程序代写代做代考 algorithm checklist.dvi

checklist.dvi

ECS130 Mini-Project Grading Check List

1. Summary (abstract)

What is the mathematical problem of this report about? What is your solution? What is
your finding?

2. Introduction of the main problem

Statement of the problem: the linear least squares problem minβ ‖Xβ − b‖

(Background) When X is nonsingular, …. normal equation and QR decomposition

(Purpose of this project) We study the solution of the LS when when X is singular.

3. Definitions (tools/theory) needed:

rank deficient, singular value decomposition, pseudo-inverse

4. Algorithms

(a) β = X\y (why not use)

(b) β = pinv(X, tol)y with different drop tolerance values tol pinv(X) uses the default
tolerance value.

(c) β = V Σ+UT y, where X = UΣV T is the SVD of X, Σ+ is defined with with a drop
tolerance value tol

Note that Algorithms (b) and (c) are essentially the same.

5. Numerical examples

Use the Problem 5.6 and shaw.m to illustrate the key finding:

The accuracy (measured in the relative error) of the computed solution strongly
depends on the drop tolerance value “tol” in pinv (or say SVD).

Here are numerical results in plots for the Shaw problem to support the key finding:

0 10 20 30 40 50 60 70 80 90 100
0

0.5

1

1.5

2

2.5
exact and computed solution

exact
computed with default tol

0 10 20 30 40 50 60 70 80 90 100
−0.5

0

0.5

1

1.5

2

2.5
exact and computed solution

exact
computed with tol = 1e−15

0 10 20 30 40 50 60 70 80 90 100
10

−6

10
−5

10
−4

10
−3

10
−2

10
−1

10
0

10
1

Relative errors

rel err with default tol
rel error with tau=1e−15

6. Conclusion

Recap of the problem, the solution method and key finding.

7. Acknowledgement if any

8. References

1