Numerical Analysis

MATH-UA0252-1, Spring 2014,

Lectures: Tuesdays and Thursdays, 12:30 - 1:45 pm. CIWWH 201
Instructor: Olof Widlund
Office hours: 30 minutes before and after each class or by appointment.

Recitation sessions: Wednesdays, 8:00 - 9:15 am. CIWWH 201.
Instructor: Juan Calvo
Office hours: Wednesdays 10-11am, WWH605.

  • Coordinates of Olof Widlund
    Office: WWH 612
    Telephone: 998-3110
    Email: widlund at

  • Coordinates of Juan Calvo
    Office: WWH 607
    Telephone: 998-3179
    Email: calvo at

  • Required Text: An Introduction to Numerical Analysis by Endre Suli and David Mayers. Cambridge University Press.

  • Exams
    There will be a final and a midterm exam and also relatively frequent and short quizzes. Scores will be made available on NYU Classes.
  • The final exam will be held on May 15, 12noon to 1:50pm, in the same class room WWH 201. To the final you can bring two regular-sized papers covered on both sides with notes written by yourself. The textbook and other notes will not be allowed.
  • Here is a collection of nine questions from earlier finals.
  • The midterm exam was held on March 6, 12:30-1:45pm. Here it is with solutions.
  • Here are five sample questions that could be given on a midterm exam.
  • You can bring one regular-sized paper covered on both sides with notes written by yourself to the midterm exam. The textbook and other notes will not be allowed. To understand what topics will be covered by the midterm and final exams, have a look at the Course Content section of this home page.
  • The first quiz was held on Thursday 2/13. Here it is with solutions.
  • The second quiz was held on Thursday 4/3. Here it is with solutions.
  • The third quiz was held on Thursday 4/24. Here it is with solutions.
  • Homework
    There will be regular homework assignments, including programming using Matlab, Octave, or any reasonable alternative. Scores will be made available on NYU Classes. When turning in programming assignments, please provide a listing of your program and output from runs with actual data that shows that your program works as intended. It is important that you do the homework yourself, but if you get stuck, I encourage you to consult with other students. However, when you get help, it's important to acknowledge it in writing. Passing off other people's work as your own is called plagiarism and is not acceptable. Please staple everything together and order the problems in the same order as given. The best way of turning in homework is to give it to me personally, in class or in my office. If I am not in my office, you can slide it under my door. Do not put homework in my mailbox.

  • Homework Assignments:
  • Homework set 1 Assigned January 31, due at 12:00 noon on Friday February 14.
  • Homework set 2 Assigned February 17, due at 12:00 noon on Friday February 28.
  • Homework set 3 Assigned March 11, due at 12:30 pm on Friday March 28.
  • Homework set 4 Assigned March 31, due at 12:30 pm on Friday April 11.
  • Homework set 5 Assigned April 13, due at 12:30 pm on Friday April 25.
  • Homework set 6 Assigned April 24, due at 12:30 pm on May 6.

