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Will a Course in Computational Matrix Theory be Useful?


cjyNel

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Hi all,

 

I am a junior undergrad at a huge public university in US. I plan to apply to Econ Phd programs in next fall. However, I am not sure which particular field I will focus on at the current moment.

 

I am wondering whether taking a course in matrix computation would be helpful for graduate study in Econ?

The reason why I think about it is as follows:

1) Though I got an A from an upper-level linear algebra course, I don't think I have a comprehensive knowledge in this subject.

For some reasons, the linear algebra course I took didn't even cover the chapter for orthogonality. (We use Friedberg's text)

I see that the topics in the matrix computation course also require a lot of linear algebra. Maybe I can enhance my linear algebra when taking this course.

 

2) It seems that numerical analysis is helpful in areas including macro and metrics. But, due to the schedule conflict, I can't take the numerical method sequence from the Math department.

I find this matrix computation course in the computer science department. I am not sure whether this will be a good substitute. (The CS department also provides another numerical analysis course in spring.)

 

The textbook used is: Numerical methods in Matrix Computations by Ake Bjork

 

The topics planned to be covered is as follows:

 

Background: Subspaces, Bases, Orthogonality, Matrices, Projectors, Norms.

Floating point arithmetic. Introduction to Matlab.

Text: Chap 1, parts of sec 1.1 and parts of sec 1.4

 

Systems of linear equations. Solution of Systems of Linear Equations: matrix

LU factorization. Special matrices: symmetric positive definite, banded.

Text: chap 2, sect. 1.2.1 – 1.2.5

 

Error analysis, condition numbers, operation counts, estimating accuracy.

Text: Chap 2 – sect. 1.2.7 – 1.2.8, sec. 1.4.3 – 1.4.6

 

Orthogonality, the Gram-Schmidt process. Classical and modified GramSchmidt.

Householder QR factorization. Givens rotations. Least-squares

systems. Rank deficient LS problem.

Text: 2.1, 2.3, 2.4

 

More on least squares problems. Regularization, Least squares problems with

constraints. Text: 2.6, parts of 2.7

 

Eigenvalues, singular values. The Singular Value Decomposition. Applications

of the SVD.

Text: Sec. 1.1.9, 2.2.1, applications 2.2.3 – 2.2.5

 

Eigenvalue problems: Background, Schur decomposition, perturbation analysis,

power and inverse power methods, subspace iteration; the QR algorithm.

Text: chap 3: 3.1, 3.2, 3.3, 3.4

 

The Symmetric Eigenvalue Problem: special properties and perturbation

theory, symmetric QR algorithm, Jacobi method. Applications.

Text: chap 3: 3.5, 3.6.1, 3.6.2

 

Sparse matrix techniques. The Lanczos algorithm. Lanczos bidiagonalization.

Sparse direct solution methods (overview). Krylov subspace methods

for linear systems (overview).

Text: parts of chap 10 and chap 11.

 

This course uses MATLAB and homework problems consist of both theoretical problems and some problems involving MATLAB computation.

 

Would you like to share your opinion?

 

Thanks!

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Sounds useful, but given that it sounds like you have a decent math background in general, it isn't a necessity. I don't know what your opportunity costs are; it definitely wouldn't hurt, but what would you take if not this?

 

Hi,

 

If taking this matrix computation course, I guess I would probably not take the intro PDE.(using Olver's text)

For the next semester, I am also taking the analysis I (baby rudin), game theory, IO, theory of Statistics I (Master level).

 

Thank you!

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Those are difficult courses. Make sure you're confident of doing well in all of them before you add another course.

 

I was actually in a similar spot once (with roughly those courses lined up) and decided whether I should take an additional computational math class, and decided against it because I felt I wouldn't want to screw around with MATLAB for one class while doing analysis/proofs for the rest of my classes. To me, there's increasing returns in working in the same type of coursework for the semester.

 

Computational matrix theory is one of those things where you only need it for some specific subfields within a few economic fields, so it's certainly not a huge benefit to have it. All the other courses you're doing, on the other hand, will be hugely useful regardless of your specialty.

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