Homework #6 Brief Solutions 2011
|
|
- Patricia Lester
- 6 years ago
- Views:
Transcription
1 Homework #6 Brief Solutions %page 95 problem 4 data=[-,;-,;,;4,] data = xk=data(:,);yk=data(:,);s=csfit(xk,yk,-,) %Using the program to find the coefficients S = S=.456*(x+).^ -.456*(x+).^-*(x+)+; S= -.5*(x+).^+.9*(x+).^ *(x+); S=.*(x-).^ -.58*(x-).^+.87*(x-)+; x=-:.:4; y=spline_eval(s(:,4),s(:,),s(:,),s(:,),xk,x); plot(xk,yk, o,x,y) % see plot below %p95 problem 5 type natspline_coeff function [a,b,c,d]=natspline_coeff(knots,data); %% function [a,b,c,d]=natspline_coeff(knots,data); %% Modification of Cheney Kincaid pseudo-code %% The natural spline for interpolating data at the knots a=x<...<x_n=b; %% System of equations is solved using the tridiagonal nature %% WARNING: THIS PROGRAM REQUIRES AT LEAST 4 KNOTS %% INPUT: %% knots - distinct points (t_j) of interpolation as a row vector %% data - data at the interpolation points as a row vector %% OUTPUT: %% a - column vector of constant terms for the spline on [t_j,t_(j+)] %% b - column vector of coefficients of x-t_j for the spline on [t_j,t_(j+)] %% c - column vector of coefficients of (x-t_j)^ for the spline on [t_j,t_(j+)] %% d - column vector of coefficients of (x-t_j)^ for the spline on [t_j,t_(j+)] % determine lengths of the intervals n=length(knots)-; for j=:n; h(j)=knots(j+)-knots(j); % determine the right hand side of the system (without first and last eqn) for i=:n; rhs(i)=6*((data(i+)-data(i))/h(i)-(data(i)-data(i-))/h(i-)); % solve the system using tri_diag.m z()=; z(n+)=;
2 dd=*(h(:n)+h(:n-)));subd=h(:n-);supd=h(:n-); z(:n)=tri_diag(dd,subd,supd,rhs(:n)); % find the other coefficients for j=:n;file://localhost/users/sherm/math4/natspline_coeff b(j,)=(data(j+)-data(j))/h(j) - h(j)*(z(j+)+*z(j))/6; d(j,)=(z(j+)-z(j))/(6*h(j)); a=data(:n) ; c=z(:n) /; %applying this [a,b,c,d]=natspline_coeff(xk,yk ) a = b = c = d = S=.47*(x+).^ -.47*(x+).^-.47*(x+)+; S= -.47*(x+).^+.48*(x+).^ -.575*(x+); S=.856*(x-).^-.77*(x-).^+.876*(x-)+; naty=spline_eval(a,b,c,d,xk,x); plot(xk,yk, o,x,naty) %p95 problem 6 %I modified natspline_coeff for this function [a,b,c,d]=nakspline_coeff(knots,data); %% function [a,b,c,d]=nakspline_coeff(knots,data); %% Modification of the natural spline code spline_coeff to give %% TheNot-A-Knot spline for interpolating data at the knots a=x<...<x_n=b; %% System of equations is solved using the tridiagonal nature %% WARNING: THIS PROGRAM REQUIRES AT LEAST 4 KNOTS %% INPUT: %% knots - distinct points (t_j) of interpolation as a row vector %% data - data at the interpolation points as a row vector %% OUTPUT: %% a - column vector of constant terms for the spline on [t_j,t_(j+)] %% b - column vector of coefficients of x-t_j for the spline on [t_j,t_(j+)] %% c - column vector of coefficients of (x-t_j)^ for the spline on [t_j,t_(j+)] %% d - column vector of coefficients of (x-t_j)^ for the spline on [t_j,t_(j+)] % determine lengths of the intervals n=length(knots)-; for j=:n; h(j)=knots(j+)-knots(j)
3 % determine the right hand side of the system for i=:n-; rhs(i)=6*((data(i+)-data(i+))/h(i+)-(data(i+)-data(i))/h(i)) % solve the system using tri_diag.m dd()=*h()+*h()+h()^/h();dd(:n-)=*(h(:n-)+h(:n-)); dd(n-)=(*h(n-)+*h(n)+h(n)^/h(n-)); subd=[h(:n-),h(n-)-h(n)^/h(n-)];supd=[h()-h()^/h(),h(:n-)]; z(:n)=tri_diag(dd,subd,supd,rhs(:n-)); z()=z()-h()*(z()-z())/h();z(n+)=z(n)+h(n)*(z(n)-z(n-))/h(n-); % find the other coefficients for j=:n; b(j,)=(data(j+)-data(j))/h(j) - h(j)*(z(j+)+*z(j))/6; d(j,)=(z(j+)-z(j))/(6*h(j)); a=data(:n) ; c=z(:n) /; %Using this to find the coefficients [a,b,c,d]=nakspline_coeff(xk,yk ) a = b = c = d = ynak=spline_eval(a,b,c,d,xk,x); %p95 # (a) Clamped Spline f=@(x) cos(x.^); df=@(x) -*x*sin(x.^); xk=[,sqrt(pi/),sqrt(*pi/),sqrt(5*pi/)]; yk=f(xk); dfa=df(xk());dfb=df(xk(4)); S=csfit(xk,yk,dfa,dfb); S = xx=:.: sqrt(5*pi/); yy=spline_eval(s(:,4),s(:,),s(:,),s(:,),xk,xx); plot(xk,yk, o,xx,yy,xx,f(xx)); % (b) Natural spline [a,b,c,d]=natspline_coeff(xk,yk )
4 a =.. -. b = c = d = natyy=spline_eval(a,b,c,d,xk,xx); plot(xk,yk, o,xx,natyy,xx,f(xx)); % see below Figure : The clamped spline of problem 4 p95 on the left and the natural spline of problem 5 p95 on the right Figure : The clamped spline of problem a p95 on the left and the natural spline of problem b p95 on the right. Since the spline problem was laid out for you in Matlab, here is how it should have looked with the figures. xti=585+[:49]*; 4
