Lab 10 summary
 M. M. Sussman
 {@(#) Thu Feb 16 19:34:00 2017 }
 
 $Id: summary.txt,v 1.11 2017/02/17 00:34:37 mike Exp $
 
 EXERCISE 1
 1. meshplot: fill in xNode=j*h, yNode=k*h
 2. plot of mesh
 3. copy tests.m, tests(75)=76, tests(80)='none'
 4. fill in below  75+1  test yielding 'none'
 5. fill in left   75-10 test yielding 'none'
 6. fill in right  75+10 test yielding 'none'
 
 EXERCISE 2
 1. poissonmatrix.m
 2. norm(A-A','fro')=0
 3. v'*A*v>0, random v
 4. det(A)=7.6579e+260
 5. norm(L*E-A*E)=roundoff, L=48.219 E(10)=-.1523
 6. min(min(inv(A)))=2.0830e-6>0
 7. norm(gallery('poisson',10)/h^2-A,'fro')=0
 
 EXERCISE 3
 1-2, anothermatrix.m
 3. norm(A-A','fro')
 4. v'*A*v > 0
 5. spy plot all red
 6. N=12;x=((1:N^2).^2)'; sum(A*x)= 129339420; N=13, sum=226538260
    (N=15, sum=617237640; N=20, sum=4633162020)
    
 EXERCISE 4
 1. cgm.m
 2. norm(y-xExact)=2.1216e-12, norm(x-xExact)=4.2730e-08
 3. norm(x-xExact)/norm(xExact)=2.4466e-16
 4. norm(x-xExact)/norm(xExact)=4.2565e-11 (after 100 iterations) (N=31)

 EXERCISE 5
 1. cg.m with plot and still has parm m
 2. semilog plot: decline to around 1e-28,
    roughly 108 its to 1.e-12
 3. add tolerance to call, etc.
 4. took me 101 iterations to get to 1e-10,  true rel error=4.2565e-11
    took me 111 iterations to get to 1e-12, true rel error=2.3981e-13
 5. anothermatrix took 162 steps to get to 1e-10, true error=1.9168e-10

 EXERCISE 6
 1,2 anotherdiags.m
 3. test as specified: 4 tests get 0 each
 4. poissondiags.m + TESTS
 
 EXERCISE 7
 1,2. multdiags.m
 3. test with x=ones(9,1). norm(multdiags(Adiags,x)-A*x)=0
 4. x=(10:18)'. norm=0
 5. N=12, random x, rel norm=roundoff
 
 EXERCISE 8
 1. cgdiags.m
 2. solve problem twice, get k=9, same ans.
 3. same ans twice, k=66.  rel err in norm 2.157e-12
 4. n=672 (vectorized on oz: 28.52 sec, loops on oz: 66.57 or 77)
 5. 2.5001e-4, tol=1.e-6, much worse
 
 EXERCISE 9
 1. Adiag has 3*250000 entries=6e6 bytes=6MB
 2. A has (2.5e5)^2 entries=5e11 bytes=500GB
    cost more than $2000 at $4/GB, more than the cost of my computer
 3. C=1(min)/(1e4)^3=1e-12 (min)
 4. T=C*(500^2)^3=15625 min = 10.85 days, compared with .5 min or 1 min
 
 EXERCISE 10
 1. Matlab sparse representation of A implies
    A(2,1)=2, A(3,2)=2, A(5,1)=0
 2. Nonzero entries column 4:
   (2,4)        5
   (3,4)        4
   (4,4)        3
   (5,4)        2
   (6,4)        1
   Nonzero entries in row 4:
   (4,2)        1
   (4,3)        2
   (4,4)        3
   (4,5)        4
   (4,6)        5
 2. representation of B would be
   (1,1)        1
   (4,1)        5
   (3,2)        4
   (2,3)        3
   (4,3)        6
   (1,4)        2

EXERCISE 11 (Extra credit 8 points)
1. precg.m, with convergence criterion based on r, not rho.
   test poissonmatrix(100), U=eye, k=16 both agree.
2. r^0=b, \gamma_1=\rho_0=b'*A^{-1}*b, \alpha_1=1, k=2 at cnvgce
3. k=2, error=roundoff, (alpha=1)
4. N=50: k(cg)=230, k(precg)=37, err(cg)=1.9609e-11, err(precg)=1.0329e-11
   (resid(cg)=9.2463e-11, resid(precg)=7.7504e-11)