  • Course Content. All page and section references are to the textbook.
  • For each week, there will be a short description of the content of the lectures.
  • Note that you should make sure that you obtain and save all handouts; they should be an important part of your reading. Request copies if you miss any of them.
  • Week 1: Chapter 1 of textbook. Finding roots of one nonlinear equation of one real variable. Brouwer's fixed point theorem. Contractions and the contraction mapping theorem. Rates of convergence. Linear and super-linear convergence. The bisection algorithm and its rate of convergence. Newton's method and its convergence rate in case of a simple root and a double root. The secant method. The Illinois method; this method will be further described in the first homework set. Sections not covered: 1.7 and 1.8.
  • Week 2: Review of linear system of algebraic equations for the case of n unknowns and n equations; linear independence of the columns (and the rows) of the matrix guarantees that the there is a unique solution. Gaussian elimination. What operations can be done at the same time? Expressing the process in terms of multiplication of the coefficient matrix from the left by (n-1) special unit lower triangular matrices. How to find the elements of the unit lower triangular matrix in the factorization PA=LU. The need for pivoting. Partial pivoting and an example of a matrix of order 2 for which partial pivoting is required in order to avoid very poor accuracy of the solution. Sections not covered: 2.8 and 2.10.
  • Week 3: How to decide if we can compute the values of a function with good relative accuracy. What matters is xdf(x)/dx/f(x) known as the condition number of the problem. Norms of vectors in R^n. A family of ell_p norms for 1 \leq p \leq infinity. A proof that all these norms satisfy the three requirements; to prove the triangle inequality is the hardest task. While doing these proofs, we develop the Cauchy-Schwarz and Hoelder inequalities. Norms of matrices for the case of p=1, p=2, and p=infinity.
  • Week 4: More on the norm of matrices defined by vector norms. The norm of a product of matrices can be bounded by the product of their norms. The condition number of the solution of a linear system of equations: the relative change of the norm of the solution as a function of the relative change of the right hand side and of the relative change of the coefficient matrix. Symmetric, positive definite matrices; no pivoting is necessary when factoring them by Cholesky's algorithm. This algorithm can also be used to decide if a symmetric matrix is positive definite. How to save arithmetic and storage when using Cholesky. Diagonally dominant matrices; another family for which pivoting is not needed. Linear least squares problems when we have more linear equations than unknowns. The normal equations. How to fit data to a model defined by a polynomial. Why to try to develop an algorithm which does not require the computation of the matrix of the normal equations; to be continued. Sections not covered: 3.4 and 3.5.
  • Week 5: Linear independence and the uniqueness of the solutions of least squares problems. Householder transformations as in Section 5.5. The orthogonality of any Householder matrix; it can most conveniently be stored by just one vector. The basic geometry of Householder transformations. The QR factorization of an m-by-n matrix in terms of a product of (n-1) Householder matrices and an m-by-n matrix with a lead-off n-by-n upper triangular matrix; this matrix is nonsingular if A has linearly independent columns. How to solve a least least squares problem using a QR factorization. How Householder matrices can be used to compute a symmetric tridiagonal matrix with the same eigenvalues as a given symmetric m-by-m matrix. Further comments on Householder transformations and on manipulating block matrices. Solving systems of non-linear equations where the number of unknowns and the number of equations are the same. Contraction mapping and a derivation of Newton's method; a linear system of equations must be solved in each iteration where the matrix elements are partial derivatives of the different functions with respect to the different variable. This matrix is known as the Jacobian. If it is well behaved close to the solution, we will have fast convergence from any starting approximation close to the solution. A review of the algebraic eigenvalue problem, in particular for symmetric matrices.
  • Week 6: Using eigenvalues and eigenvectors to solve initial value problems for simple linear systems of ordinary differential equations. The danger of using the roots of the characteristic equation to compute the eigenvalues of a matrix. Two theorems by Gerschgorin. Determinants of symmetric tridiagonal matrices. Cauchy's interlace theorem and how it can be combined with a bisection algorithm to located eigenvalues of a tridiagonal matrix.
  • Week 7: Sturm sequences and how we can use Cauchy's theorem to determine the number of eigenvalues of a tridiagonal matrix that are below a given number. Combining this idea with a bisection algorithm. An example of Wilkinson, which shows that the roots of a polynomial can be very poorly determined by its coefficients; this rules out the use of the characteristic polynomial to find the eigenvalues of a matrix numerically. Inverse iterations and the use of eigenvector expansions. Two perturbation theorems including that of Bauer and Fike. Rayleigh quotients and how to improve the approximation of an eigenvalue using an approximation of its eigenvector. Jacobi's algorithm to find all the eigenvalues of a symmetric matrix.
  • Week 8: Remarks on the midterm exam. A short introduction to the QR algorithm to find eigenvalues of matrices. Polynomial interpolation; there is a unique polynomial of degree n that matches any set of given values y_k at (n+1) distinct points x_k, k =0. 1, ...,n. The Lagrange formula. Barycentric interpolation formulas; see handout marked Chapter 5. These interpolation formulas greatly reduce the amount of arithmetic. Interpolating at equidistant point and at Chebyshev points. Runge's example and a discussion of Lebesgue constants; see handout marked Chapter 15. Hermite interpolation and a formula more complicated but similar to the Lagrange formula.
  • Week 9: Numerical differentiation using polynomial interpolation. Error bounds for the resulting formula. Numerical quadrature, in particular the trapezoidal rule and Simpson's formula. Composite numerical quadrature. Error bounds, in particular for Simpson's formula. Adaptive Simpson and how to avoid repeating function evaluations at the same points; see handout on adaptive Simpson. The trapezoidal formula with end point corrections, which can be obtained by integrating the Hermite interpolation polynomial using two points. If we use it for a composite formula and use the same subinterval lengths, only two values of the derivative are required. Extrapolation using the composite trapezoidal rule with different number of points. Spline interpolation, which provides twice continuously differentiable piece-wise cubics matching tridiagonal linear system of equations with a positive definite symmetric matrix, which allows the use of Cholesky's algorithm and results in a linear algorithm.
  • Week 10: More on linear and cubic spline interpolation; they can also be characterized by minimizing certain integrals subject to the interpolation conditions at the prescribed points. Inner products of functions defined by an integral and a weight function. Cauchy-Schwarz inequality for these inner products; the proof is almost identical to the proof for the inner products in R^n. Chebyshev polynomials defined by using cosine and arccos functions. A proof that we obtain polynomials of the correct degree. A three-term recurrence. The orthogonality of the Chebyshev polynomials with respect to a specific inner product. How to define a norm from an inner product. The best approximation of a function f(x) by polynomials and with respect to norms defined by weighted integrals; the use of orthogonal polynomials is very convenient. Three term recurrences for all sets of orthogonal polynomials. Legendre and Laguerre polynomials. A proof that all roots of the orthogonal polynomials are simple and located in the interval (a,b).
  • Week 11: Gaussian quadrature; there are different numerical quadrature rules for each set of orthogonal polynomials. A proof that by using n+1 quadrature nodes and a good choice of weights, we can integrate all polynomials of degree 2n+1 exactly. The quadrature nodes must be chosen as the zeros of the orthogonal polynomial of degree n+1. Details about the Laguerre case: The Gauss quadrature rule and the Radau numerical quadrature rule. A handout on Floating Point Arithmetic and a few first comments on different ways of representing real numbers on modern computers.
  • Week 12: Representing integers using 32 bits and the 2's complement idea. More on IEEE floating point numbers.
  • Week 13: More about IEEE floating point numbers. The outcome of simple arithmetic operations. A recollection of how a choice is made when designing Householder transformations. The subnormal numbers and how they can prevent underflow. Norms of function spaces in particular the maximum norm of continuous functions on finite intervals. Existence of best "min-max" polynomials. Two theorems characterizing the min-max polynomials. Uniqueness of the min-max polynomial. Best approximation of x^{n+1} by a polynomial of degree n.
  • Week 14: Review and problem solving including problems from old exams.