4610 HW

Week 1 Code (All due Wednesday Week 2)
1. Write your own matrix vector multiply procedure in the software of your choice.
2. Write your own matrix matrix multiply procedure in the software of your choice.
3. a) Write a function of a vector v that gives the matrix Q = Id - 2/w*w (ww*) where the w is the vector v + ||v|| e1 scaled by 1/||v|| and * indicates conjugate transpose of the vector. Note w*w is the inner product and ww* is the outer product. i.e.
```
      w = v + ||v|| e1
      w = w/||v||
      Q = Id - 2/||w||^2 w w*
      
```
  b) Build a random 6x6 matrix. Extract the first column and build Q using your function. Compute and print QA. Explain what your matrix Q does.
  c) Compute and print Q*Q. Explain what the matrix Q* is.
Week 2 Text (Due Friday Week 2)
1. 1.1, 1.2
Week 3 Code (Due Wednesday Week 3)
1. Write a function that use your "Q" Function from week one to compute a QR Factorization (an orthogonal matrix Q and an upper triangular matrix R satisfying A = QR) for a square matrix A. A rough outline is:
```
     pre-assign R=A and Q=Id
     for each column i
         * compute Qi using your Q function
         * zero the column of R using Qi 
         * accumulate the effect of Qi in Q
      return R and Q 
     
```
Week 3 HW (Due Friday Week 3)
1. 2.3 2.4, 2.6
2. 3.2, 3.3, 3.4
Week 4 Code (Due Monday Week 4)
1. Write a back substitution function to solve U x = b for an upper triangular U
2. Combine this with your QR code to write your first linear solver.
Week 4 HW (Due Wednesday Week 4)
1. 4.1, 4.2
Week 5 HW (Due Friday Week 5)
1. 5.3
2. 6.1, 6.2, 6.4
Week 6 Code (Due Monday Week 6)
1. Time your QR Factorization code on a range of matrix sizes and plot the times on a log scale. You need to make your problems large enough that it takes a few seconds for the largest size. Report the memory used for the largest size in MB.
2. Write a function called "QRIter": Input a square matrix A and an integer p; Output a matrix U of the same dimensions as A. The output is generated by
```
 	U=A
 	Do p times
 	 Compute the QR Factorization of U 
 	 update U by reversing the order of the product i.e. U = R.Q
 	return U
 	
```
3. Run your code from (2) on a number of randomly generated symmetric matrices and examine the output. Explain what happens to the output matrix. Try to guess what this might be good for?
Week 7 Code (Due Monday Week 7)
1. Implement the CGS (7.1) and MGS algorithms (8.1) from the text. For both, the input is a matrix A (unless you are writing the code in Fortran in which case you can include dimensions) and output is a pair Q and R.
2. Make timing plots that show their run times.
3. Explain what their operation counts and storage requirements are and explain any differences between their run times.
4. Recreate Figure 9.1 from the text and explain what it shows. Estimate the value of machine epsilon for your computing environment.
Week 7 HW (Due Friday Week 7)
1. 10.1
2. Write a few sentences and give examples to explain what machine epsilon is.
Week 12 Code (Due Monday Week 12)
1. Find, implement, and test the Thomas Algorithm for solving a tridiagonal linear system.
2. Give an operation and storage count for your implementation.
3. Make a simple modification to reduce the storage and computation used for your Thomas algorithm for symmetric tri-diagonal system. Again implement, test and provide operation counts etc.
4. Compare your solvers timings to a general full solver and a general sparse solver in the software of your choice for symmetric tro-diagonal systems. You need to present a labelled graph with the different algorithms for a range of problem sizes on a log-log scale.
Week 12 Project Plan (Due Friday Week 12)
Week 13 Code (Due Monday Week 13)
1. Implement and Test a reduced storage version for a QR eigenvalue algorithm for symmetric tri-diagonal systems. The converence criterion should be Max[Abs[OffDiagonal]] less than a specified and easily adjustable Tolerance.
2. Estimate the operation count per iteration and time your symmetric tridiagonal QR code on a range of probem sizes. You need to present a labelled graph for reasonably sized samples on a range of problem sizes on a log-log scale.