CSC436 Numerical Algorithms

Spring 2026 Bulletin/discussion board for csc436 -- course outline -- MarkUs (for MarkUs, login with UTorId)

[Announcements] [Material covered] [Lectures, Tutorials, Assignments (with pass)]

Announcements, Course information for current students:

Material to be covered covered in the course (Spring 2026): (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.
First couple of lectures may go a bit fast over things that were done in csc336. A few more things will be added. Later lectures go on to new material. 
2026-01-06 3 hrs
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]
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
     (we will do these next time)
1.10 Linear independence of functions/polynomials [BF 8.2, AG 10.1 pg 297]
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]
1.14 Hermite polynomial interpolation using Newton's basis [KC 6.3, BF 3.3, AG 10.7]
2026-01-13 3 hrs
1.9b Remarks on polynomial interpolation error Theorem
     Theorems relating the polynomial interpolation error with Newton's DD
1.10 Linear independence of functions/polynomials [BF 8.2, AG 10.1 pg 297]

Tutorial 1, only Q4, rest are from csc336
Tutorial 2, Taylor's and interpolation, case of no or inf solutions
Tutorial 3, Hermite, monomials, Lagrange, NDD, error, Hermite-Birkhoff

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
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
2026-01-20 3 hrs
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]
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, piecewise polynomials, splines and interpolation

2026-01-27 3 hrs
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]
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]
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]
2026-02-03 3 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]
Tutorial 6
Appendix pgs 301-308
2.x  Inner products and norms of functions
2.y  Orthogonal polynomials, orthonormal, monic, Chebyshev, Legendre
2026-02-10 3 (2.5) hrs
Discussion on A1
2.17 Infinite (and semi-infinite) integrals [H 8.4.2, BF 4.9, AG 15.3,4 (examples)]
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 7

2026-02-24 3 (2.5) hrs
term test 1
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]
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]
2026-03-03 3 hrs
     Global error bound with    round-off errors [BF 5.2]
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]
3.12 Linear multistep methods [H 9.3.8, KC 8.4, BF 5.6, AG 16.4]
Tutorial 8
2026-03-10 3 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 9

2026-03-17 3 hrs
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]
     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
Tutorial 10, Q1, Q2 only normal equations (i)
4.7  Orthogonal transformations [H 3.4.3, AG 6.2]
     Using orthogonal transformations for least squares problems
4.8  Householder reflections (orthogonal transformations) [H 3.5, AG 6.3]
     Householder reflections - definition, symmetry, orthogonality, reflection
     Constructing Householder matrices to eliminate components of vectors
4.9  QR factorization [H 3.4.3, 3.4.5, 3.5.1, 3.5.4, AG 6.2]
     The QR factorization of a matrix
skip Computing the QR factorization with Householder transformations
     Solving linear systems with the QR factorization
     Square, overdetermined (and underdetermined systems)
2026-03-24 3 hrs (- term test)
     (Square, overdetermined and) underdetermined systems
4.10 The Gram-Schmidt orthogonalization algorithm
     Solving linear systems with the Gram-Schmidt factorization
     Overdetermined and underdetermined systems

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
     Caley-Hamilton theorem
     Perron-Frobenius theorem
2026-03-31 3 hrs
5.2  Why are eigenvalues/vectors important [AG 8.1]
     -- calculation of importance score of web pages by search engines
5.3  Computing eigenvalues and eigenvectors [H 4.5.1, 4.5.6, AG 8.1]
     The power method (power iteration) [AG 8.1]
     Example of power iteration, and inverse power iteration -- start
     Example of power iteration, and inverse power iteration
     The QR iteration for computing the eigenvalues/vectors [AG 8.3]
5.4  Review of matrix decompositions

Notes and handouts: Course information
Outline

Access to the data below requires that you type in your CDF (teaching lab) username (as login) and last 5 digits of your student number (as pass). This password (for accessing the website) cannot be changed.

Notes

Tutorials Assignments

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
An example plot in matlab myplot.m.