5 yti=[ ]; yti=[ yti ]; %Plotting the points, see below plot(xti,yti,o) %Computing the interpolating polynomial xx=xti():.:xti(49); ypoly=int_poly(yti,xti,xx); %see int_poly.m for how data is input plot(xx,ypoly,xti,yti,o) %plot below % It is way off near the ends %Computing the natural spline interpolant [a,b,c,d]=spline_coeff(xti,yti);% see spline_coeff.m ynatspl=spline_eval(a,b,c,d,xti,xx); % see spline_coeff.m plot(xx,ynatspl,xti,yti,o) %plot below MUCH BETTER %A comparison where we interpolate at fewer points knots=xti(:6:49); data=yti(:6:49); yyintpoly=int_poly(data,knots,xx); [a,b,c,d]=spline_coeff(knots,data); yynatspl=spline_eval(a,b,c,d,knots,xx); plot(xx,yyintpoly,xti,yti,o,xx,yynatspl) %See plot below %Again, the spline is very much better. %Computing error estimates max(abs(yti-int_poly(data,knots,xti))) ans = max(abs(yti-spline_eval(a,b,c,d,knots,xti))) ans = % and the least squares error sum(abs(yti-int_poly(data,knots,xti)).^) ans = sum(abs(yti-spline_eval(a,b,c,d,knots,xti)).^) ans = % The spline is superior in both cases. % Looking for a true least squares approximation by splines % First discover the B-spline basis of independent functions % Setting up the knots knots=[knots(),knots(),knots(),knots(),knots()]; knots=[knots(),knots(),knots(),knots(),knots()]; knots=[knots(),knots(),knots(),knots(),knots(4)]; knots4=[knots(),knots(),knots(),knots(4),knots(5)]; knots5=[knots(),knots(),knots(4),knots(5),knots(6)]; 5
6 knots6=[knots(),knots(4),knots(5),knots(6),knots(7)]; knots7=[knots(4),knots(5),knots(6),knots(7),knots(8)]; knots8=[knots(5),knots(6),knots(7),knots(8),knots(9)]; knots9=[knots(6),knots(7),knots(8),knots(9),knots(9)]; knots=[knots(7),knots(8),knots(9),knots(9),knots(9)]; knots=[knots(8),knots(9),knots(9),knots(9),knots(9)]; %Using the program cubicbspl that gives the B-spline based on the knots cubicbspl(knots,x); cubicbspl(knots,x); cubicbspl(knots,x); cubicbspl(knots4,x); cubicbspl(knots5,x); cubicbspl(knots6,x); cubicbspl(knots7,x); cubicbspl(knots8,x); cubicbspl(knots9,x); cubicbspl(knots,x); cubicbspl(knots,x); % Plot the B-splines to get a better feel for them plot(xx,f(xx),xx,f(xx),xx,f(xx),xx,f4(xx),xx,f5(xx),xx,f6(xx),xx,f7(xx),xx,f8(xx),xx,f9(xx),xx,f(xx % see below %Plot the sum to show that the sum of the B-splines is identically (not much to see!). sumbsplines= f(xx)+f(xx)+f(xx)+f4(xx)+f5(xx)+f6(xx)+f7(xx)+f8(xx)+f9(xx)+f(xx)+f(xx); plot(xx,sumbsplines) max(sumbsplines) ans =.8 %small round off error min(sumbsplines) ans = %well, not quite % Doing the least squares fit using the B-splines as the linearly independent functions F=@(x) [f(x);f(x);f(x);f4(x);f5(x);f6(x);f7(x);f8(x);f9(x);f(x);f(x)]; FC=[]; for j=:49; y=f(xti(j));fc=[fc;y]; FC %This gives the matrix called F in the book and F_C in class. FR=FC; %its transpose B=FR*yti; %The right hand side of the matrix form of the system of normal equations for least squares ap FRFC=FR*FC %The coefficient matrix in the matrix form of the system of normal equations for least square %Notice that it has a banded property. This is because of the short support of the B-splines. FRFC = Columns through
7 Columns 5 through Columns 9 through C=FRFC\B %This uses Matlab to solve the system C = bestlsspline=@(x) C()*f(x)+C()*f(x)+C()*f(x)+C(4)*f4(x)+C(5)*f5(x)+C(6)*f6(x)+C(7)*f7(x)+... C(8)*f8(x)+C(9)*f9(x)+C()*f(x)+C()*f(x); % Making the best linear combination of the basis 7
8 plot(xx,bestlsspline(xx),xti,yti,o) %plot the best least square spline fit with the data See below %%Plotting the natural spline and the best least squares spline and comparing errors plot(xx,bestlsspline(xx),xti,yti,o,xx,yynatspl) spl_intxti=spline_eval(a,b,c,d,knots,xti); %evluate the natural spline interpolant at the data points ls_spl_xti=bestlsspline(xti); %evaluate the best least squares spline at all the data points sum((spl_intxti-yti).^) %least squares error of the natural spline interpolant ans = sum((ls_spl_xti-yti).^) %least squares error of the best least squares spline (its better of course:) ans = max(abs(spl_intxti-yti)) % max abs value error for the natural spline interpolant ans = max(abs(ls_spl_xti-yti)) %max abs value error for the least squares spline (again better) ans = x Figure : The titanium data on the left and the same data with its interpolating polynomial on the right Figure 4: The natural spline interpolation of the titanium data on the left. The natural spline and the polynomial that interpolates the data at every 6th data point on the right. 8
