CSC351S Numerical Approximation, Integration and Ordinary Differential Equat ions

Spring 2009

Course information for current students:

• 21-4-09: Assignment 4 is marked. You can pick it up this afternoon, and tomorrow during office hours.
• 9-4-09: Office hours for the exam: Tue 21 April 2009, 1-2, Wed 22 April 2009, 1-2.
• 6-4-09: Assignment 4, q3: The initial conditions are theta_0 = 1 and theta'_0 = 0.
• 22-2-09: On Tuesday, 24 Feb 2009, 3-4 p.m., we will do a lecture (not a tutorial) in the tutorial room BAB025.
  Marks for assignment 1 and the midterm test are on the cdf marks system.
• 10-2-09: For the midterm exam, you may wish to see some practice questions.
• 3-2-09: From now on, CSC351H1S, L0101 will be held on R3-5 in BA1210; the midterm will be held in this room Thu 12 Feb 2009.
  and tutorials will be held on T3 in BAB025, as usual.
  Please remember to come to the new room this Thursday 5 Feb 2009.
• 15-1-09: For the midterm on Thu 12 Feb 2009, we have a better room reserved, BA 1210.
  Assignment one is haded out. See further in this page.
• 3-1-09: The first meeting of CSC351 is Thursday, January 8, 2009, 3-5 p.m., in GB 220.
  Note the room number!
  Tutorials will start on Tuesday, January 13, 2009, in BA B025.
  Note the room number!
  Also note the room number has been corrected!
• 4-12-08: The textbook for this course is Michael Heath's, Scientific Computing: an introductory survey, McGraw-Hill Inc. 2002.
  This book does not cover everything needed for the course, but most of what is needed. I have ordered a custom copy (a reduced version) of the book, which should be available at the UT bookstore for about $60.
  You may want to consider getting the paperback (international?) edition of the book at amazon.ca, which (I believe) costs about $90. This is a complete version of the book.

Bulletin board for CSC351 Spring 2009

Material covered in the course (Spring 2009): (Textbook sections in parentheses:
H = Heath, KC = Kincaid and Cheney, BF = Burden and Faires)

8-1-2009 (2 hours)
1    Interpolation
1.1  Approximation and interpolation - Introduction [H 7.1, KC 6.0, BF 3]
1.2  Polynomial approximation - Weierstrass theorem [KC 6.1, BF 3]
1.3  Evaluating a polynomial -- Horner's rule (nested multiplication)
     [H 7.3.1, KC 6.1, BF 2.6 pgs 92-94]
1.4  Polynomial interpolation using monomial basis functions [H 7.3.1, KC 6.1, BF 3.1]
1.5  Polynomial interpolation using Lagrange basis functions [H 7.3.2, KC 6.1, BF 3.1]
1.6  Existence and uniqueness of polynomial interpolant [H 7.2, KC 6.1, BF 3.1]
1.7  Polynomial interpolation using Newton's basis functions
     and the Divided Differences Table [H 7.3.3, KC 6.1-2, BF 3.2]
13-1-2009 (1 hour) tutorial
polynomial interpolation, existence, uniqueness, monomials, Lagrange, NDD
15-1-2009 (2 hours)
     Proof that Newton's form can be computed by DDs
1.8  Comparison of the three bases
1.9  Error of the polynomial interpolant [H 7.3.5, KC 6.1, BF 3.1]
1.10 Polynomial interpolation with derivative data [KC 6.3, BF 3.3]
     Most general (Birkoff) polynomial interpolation problem
     A less general (Hermite) polynomial interpolation problem -- existence and uniqueness
     An even less general (Hermite) polynomial interpolation problem
     Standard Hermite polynomial interpolation problem
20-1-2009 (1 hour) tutorial
polynomial interpolation error, Taylor polynomial approximation,
interpolation with some derivative data, existence, uniqueness
22-1-2009 (2 hours)
1.11 Hermite polynomial interpolation using monomial basis
1.12 Hermite polynomial interpolation using Lagrange basis [KC 6.3, BF 3.3]
1.13 Hermite polynomial interpolation using Newton's basis [KC 6.3, BF 3.3]
1.14 Existence and uniqueness of Hermite polynomial interpolant [KC 6.3, BF 3.3]
1.15 Error of the Hermite polynomial interpolant [KC 6.3, BF 3.3]
1.16 Pitfalls of polynomial interpolation [H 7.3.5]
     [KC 6.1, pg 319, comp. probl pg 338] [BF 3.4, Figures 3.9, 3.10, 3.12]
1.17 Chebyshev polynomials [H 7.3.4, KC 6.1, BF 8.3]
     The optimal placing of data points in polynomial interpolation
1.18 Piecewise polynomials and splines [H 7.4, KC 6.4, BF 3.4]
1.19 Linear spline interpolation (Lagrange form)
     Error in linear spline interpolation
27-1-2009 (1 hour) tutorial
Hermite polynomial interpolation, monomials, Lagrange, NDD
Hermite polynomial interpolation error
29-1-2009 (2 hours)
     Error in linear spline interpolation (end)
1.20 Cubic spline interpolation -- choice of end-conditions [H 7.4.2, KC 6.4, BF 3.4]
     Error in cubic spline interpolation
     Construction of a clamped cubic spline interpolant
1.21 Piecewise polynomial basis functions [H 7.4.3, KC 6.5]
1.22 B-splines [H 7.4.3, KC 6.5]
1.23 Linear B-spline basis functions [KC 6.5]
1.24 Linear spline interpolation using B-splines as basis functions [KC 6.5-6]
3-2-2009 (1 hour) tutorial
piecewise polynomials and splines
5-2-2009 (2 hours)
1.25 Cubic B-spline basis functions [KC 6.5]
1.26 Cubic spline interpolation using B-splines as basis functions [KC 6.5-6]
1.27 Piecewise cubic Hermite interpolation [H 7.4.1]

