Test case: solving some simultaneous equations

Now we assemble a module cgmeth for solving simultaneous equations. Starting with the conjugate-direction module cgstep_mod we insert the module matmult as the linear operator.  

module cgmeth {
  use matmult
  use cgstep_mod
  use simple_solver
contains
  # setup of conjugate gradient descent, minimize  SUM rr(i)**2
  #             nx
  # rr(i)  =   sum fff(i,j) * x(j)  -  yy(i) 
  #            j=1
  subroutine cgtest( x, yy, rr, fff, niter) {
    real, dimension (:), intent (out) :: x, rr
    real, dimension (:), intent (in)  :: yy
    real, dimension (:,:), pointer    :: fff
    integer,             intent (in)  :: niter

call matmult_init( fff)

call solver( matmult_lop, cgstep, x, yy, niter, res = rr)

call cgstep_close ()
  }
}

Below shows the solution to $5 \times 4$ set of simultaneous equations. Observe that the ``exact'' solution is obtained in the last step. Because the data and answers are integers, it is quick to check the result manually.

d transpose
      3.00      3.00      5.00      7.00      9.00

F transpose
      1.00      1.00      1.00      1.00      1.00
      1.00      2.00      3.00      4.00      5.00
      1.00      0.00      1.00      0.00      1.00
      0.00      0.00      0.00      1.00      1.00

for iter = 0, 4
x    0.43457383  1.56124675  0.27362058  0.25752524
res -0.73055887  0.55706739  0.39193487 -0.06291389 -0.22804642
x    0.51313990  1.38677299  0.87905121  0.56870615
res -0.22103602  0.28668585  0.55251014 -0.37106210 -0.10523783
x    0.39144871  1.24044561  1.08974111  1.46199656
res -0.27836466 -0.12766013  0.20252672 -0.18477242  0.14541438
x    1.00001287  1.00004792  1.00000811  2.00000739
res  0.00006878  0.00010860  0.00016473  0.00021179  0.00026788
x    1.00000024  0.99999994  0.99999994  2.00000024
res -0.00000001 -0.00000001  0.00000001  0.00000002 -0.00000001

EXERCISES:

1. One way to remove a mean value m from signal $s(t)= \bold s$is with the fitting goal $\bold 0 \approx \bold s - m$.What operator matrix is involved?
2. What linear operator subroutine from Chapter can be used for finding the mean?
3. How many CD iterations should be required to get the exact mean value?
4. Write a mathematical expression for finding the mean by the CG method.