Course information for current students:
Bulletin board for CSC436 Fall 2022
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Material to be covered covered in the course (Fall 2022):
(Textbook sections in parentheses:
H = Heath, AG = Ascher and Greif, KC = Kincaid and Cheney, BF = Burden and Faires)
You may consult any of the following references.
The primary textbook is Heath's.
2022-09-09 1 hr 1 Interpolation 1.1 Approximation and interpolation - Introduction [H 7.1, KC 6.0, BF 3, AG 10.1] 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, AG 1.3 pgs 10-11] 1.4 Polynomial interpolation using monomial basis functions [H 7.3.1, KC 6.1, BF 3.1, AG 10.2] 2022-09-14 2 hrs 1.5 Polynomial interpolation using Lagrange basis functions [H 7.3.2, KC 6.1, BF 3.1, AG 10.3] 1.6 Existence and uniqueness of polynomial interpolant [H 7.2, KC 6.1, BF 3.1, AG 10.1+2] 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, AG 10.4] 1.8 Comparison of the three bases 1.9 Error of the polynomial interpolant [H 7.3.5, KC 6.1, BF 3.1, AG 10.5] Proof of Theorem Remarks on Theorem Theorems relating the polynomial interpolation error with Newton's DD 2022-09-16 1 hr 1.10 Linear independence of functions/polynomials [BF 8.2, AG 10.1 pg 297] Tutorial 1, Q7 1.11 Polynomial interpolation with derivative data [KC 6.3, BF 3.3, AG 10.7] 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 1.12 Hermite polynomial interpolation using monomial basis 1.13 Hermite polynomial interpolation using Lagrange basis [KC 6.3, BF 3.3] 2022-09-21 2 hrs 1.14 Hermite polynomial interpolation using Newton's basis [KC 6.3, BF 3.3, AG 10.7] 1.15 Existence and uniqueness of Hermite polynomial interpolant [KC 6.3, BF 3.3] 1.16 Error of the Hermite polynomial interpolant [KC 6.3, BF 3.3] Tutorial 2, Tutorial 3 1.17 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] [AG 10.6, 11.1] 1.18 Piecewise polynomials and splines [H 7.4, KC 6.4, BF 3.4, AG 11.1] 1.19 Linear spline interpolation (Lagrange form) [AG 11.2] Error in linear spline interpolation 2022-09-23 1 hr 1.20 Cubic spline interpolation -- choice of end-conditions [H 7.4.2, KC 6.4, BF 3.4, AG 11.2-3] Error in cubic spline interpolation Spline interpolation in MATLAB 1.21 Construction of a clamped cubic spline interpolant [AG 11.3] 1.22 Piecewise polynomial basis functions [H 7.4.3, KC 6.5, AG 11.4] 2022-09-28 2 hrs 1.23 B-splines [H 7.4.3, KC 6.5, AG 11.4] 1.24 Linear B-spline basis functions [KC 6.5, AG 11.4] 1.25 Linear spline interpolation using B-splines as basis functions [KC 6.5-6, AG 11.4] 1.26 Cubic B-spline basis functions [KC 6.5, AG 11.4] 1.27 Cubic spline interpolation using B-splines as basis functions [KC 6.5-6] 1.28 Piecewise cubic Hermite interpolation [H 7.4.1] Tutorial 4, Q1, Q2, Q3 2022-09-29 1 hr Tutorial 4, Q4 2 Numerical Integration [H Ch 8, KC Ch 7, BF Ch 4, AG Ch 15] 2.1 Introduction [H 8.1, KC 7.2, BF 4.3, AG 15.1] 2022-10-05 2 hrs 2.1 Introduction [H 8.1, KC 7.2, BF 4.3, AG 15.1] end 2.2 Midpoint rule and error formula [H 8.3.1, KC 7.2, BF 4.3, AG 15.1] 2.3 Composite midpoint rule and error formula [H 8.3.5, KC 7.2, BF 4.4, AG 15.2] 2.4 Trapezoidal rule and error formula [H 8.3.1, KC 7.2, BF 4.3, AG 15.1] 2.5 Alternative derivation of quadrature rules based on model intervals [H 8.2] 2.6 Transforming quadrature rules to other intervals [H 8.3.3, KC 7.2, BF 4.7] 2.7 Simpson's rule and error formula [H 8.3.1, KC 7.2, BF 4.3, AG 15.1] 2.8 Corrected trapezoidal rule and error formula 2.9 Convergence of polynomial interpolatory quadrature rules 2.10 Newton-Cotes quadrature rules [H 8.3.1, KC 7.2, BF 4.3] 2.11 Composite quadrature rules and error formulae [H 8.3.5, KC 7.2, BF 4.4, AG 15.2] 2022-10-07 1 hr 2.12 Error estimators for quadrature rules [H 8.3.1, KC 7.5, parts of 7.4, BF 4.6, AG 15.4] 2.13 Adaptive