9 Figure 5: The cubic B-splines on the left the left and their sum on the interval on the right Figure 6: The best least squares approximation to the titanium data using the B-splines on the left. The natural spline interpolant on every 6th point plotted with the best least squares approximation by cubic splines on the same knot set on the right. 9
Homework #6 Brief Solutions 2012
Homework #6 Brief Solutions %page 95 problem 4 data=[-,;-,;,;4,] data = - - 4 xk=data(:,);yk=data(:,);s=csfit(xk,yk,-,) %Using the program to find the coefficients S =.456 -.456 -.. -.5.9 -.5484. -.58.87.
More informationRunge Example Revisited for Splines
Runge Example Revisited for Splines The Runge function f(x) = +25x on [, ] provided an very nice function that was not wellapproximated by its polynomials of interpolation. In fact, a higher degree (more
More informationHandout 4 - Interpolation Examples
Handout 4 - Interpolation Examples Middle East Technical University Example 1: Obtaining the n th Degree Newton s Interpolating Polynomial Passing through (n+1) Data Points Obtain the 4 th degree Newton
More informationMath 226A Homework 4 Due Monday, December 11th
Math 226A Homework 4 Due Monday, December 11th 1. (a) Show that the polynomial 2 n (T n+1 (x) T n 1 (x)), is the unique monic polynomial of degree n + 1 with roots at the Chebyshev points x k = cos ( )
More informationInterpolation & Polynomial Approximation. Cubic Spline Interpolation II
Interpolation & Polynomial Approximation Cubic Spline Interpolation II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More informationCubic spline interpolation
Cubic spline interpolation In the following, we want to derive the collocation matrix for cubic spline interpolation. Let us assume that we have equidistant knots. To fulfill the Schoenberg-Whitney condition
More informationSee the course website for important information about collaboration and late policies, as well as where and when to turn in assignments.
COS Homework # Due Tuesday, February rd See the course website for important information about collaboration and late policies, as well as where and when to turn in assignments. Data files The questions
More informationAPPM/MATH Problem Set 4 Solutions
APPM/MATH 465 Problem Set 4 Solutions This assignment is due by 4pm on Wednesday, October 16th. You may either turn it in to me in class on Monday or in the box outside my office door (ECOT 35). Minimal
More informationPiecewise Polynomial Interpolation, cont d
Jim Lambers MAT 460/560 Fall Semester 2009-0 Lecture 2 Notes Tese notes correspond to Section 4 in te text Piecewise Polynomial Interpolation, cont d Constructing Cubic Splines, cont d Having determined
More informationNatural Quartic Spline
Natural Quartic Spline Rafael E Banchs INTRODUCTION This report describes the natural quartic spline algorithm developed for the enhanced solution of the Time Harmonic Field Electric Logging problem As
More informationFour equations are necessary to evaluate these coefficients. Eqn
1.2 Splines 11 A spline function is a piecewise defined function with certain smoothness conditions [Cheney]. A wide variety of functions is potentially possible; polynomial functions are almost exclusively
More informationFebruary 2017 (1/20) 2 Piecewise Polynomial Interpolation 2.2 (Natural) Cubic Splines. MA378/531 Numerical Analysis II ( NA2 )
f f f f f (/2).9.8.7.6.5.4.3.2. S Knots.7.6.5.4.3.2. 5 5.2.8.6.4.2 S Knots.2 5 5.9.8.7.6.5.4.3.2..9.8.7.6.5.4.3.2. S Knots 5 5 S Knots 5 5 5 5.35.3.25.2.5..5 5 5.6.5.4.3.2. 5 5 4 x 3 3.5 3 2.5 2.5.5 5
More informationLecture 8. Divided Differences,Least-Squares Approximations. Ceng375 Numerical Computations at December 9, 2010
Lecture 8, Ceng375 Numerical Computations at December 9, 2010 Computer Engineering Department Çankaya University 8.1 Contents 1 2 3 8.2 : These provide a more efficient way to construct an interpolating
More informationPolynomials tend to oscillate (wiggle) a lot, even when our true function does not.
