Class meetings: TuesThurs, 3:30  4:45pm, in Warren Weaver Hall (CIWW) 102.
Last day of class: Thursday, May 3, 2018.
Final examination: Thursday, May 10, 2018, 4:005:50pm, in Warren Weaver 317.
**Note the different room for the final exam!**
This class will cover several topics, including: onedimensional nonlinear equations; understanding and dealing with sources of error; linear equations and linear leastsquares; data fitting; splines; numerical integration; ordinary differential equations; and elementary eigenvalue problems. As much as possible, numerical methods will be presented in the context of realworld applications.
The final grade will be calculated by averaging the three elements (Homework, Midterm, Final Project/Final Examination), with weights of 40%, 35%, 25%, where the weighting will be chosen individually to maximize each student's grade.
Students without this background should check with the instructor for permission to take the class.
Professor Gilbert Strang's famous linear algebra courses at MIT can be found on the MIT open courseware website or on YouTube, with search terms ``Strang linear algebra MIT''.
Matlab tutorials are available online from several sites. For example,
there is an array of tutorial and other educational resources
for students at
the MathWorks website .
HW2, due February 23, 2018 (24hour extension of original deadline).
HW3, due March 4, 2018.
HW4, due April 5, 2018.
HW5, due April 17, 2018.
HW6,
due May 1, 2018.
A brief LaTeX document containing a template for projects is
here.
January 23 (Lecture 1) 
Introduction. Overview of numerical computing. Nonlinear equations in one dimension. Bisection. Chapter 3 of textbook (Ascher and Greif). 
January 25 (Lecture 2) 
Newton's method. Rate of convergence. Pure secant method. 
January 30 (Lecture 3) 
Properties of the secant method. Regula falsi and Wheeler's method. Inverse quadratic interpolation. Brent's method (zeroin). Starting and stopping. 
February 1 (Lecture 4) 
More details about Wheeler's method, inverse quadratic interpolation, and Brent's method (fzero). Starting and stopping. 
February 6 (Lecture 5) 
Chapters 1 and 2 of Ascher and Greif. General ideas in numerical computing. A generic model of floatingpoint computation. IEEE arithmetic. 
February 8 (Lecture 6) 
IEEE doubleprecision arithmetic. 
February 13 (Lecture 7) 
Review of HW1. More on IEEE doubleprecision arithmetic. Introduction to numerical linear algebra. Ascher and Greif, Chapters 4 and 5. 
February 15 (Lecture 8) 
More discussion of issues in graded HW1. Matrix norms. Diagonal linear systems. Triangular linear systems. Analysis of solving a lower triangular system, showing backward stability. Condition number of a matrix. 
February 20 (Lecture 9) 
Gaussian elimination with elementary matrices. LU factorization. Pivot size and its effects. Growth. 
February 22 (Lecture 10) 
Variations on partial pivoting. Symmetric positive definite matrices. The Cholesky factorization and its favorable numerical properties. Failure when the matrix is not ``sufficiently'' positive definite. 
February 27 (Lecture 11) 
The singular value decomposition (SVD) and what it reveals. Connections with the 2norm of Ax and (A inverse); maximizing and minimizing the 2norm. Significance of the columns of V and the columns of U. Estimating the rank of a matrix. Liberal and conservative rankestimation strategies and their consequences. 
March 1 (Lecture 12) 
Linear leastsquares problems. Linear least squares in mathematical modeling. Properties of the leastsquares residual. The normal equations and concerns about conditioning. Orthogonal triangularization and its differences from Gaussian elimination. 
March 6 (Lecture 13) 
Householder transformations and their properties. Using Householder transformations to reduce an m times n matrix to upper triangular form. QR factorization. 
March 8 (Midterm) 

March 1216 (Spring break)  
March 20 (Lecture 15) 
Solving leastsquares problems using the QR factorization. Backward error analysis of the QR factorization. Some effects of perturbations. QR with column interchanges. Estimating rank with the QR factorization. Condition of the leastsquares problem. 
March 22 (Lecture 16) 
Introduction to interpolation and approximation in one dimension. Ascher and Greif, Chapters 10 and 11. Functional norms. Horner's method for evaluating a polynomial. Uniqueness of the interpolating polynomial. The Vandermonde matrix. Lagrange and Newton forms of the interpolating polynomial. Weierstrass's Theorem. 
March 27 (Lecture 17) 
The error factor polynomial. Runge's function. The Runge phenomenon. Polynomial approximations based on function norms. Quick look at Chebyshev polynomials and Chebyshev points. Hermite osculating polynomials. 
March 29 (Lecture 18) 
Splines. Piecewise linear interpolation. Piecewise cubic splines. Conditions that define a spline. Natural splines, `clamped' splines and `not a knot' splines. polynomial. Piecewise cubic Hermite a la Matlab (pchip). Comparing different interpolants. 
April 3 (Lecture 19) 
Numerical integration (quadrature). The midpoint and trapezoid rules, and their connection with polynomial interpolation. Error estimates. Simpson's rule. The roles of Legendre polynomials and the Lagrange form of the interpolating polynomial. ClenshawCurtis quadrature. 
April 5 (Lecture 20) 
Composite quadrature; adaptive quadrature. Romberg quadrature. Introduction to initial value problems in ordinary differential equations. The test equation. The forward Euler method and its error estimate. Introduction to RungeKutta methods. 
April 10 (Lecture 21) 
RungeKutta methods. Transforming problems with higher serivatives into systems of firstorder equations. Implicit methods. Backward Euler. Multistep methods. AdamsBashforth and AdamsMoulton methods. Backward differentiation methods. 
April 12 (Lecture 22) 
Derivation of backward differentiation methods. Predictorcorrector methods. Stiffness. Stability. The MilneSimpson method. Linear difference equations. Numerical issues. ODE software. 
April 17 (Lecture 23) 
Background theorems and results about eigenvalues and eigenvectors. The power method. 
April 19 (Lecture 24) 
The power method with shifts. The inverse power method (inverse iteration). The Rayleigh quotient; Rayleigh quotient iteration. The QR method. The pedagogical QR method. 
April 24 (Lecture 25) 
More on the pedagogical QR method. A diversion about PageRank and its connection with eigenvalues. 
April 26 (Lecture 26) 
Example of the pedagogical QR method. Finding the eigenvectors of a triangular matrix. Reduction to upper Hessenberg form. 
May 1 (Lecture 27) 
The implicit QR method; the implicit Q theorem. Plane rotations. Chasing the bulge. Some useful theory. The QR algorithm for symmetric matrices. Using bisection to find eigenvalues. 
May 3 (Lecture 28) 
Review of material covered in the course. 