2071 Lab 5 Summary
$Id: summary.txt,v 1.10 2016/12/22 21:03:19 mike Exp $
M. M. Sussman
{@(#) Wed Jan 13 14:58:56 2016 }

EXERCISE 1
      L1     L2     L Infinity
x1    17   10.050      7  
x2    18   10.488      7
x3    10    6.4807     5

EXERCISE 2.

Matrix  Vector
 norm    norm  S/U r(A1,x1)  r(A1,x2)  r(A1,x3)  r(A2,x1)  r(A2,x2)  r(A2,x3)
  p=1,   q=1,   S  0.909916  0.858095  0.953714  0.867185  0.878756  0.879874
  p=1,   q=2,   U  0.889431  0.850695  0.854103  0.979209  1.00485   0.902066
  p=1,   q=Inf, U  0.762449  0.76      0.722286  1.24708   1.30458   1.04906

  p=2,   q=1,   U  1.02097   0.962827  1.07012   0.850889  0.862243  0.86334
  p=2,   q=2,   S  0.997988  0.954524  0.958348  0.960808  0.985971  0.885115
  p=2,   q=Inf, U  0.855507  0.85276   0.810442  1.22365   1.28007   1.02934

  p=Inf, q=1,   U  1.02732   0.968817  1.07677   0.565092  0.572632  0.573361
  p=Inf, q=2,   U  1.0042    0.960462  0.96431   0.638091  0.654802  0.587822
  p=Inf, q=Inf, S  0.860829  0.858065  0.815484  0.812646  0.850117  0.683607

  p=fro, q=1,   U  0.974614  0.919108  1.02153   0.832628  0.843739  0.844812
  p=fro, q=2,   S  0.952672  0.911182  0.914832  0.940189  0.964811  0.86612
  p=fro, q=Inf, U  0.816661  0.814038  0.773642  1.19739   1.2526    1.00725


EXERCISE 3
1.  all e-vals same=0.5, e-vec=[1 0 0 0 0 0 0]', all equal
2.  spec radius= 0.5
3.  norm(x)=2.6548, norm(A*x)=3.7081, 3.7081< .5*2.6548=1.3229 ?
4.  NO

EXERCISE 4
1.  start small, hump peaks around 28 at A^10, then dies off to zero
2.  norm(A^(20)*x)<norm(A*x) 
3.  max=28.597

EXERCISE 5
1-4. 2-norm takes 4.5477 sec, longer than 'fro' norm 0.0256 at GSCC.
     0.6057 for 2-norm, .0165 for 'fro' Matlab on my Dell
     takes 0.486 sec on computer in Posvar 1200A, vs .0442 sec.

EXERCISE 6
   Residual                   xTrue   Error
1. 9.9998e-13    small        [0;0]   1.414     large
2. 1.4142e-5     small        [1;0]   .00001    small
3. 281.43        large        [1;0]   140.72    large
4. 1e9           large        [1;1]   .001      small

EXERCISE 7
1. bvp.m from rope_bvp.m
   LEFT_X=0; LEFT_U=0; RIGHT_X=1; RIGHT_U=1;
   b = -dx^2*pi^2/4*sin(pi*x(2:N+1)/2)';
2. test plot: essentially one curve not required.
3. two norm
  error rat  rel err rat
    2.6397   3.5741
    2.7286   3.7701
    2.7770   3.8802
    2.8023   3.9388
    2.8153   3.9691
    2.8218   3.9845
4. one norm
  error rat  rel err rat
   1.8995   3.5152
   1.9497   3.7408
   1.9749   3.8657
   1.9875   3.9316
   1.9937   3.9655
   1.9969   3.9827
5. inf norm
  error rat  rel err rat
   3.6455   3.6455
   3.8106   3.8106
   3.9037   3.9037
   3.9504   3.9504
   3.9752   3.9752
   3.9876   3.9876

6.             Norm  Relative norm
Euclidean       no     yes
one             no     yes
infinity        yes    yes

EXERCISE 8
1. Compute formula for eigenvalue dif2 of tridiag, knowing eigenvectors.
   write x=j*pi*k/(N+1) and dx=j*pi/(N+1)
   sin(x+dx)+sin(x-dx)=2*sin(x)cos(dx)
   hence A*v=(2-2*cos(dx))*v
2. tridiagonal.m (dif2 tridiag)
3. visually check it for N=5 against matrix in lab equation (3)
4. norm(A*v-(2-2*cos(j*pi/(N+1)))*v)= roundoff for j=1,10

EXERCISE 9
1. rrs.m (one row reduction step)
2. N=25, det(A)=26
3. 1 rrs, det(A)=26 still
4. use script m-file to complete row reduction, det hasn't changed
5. j=(1:N);dd=(j+1)'./j'; d=diag(A); norm(d-dd)=roundoff
   prod(diag(A))=26.

EXERCISE 10 (Extra)
compute determinant using Laplace rule
1. det(magic(5))=5070000, agrees with Matlab det.
2. times: magic(7),8,9,10 = .209,1.634,14.42,144.43
  8*.209=1.672, 1.634*9=14.706, 14.42*10=144.2
  confirms O(N!)
  .538, 4.157, 37.53, 390.1 (much longer!)