2    Least squares approximation and orthogonal polynomials
2.1  Least squares approximation [KC 6.8. BF 8.2]
2.2  Inner products and norms of functions [KC 6.8]
2.3  Linear independence of functions/polynomials [KC 6.8, BF 8.2]
2.4  Orthogonal polynomials -- Monic polynomials [KC 6.8, BF 8.2]
10-2-2009 (1 hour) tutorial
Orthogonal polynomials
12-2-2009 (2 hours) midterm + short lecture
2.5  Constructing the least squares polynomial approximation [KC 6.8, 8.2]
     The normal equations -- Gram matrix -- Hilbert matrix
2.6  Constructing the least squares polynomial approximation [KC 6.8, 8.2]
     using orthogonal polynomials
     Proof of orthogonality of error to the space of q_i's
Spring break
24-2-2009 (1 hour lecture)
2.7  Constructing a set of orthogonal (or orthonormal) polynomials
-.-. The method of undetermined coefficients
-.-. The Gram-Schmidt three-term recurrence relation algorithm
26-2-2009 (2 hours)
2.8  Some famous orthogonal polynomials [KC 6.1, BF 8.3]
     Chebyshev, Legendre

3    Numerical Integration [H Ch 8, KC Ch 7, BF Ch 4]
3.1  Introduction [H 8.1, KC 7.2, BF 4.3]
3.2  Midpoint rule and error formula [H 8.3.1, KC 7.2, BF 4.3]
3.3  Composite midpoint rule and error formula [H 8.3.5, KC 7.2, BF 4.4]
3.4  Trapezoidal rule and error formula [H 8.3.1, KC 7.2, BF 4.3]
3.5  Alternative derivation of quadrature rules based on model intervals [H 8.2]
3.6  Transforming quadrature rules to other intervals [H 8.3.3, KC 7.2, BF 4.7]
3-3-2009 (1 hour) tutorial not done -- explained in class
5-3-2009 (2 hours)
3.7  Simpson's rule and error formula [H 8.3.1, KC 7.2, BF 4.3]
3.8  Corrected trapezoidal rule and error formula
3.9  Convergence of polynomial interpolatory quadrature rules
3.10 Newton-Cotes quadrature rules [H 8.3.1, KC 7.2, BF 4.3]
3.11 Composite quadrature rules and error formulae [H 8.3.5, KC 7.2, BF 4.4]
3.12 Error estimators for quadrature rules [H 8.3.1, KC 7.5, parts of 7.4, BF 4.6]
3.13 Adaptive quadrature [H 8.3.6, KC 7.5, BF 4.6]
10-3-2009 (1 hour) tutorial
Three term recurrence relation Gram-Schmidt,
least squares polynomial approximation
Quadrature: midtpoint, trapezoid, Simpson's, corrected trapezoid,
composite rules, degree of exactness, error formulae
12-3-2009 (2 hours)
3.14 Gauss quadrature rules [H 8.3.3, KC 7.3, BF 4.7]
3.15 Comparison of NC and Gauss rules
3.16 Romberg integration [H 8.7, KC 7.4, BF ~4.2, 4.5]
3.17 Infinite (and semi-infinite) integrals [H 8.4.2, BF 4.9]
3.18 Singular integrals [H 8.4.2, BF 4.9]
17-3-2009 (1 hour) tutorial
Adaptive quadrature, construction of rules, Gauss rules,
degree of exactness, error formulae, order of convergence
19-3-2009 (2 hours)
3.18 Singular integrals [H 8.4.2, BF 4.9]
(end)