AMSC/CMSC 460 Computational Methods, Fall 2007 UNIT 2: Spline Approximations Dianne P O Leary c 2001, 2002, 2007 Piecewise polynomial interpolation Piecewise polynomial interpolation Read: Chapter 3 Skip:
More informationME 261: Numerical Analysis Lecture-12: Numerical Interpolation
1 ME 261: Numerical Analysis Lecture-12: Numerical Interpolation Md. Tanver Hossain Department of Mechanical Engineering, BUET http://tantusher.buet.ac.bd 2 Inverse Interpolation Problem : Given a table
More informationInterpolation. TANA09 Lecture 7. Error analysis for linear interpolation. Linear Interpolation. Suppose we have a table x x 1 x 2...
TANA9 Lecture 7 Interpolation Suppose we have a table x x x... x n+ Interpolation Introduction. Polynomials. Error estimates. Runge s phenomena. Application - Equation solving. Spline functions and interpolation.
More informationConsider functions such that then satisfies these properties: So is represented by the cubic polynomials on on and on.
1 of 9 3/1/2006 2:28 PM ne previo Next: Trigonometric Interpolation Up: Spline Interpolation Previous: Piecewise Linear Case Cubic Splines A piece-wise technique which is very popular. Recall the philosophy
More informationlecture 10: B-Splines
9 lecture : -Splines -Splines: a basis for splines Throughout our discussion of standard polynomial interpolation, we viewed P n as a linear space of dimension n +, and then expressed the unique interpolating
More informationHandout 2 - Root Finding using MATLAB
Handout 2 - Root Finding using MATLAB Middle East Technical University MATLAB has couple of built-in root finding functions. In this handout we ll have a look at fzero, roots and solve functions. It is
More informationMATLAB NOTES. Matlab designed for numerical computing. Strongly oriented towards use of arrays, one and two dimensional.
MATLAB NOTES Matlab designed for numerical computing. Strongly oriented towards use of arrays, one and two dimensional. Excellent graphics that are easy to use. Powerful interactive facilities; and programs
More information8 Piecewise Polynomial Interpolation
Applied Math Notes by R. J. LeVeque 8 Piecewise Polynomial Interpolation 8. Pitfalls of high order interpolation Suppose we know the value of a function at several points on an interval and we wish to
More informationMatlab and Octave: Quick Introduction and Examples 1 Basics
Matlab and Octave: Quick Introduction and Examples 1 Basics 1.1 Syntax and m-files There is a shell where commands can be written in. All commands must either be built-in commands, functions, names of
More informationComputational Physics PHYS 420
Computational Physics PHYS 420 Dr Richard H. Cyburt Assistant Professor of Physics My office: 402c in the Science Building My phone: (304) 384-6006 My email: rcyburt@concord.edu My webpage: www.concord.edu/rcyburt
More informationCubic Splines and Matlab
Cubic Splines and Matlab October 7, 2006 1 Introduction In this section, we introduce the concept of the cubic spline, and how they are implemented in Matlab. Of particular importance are the new Matlab
More informationLinear Interpolating Splines
Jim Lambers MAT 772 Fall Semester 2010-11 Lecture 17 Notes Tese notes correspond to Sections 112, 11, and 114 in te text Linear Interpolating Splines We ave seen tat ig-degree polynomial interpolation
More information[100] 091 News, Tutorial by Dec. 10, 2012 =======================================
[100] 091 revised on 2012.12.10 cemmath The Simple is the Best News Dec. 10, 2012 ======================================= Cemmath 2.22 (a new name of Msharpmath) is newly upgraded. indefinite integrals
More informationConcept of Curve Fitting Difference with Interpolation
Curve Fitting Content Concept of Curve Fitting Difference with Interpolation Estimation of Linear Parameters by Least Squares Curve Fitting by Polynomial Least Squares Estimation of Non-linear Parameters
More informationSplines. Parameterization of a Curve. Curve Representations. Roller coaster. What Do We Need From Curves in Computer Graphics? Modeling Complex Shapes
CSCI 420 Computer Graphics Lecture 8 Splines Jernej Barbic University of Southern California Hermite Splines Bezier Splines Catmull-Rom Splines Other Cubic Splines [Angel Ch 12.4-12.12] Roller coaster
More informationCubic Spline Questions
Cubic Spline Questions. Find natural cubic splines which interpolate the following dataset of, points:.0,.,.,.0, 7.0,.,.0,0.; estimate the value for. Solution: Step : Use the n- cubic spline equations
More informationFriday, 11 January 13. Interpolation
Interpolation Interpolation Interpolation is not a branch of mathematic but a collection of techniques useful for solving computer graphics problems Basically an interpolant is a way of changing one number
More informationAssignment 2. with (a) (10 pts) naive Gauss elimination, (b) (10 pts) Gauss with partial pivoting
Assignment (Be sure to observe the rules about handing in homework). Solve: with (a) ( pts) naive Gauss elimination, (b) ( pts) Gauss with partial pivoting *You need to show all of the steps manually.
