#    discrep.map  (MAPLE file)
#  To execute, enter "read `discrep.map`". 
#  Be sure to use "backward quotes" (`).
#
#  Derive asymptotic expressions related to the discrepancy principle
#  for Tikhonov regularization.

#  Asymptotic expected squared data residual norm for Tikhonov regularization.
#  alpha is the regularization parameter.

  restart;
  assume(b>0);
  assume(c>0);
  assume(p>1);
  assume(q>1);
  assume(sigma>0);
  assume(n>1);
  assume(I1>0);
  assume(I2>0);
  assume(I3>0);
  assume(alpha>0);
  S1 := (alpha/c)^((-p+q-1)/p) * I1;
  S2 := (alpha/c)^((-p-1)/p) * I2;
  S3 := (alpha/c)^(-1/p) * I3;
  FD := alpha^2*b/c*S1 - sigdn * (2*alpha/c*S2 + S3);
  FD := simplify(expand(FD));
  alphaD := solve(FD=0, alpha);
  alphaD := simplify(expand(alphaD),exp);
  alphaD := simplify(combine(alphaD,'power'));