis Quiz #2
Applied Numerical Analysis Quiz #2 Modules 3 and 4 Name: Student number: DO NOT OPEN UNTIL ASKED Instructions: Make sure you have a machine-readable answer form. Write your name and student number in the
Lagrange Interpolation and Neville s Algorithm. Ron Goldman Department of Computer Science Rice University
Lagrange Interpolation and Neville s Algorithm Ron Goldman Department of Computer Science Rice University Tension between Mathematics and Engineering 1. How do Mathematicians actually represent curves
QUINTIC SPLINE SOLUTIONS OF FOURTH ORDER BOUNDARY-VALUE PROBLEMS
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 5, Number, Pages 0 c 2008 Institute for Scientific Computing Information QUINTIC SPLINE SOLUTIONS OF FOURTH ORDER BOUNDARY-VALUE PROBLEMS
College Algebra. Course Text. Course Description. Course Objectives. StraighterLine MAT101: College Algebra
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 9780072867381 [This text is available as an etextbook at
Memoirs on Differential Equations and Mathematical Physics
Memoirs on Differential Equations and Mathematical Physics Volume 34, 005, 1 76 A. Dzhishariani APPROXIMATE SOLUTION OF ONE CLASS OF SINGULAR INTEGRAL EQUATIONS BY MEANS OF PROJECTIVE AND PROJECTIVE-ITERATIVE
G: Uniform Convergence of Fourier Series
G: Uniform Convergence of Fourier Series From previous work on the prototypical problem (and other problems) u t = Du xx 0 < x < l, t > 0 u(0, t) = 0 = u(l, t) t > 0 u(x, 0) = f(x) 0 < x < l () we developed
5.1 introduction problem : Given a function f(x), find a polynomial approximation p n (x).
capter 5 : polynomial approximation and interpolation 5 introduction problem : Given a function f(x), find a polynomial approximation p n (x) Z b Z application : f(x)dx b p n(x)dx, a a one solution : Te
OBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph.
4.1 The Area under a Graph OBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph. 4.1 The Area Under a Graph Riemann Sums (continued): In the following
Solving Quadratic Equations Using the Quadratic Formula
Section 9 : Solving Quadratic Equations Using the Quadratic Fmula Quadratic Equations are equations that have an x term as the highest powered term. They are also called Second Degree Equations. The Standard
CS 450 Numerical Analysis. Chapter 8: Numerical Integration and Differentiation
Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80
Representation: Fractional Splines, Wavelets and related Basis Function Expansions. Felix Herrmann and Jonathan Kane, ERL-MIT
Representation: Fractional Splines, Wavelets and related Basis Function Expansions Felix Herrmann and Jonathan Kane, ERL-MIT Objective: Build a representation with a regularity that is consistent with
2017 VCE Mathematical Methods 2 examination report
7 VCE Mathematical Methods examination report General comments There were some excellent responses to the 7 Mathematical Methods examination and most students were able to attempt the four questions in
THE USE OF KALMAN FILTRATION TO ESTIMATE CHANGES OF TRUNK INCLINATION ANGLE DURING WEIGHTLIFTING 1. INTRODUCTION
JOURNAL OF MEDICAL INFORMATICS & TECHNOLOGIES Vol. 15/2010, ISSN 1642-6037 Kalman filtration, filter algorithm, accelerometric sensor Grzegorz SAPOTA 1, Anna SAPOTA 1, Zygmunt WRÓBEL 1 THE USE OF KALMAN
Systems of Linear Equations
4 Systems of Linear Equations Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.1, Slide 1 1-1 4.1 Systems of Linear Equations in Two Variables R.1 Fractions Objectives 1. Decide whether an
Applied Numerical Analysis Homework #3
Applied Numerical Analysis Homework #3 Interpolation: Splines, Multiple dimensions, Radial Bases, Least-Squares Splines Question Consider a cubic spline interpolation of a set of data points, and derivatives
NONLINEAR CIRCUITS XXI. Prof. E. A. Guillemin Prof. H. J. Zimmermann T. E. Stern Prof. S. J. Mason J. Gross R. D. Thornton RESEARCH OBJECTIVES
XXI. NONLINEAR CIRCUITS Prof. E. A. Guillemin Prof. H. J. Zimmermann T. E. Stern Prof. S. J. Mason J. Gross R. D. Thornton RESEARCH OBJECTIVES This research is aimed at a variety of problems, including
MATH 250 REVIEW TOPIC 3 Partial Fraction Decomposition and Irreducible Quadratics. B. Decomposition with Irreducible Quadratics
Math 250 Partial Fraction Decomposition Topic 3 Page MATH 250 REVIEW TOPIC 3 Partial Fraction Decomposition and Irreducible Quadratics I. Decomposition with Linear Factors Practice Problems II. A. Irreducible
Chapter 3 Interpolation and Polynomial Approximation
Chapter 3 Interpolation and Polynomial Approximation Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 128A Numerical Analysis Polynomial Interpolation
The integral test and estimates of sums
The integral test Suppose f is a continuous, positive, decreasing function on [, ) and let a n = f (n). Then the series n= a n is convergent if and only if the improper integral f (x)dx is convergent.
Ch. 7 Absolute Value and Reciprocal Functions Notes
First Name: Last Name: Block: Ch. 7 Absolute Value and Reciprocal Functions Notes 7. ABSOLUTE VALUE Ch. 7. HW: p. 364 # 7 odd letters, 9, 3 7. PRE-REQUISITES - GRAPH OF LINEAR FUNCTIONS 4 7. PRE-REQUISITES
MCPS Algebra 2 and Precalculus Standards, Categories, and Indicators*
Content Standard 1.0 (HS) Patterns, Algebra and Functions Students will algebraically represent, model, analyze, and solve mathematical and real-world problems involving functional patterns and relationships.
Quantum Chemistry Exam 2 Solutions
Chemistry 46 Fall 17 Dr. Jean M. Standard November 8, 17 Name KEY Quantum Chemistry Exam Solutions 1.) ( points) Answer the following questions by selecting the correct answer from the choices provided.
SOLUTIONS TO EXAM 2, MATH 10550
SOLUTIONS TO EXAM 2, MATH 0550. Find the critical numbers of f(x) = 6 x2 x /3. We have f (x) = 3 x 3 x 2/3 = [ x 5/3 ] 3 x 2/3. So x = 0 is a critical point. For x 0, the equation f (x) = 0 can be written
(0, 0), (1, ), (2, ), (3, ), (4, ), (5, ), (6, ).
1 Interpolation: The method of constructing new data points within the range of a finite set of known data points That is if (x i, y i ), i = 1, N are known, with y i the dependent variable and x i [x