More informationMath F302: Octave Miscellany September 28, e 1 x3 dx. Here s how to find a numerical approximation with Octave
Definite Integrals Despite your training in Calculus, most definite integrals cannot be computed exactly, and must be approximated numerically. You learned a number of rules for doing this: the trapezoidal
More informationA Brief Introduction to MATLAB
A Brief Introduction to MATLAB MATLAB (Matrix Laboratory) is an interactive software system for numerical computations and graphics. As the name suggests, MATLAB was first designed for matrix computations:
More informationFall CSCI 420: Computer Graphics. 4.2 Splines. Hao Li.
Fall 2014 CSCI 420: Computer Graphics 4.2 Splines Hao Li http://cs420.hao-li.com 1 Roller coaster Next programming assignment involves creating a 3D roller coaster animation We must model the 3D curve
More informationSplines and Piecewise Interpolation. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan
Splines and Piecewise Interpolation Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan chanhl@mail.cgu.edu.tw Splines n 1 intervals and n data points 2 Splines (cont.) Go through
More informationFrom: Robert Sharpley Subject: Homeworks #6 Date: February 22, :09:53 AM EST Cc: Robert Sharpley
From: Robert Sharpley Subject: Homeworks #6 Date: February 22, 2006 9:09:53 AM EST Cc: Robert Sharpley %% Homework #5 - Solutions %% Here is a matlab code
More informationand r 23 e iθ 23 function [CornersFound] = FindCorners(x,y,threshhold,closed)
Math 28a - Programming Project 2 - Answers We provide. our implementations of the subroutines you had to supply, 2. a discussion of periodic splines, and 3. outputs and discussion. FindCorners.m is self-explanatory.
More informationInterpolation by Spline Functions
Interpolation by Spline Functions Com S 477/577 Sep 0 007 High-degree polynomials tend to have large oscillations which are not the characteristics of the original data. To yield smooth interpolating curves
More informationJustify all your answers and write down all important steps. Unsupported answers will be disregarded.
Numerical Analysis FMN011 2017/05/30 The exam lasts 5 hours and has 15 questions. A minimum of 35 points out of the total 70 are required to get a passing grade. These points will be added to those you
More informationMATH 51: MATLAB HOMEWORK 3
MATH 5: MATLAB HOMEWORK Experimental data generally suffers from imprecision, though frequently one can predict how data should behave by graphing results collected from experiments. For instance, suppose
More informationSet 5, Total points: 100 Issued: week of
Prof. P. Koumoutsakos Prof. Dr. Jens Walther ETH Zentrum, CLT F 1, E 11 CH-809 Zürich Models, Algorithms and Data (MAD): Introduction to Computing Spring semester 018 Set 5, Total points: 100 Issued: week
More informationMaximizing an interpolating quadratic
Week 11: Monday, Apr 9 Maximizing an interpolating quadratic Suppose that a function f is evaluated on a reasonably fine, uniform mesh {x i } n i=0 with spacing h = x i+1 x i. How can we find any local
More informationMath Numerical Analysis
... Math 541 - Numerical Analysis Interpolation and Polynomial Approximation Piecewise Polynomial Approximation; Joseph M. Mahaffy, jmahaffy@mail.sdsu.edu Department of Mathematics and Statistics Dynamical
More informationVector: A series of scalars contained in a column or row. Dimensions: How many rows and columns a vector or matrix has.
