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Math 3NA3

Information about the final

The midterm exam will be held on Monday December 20th from 7:30 pm - 10:00 pm on Crowd-mark. The exam will be available on Crowdmark at 7:30 pm. You will have 2 hours and 30 minutes to fifinish the test and another 1 hour to organize and upload solutions on Crowdmark.

The exam will be of similar format to the midterm exam with 3 long answer questions. You can upload written answers as scanned pdfs or pictures. There will be computational questions involving Matlab and you can use the code posted on Avenue and from homework. You are supposed to submit both code and results for such questions.

During the test you may use only:

1. The lecture notes.

2. The textbook: Numerical Mathematics, 2nd edition by M.R. Grasselli & D.E. Pelinovsky.

3. Matlab.

4. Matlab code posted on Avenue and from homework

No other aids are permitted, including internet searches. You must not communicate with anyone else during the test. You are expected to exhibit honesty and behave ethically during this test. Any evidence of academic dishonesty will be investigated fully. If you have any questions during the exam, you can either reach me through email [email protected] or the lecture’s zoom link. I will use the lecture’s zoom link to post announcement related to the exam so it is recommended to join the lecture’s zoom link during the exam.

Content of the exam:

It is recommended to review the homework and lecture notes posted on Avenue to prepare for the final. There will be computational questions involving Matlab and you can use the code posted on Avenue and from homework. You are supposed to submit both code and results for such questions.

• Floating point arithmetic: Floating point number system, properties of floating point number system, machine epsilon and maxRRE, floating point approximations and rounding error, cancellation error.

• Condition number and error analysis: condition number, truncation error, rounding error and total computational error, forward and backward error, backward stability.

• Linear algebra and matrix norms: basic linear algebra knowledge, vector and matrix norms and their properties, condition number of a matrix, SPD matrices and their properties, orthogonal matrices and their properties, strictly diagonally dominant matrices and their properties.

• Singular Value Decomposition: definition, properties, low rank approximation (i.e. best k-rank approximation to a matrix), pseudo-inverse and its applications in least squares problem.

• Direct solution of linear systems: triangular systems (forward and backward substitution), Gaus-sian elimination, LU factorization, LU factorization with partial pivoting, Cholesky factorization, com-putational complexity and backward stability of LU factorization, LU factorization with partial pivoting and Cholesky factorization.

• Iterative methods for linear systems: Jacobi, Gauss-Seidel, SOR, convergence conditions of iter-ative methods, optimal choice of relaxation parameter for SOR.

• Least squares problem: definition, solutions of a least squares problem, normal equation, SVD and pseudo-inverse, Householder transformation, QR factorization.