4    Ordinary Differential Equations [H Ch 9, KC Ch 8, BF Ch 5]
4.1  Introduction: DEs, ODEs, PDEs and IVPs [H 9.1, KC 8.1, BF 5.1]
4.2  Existence and uniqueness of solution of an IVP-ODE [H 9.2, KC 8.1, BF 5.1]
4.3  Second order ODEs and BVPs [H 9.1, KC 8.6, BF Ch 11, intro. pgs 624-625]
4.4  nth order ODEs and IVPs for ODEs [H 9.1, KC 8.6, BF 5.9]
     Systems of ODEs [KC 8.12, BF 5.13]
4.5  Stability of ODEs -- Jacobian [H 9.2]
4.6  Stiff ODEs [KC 8.12]
4.7  Numerical methods for first order IVPs for ODEs
4.8  Forward Euler's method [H 9.3.1, KC 8.2, BF 5.2]
     Global and local truncation errors [H 9.3.2, KC 8.2, 8.5]
     Order of a numerical method for IVPs-ODEs [H 9.3.2, KC 8.2, BF 5.3 pgs 269-270]
24-3-2009 (1 hour) tutorial
Improper integrals
26-3-2009 (2 hours)
     Stability of the numerical method [H 9.3.2, KC 8.5, BF ~5.10]
     Region of absolute stability [H 9.3.2, KC 8.12, BF 5.11]
     Stiff ODEs [H 9.3.4, KC 8.12, BF ~5.11]
     Systems of ODEs [KC 8.12]
     Control of magnitude of LTE
     Global error bound without round-off errors [KC 8.5, BF 5.2]
     Global error bound with    round-off errors [BF 5.2]
4.9  Backward Euler's method [H 9.3.3, KC 8.4, BF 5.6]
4.10 Explicit versus implicit
4.11 Trapezoid method [H 9.6, BF 5.11]
31-3-2009 (1 hour) tutorial
ODEs, Lipschitz condition, conversion of higher order ODEs to systems of first order ODEs, stability of ODEs
2-4-2009 (2 hours)
4.12 Linear multistep methods [H 9.3.8, KC 8.4, BF 5.6]
4.13 Runge-Kutta methods [H 9.3.6, KC 8.3, BF 5.4, 5.5]
9-4-2009 (1+ hours)
Summary

No time left:
4.14 BVPs for ODEs
4.15 Existence and uniqueness of solution of a BVP for an ODE
4.16 Numerical methods for solving BVPs for ODEs
4.17 Finite Difference Approximations to Derivatives
4.18 A finite difference method for a model BVP for ODE

Notes and handouts:
Access to these data requires that you type in your CDF username and last 5 digits of your student id.
Introduction, Polynomial approximation, polynomial interpolation with monomials (pgs 1-16)
Lagrange basis functions, existence, uniqueness, NDD, comparison (pgs 17-32)
Error of polynomial interpolation and related theorems (pgs 33-40)
Polynomial interpolation with derivative data, Hermite interpolation with monomials, Lagrange, NDD, existence, uniqueness, error, pitfalls of poly. interpolation (pgs 41-56)
Assignment 1
Chebyshev polynomials, interpolation at Chebyshev points (pgs 57-68)
Piecewise polynomial and spline interpolation (pgs 69-99)
Assignment 2 , file testQ.m
Least squares approximation (pgs 105-119)
Assignment 3 ,
Numerical integration (Quadrature) (pgs 129-188)
Numerical ODEs 0 (pgs 201-212)
Assignment 4 ,
Numerical ODEs 1 (pgs 213-255)
Summary (pgs 1-16)

Tutorial 1 , Tutorial 2 , Tutorial 3 , Tutorial 4 , Tutorial 5 , Tutorial 6 , Tutorial 7 , Tutorial 8 , Tutorial 9 , Tutorial 10 ,

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