quadrature [H 8.3.6, KC 7.5, BF 4.6, AG 15.4] 2022-10-12 2 hrs Tutorial 5 2.14 Gauss quadrature rules [H 8.3.3, KC 7.3, BF 4.7, AG 15.3] 2.15 Comparison of NC and Gauss rules 2.16 Romberg integration [H 8.7, KC 7.4, BF ~4.2, 4.5, AG 15.5] 2022-10-14 1 hr 2.17 Infinite (and semi-infinite) integrals [H 8.4.2, BF 4.9, AG 15.3,4 (examples)] 2022-10-19 2 hrs 2.18 Singular integrals [H 8.4.2, BF 4.9, AG 15.3,4 (examples)] 2.19 Numerical integration in multiple dimensions [H 8.4.2, 8.4.4, AG 15.6] Tutorial 6 Tutorial 7 2022-10-21 1 hr discussion on A1 3 Ordinary Differential Equations [H Ch 9, KC Ch 8, BF Ch 5, AG Ch 16] 3.1 Introduction: DEs, ODEs, PDEs and IVPs [H 9.1, KC 8.1, BF 5.1, AG 16.1] 2022-10-26 2 hrs 3.1 Introduction: DEs, ODEs, PDEs and IVPs [H 9.1, KC 8.1, BF 5.1, AG 16.1] end 3.2 Existence and uniqueness of solution of an IVP-ODE [H 9.2, KC 8.1, BF 5.1] 3.3 Second order ODEs and BVPs [H 9.1, KC 8.6, BF Ch 11, intro. pgs 624-625, AG 16.1] 3.4 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, AG 16.1] 3.5 Stability of ODEs -- Jacobian [H 9.2] 3.6 Stiff ODEs [KC 8.12] 3.7 Numerical methods for first order IVPs for ODEs 3.8 Forward Euler's method [H 9.3.1, KC 8.2, BF 5.2, AG 16.2] Global and local truncation errors [H 9.3.2, KC 8.2, 8.5, AG 16.2] Order of a numerical method for IVPs-ODEs [H 9.3.2, KC 8.2, BF 5.3 pgs 269-270, AG 16.2] 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, AG 16.2] 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] 2022-10-28 1 hr term test 1 2022-11-02 2 hrs 3.9 Backward Euler's method [H 9.3.3, KC 8.4, BF 5.6, AG 16.2, 16.5] Accuracy and stability of BE 3.10 Explicit versus implicit 3.11 Trapezoid method [H 9.6, BF 5.11] 2022-11-04 1 hr 3.12 Linear multistep methods [H 9.3.8, KC 8.4, BF 5.6, AG 16.4] Fall break (reading week) 2022-11-16 2 hrs 3.13 Runge-Kutta methods [H 9.3.6, KC 8.3, BF 5.4, 5.5, AG 16.3, 16.6] 3.14 Software for IVP-ODEs Other methods for IVPs Some notes on numerical stability of methods for IVPs Tutorial 8, Q1, Q2 2022-11-18 1 hr Tutorial 8, Q3, Q4, Q5 Tutorial 9, Q1, Q2, Q3 first part) 2022-11-23 2 hrs Tutorial 9, Q3 (second part), Q4, Q5 discussion on A2 4 Least squares approximation 4.1 Least squares approximation [H 3.1, AG 6] 4.2 Overdetermined linear systems [H 3.1-2, 3.4.1, 3.7, AG 6.1] 2022-11-25 1 hr term test 2 2022-11-30 2 hrs The normal equations for overdetermined systems 4.3 Underdetermined linear systems [H 3.5.4] The normal equations for underdetermined systems 4.4 MATLAB and over- or under-determined linear systems [H 3.8] 4.5 Data fitting, curve fitting [H 3.1, AG 6.1] 4.6 MATLAB, data fitting and least squares problems 5 Computing eigenvalues and eigenvectors 5.1 Eigenvalues and eigenvectors [H 4.1-2, 4.4] Definition, characteristic equation/polynomial of a matrix, multiplicity of eigenvalue, matrix polynomial, similarity transformation, diagonalization of matrix, Jordan decomposition, Schur decomposition, singular value decomposition, symmetric, orthogonal, [triangular matrices,] [Gerschgorin disks, examples of locating eigenvalues with Gerschgorin disks] 2022-12-02 1 hr 5.2 Why are eigenvalues/vectors important [AG 8.1] -- calculation of importance score of web pages by search engines 2022-12-07 2 hrs 5.3 Computing eigenvalues and eigenvectors [H 4.5.1, 4.5.6, AG 8.1] The power method (power iteration) [H 4.5.1, 4.5.2, AG 8.1] Orthogonal matrices and Householder reflections [H 3.4.3, 3.5.1, AG 6.2, 6.3] The QR factorization and the QR iteration [H 4.5.6, AG 6.2, 8.3] Examples of power iteration and inverse power iteration Tutorial 10 Q1
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Notes
If you are new to MATLAB, you may be interested in
A brief introduction to MATLAB,
Christina C. Christara and Winky Wai
Tutorial on MATLAB,
Christina C. Christara