MATH 1080: Numerical Linear AlgebraLectures: MWF 10:00-10:50AM 524 Thackeray Hall Office HoursMW 2:00-3:50PM, and by appointment (also via zoom) Office: 612 Thackeray HallE-mail: trenchea@pitt.eduTextbook Numerical Mathematics and Computing fifthedition, by W. Cheney and D. Kincaid. Available from Pitt Bookstore.ContentThis course is an introduction to numerical linear algebra which addresses numerical methods for solving linear algebraic systems and matrix eigenvalue problems and applications to partial differential equations. Although the course will stress a computational viewpoint, analysis of the convergences and stability of the algorithms will be investigated.We will cover the following chapters in the textbook: systems of linear equations, ordinary differentialequations, least-squares method, boundary value problem, partial differential equationsand minimization of functions.This course fulfills the requirements for the following Math majors: The Bachelor of Science in Mathematics, The Bachelor of Science in Applied Mathematics, The Bachelor of Science in Actuarial Mathematics, and The Bachelor of Science in Mathematical Biology, offered by the Department of Mathematics. Grading Policy The final grade will be based onhomeworks (40%), and exams (60%).Late homework will be accepted only by special permission of theinstructor.Additional references: Numerical Mathematics, second edition, by A. Quarteroni, R. Sacco, F. Faleri.Book's ProgramsNumerical Methods in Scientific Computing, volume I, by G. Dahlquist, A. Bjorck. SIAM.
Numerical Methods, by G. Dahlquist, A. Bjorck. Dover.Numerical Linear Algebra, by Lloyd N. Trefethen, David Bau, III. SIAM.Numerical Analysis, by Timothy Sauer. Pearson.Introduction to linear algebra, fourth edition, by Gilbert Strang. Wellesley Cambridge Press.Homework AssignmentsThe printouts of the codes should be included. Hwk 1 (due 01/20/23): problems 2, 9, 15 pages 292-294 (respectively pages 272-275 in the 6th edition, or pages 98-100 in the 7th edition); problems: 2, 9 page 296 (respectively page 276 in the 6th edition, or pages 100-101 in the 7th edition), and problem: Count the number of additions in naive Gaussian elimination. Hwk 2 (due 01/31/23): problem 5 page 307 (respectively page 287 in the 6th edition, or page 110 in the 7th edition), computer problems 3, 6, 18, and count the number of additions and multiplications in naive Gaussian elimination of tridiagonal matrices. Hwk 3 (due 02/08/23): problems 4, 9, page 332 (respectively page 311 in the 6th edition, or page 376 in the 7th edition); problems 1, 4 page 336 (respectively page 316 in the 6th edition, or page 378 in the 7th edition), and two more problems. Extra credit: exercise 22 , page 336 (respectively 315 in the sixth edition). Hwk 4 (due 02/15/23): problems 3, 8 page 353 (respectively page 337 in the 6th edition, or page 422 in the 7th edition); computer problems 2, 3, 7, 8 page 355 (respectively page 339 in the 6th edition, or page 311 in the 7th edition). Hwk 5 (due 02/22/23): problems 12 page 370 (respectively page 357 in the 6th edition, or page 394 in the 7th edition), computer problem 1 a,b,c,d page 371 (respectively pages 358 in the 6th edition, or page 395 in the 7th edition), computer problem 4 page 381 (respectively page 369 in the 6th edition, or page 404 in the 7th edition), and one problem. Hwk 6 (due 03/01/23): problems 5,6 page 455 (respectively page 436 in the 6th edition, or page 309 in the 7th edition), and computer problems 3,5,6 page 457 (respectively page 438 in the 6th edition, or page 310 in the 7th edition). Hwk 7 (due 03/15/23): problems 13, 18 pages 466-467 (respectively page 446 in the 6th edition, or page 317 in the 7th edition), computer problem 7 (plot the numerical solution), 10 page 468 (respectively page 448 in the 6th edition,or page 318 in the 7th edition), problem 5 page 481 (respectively page 460 in the 6th edition, or page 329 in the 7th edition), computer problem 5 page 483 (respectively page 462 in the 6th edition, or page 329 in the 7th edition), and computer problem 3 page 499 (respectively page 475 in the 6th edition, or page 346 in the 7th edition). Hwk 8 (due 03/24/23): problems 3, 19 page 529 (respectively pages 502-504 in the 6th edition, or page 433 in the 7th edition), problems 3, 4, 9 page 543, problem 22 page 555 (respectively page 529 in the 6th edition, or page 446 in the 7th edition) and one problem. Hwk 9 (due 03/29/23): problems 4, 5, 7 page 610 (respectively pages 578-579 in the 6th edition, or page 521 in the 7th edition), and computer problem 2a page 611 (respectively page 580 in the 6th edition, or page 522 in the 7th edition) using finite difference methodfor N=10, 20, 40; and plot the numerical solution and error. Hwk 10 (due 04/04/23): problems 1a, 1b, 1c, 1d, 3, 5 page 627 (respectively page 594 in the 6th edition, or page 535 in the 7th edition),and problem 4 page 636 (respectively page 604 in the 6th edition, or page 543 in the 7th edition). Midterm in class: February 24, 2023.Take-home final.Matlab Tutorial: ps; HTML
University of Pittsburgh offers many academic skills workshops.Disability Resource ServicesIf you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both your instructor and Disability Resources and Services (DRS), 140 William Pitt Union, 412-648-7890, drsrecep@pitt.edu, (412) 228-5347 for P3 ASL users, as early as possible in the term. DRS will verify your disability and determine reasonable accommodations for this course.Academic IntegrityThe University of Pittsburgh Academic Integrity Code is available at -integrity-freedom/academic-integrity-guidelines. The code states that "A student has an obligation to exhibit honesty and to respect the ethical standards of the academy in carrying out his or her academic assignments."The website lists examples of actions that violate this code. Students are expected to adhere to the Academic Integrity Code, and violations of the code will be dealt with seriously.On homework, you may work with other students or use library resources,but each student must write up his or her solutions independently.Copying solutions from other students will be considered cheating,and handled accordingly.
Numerical Mathematics and Computing, Sixth Edition
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We develop a stable finite difference method for the elastic wave equation in bounded media, where the material properties can be discontinuous at curved interfaces. The governing equation is discretized in second order form by a fourth or sixth order accurate summation-by-parts operator. The mesh size is determined by the velocity structure of the material, resulting in nonconforming grid interfaces with hanging nodes. We use order-preserving interpolation and the ghost point technique to couple adjacent mesh blocks in an energy-conserving manner, which is supported by a fully discrete stability analysis. In our previous work for the wave equation, two pairs of order-preserving interpolation operators were needed when imposing the interface conditions weakly by a penalty technique. Here, we only use one pair in the ghost point method. In numerical experiments, we demonstrate that the convergence rate is optimal and is the same as when a globally uniform mesh is used in a single domain. In addition, with a predictor-corrector time integration method, we obtain time stepping stability with stepsize almost the same as that given by the usual Courant-Friedrichs-Lewy condition. 2ff7e9595c
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