ASSIGNMENT 0 Introduction to Linear Algebra (Basics of vectors and matrices) Due 3:30 PM, Tuesday, October 10 th. Assignments should be submitted via e-mail to: matlabfun.ucsd@gmail.com You can also submit
More informationChapter 19 Interpolation
19.1 One-Dimensional Interpolation Chapter 19 Interpolation Empirical data obtained experimentally often times conforms to a fixed (deterministic) but unkown functional relationship. When estimates of
More informationPolynomial Functions Graphing Investigation Unit 3 Part B Day 1. Graph 1: y = (x 1) Graph 2: y = (x 1)(x + 2) Graph 3: y =(x 1)(x + 2)(x 3)
Part I: Polynomial Functions when a = 1 Directions: Polynomial Functions Graphing Investigation Unit 3 Part B Day 1 1. For each set of factors, graph the zeros first, then use your calculator to determine
More informationMATLAB QUICK START TUTORIAL
MATLAB QUICK START TUTORIAL This tutorial is a brief introduction to MATLAB which is considered one of the most powerful languages of technical computing. In the following sections, the basic knowledge
More information1) Generate a vector of the even numbers between 5 and 50.
MATLAB Sheet 1) Generate a vector of the even numbers between 5 and 50. 2) Let x = [3 5 4 2 8 9]. a) Add 20 to each element. b) Subtract 2 from each element. c) Add 3 to just the odd index elements. d)
More informationModule 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 6:
file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_1.htm 1 of 1 6/20/2012 12:24 PM The Lecture deals with: ADI Method file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_2.htm 1 of 2 6/20/2012
More informationPolymath 6. Overview
Polymath 6 Overview Main Polymath Menu LEQ: Linear Equations Solver. Enter (in matrix form) and solve a new system of simultaneous linear equations. NLE: Nonlinear Equations Solver. Enter and solve a new
More informationIntroduction to MATLAB 7 for Engineers
PowerPoint to accompany Introduction to MATLAB 7 for Engineers William J. Palm III Chapter 2 Numeric, Cell, and Structure Arrays Copyright 2005. The McGraw-Hill Companies, Inc. Permission required for
More informationUNIVERSITY OF CALIFORNIA COLLEGE OF ENGINEERING
UNIVERSITY OF CALIFORNIA COLLEGE OF ENGINEERING E7: INTRODUCTION TO COMPUTER PROGRAMMING FOR SCIENTISTS AND ENGINEERS Professor Raja Sengupta Spring 2010 Second Midterm Exam April 14, 2010 [30 points ~
More informationPart III Functions and Data
Part III Functions and Data c Copyright, Todd Young and Martin Mohlenkamp, Mathematics Department, Ohio University, 2017 Lecture 19 Polynomial and Spline Interpolation A Chemical Reaction In a chemical
More information3.2 - Interpolation and Lagrange Polynomials
3. - Interpolation and Lagrange Polynomials. Polynomial Interpolation: Problem: Givenn pairs of data points x i, y i,wherey i fx i, i 0,,...,n for some function fx, we want to find a polynomial P x of
More informationMaster Thesis. Comparison and Evaluation of Didactic Methods in Numerical Analysis for the Teaching of Cubic Spline Interpolation
Master Thesis Comparison and Evaluation of Didactic Methods in Numerical Analysis for the Teaching of Cubic Spline Interpolation Abtihal Jaber Chitheer supervised by Prof. Dr. Carmen Arévalo May 17, 2017
More informationNeed for Parametric Equations
Curves and Surfaces Curves and Surfaces Need for Parametric Equations Affine Combinations Bernstein Polynomials Bezier Curves and Surfaces Continuity when joining curves B Spline Curves and Surfaces Need
More informationLECTURE NOTES - SPLINE INTERPOLATION. 1. Introduction. Problems can arise when a single high-degree polynomial is fit to a large number
LECTURE NOTES - SPLINE INTERPOLATION DR MAZHAR IQBAL 1 Introduction Problems can arise when a single high-degree polynomial is fit to a large number of points High-degree polynomials would obviously pass
More informationCourse Number 432/433 Title Algebra II (A & B) H Grade # of Days 120
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
More informationAn Introduction to Numerical Methods
An Introduction to Numerical Methods Using MATLAB Khyruddin Akbar Ansari, Ph.D., P.E. Bonni Dichone, Ph.D. SDC P U B L I C AT I O N S Better Textbooks. Lower Prices. www.sdcpublications.com Powered by
More informationMath 3 Coordinate Geometry Part 2 Graphing Solutions
Math 3 Coordinate Geometry Part 2 Graphing Solutions 1 SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY The solution of two linear equations is the point where the two lines intersect. For example, in the graph
More informationRemark. Jacobs University Visualization and Computer Graphics Lab : ESM4A - Numerical Methods 331
Remark Reconsidering the motivating example, we observe that the derivatives are typically not given by the problem specification. However, they can be estimated in a pre-processing step. A good estimate
More informationEngineering Analysis ENG 3420 Fall Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00
Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00 1 Lecture 24 Attention: The last homework HW5 and the last project are due on Tuesday November
More informationTable for Third-Degree Spline Interpolation Using Equi-Spaced Knots. By W. D. Hoskins
