Numerical integration and differentiation
In the Reading column, the indicated sections or subsections are from the textbook, Scientific Computing: An Introductory Survey by Heath.
Lecture
Date
Topics
Assignments
Reading
1
Jun 22
Scientific computing
Sources of errors
Well-posed problems
Conditioning,
Stability,
Accuracy
(Slides: PDF (4 in 1 grey),
PDF(full color))
§ 1.1, 1.2, 1.4
2
Jun 24
Floating-point arithmetic
Cancellation
Vector and matrix norms
Solving linear systems: well-posedness, conditioning, error analysis
(Slides: PDF (4 in 1 grey),
PDF(full color)) Homework 1 assigned
§ 1.3, 2.1, 2.2, 2.3
3
Jun 29
Matrix-vector multiplications
Gaussian Elimination
LU factorization
Pivoting
(Slides: PDF(full clean)
PDF(full color))
§ 2.4, 2.5
4
Jul 1
Gaussian elimination with pivoting
Error analysis for LU factorization
Cholesky decomposition
Banded matrices
Sherman-Morrison formula
Iterative methods: Jacobi, Gauss-Seidel, SOR
(Slides: PDF(full clean)
PDF(full color)) Homework 1 due
Homework 2 assigned§ 2.5, 2.6, 11.5
5
Jul 6
Conjugate Gradient method
QR decomposition
Singular value decomposition
Linear least squares
(Slides: PDF(full clean)
PDF(full color))
§ 11.5, 3.1, 3.2, 3.3
6
Jul 8
Normal equation method
Least squares solution via SVD
Modified Gram-Schmidt Orthogonalization
Householder transformations
Givens rotations
(Slides: PDF(full clean)
PDF(full color))
Homework 2 due
Homework 3 assigned
Homework 1 returned and solution posted
§ 3.4, 3.5
7
Jul 13
Lagrange interpolation
Newton interpolation
Chebyshev Interpolation
Splines
(Slides: PDF
§ 7.3.1 - 7.3.5 7.4.2
8
Jul 15
Exercise problems
(Slides: PDF)
Homework 3 due
No Homework this week
Homework 2 to be returned & solution posted§ Chapter 7
9
Jul 20
Root finding for non-linear equations
Bisection
Fixed-point iteration
Newton's method and secant method
Inverse quadratic interpolation
Newton's method and secant method for higher dimensions
(Slides: PDF (full clean),
Slides: PDF (full color))
Homework #4 assigned
§ 5.1, 5.2, 5.3 5.5.1-5, 5.6.1-3
10
Jul 22
Rectangle, trapezoid and Simpson's rules
Composite quadrature
Adaptive quadrature
Gauss quadrature
finite difference formulas
Richardson's extrapolation
(Slides: PDF (full clean),
Slides: PDF (full color))
§ 8.1, 8.2, 8.3, 8.4.2, 8.6.1
11
Jul 27
Lecture postponed
11
Jul 29
Numerical solution to initial value problems
Euler's method
Implicit vs. Explicit methods
Truncation error, stability,
consistency, convergence
(Slides: PDF (full clean),
Slides: PDF (full color))
§ 9.1, 9.2.1, 9.3.1-3
12
Aug 3
Numerical solution to initial value problems
Stiff ODEs
Runge-Kutta methods
Multi-step methods
(Slides: PDF (full clean),
Slides: PDF (full color))
§ 9.3.4-8
13
Aug 5
Numerical solution to eigenvalue problems
Characteristic polynomial
Absolute condition number of eigenvalues
Power method
Inverse power method
Rayleigh quotient iteration
Deflation
QR iteration in 2 stages with shift
(Slides: PDF (full clean),
Slides: PDF (full color))
§ 4.1-4.4, 4.5.1-6