Math 614-002 Numerical Methods I

Prerequisites:

• Linear Algebra and Advanced Calculus

Textbook:

• Kendall E. Atkinson, "An Introduction to Numerical Analysis".

General References:

1. William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, "Numerical Recipes in C (or Fortran 77): The Art of Scientific Computing".
2. J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis".
3. Eugene Isaacson and Herbert Bishop Keller, "Analysis of Numerical Methods".
4. Arieh Iserles, "A First Course in the Numerical Analysis of Differential Equations".
5. Gene H. Golub and Charles F. Van Loan, "Matrix Computation".
6. James W. Demmel, "Applied Numerical Linear Algebra".
7. Lloyd N. Trefethen and David Bau, "Numerical Linear Algebra".

Suggested Readings:

• Top Ten Algorithms of the 20th Century

• Lloyd N. Trefethen, " The Definition of Numerical Analysis".

• Kendall E. Atkinson, " Numerical Analysis".

• Kendall E. Atkinson, " Resources in Numerical Analysis".

• George E. Forsythe, " How do you solve a quadratic equation?".

• Jonathan Shewchuk, " An Introduction to the Conjugate Gradient Method Without the Agonizing Pain ".

• Rick Beatson and Leslie Greengard, " A Short Course on Fast Multipole Methods ".

• Michael Overton, " Floating Point Representation and the IEEE Standard ".

• J.W. Cooley and J.W. Tukey, "An algorithm for the machine calculation of complex Fourier series," Math. Comp., 19 (1965), pp. 297–301.

• R. Courant, K. Friedrichs, and H. Lewy, "On the partial difference equations of mathematical physics," IBM J. Res. Develop., 11 (1967), pp. 215–234.

• A.S. Householder, "Unitary triangularization of a nonsymmetric matrix," J. Assoc. Comput. Mach., 5 (1958), pp. 339–342.

• C.F. Curtiss and J.O. Hirschfelder, "Integration of stiff equations," Proc. Nat. Acad. Sci. U.S.A., 38 (1952), pp. 235–243.

• C. de Boor, "On calculating with B-splines," J. Approximation Theory, 6 (1972), pp. 50–62.

• R. Courant, "Variational methods for the solution of problems of equilibrium and vibrations," Bull. Amer. Math. Soc., 49 (1943), pp. 1–23.

• G. Golub and W. Kahan, "Calculating the singular values and pseudo-inverse of a matrix," J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), pp. 205–224.

• A. Brandt, "Multi-level adaptive solutions to boundary-value problems," Math. Comp., 31 (1977), no. 138, pp. 333–390.

• M.R. Hestenes and E. Stiefel, "Methods of conjugate gradients for solving linear systems," J. Research Nat. Bur. Standards, 49 (1952), pp. 409–436.

• R. Fletcher and M.J.D. Powell, "A rapidly convergent descent method for minimization," Comput. J., 6 (1963/1964), pp. 163–168.

• Germund Dahlquist, "Convergence and stability in the numerical integration of ordinary differential equations".

• G. Wanner, E. Hairer, and S.P. Nørsett, "Order stars and stability theorems," BIT, 18 (1978), no. 4, pp. 475–489.

• N. Karmarkar, "A new polynomial-time algorithm for linear programming," Combinatorica, 4 (1984), no. 4, pp. 373–395.

• L. Greengard and V. Rokhlin, "A fast algorithm for particle simulations," J. Comput. Phys., 73 (1987), no. 2, pp. 325–348.

Web Resources

1. Codes

   • Netlib
   • GAMS (Guide to Available Mathematical Software)



2. Books

   • Abramowitz and Stegun, " Handbook of Mathematical Functions " (Bible).
   • Numerical Recipes in Fortran 77.
   • Lloyd N. Trefethen, " Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations " .
   • Yousef Saad, " Iterative methods for sparse linear systems ".

Homework Assignments

• Homework Assignment 1.
• Homework Assignment 1 problem 19.
• Homework Assignment 1 project.
• Newton Fractals.
• Homework Assignment 2.
• Homework Assignment 2 project.