MATHEMATICS OF COMPUTATION, VOLUME 25, NUMBER 116, OCTOBER, 1971 Table for Third-Degree Spline Interpolation Using Equi-Spaced Knots By W. D. Hoskins Abstract. A table is given for the calculation of the
More informationMid-Chapter Quiz: Lessons 2-1 through 2-3
Graph and analyze each function. Describe its domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 2x 3 2 16 1.5 6.75 1 2 0 0 1 2 1.5 6.75
More informationCubic smoothing spline
Cubic smooting spline Menu: QCExpert Regression Cubic spline e module Cubic Spline is used to fit any functional regression curve troug data wit one independent variable x and one dependent random variable
More informationFondamenti di Informatica
Fondamenti di Informatica Scripts and Functions: examples lesson 9 2012/04/16 Prof. Emiliano Casalicchio emiliano.casalicchio@uniroma2.it Agenda Examples Bisection method Locating roots Secant methods
More informationLecture 9. Curve fitting. Interpolation. Lecture in Numerical Methods from 28. April 2015 UVT. Lecture 9. Numerical. Interpolation his o
Curve fitting. Lecture in Methods from 28. April 2015 to ity Interpolation FIGURE A S Splines Piecewise relat UVT Agenda of today s lecture 1 Interpolation Idea 2 3 4 5 6 Splines Piecewise Interpolation
More informationMATLAB Examples. Interpolation and Curve Fitting. Hans-Petter Halvorsen
MATLAB Examples Interpolation and Curve Fitting Hans-Petter Halvorsen Interpolation Interpolation is used to estimate data points between two known points. The most common interpolation technique is Linear
More informationThe use of the Spectral Properties of the Basis Splines in Problems of Signal Processing
The use of the Spectral Properties of the Basis Splines in Problems of Signal Processing Zaynidinov Hakim Nasiritdinovich, MirzayevAvazEgamberdievich, KhalilovSirojiddinPanjievich Doctor of Science, professor,
More informationInterpolation and Splines
Interpolation and Splines Anna Gryboś October 23, 27 1 Problem setting Many of physical phenomenona are described by the functions that we don t know exactly. Often we can calculate or measure the values
More informationMATLAB Modul 3. Introduction
MATLAB Modul 3 Introduction to Computational Science: Modeling and Simulation for the Sciences, 2 nd Edition Angela B. Shiflet and George W. Shiflet Wofford College 2014 by Princeton University Press Introduction
More informationIntroduction to MATLAB for Engineers, Third Edition
PowerPoint to accompany Introduction to MATLAB for Engineers, Third Edition William J. Palm III Chapter 2 Numeric, Cell, and Structure Arrays Copyright 2010. The McGraw-Hill Companies, Inc. This work is
More informationAn introduction to interpolation and splines
An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve
More informationUniversity of Alberta
A Brief Introduction to MATLAB University of Alberta M.G. Lipsett 2008 MATLAB is an interactive program for numerical computation and data visualization, used extensively by engineers for analysis of systems.
More informationPolynomial Approximation and Interpolation Chapter 4
4.4 LAGRANGE POLYNOMIALS The direct fit polynomial presented in Section 4.3, while quite straightforward in principle, has several disadvantages. It requires a considerable amount of effort to solve the
More informationVariable Definition and Statement Suppression You can create your own variables, and assign them values using = >> a = a = 3.
MATLAB Introduction Accessing Matlab... Matlab Interface... The Basics... 2 Variable Definition and Statement Suppression... 2 Keyboard Shortcuts... More Common Functions... 4 Vectors and Matrices... 4
More informationFitting to a set of data. Lecture on fitting
Fitting to a set of data Lecture on fitting Linear regression Linear regression Residual is the amount difference between a real data point and a modeled data point Fitting a polynomial to data Could use
More information(Creating Arrays & Matrices) Applied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB (Creating Arrays & Matrices) Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional
More informationProf. Manoochehr Shirzaei. RaTlab.asu.edu
RaTlab.asu.edu Introduction To MATLAB Introduction To MATLAB This lecture is an introduction of the basic MATLAB commands. We learn; Functions Procedures for naming and saving the user generated files
More informationFreeMat Tutorial. 3x + 4y 2z = 5 2x 5y + z = 8 x x + 3y = -1 xx
1 of 9 FreeMat Tutorial FreeMat is a general purpose matrix calculator. It allows you to enter matrices and then perform operations on them in the same way you would write the operations on paper. This
More informationPoints Lines Connected points X-Y Scatter. X-Y Matrix Star Plot Histogram Box Plot. Bar Group Bar Stacked H-Bar Grouped H-Bar Stacked
Plotting Menu: QCExpert Plotting Module graphs offers various tools for visualization of uni- and multivariate data. Settings and options in different types of graphs allow for modifications and customizations
More informationPerforming Matrix Operations on the TI-83/84
Page1 Performing Matrix Operations on the TI-83/84 While the layout of most TI-83/84 models are basically the same, of the things that can be different, one of those is the location of the Matrix key.
More informationInteractive Graphics. Lecture 9: Introduction to Spline Curves. Interactive Graphics Lecture 9: Slide 1
Interactive Graphics Lecture 9: Introduction to Spline Curves Interactive Graphics Lecture 9: Slide 1 Interactive Graphics Lecture 13: Slide 2 Splines The word spline comes from the ship building trade
More informationTHE STUDY OF NEW APPROACHES IN CUBIC SPLINE INTERPOLATION FOR AUTO MOBILE DATA
Journal of Science and Arts Year 17, No. 3(4), pp. 41-46, 217 ORIGINAL PAPER THE STUDY OF NEW APPROACHES IN CUBIC SPLINE INTERPOLATION FOR AUTO MOBILE DATA NAJMUDDIN AHMAD 1, KHAN FARAH DEEBA 1 Manuscript
More information4.3 Quadratic functions and their properties
4.3 Quadratic functions and their properties A quadratic function is a function defined as f(x) = ax + x + c, a 0 Domain: the set of all real numers x-intercepts: Solutions of ax + x + c = 0 y-intercept:
More informationx = 12 x = 12 1x = 16
2.2 - The Inverse of a Matrix We've seen how to add matrices, multiply them by scalars, subtract them, and multiply one matrix by another. The question naturally arises: Can we divide one matrix by another?
More informationMatlab and Coordinate Systems
Matlab and Coordinate Systems Math 45 Linear Algebra David Arnold David-Arnold@Eureka.redwoods.cc.ca.us Abstract In this exercise we will introduce the concept of a coordinate system for a vector space.
More informationARRAY VARIABLES (ROW VECTORS)
11 ARRAY VARIABLES (ROW VECTORS) % Variables in addition to being singular valued can be set up as AN ARRAY of numbers. If we have an array variable as a row of numbers we call it a ROW VECTOR. You can
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
More informationIntroduction to Matlab
Introduction to Matlab Enrique Muñoz Ballester Dipartimento di Informatica via Bramante 65, 26013 Crema (CR), Italy enrique.munoz@unimi.it Contact Email: enrique.munoz@unimi.it Office: Room BT-43 Industrial,
More informationMultiple-Choice Test Spline Method Interpolation COMPLETE SOLUTION SET
Multiple-Choice Test Spline Method Interpolation COMPLETE SOLUTION SET 1. The ollowing n data points, ( x ), ( x ),.. ( x, ) 1, y 1, y n y n quadratic spline interpolation the x-data needs to be (A) equally
More information1 A Section A BRIEF TOUR. 1.1 Subsection Subsection 2
A BRIEF TOUR Maple V is a complete mathematical problem-solving environment that supports a wide variety of mathematical operations such as numerical analysis, symbolic algebra, and graphics. This worksheet
More informationAlgebra 1. Standard 11 Operations of Expressions. Categories Combining Expressions Multiply Expressions Multiple Operations Function Knowledge
Algebra 1 Standard 11 Operations of Expressions Categories Combining Expressions Multiply Expressions Multiple Operations Function Knowledge Summative Assessment Date: Wednesday, February 13 th Page 1
More informationLeast-Squares Fitting of Data with B-Spline Curves
Least-Squares Fitting of Data with B-Spline Curves David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International
More informationTeaching Manual Math 2131
Math 2131 Linear Algebra Labs with MATLAB Math 2131 Linear algebra with Matlab Teaching Manual Math 2131 Contents Week 1 3 1 MATLAB Course Introduction 5 1.1 The MATLAB user interface...........................
More information. As x gets really large, the last terms drops off and f(x) ½x
Pre-AP Algebra 2 Unit 8 -Lesson 3 End behavior of rational functions Objectives: Students will be able to: Determine end behavior by dividing and seeing what terms drop out as x Know that there will be
More informationInterpolation - 2D mapping Tutorial 1: triangulation
Tutorial 1: triangulation Measurements (Zk) at irregular points (xk, yk) Ex: CTD stations, mooring, etc... The known Data How to compute some values on the regular spaced grid points (+)? The unknown data
More informationMar. 20 Math 2335 sec 001 Spring 2014
Mar. 20 Math 2335 sec 001 Spring 2014 Chebyshev Polynomials Definition: For an integer n 0 define the function ( ) T n (x) = cos n cos 1 (x), 1 x 1. It can be shown that T n is a polynomial of degree n.
More informationGeneralised Mean Averaging Interpolation by Discrete Cubic Splines
Publ. RIMS, Kyoto Univ. 30 (1994), 89-95 Generalised Mean Averaging Interpolation by Discrete Cubic Splines By Manjulata SHRIVASTAVA* Abstract The aim of this work is to introduce for a discrete function,
More informationHybrid Newton-Cotes Integrals
Hybrid Newton-Cotes Integrals 1 Hybrid Newton-Cotes Integrals By Namir C. Shammas Introduction Newton-Cotes integration methods are numerical methods for integration. These methods calculate the estimate
More information